Integrand size = 26, antiderivative size = 337 \[ \int \frac {\left (d+c^2 d x^2\right )^3 (a+b \text {arcsinh}(c x))^2}{x} \, dx=\frac {71}{144} b^2 c^2 d^3 x^2+\frac {7}{144} b^2 c^4 d^3 x^4+\frac {1}{108} b^2 d^3 \left (1+c^2 x^2\right )^3-\frac {19}{24} b c d^3 x \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))-\frac {7}{36} b c d^3 x \left (1+c^2 x^2\right )^{3/2} (a+b \text {arcsinh}(c x))-\frac {1}{18} b c d^3 x \left (1+c^2 x^2\right )^{5/2} (a+b \text {arcsinh}(c x))-\frac {19}{48} d^3 (a+b \text {arcsinh}(c x))^2+\frac {1}{2} d^3 \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))^2+\frac {1}{4} d^3 \left (1+c^2 x^2\right )^2 (a+b \text {arcsinh}(c x))^2+\frac {1}{6} d^3 \left (1+c^2 x^2\right )^3 (a+b \text {arcsinh}(c x))^2+\frac {d^3 (a+b \text {arcsinh}(c x))^3}{3 b}+d^3 (a+b \text {arcsinh}(c x))^2 \log \left (1-e^{-2 \text {arcsinh}(c x)}\right )-b d^3 (a+b \text {arcsinh}(c x)) \operatorname {PolyLog}\left (2,e^{-2 \text {arcsinh}(c x)}\right )-\frac {1}{2} b^2 d^3 \operatorname {PolyLog}\left (3,e^{-2 \text {arcsinh}(c x)}\right ) \]
71/144*b^2*c^2*d^3*x^2+7/144*b^2*c^4*d^3*x^4+1/108*b^2*d^3*(c^2*x^2+1)^3-7 /36*b*c*d^3*x*(c^2*x^2+1)^(3/2)*(a+b*arcsinh(c*x))-1/18*b*c*d^3*x*(c^2*x^2 +1)^(5/2)*(a+b*arcsinh(c*x))-19/48*d^3*(a+b*arcsinh(c*x))^2+1/2*d^3*(c^2*x ^2+1)*(a+b*arcsinh(c*x))^2+1/4*d^3*(c^2*x^2+1)^2*(a+b*arcsinh(c*x))^2+1/6* d^3*(c^2*x^2+1)^3*(a+b*arcsinh(c*x))^2+1/3*d^3*(a+b*arcsinh(c*x))^3/b+d^3* (a+b*arcsinh(c*x))^2*ln(1-1/(c*x+(c^2*x^2+1)^(1/2))^2)-b*d^3*(a+b*arcsinh( c*x))*polylog(2,1/(c*x+(c^2*x^2+1)^(1/2))^2)-1/2*b^2*d^3*polylog(3,1/(c*x+ (c^2*x^2+1)^(1/2))^2)-19/24*b*c*d^3*x*(a+b*arcsinh(c*x))*(c^2*x^2+1)^(1/2)
Time = 0.69 (sec) , antiderivative size = 419, normalized size of antiderivative = 1.24 \[ \int \frac {\left (d+c^2 d x^2\right )^3 (a+b \text {arcsinh}(c x))^2}{x} \, dx=\frac {d^3 \left (5184 a^2 c^2 x^2+2592 a^2 c^4 x^4+576 a^2 c^6 x^6-3600 a b c x \sqrt {1+c^2 x^2}-1056 a b c^3 x^3 \sqrt {1+c^2 x^2}-192 a b c^5 x^5 \sqrt {1+c^2 x^2}+10368 a b c^2 x^2 \text {arcsinh}(c x)+5184 a b c^4 x^4 \text {arcsinh}(c x)+1152 a b c^6 x^6 \text {arcsinh}(c x)-3456 a b \text {arcsinh}(c x)^2-1152 b^2 \text {arcsinh}(c x)^3+783 b^2 \cosh (2 \text {arcsinh}(c x))+1566 b^2 \text {arcsinh}(c x)^2 \cosh (2 \text {arcsinh}(c x))+27 b^2 \cosh (4 \text {arcsinh}(c x))+216 b^2 \text {arcsinh}(c x)^2 \cosh (4 \text {arcsinh}(c x))+b^2 \cosh (6 \text {arcsinh}(c x))+18 b^2 \text {arcsinh}(c x)^2 \cosh (6 \text {arcsinh}(c x))+6912 a b \text {arcsinh}(c x) \log \left (1-e^{2 \text {arcsinh}(c x)}\right )+3456 b^2 \text {arcsinh}(c x)^2 \log \left (1-e^{2 \text {arcsinh}(c x)}\right )+3456 a^2 \log (c x)-3600 a b \log \left (-c x+\sqrt {1+c^2 x^2}\right )+3456 b (a+b \text {arcsinh}(c x)) \operatorname {PolyLog}\left (2,e^{2 \text {arcsinh}(c x)}\right )-1728 b^2 \operatorname {PolyLog}\left (3,e^{2 \text {arcsinh}(c x)}\right )-1566 b^2 \text {arcsinh}(c x) \sinh (2 \text {arcsinh}(c x))-108 b^2 \text {arcsinh}(c x) \sinh (4 \text {arcsinh}(c x))-6 b^2 \text {arcsinh}(c x) \sinh (6 \text {arcsinh}(c x))\right )}{3456} \]
(d^3*(5184*a^2*c^2*x^2 + 2592*a^2*c^4*x^4 + 576*a^2*c^6*x^6 - 3600*a*b*c*x *Sqrt[1 + c^2*x^2] - 1056*a*b*c^3*x^3*Sqrt[1 + c^2*x^2] - 192*a*b*c^5*x^5* Sqrt[1 + c^2*x^2] + 10368*a*b*c^2*x^2*ArcSinh[c*x] + 5184*a*b*c^4*x^4*ArcS inh[c*x] + 1152*a*b*c^6*x^6*ArcSinh[c*x] - 3456*a*b*ArcSinh[c*x]^2 - 1152* b^2*ArcSinh[c*x]^3 + 783*b^2*Cosh[2*ArcSinh[c*x]] + 1566*b^2*ArcSinh[c*x]^ 2*Cosh[2*ArcSinh[c*x]] + 27*b^2*Cosh[4*ArcSinh[c*x]] + 216*b^2*ArcSinh[c*x ]^2*Cosh[4*ArcSinh[c*x]] + b^2*Cosh[6*ArcSinh[c*x]] + 18*b^2*ArcSinh[c*x]^ 2*Cosh[6*ArcSinh[c*x]] + 6912*a*b*ArcSinh[c*x]*Log[1 - E^(2*ArcSinh[c*x])] + 3456*b^2*ArcSinh[c*x]^2*Log[1 - E^(2*ArcSinh[c*x])] + 3456*a^2*Log[c*x] - 3600*a*b*Log[-(c*x) + Sqrt[1 + c^2*x^2]] + 3456*b*(a + b*ArcSinh[c*x])* PolyLog[2, E^(2*ArcSinh[c*x])] - 1728*b^2*PolyLog[3, E^(2*ArcSinh[c*x])] - 1566*b^2*ArcSinh[c*x]*Sinh[2*ArcSinh[c*x]] - 108*b^2*ArcSinh[c*x]*Sinh[4* ArcSinh[c*x]] - 6*b^2*ArcSinh[c*x]*Sinh[6*ArcSinh[c*x]]))/3456
Result contains complex when optimal does not.
Time = 4.09 (sec) , antiderivative size = 570, normalized size of antiderivative = 1.69, number of steps used = 31, number of rules used = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.154, Rules used = {6223, 27, 6201, 241, 6201, 244, 2009, 6200, 15, 6198, 6223, 6201, 244, 2009, 6200, 15, 6198, 6223, 6190, 25, 3042, 26, 4201, 2620, 3011, 2720, 6200, 15, 6198, 7143}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {\left (c^2 d x^2+d\right )^3 (a+b \text {arcsinh}(c x))^2}{x} \, dx\) |
| \(\Big \downarrow \) 6223 |
| \(\displaystyle -\frac {1}{3} b c d^3 \int \left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))dx+d \int \frac {d^2 \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))^2}{x}dx+\frac {1}{6} d^3 \left (c^2 x^2+1\right )^3 (a+b \text {arcsinh}(c x))^2\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {1}{3} b c d^3 \int \left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))dx+d^3 \int \frac {\left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))^2}{x}dx+\frac {1}{6} d^3 \left (c^2 x^2+1\right )^3 (a+b \text {arcsinh}(c x))^2\) |
| \(\Big \downarrow \) 6201 |
| \(\displaystyle -\frac {1}{3} b c d^3 \left (\frac {5}{6} \int \left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))dx-\frac {1}{6} b c \int x \left (c^2 x^2+1\right )^2dx+\frac {1}{6} x \left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))\right )+d^3 \int \frac {\left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))^2}{x}dx+\frac {1}{6} d^3 \left (c^2 x^2+1\right )^3 (a+b \text {arcsinh}(c x))^2\) |
| \(\Big \downarrow \) 241 |
| \(\displaystyle -\frac {1}{3} b c d^3 \left (\frac {5}{6} \int \left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))dx+\frac {1}{6} x \left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))-\frac {b \left (c^2 x^2+1\right )^3}{36 c}\right )+d^3 \int \frac {\left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))^2}{x}dx+\frac {1}{6} d^3 \left (c^2 x^2+1\right )^3 (a+b \text {arcsinh}(c x))^2\) |
| \(\Big \downarrow \) 6201 |
| \(\displaystyle -\frac {1}{3} b c d^3 \left (\frac {5}{6} \left (\frac {3}{4} \int \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))dx-\frac {1}{4} b c \int x \left (c^2 x^2+1\right )dx+\frac {1}{4} x \left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))\right )+\frac {1}{6} x \left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))-\frac {b \left (c^2 x^2+1\right )^3}{36 c}\right )+d^3 \int \frac {\left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))^2}{x}dx+\frac {1}{6} d^3 \left (c^2 x^2+1\right )^3 (a+b \text {arcsinh}(c x))^2\) |
| \(\Big \downarrow \) 244 |
| \(\displaystyle d^3 \int \frac {\left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))^2}{x}dx-\frac {1}{3} b c d^3 \left (\frac {5}{6} \left (\frac {3}{4} \int \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))dx-\frac {1}{4} b c \int \left (c^2 x^3+x\right )dx+\frac {1}{4} x \left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))\right )+\frac {1}{6} x \left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))-\frac {b \left (c^2 x^2+1\right )^3}{36 c}\right )+\frac {1}{6} d^3 \left (c^2 x^2+1\right )^3 (a+b \text {arcsinh}(c x))^2\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle d^3 \int \frac {\left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))^2}{x}dx-\frac {1}{3} b c d^3 \left (\frac {5}{6} \left (\frac {3}{4} \int \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))dx+\frac {1}{4} x \left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))-\frac {1}{4} b c \left (\frac {c^2 x^4}{4}+\frac {x^2}{2}\right )\right )+\frac {1}{6} x \left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))-\frac {b \left (c^2 x^2+1\right )^3}{36 c}\right )+\frac {1}{6} d^3 \left (c^2 x^2+1\right )^3 (a+b \text {arcsinh}(c x))^2\) |
| \(\Big \downarrow \) 6200 |
| \(\displaystyle d^3 \int \frac {\left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))^2}{x}dx-\frac {1}{3} b c d^3 \left (\frac {5}{6} \left (\frac {3}{4} \left (\frac {1}{2} \int \frac {a+b \text {arcsinh}(c x)}{\sqrt {c^2 x^2+1}}dx-\frac {1}{2} b c \int xdx+\frac {1}{2} x \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))\right )+\frac {1}{4} x \left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))-\frac {1}{4} b c \left (\frac {c^2 x^4}{4}+\frac {x^2}{2}\right )\right )+\frac {1}{6} x \left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))-\frac {b \left (c^2 x^2+1\right )^3}{36 c}\right )+\frac {1}{6} d^3 \left (c^2 x^2+1\right )^3 (a+b \text {arcsinh}(c x))^2\) |
| \(\Big \downarrow \) 15 |
| \(\displaystyle d^3 \int \frac {\left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))^2}{x}dx-\frac {1}{3} b c d^3 \left (\frac {5}{6} \left (\frac {3}{4} \left (\frac {1}{2} \int \frac {a+b \text {arcsinh}(c x)}{\sqrt {c^2 x^2+1}}dx+\frac {1}{2} x \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))-\frac {1}{4} b c x^2\right )+\frac {1}{4} x \left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))-\frac {1}{4} b c \left (\frac {c^2 x^4}{4}+\frac {x^2}{2}\right )\right )+\frac {1}{6} x \left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))-\frac {b \left (c^2 x^2+1\right )^3}{36 c}\right )+\frac {1}{6} d^3 \left (c^2 x^2+1\right )^3 (a+b \text {arcsinh}(c x))^2\) |
| \(\Big \downarrow \) 6198 |
| \(\displaystyle d^3 \int \frac {\left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))^2}{x}dx+\frac {1}{6} d^3 \left (c^2 x^2+1\right )^3 (a+b \text {arcsinh}(c x))^2-\frac {1}{3} b c d^3 \left (\frac {1}{6} x \left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))+\frac {5}{6} \left (\frac {1}{4} x \left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))+\frac {3}{4} \left (\frac {1}{2} x \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))+\frac {(a+b \text {arcsinh}(c x))^2}{4 b c}-\frac {1}{4} b c x^2\right )-\frac {1}{4} b c \left (\frac {c^2 x^4}{4}+\frac {x^2}{2}\right )\right )-\frac {b \left (c^2 x^2+1\right )^3}{36 c}\right )\) |
| \(\Big \downarrow \) 6223 |
| \(\displaystyle d^3 \left (-\frac {1}{2} b c \int \left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))dx+\int \frac {\left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))^2}{x}dx+\frac {1}{4} \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))^2\right )+\frac {1}{6} d^3 \left (c^2 x^2+1\right )^3 (a+b \text {arcsinh}(c x))^2-\frac {1}{3} b c d^3 \left (\frac {1}{6} x \left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))+\frac {5}{6} \left (\frac {1}{4} x \left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))+\frac {3}{4} \left (\frac {1}{2} x \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))+\frac {(a+b \text {arcsinh}(c x))^2}{4 b c}-\frac {1}{4} b c x^2\right )-\frac {1}{4} b c \left (\frac {c^2 x^4}{4}+\frac {x^2}{2}\right )\right )-\frac {b \left (c^2 x^2+1\right )^3}{36 c}\right )\) |
| \(\Big \downarrow \) 6201 |
| \(\displaystyle d^3 \left (-\frac {1}{2} b c \left (\frac {3}{4} \int \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))dx-\frac {1}{4} b c \int x \left (c^2 x^2+1\right )dx+\frac {1}{4} x \left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))\right )+\int \frac {\left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))^2}{x}dx+\frac {1}{4} \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))^2\right )+\frac {1}{6} d^3 \left (c^2 x^2+1\right )^3 (a+b \text {arcsinh}(c x))^2-\frac {1}{3} b c d^3 \left (\frac {1}{6} x \left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))+\frac {5}{6} \left (\frac {1}{4} x \left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))+\frac {3}{4} \left (\frac {1}{2} x \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))+\frac {(a+b \text {arcsinh}(c x))^2}{4 b c}-\frac {1}{4} b c x^2\right )-\frac {1}{4} b c \left (\frac {c^2 x^4}{4}+\frac {x^2}{2}\right )\right )-\frac {b \left (c^2 x^2+1\right )^3}{36 c}\right )\) |
| \(\Big \downarrow \) 244 |
| \(\displaystyle d^3 \left (\int \frac {\left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))^2}{x}dx-\frac {1}{2} b c \left (\frac {3}{4} \int \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))dx-\frac {1}{4} b c \int \left (c^2 x^3+x\right )dx+\frac {1}{4} x \left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))\right )+\frac {1}{4} \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))^2\right )+\frac {1}{6} d^3 \left (c^2 x^2+1\right )^3 (a+b \text {arcsinh}(c x))^2-\frac {1}{3} b c d^3 \left (\frac {1}{6} x \left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))+\frac {5}{6} \left (\frac {1}{4} x \left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))+\frac {3}{4} \left (\frac {1}{2} x \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))+\frac {(a+b \text {arcsinh}(c x))^2}{4 b c}-\frac {1}{4} b c x^2\right )-\frac {1}{4} b c \left (\frac {c^2 x^4}{4}+\frac {x^2}{2}\right )\right )-\frac {b \left (c^2 x^2+1\right )^3}{36 c}\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle d^3 \left (\int \frac {\left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))^2}{x}dx-\frac {1}{2} b c \left (\frac {3}{4} \int \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))dx+\frac {1}{4} x \left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))-\frac {1}{4} b c \left (\frac {c^2 x^4}{4}+\frac {x^2}{2}\right )\right )+\frac {1}{4} \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))^2\right )+\frac {1}{6} d^3 \left (c^2 x^2+1\right )^3 (a+b \text {arcsinh}(c x))^2-\frac {1}{3} b c d^3 \left (\frac {1}{6} x \left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))+\frac {5}{6} \left (\frac {1}{4} x \left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))+\frac {3}{4} \left (\frac {1}{2} x \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))+\frac {(a+b \text {arcsinh}(c x))^2}{4 b c}-\frac {1}{4} b c x^2\right )-\frac {1}{4} b c \left (\frac {c^2 x^4}{4}+\frac {x^2}{2}\right )\right )-\frac {b \left (c^2 x^2+1\right )^3}{36 c}\right )\) |
| \(\Big \downarrow \) 6200 |
| \(\displaystyle d^3 \left (\int \frac {\left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))^2}{x}dx-\frac {1}{2} b c \left (\frac {3}{4} \left (\frac {1}{2} \int \frac {a+b \text {arcsinh}(c x)}{\sqrt {c^2 x^2+1}}dx-\frac {1}{2} b c \int xdx+\frac {1}{2} x \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))\right )+\frac {1}{4} x \left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))-\frac {1}{4} b c \left (\frac {c^2 x^4}{4}+\frac {x^2}{2}\right )\right )+\frac {1}{4} \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))^2\right )+\frac {1}{6} d^3 \left (c^2 x^2+1\right )^3 (a+b \text {arcsinh}(c x))^2-\frac {1}{3} b c d^3 \left (\frac {1}{6} x \left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))+\frac {5}{6} \left (\frac {1}{4} x \left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))+\frac {3}{4} \left (\frac {1}{2} x \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))+\frac {(a+b \text {arcsinh}(c x))^2}{4 b c}-\frac {1}{4} b c x^2\right )-\frac {1}{4} b c \left (\frac {c^2 x^4}{4}+\frac {x^2}{2}\right )\right )-\frac {b \left (c^2 x^2+1\right )^3}{36 c}\right )\) |
| \(\Big \downarrow \) 15 |
| \(\displaystyle d^3 \left (\int \frac {\left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))^2}{x}dx-\frac {1}{2} b c \left (\frac {3}{4} \left (\frac {1}{2} \int \frac {a+b \text {arcsinh}(c x)}{\sqrt {c^2 x^2+1}}dx+\frac {1}{2} x \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))-\frac {1}{4} b c x^2\right )+\frac {1}{4} x \left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))-\frac {1}{4} b c \left (\frac {c^2 x^4}{4}+\frac {x^2}{2}\right )\right )+\frac {1}{4} \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))^2\right )+\frac {1}{6} d^3 \left (c^2 x^2+1\right )^3 (a+b \text {arcsinh}(c x))^2-\frac {1}{3} b c d^3 \left (\frac {1}{6} x \left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))+\frac {5}{6} \left (\frac {1}{4} x \left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))+\frac {3}{4} \left (\frac {1}{2} x \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))+\frac {(a+b \text {arcsinh}(c x))^2}{4 b c}-\frac {1}{4} b c x^2\right )-\frac {1}{4} b c \left (\frac {c^2 x^4}{4}+\frac {x^2}{2}\right )\right )-\frac {b \left (c^2 x^2+1\right )^3}{36 c}\right )\) |
| \(\Big \downarrow \) 6198 |
| \(\displaystyle d^3 \left (\int \frac {\left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))^2}{x}dx+\frac {1}{4} \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))^2-\frac {1}{2} b c \left (\frac {1}{4} x \left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))+\frac {3}{4} \left (\frac {1}{2} x \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))+\frac {(a+b \text {arcsinh}(c x))^2}{4 b c}-\frac {1}{4} b c x^2\right )-\frac {1}{4} b c \left (\frac {c^2 x^4}{4}+\frac {x^2}{2}\right )\right )\right )+\frac {1}{6} d^3 \left (c^2 x^2+1\right )^3 (a+b \text {arcsinh}(c x))^2-\frac {1}{3} b c d^3 \left (\frac {1}{6} x \left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))+\frac {5}{6} \left (\frac {1}{4} x \left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))+\frac {3}{4} \left (\frac {1}{2} x \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))+\frac {(a+b \text {arcsinh}(c x))^2}{4 b c}-\frac {1}{4} b c x^2\right )-\frac {1}{4} b c \left (\frac {c^2 x^4}{4}+\frac {x^2}{2}\right )\right )-\frac {b \left (c^2 x^2+1\right )^3}{36 c}\right )\) |
| \(\Big \downarrow \) 6223 |
| \(\displaystyle d^3 \left (-b c \int \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))dx+\int \frac {(a+b \text {arcsinh}(c x))^2}{x}dx+\frac {1}{4} \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))^2+\frac {1}{2} \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))^2-\frac {1}{2} b c \left (\frac {1}{4} x \left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))+\frac {3}{4} \left (\frac {1}{2} x \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))+\frac {(a+b \text {arcsinh}(c x))^2}{4 b c}-\frac {1}{4} b c x^2\right )-\frac {1}{4} b c \left (\frac {c^2 x^4}{4}+\frac {x^2}{2}\right )\right )\right )+\frac {1}{6} d^3 \left (c^2 x^2+1\right )^3 (a+b \text {arcsinh}(c x))^2-\frac {1}{3} b c d^3 \left (\frac {1}{6} x \left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))+\frac {5}{6} \left (\frac {1}{4} x \left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))+\frac {3}{4} \left (\frac {1}{2} x \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))+\frac {(a+b \text {arcsinh}(c x))^2}{4 b c}-\frac {1}{4} b c x^2\right )-\frac {1}{4} b c \left (\frac {c^2 x^4}{4}+\frac {x^2}{2}\right )\right )-\frac {b \left (c^2 x^2+1\right )^3}{36 c}\right )\) |
| \(\Big \downarrow \) 6190 |
| \(\displaystyle d^3 \left (-b c \int \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))dx+\frac {\int -(a+b \text {arcsinh}(c x))^2 \coth \left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c x)}{b}\right )d(a+b \text {arcsinh}(c x))}{b}+\frac {1}{4} \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))^2+\frac {1}{2} \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))^2-\frac {1}{2} b c \left (\frac {1}{4} x \left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))+\frac {3}{4} \left (\frac {1}{2} x \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))+\frac {(a+b \text {arcsinh}(c x))^2}{4 b c}-\frac {1}{4} b c x^2\right )-\frac {1}{4} b c \left (\frac {c^2 x^4}{4}+\frac {x^2}{2}\right )\right )\right )+\frac {1}{6} d^3 \left (c^2 x^2+1\right )^3 (a+b \text {arcsinh}(c x))^2-\frac {1}{3} b c d^3 \left (\frac {1}{6} x \left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))+\frac {5}{6} \left (\frac {1}{4} x \left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))+\frac {3}{4} \left (\frac {1}{2} x \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))+\frac {(a+b \text {arcsinh}(c x))^2}{4 b c}-\frac {1}{4} b c x^2\right )-\frac {1}{4} b c \left (\frac {c^2 x^4}{4}+\frac {x^2}{2}\right )\right )-\frac {b \left (c^2 x^2+1\right )^3}{36 c}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle d^3 \left (-b c \int \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))dx-\frac {\int (a+b \text {arcsinh}(c x))^2 \coth \left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c x)}{b}\right )d(a+b \text {arcsinh}(c x))}{b}+\frac {1}{4} \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))^2+\frac {1}{2} \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))^2-\frac {1}{2} b c \left (\frac {1}{4} x \left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))+\frac {3}{4} \left (\frac {1}{2} x \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))+\frac {(a+b \text {arcsinh}(c x))^2}{4 b c}-\frac {1}{4} b c x^2\right )-\frac {1}{4} b c \left (\frac {c^2 x^4}{4}+\frac {x^2}{2}\right )\right )\right )+\frac {1}{6} d^3 \left (c^2 x^2+1\right )^3 (a+b \text {arcsinh}(c x))^2-\frac {1}{3} b c d^3 \left (\frac {1}{6} x \left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))+\frac {5}{6} \left (\frac {1}{4} x \left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))+\frac {3}{4} \left (\frac {1}{2} x \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))+\frac {(a+b \text {arcsinh}(c x))^2}{4 b c}-\frac {1}{4} b c x^2\right )-\frac {1}{4} b c \left (\frac {c^2 x^4}{4}+\frac {x^2}{2}\right )\right )-\frac {b \left (c^2 x^2+1\right )^3}{36 c}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle d^3 \left (-b c \int \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))dx-\frac {\int -i (a+b \text {arcsinh}(c x))^2 \tan \left (\frac {i a}{b}-\frac {i (a+b \text {arcsinh}(c x))}{b}+\frac {\pi }{2}\right )d(a+b \text {arcsinh}(c x))}{b}+\frac {1}{4} \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))^2+\frac {1}{2} \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))^2-\frac {1}{2} b c \left (\frac {1}{4} x \left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))+\frac {3}{4} \left (\frac {1}{2} x \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))+\frac {(a+b \text {arcsinh}(c x))^2}{4 b c}-\frac {1}{4} b c x^2\right )-\frac {1}{4} b c \left (\frac {c^2 x^4}{4}+\frac {x^2}{2}\right )\right )\right )+\frac {1}{6} d^3 \left (c^2 x^2+1\right )^3 (a+b \text {arcsinh}(c x))^2-\frac {1}{3} b c d^3 \left (\frac {1}{6} x \left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))+\frac {5}{6} \left (\frac {1}{4} x \left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))+\frac {3}{4} \left (\frac {1}{2} x \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))+\frac {(a+b \text {arcsinh}(c x))^2}{4 b c}-\frac {1}{4} b c x^2\right )-\frac {1}{4} b c \left (\frac {c^2 x^4}{4}+\frac {x^2}{2}\right )\right )-\frac {b \left (c^2 x^2+1\right )^3}{36 c}\right )\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle d^3 \left (-b c \int \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))dx+\frac {i \int (a+b \text {arcsinh}(c x))^2 \tan \left (\frac {1}{2} \left (\frac {2 i a}{b}+\pi \right )-\frac {i (a+b \text {arcsinh}(c x))}{b}\right )d(a+b \text {arcsinh}(c x))}{b}+\frac {1}{4} \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))^2+\frac {1}{2} \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))^2-\frac {1}{2} b c \left (\frac {1}{4} x \left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))+\frac {3}{4} \left (\frac {1}{2} x \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))+\frac {(a+b \text {arcsinh}(c x))^2}{4 b c}-\frac {1}{4} b c x^2\right )-\frac {1}{4} b c \left (\frac {c^2 x^4}{4}+\frac {x^2}{2}\right )\right )\right )+\frac {1}{6} d^3 \left (c^2 x^2+1\right )^3 (a+b \text {arcsinh}(c x))^2-\frac {1}{3} b c d^3 \left (\frac {1}{6} x \left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))+\frac {5}{6} \left (\frac {1}{4} x \left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))+\frac {3}{4} \left (\frac {1}{2} x \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))+\frac {(a+b \text {arcsinh}(c x))^2}{4 b c}-\frac {1}{4} b c x^2\right )-\frac {1}{4} b c \left (\frac {c^2 x^4}{4}+\frac {x^2}{2}\right )\right )-\frac {b \left (c^2 x^2+1\right )^3}{36 c}\right )\) |
| \(\Big \downarrow \) 4201 |
| \(\displaystyle d^3 \left (-b c \int \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))dx+\frac {i \left (2 i \int \frac {e^{\frac {2 a}{b}-\frac {2 (a+b \text {arcsinh}(c x))}{b}-i \pi } (a+b \text {arcsinh}(c x))^2}{1+e^{\frac {2 a}{b}-\frac {2 (a+b \text {arcsinh}(c x))}{b}-i \pi }}d(a+b \text {arcsinh}(c x))-\frac {1}{3} i (a+b \text {arcsinh}(c x))^3\right )}{b}+\frac {1}{4} \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))^2+\frac {1}{2} \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))^2-\frac {1}{2} b c \left (\frac {1}{4} x \left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))+\frac {3}{4} \left (\frac {1}{2} x \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))+\frac {(a+b \text {arcsinh}(c x))^2}{4 b c}-\frac {1}{4} b c x^2\right )-\frac {1}{4} b c \left (\frac {c^2 x^4}{4}+\frac {x^2}{2}\right )\right )\right )+\frac {1}{6} d^3 \left (c^2 x^2+1\right )^3 (a+b \text {arcsinh}(c x))^2-\frac {1}{3} b c d^3 \left (\frac {1}{6} x \left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))+\frac {5}{6} \left (\frac {1}{4} x \left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))+\frac {3}{4} \left (\frac {1}{2} x \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))+\frac {(a+b \text {arcsinh}(c x))^2}{4 b c}-\frac {1}{4} b c x^2\right )-\frac {1}{4} b c \left (\frac {c^2 x^4}{4}+\frac {x^2}{2}\right )\right )-\frac {b \left (c^2 x^2+1\right )^3}{36 c}\right )\) |
| \(\Big \downarrow \) 2620 |
| \(\displaystyle d^3 \left (-b c \int \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))dx+\frac {i \left (2 i \left (b \int (a+b \text {arcsinh}(c x)) \log \left (1+e^{\frac {2 a}{b}-\frac {2 (a+b \text {arcsinh}(c x))}{b}-i \pi }\right )d(a+b \text {arcsinh}(c x))-\frac {1}{2} b (a+b \text {arcsinh}(c x))^2 \log \left (1+e^{-\frac {2 (a+b \text {arcsinh}(c x))}{b}+\frac {2 a}{b}-i \pi }\right )\right )-\frac {1}{3} i (a+b \text {arcsinh}(c x))^3\right )}{b}+\frac {1}{4} \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))^2+\frac {1}{2} \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))^2-\frac {1}{2} b c \left (\frac {1}{4} x \left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))+\frac {3}{4} \left (\frac {1}{2} x \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))+\frac {(a+b \text {arcsinh}(c x))^2}{4 b c}-\frac {1}{4} b c x^2\right )-\frac {1}{4} b c \left (\frac {c^2 x^4}{4}+\frac {x^2}{2}\right )\right )\right )+\frac {1}{6} d^3 \left (c^2 x^2+1\right )^3 (a+b \text {arcsinh}(c x))^2-\frac {1}{3} b c d^3 \left (\frac {1}{6} x \left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))+\frac {5}{6} \left (\frac {1}{4} x \left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))+\frac {3}{4} \left (\frac {1}{2} x \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))+\frac {(a+b \text {arcsinh}(c x))^2}{4 b c}-\frac {1}{4} b c x^2\right )-\frac {1}{4} b c \left (\frac {c^2 x^4}{4}+\frac {x^2}{2}\right )\right )-\frac {b \left (c^2 x^2+1\right )^3}{36 c}\right )\) |
| \(\Big \downarrow \) 3011 |
| \(\displaystyle d^3 \left (-b c \int \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))dx+\frac {i \left (2 i \left (b \left (\frac {1}{2} b (a+b \text {arcsinh}(c x)) \operatorname {PolyLog}\left (2,-e^{\frac {2 a}{b}-\frac {2 (a+b \text {arcsinh}(c x))}{b}-i \pi }\right )-\frac {1}{2} b \int \operatorname {PolyLog}\left (2,-e^{\frac {2 a}{b}-\frac {2 (a+b \text {arcsinh}(c x))}{b}-i \pi }\right )d(a+b \text {arcsinh}(c x))\right )-\frac {1}{2} b (a+b \text {arcsinh}(c x))^2 \log \left (1+e^{-\frac {2 (a+b \text {arcsinh}(c x))}{b}+\frac {2 a}{b}-i \pi }\right )\right )-\frac {1}{3} i (a+b \text {arcsinh}(c x))^3\right )}{b}+\frac {1}{4} \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))^2+\frac {1}{2} \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))^2-\frac {1}{2} b c \left (\frac {1}{4} x \left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))+\frac {3}{4} \left (\frac {1}{2} x \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))+\frac {(a+b \text {arcsinh}(c x))^2}{4 b c}-\frac {1}{4} b c x^2\right )-\frac {1}{4} b c \left (\frac {c^2 x^4}{4}+\frac {x^2}{2}\right )\right )\right )+\frac {1}{6} d^3 \left (c^2 x^2+1\right )^3 (a+b \text {arcsinh}(c x))^2-\frac {1}{3} b c d^3 \left (\frac {1}{6} x \left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))+\frac {5}{6} \left (\frac {1}{4} x \left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))+\frac {3}{4} \left (\frac {1}{2} x \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))+\frac {(a+b \text {arcsinh}(c x))^2}{4 b c}-\frac {1}{4} b c x^2\right )-\frac {1}{4} b c \left (\frac {c^2 x^4}{4}+\frac {x^2}{2}\right )\right )-\frac {b \left (c^2 x^2+1\right )^3}{36 c}\right )\) |
| \(\Big \downarrow \) 2720 |
| \(\displaystyle d^3 \left (\frac {i \left (2 i \left (b \left (\frac {1}{4} b^2 \int e^{-\frac {2 a}{b}+\frac {2 (a+b \text {arcsinh}(c x))}{b}+i \pi } \operatorname {PolyLog}(2,-a-b \text {arcsinh}(c x))de^{\frac {2 a}{b}-\frac {2 (a+b \text {arcsinh}(c x))}{b}-i \pi }+\frac {1}{2} b (a+b \text {arcsinh}(c x)) \operatorname {PolyLog}\left (2,-e^{\frac {2 a}{b}-\frac {2 (a+b \text {arcsinh}(c x))}{b}-i \pi }\right )\right )-\frac {1}{2} b (a+b \text {arcsinh}(c x))^2 \log \left (1+e^{-\frac {2 (a+b \text {arcsinh}(c x))}{b}+\frac {2 a}{b}-i \pi }\right )\right )-\frac {1}{3} i (a+b \text {arcsinh}(c x))^3\right )}{b}-b c \int \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))dx+\frac {1}{4} \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))^2+\frac {1}{2} \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))^2-\frac {1}{2} b c \left (\frac {1}{4} x \left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))+\frac {3}{4} \left (\frac {1}{2} x \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))+\frac {(a+b \text {arcsinh}(c x))^2}{4 b c}-\frac {1}{4} b c x^2\right )-\frac {1}{4} b c \left (\frac {c^2 x^4}{4}+\frac {x^2}{2}\right )\right )\right )+\frac {1}{6} d^3 \left (c^2 x^2+1\right )^3 (a+b \text {arcsinh}(c x))^2-\frac {1}{3} b c d^3 \left (\frac {1}{6} x \left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))+\frac {5}{6} \left (\frac {1}{4} x \left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))+\frac {3}{4} \left (\frac {1}{2} x \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))+\frac {(a+b \text {arcsinh}(c x))^2}{4 b c}-\frac {1}{4} b c x^2\right )-\frac {1}{4} b c \left (\frac {c^2 x^4}{4}+\frac {x^2}{2}\right )\right )-\frac {b \left (c^2 x^2+1\right )^3}{36 c}\right )\) |
| \(\Big \downarrow \) 6200 |
| \(\displaystyle d^3 \left (\frac {i \left (2 i \left (b \left (\frac {1}{4} b^2 \int e^{-\frac {2 a}{b}+\frac {2 (a+b \text {arcsinh}(c x))}{b}+i \pi } \operatorname {PolyLog}(2,-a-b \text {arcsinh}(c x))de^{\frac {2 a}{b}-\frac {2 (a+b \text {arcsinh}(c x))}{b}-i \pi }+\frac {1}{2} b (a+b \text {arcsinh}(c x)) \operatorname {PolyLog}\left (2,-e^{\frac {2 a}{b}-\frac {2 (a+b \text {arcsinh}(c x))}{b}-i \pi }\right )\right )-\frac {1}{2} b (a+b \text {arcsinh}(c x))^2 \log \left (1+e^{-\frac {2 (a+b \text {arcsinh}(c x))}{b}+\frac {2 a}{b}-i \pi }\right )\right )-\frac {1}{3} i (a+b \text {arcsinh}(c x))^3\right )}{b}-b c \left (\frac {1}{2} \int \frac {a+b \text {arcsinh}(c x)}{\sqrt {c^2 x^2+1}}dx-\frac {1}{2} b c \int xdx+\frac {1}{2} x \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))\right )+\frac {1}{4} \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))^2+\frac {1}{2} \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))^2-\frac {1}{2} b c \left (\frac {1}{4} x \left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))+\frac {3}{4} \left (\frac {1}{2} x \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))+\frac {(a+b \text {arcsinh}(c x))^2}{4 b c}-\frac {1}{4} b c x^2\right )-\frac {1}{4} b c \left (\frac {c^2 x^4}{4}+\frac {x^2}{2}\right )\right )\right )+\frac {1}{6} d^3 \left (c^2 x^2+1\right )^3 (a+b \text {arcsinh}(c x))^2-\frac {1}{3} b c d^3 \left (\frac {1}{6} x \left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))+\frac {5}{6} \left (\frac {1}{4} x \left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))+\frac {3}{4} \left (\frac {1}{2} x \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))+\frac {(a+b \text {arcsinh}(c x))^2}{4 b c}-\frac {1}{4} b c x^2\right )-\frac {1}{4} b c \left (\frac {c^2 x^4}{4}+\frac {x^2}{2}\right )\right )-\frac {b \left (c^2 x^2+1\right )^3}{36 c}\right )\) |
| \(\Big \downarrow \) 15 |
| \(\displaystyle d^3 \left (\frac {i \left (2 i \left (b \left (\frac {1}{4} b^2 \int e^{-\frac {2 a}{b}+\frac {2 (a+b \text {arcsinh}(c x))}{b}+i \pi } \operatorname {PolyLog}(2,-a-b \text {arcsinh}(c x))de^{\frac {2 a}{b}-\frac {2 (a+b \text {arcsinh}(c x))}{b}-i \pi }+\frac {1}{2} b (a+b \text {arcsinh}(c x)) \operatorname {PolyLog}\left (2,-e^{\frac {2 a}{b}-\frac {2 (a+b \text {arcsinh}(c x))}{b}-i \pi }\right )\right )-\frac {1}{2} b (a+b \text {arcsinh}(c x))^2 \log \left (1+e^{-\frac {2 (a+b \text {arcsinh}(c x))}{b}+\frac {2 a}{b}-i \pi }\right )\right )-\frac {1}{3} i (a+b \text {arcsinh}(c x))^3\right )}{b}-b c \left (\frac {1}{2} \int \frac {a+b \text {arcsinh}(c x)}{\sqrt {c^2 x^2+1}}dx+\frac {1}{2} x \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))-\frac {1}{4} b c x^2\right )+\frac {1}{4} \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))^2+\frac {1}{2} \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))^2-\frac {1}{2} b c \left (\frac {1}{4} x \left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))+\frac {3}{4} \left (\frac {1}{2} x \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))+\frac {(a+b \text {arcsinh}(c x))^2}{4 b c}-\frac {1}{4} b c x^2\right )-\frac {1}{4} b c \left (\frac {c^2 x^4}{4}+\frac {x^2}{2}\right )\right )\right )+\frac {1}{6} d^3 \left (c^2 x^2+1\right )^3 (a+b \text {arcsinh}(c x))^2-\frac {1}{3} b c d^3 \left (\frac {1}{6} x \left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))+\frac {5}{6} \left (\frac {1}{4} x \left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))+\frac {3}{4} \left (\frac {1}{2} x \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))+\frac {(a+b \text {arcsinh}(c x))^2}{4 b c}-\frac {1}{4} b c x^2\right )-\frac {1}{4} b c \left (\frac {c^2 x^4}{4}+\frac {x^2}{2}\right )\right )-\frac {b \left (c^2 x^2+1\right )^3}{36 c}\right )\) |
| \(\Big \downarrow \) 6198 |
| \(\displaystyle d^3 \left (\frac {i \left (2 i \left (b \left (\frac {1}{4} b^2 \int e^{-\frac {2 a}{b}+\frac {2 (a+b \text {arcsinh}(c x))}{b}+i \pi } \operatorname {PolyLog}(2,-a-b \text {arcsinh}(c x))de^{\frac {2 a}{b}-\frac {2 (a+b \text {arcsinh}(c x))}{b}-i \pi }+\frac {1}{2} b (a+b \text {arcsinh}(c x)) \operatorname {PolyLog}\left (2,-e^{\frac {2 a}{b}-\frac {2 (a+b \text {arcsinh}(c x))}{b}-i \pi }\right )\right )-\frac {1}{2} b (a+b \text {arcsinh}(c x))^2 \log \left (1+e^{-\frac {2 (a+b \text {arcsinh}(c x))}{b}+\frac {2 a}{b}-i \pi }\right )\right )-\frac {1}{3} i (a+b \text {arcsinh}(c x))^3\right )}{b}+\frac {1}{4} \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))^2+\frac {1}{2} \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))^2-b c \left (\frac {1}{2} x \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))+\frac {(a+b \text {arcsinh}(c x))^2}{4 b c}-\frac {1}{4} b c x^2\right )-\frac {1}{2} b c \left (\frac {1}{4} x \left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))+\frac {3}{4} \left (\frac {1}{2} x \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))+\frac {(a+b \text {arcsinh}(c x))^2}{4 b c}-\frac {1}{4} b c x^2\right )-\frac {1}{4} b c \left (\frac {c^2 x^4}{4}+\frac {x^2}{2}\right )\right )\right )+\frac {1}{6} d^3 \left (c^2 x^2+1\right )^3 (a+b \text {arcsinh}(c x))^2-\frac {1}{3} b c d^3 \left (\frac {1}{6} x \left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))+\frac {5}{6} \left (\frac {1}{4} x \left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))+\frac {3}{4} \left (\frac {1}{2} x \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))+\frac {(a+b \text {arcsinh}(c x))^2}{4 b c}-\frac {1}{4} b c x^2\right )-\frac {1}{4} b c \left (\frac {c^2 x^4}{4}+\frac {x^2}{2}\right )\right )-\frac {b \left (c^2 x^2+1\right )^3}{36 c}\right )\) |
| \(\Big \downarrow \) 7143 |
| \(\displaystyle d^3 \left (\frac {i \left (2 i \left (b \left (\frac {1}{4} b^2 \operatorname {PolyLog}(3,-a-b \text {arcsinh}(c x))+\frac {1}{2} b (a+b \text {arcsinh}(c x)) \operatorname {PolyLog}\left (2,-e^{\frac {2 a}{b}-\frac {2 (a+b \text {arcsinh}(c x))}{b}-i \pi }\right )\right )-\frac {1}{2} b (a+b \text {arcsinh}(c x))^2 \log \left (1+e^{-\frac {2 (a+b \text {arcsinh}(c x))}{b}+\frac {2 a}{b}-i \pi }\right )\right )-\frac {1}{3} i (a+b \text {arcsinh}(c x))^3\right )}{b}+\frac {1}{4} \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))^2+\frac {1}{2} \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))^2-b c \left (\frac {1}{2} x \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))+\frac {(a+b \text {arcsinh}(c x))^2}{4 b c}-\frac {1}{4} b c x^2\right )-\frac {1}{2} b c \left (\frac {1}{4} x \left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))+\frac {3}{4} \left (\frac {1}{2} x \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))+\frac {(a+b \text {arcsinh}(c x))^2}{4 b c}-\frac {1}{4} b c x^2\right )-\frac {1}{4} b c \left (\frac {c^2 x^4}{4}+\frac {x^2}{2}\right )\right )\right )+\frac {1}{6} d^3 \left (c^2 x^2+1\right )^3 (a+b \text {arcsinh}(c x))^2-\frac {1}{3} b c d^3 \left (\frac {1}{6} x \left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))+\frac {5}{6} \left (\frac {1}{4} x \left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))+\frac {3}{4} \left (\frac {1}{2} x \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))+\frac {(a+b \text {arcsinh}(c x))^2}{4 b c}-\frac {1}{4} b c x^2\right )-\frac {1}{4} b c \left (\frac {c^2 x^4}{4}+\frac {x^2}{2}\right )\right )-\frac {b \left (c^2 x^2+1\right )^3}{36 c}\right )\) |
(d^3*(1 + c^2*x^2)^3*(a + b*ArcSinh[c*x])^2)/6 - (b*c*d^3*(-1/36*(b*(1 + c ^2*x^2)^3)/c + (x*(1 + c^2*x^2)^(5/2)*(a + b*ArcSinh[c*x]))/6 + (5*(-1/4*( b*c*(x^2/2 + (c^2*x^4)/4)) + (x*(1 + c^2*x^2)^(3/2)*(a + b*ArcSinh[c*x]))/ 4 + (3*(-1/4*(b*c*x^2) + (x*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]))/2 + (a + b*ArcSinh[c*x])^2/(4*b*c)))/4))/6))/3 + d^3*(((1 + c^2*x^2)*(a + b*ArcS inh[c*x])^2)/2 + ((1 + c^2*x^2)^2*(a + b*ArcSinh[c*x])^2)/4 - b*c*(-1/4*(b *c*x^2) + (x*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]))/2 + (a + b*ArcSinh[c* x])^2/(4*b*c)) - (b*c*(-1/4*(b*c*(x^2/2 + (c^2*x^4)/4)) + (x*(1 + c^2*x^2) ^(3/2)*(a + b*ArcSinh[c*x]))/4 + (3*(-1/4*(b*c*x^2) + (x*Sqrt[1 + c^2*x^2] *(a + b*ArcSinh[c*x]))/2 + (a + b*ArcSinh[c*x])^2/(4*b*c)))/4))/2 + (I*((- 1/3*I)*(a + b*ArcSinh[c*x])^3 + (2*I)*(-1/2*(b*(a + b*ArcSinh[c*x])^2*Log[ 1 + E^((2*a)/b - I*Pi - (2*(a + b*ArcSinh[c*x]))/b)]) + b*((b*(a + b*ArcSi nh[c*x])*PolyLog[2, -E^((2*a)/b - I*Pi - (2*(a + b*ArcSinh[c*x]))/b)])/2 + (b^2*PolyLog[3, -a - b*ArcSinh[c*x]])/4))))/b)
3.3.21.3.1 Defintions of rubi rules used
Int[(a_.)*(x_)^(m_.), x_Symbol] :> Simp[a*(x^(m + 1)/(m + 1)), x] /; FreeQ[ {a, m}, x] && NeQ[m, -1]
Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a]) I nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[(x_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(a + b*x^2)^(p + 1)/ (2*b*(p + 1)), x] /; FreeQ[{a, b, p}, x] && NeQ[p, -1]
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Int[Expand Integrand[(c*x)^m*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, m}, x] && IGtQ[p , 0]
Int[(((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.))/ ((a_) + (b_.)*((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)), x_Symbol] :> Simp [((c + d*x)^m/(b*f*g*n*Log[F]))*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x] - Si mp[d*(m/(b*f*g*n*Log[F])) Int[(c + d*x)^(m - 1)*Log[1 + b*((F^(g*(e + f*x )))^n/a)], x], x] /; FreeQ[{F, a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]
Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Simp[v/D[v, x] Subst[Int[FunctionOfExponentialFunction[u, x]/x, x], x, v], x]] /; Funct ionOfExponentialQ[u, x] && !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; FreeQ [{a, m, n}, x] && IntegerQ[m*n]] && !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x)) *(F_)[v_] /; FreeQ[{a, b, c}, x] && InverseFunctionQ[F[x]]]
Int[Log[1 + (e_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.)]*((f_.) + (g_.) *(x_))^(m_.), x_Symbol] :> Simp[(-(f + g*x)^m)*(PolyLog[2, (-e)*(F^(c*(a + b*x)))^n]/(b*c*n*Log[F])), x] + Simp[g*(m/(b*c*n*Log[F])) Int[(f + g*x)^( m - 1)*PolyLog[2, (-e)*(F^(c*(a + b*x)))^n], x], x] /; FreeQ[{F, a, b, c, e , f, g, n}, x] && GtQ[m, 0]
Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + (Complex[0, fz_])*(f_.)*(x_)], x _Symbol] :> Simp[(-I)*((c + d*x)^(m + 1)/(d*(m + 1))), x] + Simp[2*I Int[ (c + d*x)^m*(E^(2*((-I)*e + f*fz*x))/(1 + E^(2*((-I)*e + f*fz*x)))), x], x] /; FreeQ[{c, d, e, f, fz}, x] && IGtQ[m, 0]
Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)/(x_), x_Symbol] :> Simp[1/b Subst[Int[x^n*Coth[-a/b + x/b], x], x, a + b*ArcSinh[c*x]], x] /; FreeQ[{a , b, c}, x] && IGtQ[n, 0]
Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)/Sqrt[(d_) + (e_.)*(x_)^2], x_ Symbol] :> Simp[(1/(b*c*(n + 1)))*Simp[Sqrt[1 + c^2*x^2]/Sqrt[d + e*x^2]]*( a + b*ArcSinh[c*x])^(n + 1), x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[e, c ^2*d] && NeQ[n, -1]
Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*Sqrt[(d_) + (e_.)*(x_)^2], x_ Symbol] :> Simp[x*Sqrt[d + e*x^2]*((a + b*ArcSinh[c*x])^n/2), x] + (Simp[(1 /2)*Simp[Sqrt[d + e*x^2]/Sqrt[1 + c^2*x^2]] Int[(a + b*ArcSinh[c*x])^n/Sq rt[1 + c^2*x^2], x], x] - Simp[b*c*(n/2)*Simp[Sqrt[d + e*x^2]/Sqrt[1 + c^2* x^2]] Int[x*(a + b*ArcSinh[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e }, x] && EqQ[e, c^2*d] && GtQ[n, 0]
Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[x*(d + e*x^2)^p*((a + b*ArcSinh[c*x])^n/(2*p + 1)), x] + (Simp[2*d*(p/(2*p + 1)) Int[(d + e*x^2)^(p - 1)*(a + b*ArcSinh[c*x])^n, x ], x] - Simp[b*c*(n/(2*p + 1))*Simp[(d + e*x^2)^p/(1 + c^2*x^2)^p] Int[x* (1 + c^2*x^2)^(p - 1/2)*(a + b*ArcSinh[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && GtQ[n, 0] && GtQ[p, 0]
Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d_) + (e_ .)*(x_)^2)^(p_.), x_Symbol] :> Simp[(f*x)^(m + 1)*(d + e*x^2)^p*((a + b*Arc Sinh[c*x])^n/(f*(m + 2*p + 1))), x] + (Simp[2*d*(p/(m + 2*p + 1)) Int[(f* x)^m*(d + e*x^2)^(p - 1)*(a + b*ArcSinh[c*x])^n, x], x] - Simp[b*c*(n/(f*(m + 2*p + 1)))*Simp[(d + e*x^2)^p/(1 + c^2*x^2)^p] Int[(f*x)^(m + 1)*(1 + c^2*x^2)^(p - 1/2)*(a + b*ArcSinh[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[e, c^2*d] && GtQ[n, 0] && GtQ[p, 0] && !LtQ[m, -1]
Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_S ymbol] :> Simp[PolyLog[n + 1, c*(a + b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d , e, n, p}, x] && EqQ[b*d, a*e]
Leaf count of result is larger than twice the leaf count of optimal. \(686\) vs. \(2(334)=668\).
Time = 0.33 (sec) , antiderivative size = 687, normalized size of antiderivative = 2.04
| method | result | size |
| parts | \(-\frac {d^{3} a b \,c^{5} x^{5} \sqrt {c^{2} x^{2}+1}}{18}-\frac {11 d^{3} a b \,c^{3} x^{3} \sqrt {c^{2} x^{2}+1}}{36}-\frac {25 d^{3} a b c x \sqrt {c^{2} x^{2}+1}}{24}-2 d^{3} b^{2} \operatorname {polylog}\left (3, -c x -\sqrt {c^{2} x^{2}+1}\right )-\frac {d^{3} b^{2} \operatorname {arcsinh}\left (c x \right )^{3}}{3}+\frac {25 d^{3} b^{2} \operatorname {arcsinh}\left (c x \right )^{2}}{48}-2 d^{3} b^{2} \operatorname {polylog}\left (3, c x +\sqrt {c^{2} x^{2}+1}\right )+2 d^{3} a b \,\operatorname {arcsinh}\left (c x \right ) \ln \left (1+c x +\sqrt {c^{2} x^{2}+1}\right )+2 d^{3} a b \,\operatorname {arcsinh}\left (c x \right ) \ln \left (1-c x -\sqrt {c^{2} x^{2}+1}\right )+\frac {d^{3} b^{2} \operatorname {arcsinh}\left (c x \right )^{2} c^{6} x^{6}}{6}+\frac {3 d^{3} b^{2} \operatorname {arcsinh}\left (c x \right )^{2} c^{4} x^{4}}{4}+\frac {3 d^{3} b^{2} \operatorname {arcsinh}\left (c x \right )^{2} c^{2} x^{2}}{2}+\frac {d^{3} a b \,\operatorname {arcsinh}\left (c x \right ) c^{6} x^{6}}{3}+\frac {3 d^{3} a b \,\operatorname {arcsinh}\left (c x \right ) c^{4} x^{4}}{2}+3 d^{3} a b \,\operatorname {arcsinh}\left (c x \right ) c^{2} x^{2}-\frac {d^{3} b^{2} \operatorname {arcsinh}\left (c x \right ) \sqrt {c^{2} x^{2}+1}\, c^{5} x^{5}}{18}-\frac {11 d^{3} b^{2} \operatorname {arcsinh}\left (c x \right ) \sqrt {c^{2} x^{2}+1}\, c^{3} x^{3}}{36}-\frac {25 d^{3} b^{2} \operatorname {arcsinh}\left (c x \right ) \sqrt {c^{2} x^{2}+1}\, c x}{24}+\frac {811 d^{3} b^{2}}{3456}+d^{3} a^{2} \left (\frac {c^{6} x^{6}}{6}+\frac {3 c^{4} x^{4}}{4}+\frac {3 c^{2} x^{2}}{2}+\ln \left (x \right )\right )+\frac {25 b^{2} c^{2} d^{3} x^{2}}{48}+\frac {11 b^{2} c^{4} d^{3} x^{4}}{144}+d^{3} b^{2} \operatorname {arcsinh}\left (c x \right )^{2} \ln \left (1-c x -\sqrt {c^{2} x^{2}+1}\right )+2 d^{3} b^{2} \operatorname {arcsinh}\left (c x \right ) \operatorname {polylog}\left (2, -c x -\sqrt {c^{2} x^{2}+1}\right )+\frac {25 d^{3} a b \,\operatorname {arcsinh}\left (c x \right )}{24}-d^{3} a b \operatorname {arcsinh}\left (c x \right )^{2}+2 d^{3} a b \operatorname {polylog}\left (2, c x +\sqrt {c^{2} x^{2}+1}\right )+2 d^{3} a b \operatorname {polylog}\left (2, -c x -\sqrt {c^{2} x^{2}+1}\right )+\frac {d^{3} b^{2} c^{6} x^{6}}{108}+d^{3} b^{2} \operatorname {arcsinh}\left (c x \right )^{2} \ln \left (1+c x +\sqrt {c^{2} x^{2}+1}\right )+2 d^{3} b^{2} \operatorname {arcsinh}\left (c x \right ) \operatorname {polylog}\left (2, c x +\sqrt {c^{2} x^{2}+1}\right )\) | \(687\) |
| derivativedivides | \(d^{3} a^{2} \left (\frac {c^{6} x^{6}}{6}+\frac {3 c^{4} x^{4}}{4}+\frac {3 c^{2} x^{2}}{2}+\ln \left (c x \right )\right )-\frac {d^{3} a b \,c^{5} x^{5} \sqrt {c^{2} x^{2}+1}}{18}-\frac {11 d^{3} a b \,c^{3} x^{3} \sqrt {c^{2} x^{2}+1}}{36}-\frac {25 d^{3} a b c x \sqrt {c^{2} x^{2}+1}}{24}-2 d^{3} b^{2} \operatorname {polylog}\left (3, -c x -\sqrt {c^{2} x^{2}+1}\right )-\frac {d^{3} b^{2} \operatorname {arcsinh}\left (c x \right )^{3}}{3}+\frac {25 d^{3} b^{2} \operatorname {arcsinh}\left (c x \right )^{2}}{48}-2 d^{3} b^{2} \operatorname {polylog}\left (3, c x +\sqrt {c^{2} x^{2}+1}\right )+2 d^{3} a b \,\operatorname {arcsinh}\left (c x \right ) \ln \left (1+c x +\sqrt {c^{2} x^{2}+1}\right )+2 d^{3} a b \,\operatorname {arcsinh}\left (c x \right ) \ln \left (1-c x -\sqrt {c^{2} x^{2}+1}\right )+\frac {d^{3} b^{2} \operatorname {arcsinh}\left (c x \right )^{2} c^{6} x^{6}}{6}+\frac {3 d^{3} b^{2} \operatorname {arcsinh}\left (c x \right )^{2} c^{4} x^{4}}{4}+\frac {3 d^{3} b^{2} \operatorname {arcsinh}\left (c x \right )^{2} c^{2} x^{2}}{2}+\frac {d^{3} a b \,\operatorname {arcsinh}\left (c x \right ) c^{6} x^{6}}{3}+\frac {3 d^{3} a b \,\operatorname {arcsinh}\left (c x \right ) c^{4} x^{4}}{2}+3 d^{3} a b \,\operatorname {arcsinh}\left (c x \right ) c^{2} x^{2}-\frac {d^{3} b^{2} \operatorname {arcsinh}\left (c x \right ) \sqrt {c^{2} x^{2}+1}\, c^{5} x^{5}}{18}-\frac {11 d^{3} b^{2} \operatorname {arcsinh}\left (c x \right ) \sqrt {c^{2} x^{2}+1}\, c^{3} x^{3}}{36}-\frac {25 d^{3} b^{2} \operatorname {arcsinh}\left (c x \right ) \sqrt {c^{2} x^{2}+1}\, c x}{24}+\frac {811 d^{3} b^{2}}{3456}+\frac {25 b^{2} c^{2} d^{3} x^{2}}{48}+\frac {11 b^{2} c^{4} d^{3} x^{4}}{144}+d^{3} b^{2} \operatorname {arcsinh}\left (c x \right )^{2} \ln \left (1-c x -\sqrt {c^{2} x^{2}+1}\right )+2 d^{3} b^{2} \operatorname {arcsinh}\left (c x \right ) \operatorname {polylog}\left (2, -c x -\sqrt {c^{2} x^{2}+1}\right )+\frac {25 d^{3} a b \,\operatorname {arcsinh}\left (c x \right )}{24}-d^{3} a b \operatorname {arcsinh}\left (c x \right )^{2}+2 d^{3} a b \operatorname {polylog}\left (2, c x +\sqrt {c^{2} x^{2}+1}\right )+2 d^{3} a b \operatorname {polylog}\left (2, -c x -\sqrt {c^{2} x^{2}+1}\right )+\frac {d^{3} b^{2} c^{6} x^{6}}{108}+d^{3} b^{2} \operatorname {arcsinh}\left (c x \right )^{2} \ln \left (1+c x +\sqrt {c^{2} x^{2}+1}\right )+2 d^{3} b^{2} \operatorname {arcsinh}\left (c x \right ) \operatorname {polylog}\left (2, c x +\sqrt {c^{2} x^{2}+1}\right )\) | \(689\) |
| default | \(d^{3} a^{2} \left (\frac {c^{6} x^{6}}{6}+\frac {3 c^{4} x^{4}}{4}+\frac {3 c^{2} x^{2}}{2}+\ln \left (c x \right )\right )-\frac {d^{3} a b \,c^{5} x^{5} \sqrt {c^{2} x^{2}+1}}{18}-\frac {11 d^{3} a b \,c^{3} x^{3} \sqrt {c^{2} x^{2}+1}}{36}-\frac {25 d^{3} a b c x \sqrt {c^{2} x^{2}+1}}{24}-2 d^{3} b^{2} \operatorname {polylog}\left (3, -c x -\sqrt {c^{2} x^{2}+1}\right )-\frac {d^{3} b^{2} \operatorname {arcsinh}\left (c x \right )^{3}}{3}+\frac {25 d^{3} b^{2} \operatorname {arcsinh}\left (c x \right )^{2}}{48}-2 d^{3} b^{2} \operatorname {polylog}\left (3, c x +\sqrt {c^{2} x^{2}+1}\right )+2 d^{3} a b \,\operatorname {arcsinh}\left (c x \right ) \ln \left (1+c x +\sqrt {c^{2} x^{2}+1}\right )+2 d^{3} a b \,\operatorname {arcsinh}\left (c x \right ) \ln \left (1-c x -\sqrt {c^{2} x^{2}+1}\right )+\frac {d^{3} b^{2} \operatorname {arcsinh}\left (c x \right )^{2} c^{6} x^{6}}{6}+\frac {3 d^{3} b^{2} \operatorname {arcsinh}\left (c x \right )^{2} c^{4} x^{4}}{4}+\frac {3 d^{3} b^{2} \operatorname {arcsinh}\left (c x \right )^{2} c^{2} x^{2}}{2}+\frac {d^{3} a b \,\operatorname {arcsinh}\left (c x \right ) c^{6} x^{6}}{3}+\frac {3 d^{3} a b \,\operatorname {arcsinh}\left (c x \right ) c^{4} x^{4}}{2}+3 d^{3} a b \,\operatorname {arcsinh}\left (c x \right ) c^{2} x^{2}-\frac {d^{3} b^{2} \operatorname {arcsinh}\left (c x \right ) \sqrt {c^{2} x^{2}+1}\, c^{5} x^{5}}{18}-\frac {11 d^{3} b^{2} \operatorname {arcsinh}\left (c x \right ) \sqrt {c^{2} x^{2}+1}\, c^{3} x^{3}}{36}-\frac {25 d^{3} b^{2} \operatorname {arcsinh}\left (c x \right ) \sqrt {c^{2} x^{2}+1}\, c x}{24}+\frac {811 d^{3} b^{2}}{3456}+\frac {25 b^{2} c^{2} d^{3} x^{2}}{48}+\frac {11 b^{2} c^{4} d^{3} x^{4}}{144}+d^{3} b^{2} \operatorname {arcsinh}\left (c x \right )^{2} \ln \left (1-c x -\sqrt {c^{2} x^{2}+1}\right )+2 d^{3} b^{2} \operatorname {arcsinh}\left (c x \right ) \operatorname {polylog}\left (2, -c x -\sqrt {c^{2} x^{2}+1}\right )+\frac {25 d^{3} a b \,\operatorname {arcsinh}\left (c x \right )}{24}-d^{3} a b \operatorname {arcsinh}\left (c x \right )^{2}+2 d^{3} a b \operatorname {polylog}\left (2, c x +\sqrt {c^{2} x^{2}+1}\right )+2 d^{3} a b \operatorname {polylog}\left (2, -c x -\sqrt {c^{2} x^{2}+1}\right )+\frac {d^{3} b^{2} c^{6} x^{6}}{108}+d^{3} b^{2} \operatorname {arcsinh}\left (c x \right )^{2} \ln \left (1+c x +\sqrt {c^{2} x^{2}+1}\right )+2 d^{3} b^{2} \operatorname {arcsinh}\left (c x \right ) \operatorname {polylog}\left (2, c x +\sqrt {c^{2} x^{2}+1}\right )\) | \(689\) |
-1/18*d^3*a*b*c^5*x^5*(c^2*x^2+1)^(1/2)-11/36*d^3*a*b*c^3*x^3*(c^2*x^2+1)^ (1/2)-25/24*d^3*a*b*c*x*(c^2*x^2+1)^(1/2)-2*d^3*b^2*polylog(3,-c*x-(c^2*x^ 2+1)^(1/2))-1/3*d^3*b^2*arcsinh(c*x)^3+25/48*d^3*b^2*arcsinh(c*x)^2-2*d^3* b^2*polylog(3,c*x+(c^2*x^2+1)^(1/2))+2*d^3*a*b*arcsinh(c*x)*ln(1+c*x+(c^2* x^2+1)^(1/2))+2*d^3*a*b*arcsinh(c*x)*ln(1-c*x-(c^2*x^2+1)^(1/2))+1/6*d^3*b ^2*arcsinh(c*x)^2*c^6*x^6+3/4*d^3*b^2*arcsinh(c*x)^2*c^4*x^4+3/2*d^3*b^2*a rcsinh(c*x)^2*c^2*x^2+1/3*d^3*a*b*arcsinh(c*x)*c^6*x^6+3/2*d^3*a*b*arcsinh (c*x)*c^4*x^4+3*d^3*a*b*arcsinh(c*x)*c^2*x^2-1/18*d^3*b^2*arcsinh(c*x)*(c^ 2*x^2+1)^(1/2)*c^5*x^5-11/36*d^3*b^2*arcsinh(c*x)*(c^2*x^2+1)^(1/2)*c^3*x^ 3-25/24*d^3*b^2*arcsinh(c*x)*(c^2*x^2+1)^(1/2)*c*x+811/3456*d^3*b^2+d^3*a^ 2*(1/6*c^6*x^6+3/4*c^4*x^4+3/2*c^2*x^2+ln(x))+25/48*b^2*c^2*d^3*x^2+11/144 *b^2*c^4*d^3*x^4+d^3*b^2*arcsinh(c*x)^2*ln(1-c*x-(c^2*x^2+1)^(1/2))+2*d^3* b^2*arcsinh(c*x)*polylog(2,-c*x-(c^2*x^2+1)^(1/2))+25/24*d^3*a*b*arcsinh(c *x)-d^3*a*b*arcsinh(c*x)^2+2*d^3*a*b*polylog(2,c*x+(c^2*x^2+1)^(1/2))+2*d^ 3*a*b*polylog(2,-c*x-(c^2*x^2+1)^(1/2))+1/108*d^3*b^2*c^6*x^6+d^3*b^2*arcs inh(c*x)^2*ln(1+c*x+(c^2*x^2+1)^(1/2))+2*d^3*b^2*arcsinh(c*x)*polylog(2,c* x+(c^2*x^2+1)^(1/2))
\[ \int \frac {\left (d+c^2 d x^2\right )^3 (a+b \text {arcsinh}(c x))^2}{x} \, dx=\int { \frac {{\left (c^{2} d x^{2} + d\right )}^{3} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2}}{x} \,d x } \]
integral((a^2*c^6*d^3*x^6 + 3*a^2*c^4*d^3*x^4 + 3*a^2*c^2*d^3*x^2 + a^2*d^ 3 + (b^2*c^6*d^3*x^6 + 3*b^2*c^4*d^3*x^4 + 3*b^2*c^2*d^3*x^2 + b^2*d^3)*ar csinh(c*x)^2 + 2*(a*b*c^6*d^3*x^6 + 3*a*b*c^4*d^3*x^4 + 3*a*b*c^2*d^3*x^2 + a*b*d^3)*arcsinh(c*x))/x, x)
\[ \int \frac {\left (d+c^2 d x^2\right )^3 (a+b \text {arcsinh}(c x))^2}{x} \, dx=d^{3} \left (\int \frac {a^{2}}{x}\, dx + \int 3 a^{2} c^{2} x\, dx + \int 3 a^{2} c^{4} x^{3}\, dx + \int a^{2} c^{6} x^{5}\, dx + \int \frac {b^{2} \operatorname {asinh}^{2}{\left (c x \right )}}{x}\, dx + \int \frac {2 a b \operatorname {asinh}{\left (c x \right )}}{x}\, dx + \int 3 b^{2} c^{2} x \operatorname {asinh}^{2}{\left (c x \right )}\, dx + \int 3 b^{2} c^{4} x^{3} \operatorname {asinh}^{2}{\left (c x \right )}\, dx + \int b^{2} c^{6} x^{5} \operatorname {asinh}^{2}{\left (c x \right )}\, dx + \int 6 a b c^{2} x \operatorname {asinh}{\left (c x \right )}\, dx + \int 6 a b c^{4} x^{3} \operatorname {asinh}{\left (c x \right )}\, dx + \int 2 a b c^{6} x^{5} \operatorname {asinh}{\left (c x \right )}\, dx\right ) \]
d**3*(Integral(a**2/x, x) + Integral(3*a**2*c**2*x, x) + Integral(3*a**2*c **4*x**3, x) + Integral(a**2*c**6*x**5, x) + Integral(b**2*asinh(c*x)**2/x , x) + Integral(2*a*b*asinh(c*x)/x, x) + Integral(3*b**2*c**2*x*asinh(c*x) **2, x) + Integral(3*b**2*c**4*x**3*asinh(c*x)**2, x) + Integral(b**2*c**6 *x**5*asinh(c*x)**2, x) + Integral(6*a*b*c**2*x*asinh(c*x), x) + Integral( 6*a*b*c**4*x**3*asinh(c*x), x) + Integral(2*a*b*c**6*x**5*asinh(c*x), x))
\[ \int \frac {\left (d+c^2 d x^2\right )^3 (a+b \text {arcsinh}(c x))^2}{x} \, dx=\int { \frac {{\left (c^{2} d x^{2} + d\right )}^{3} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2}}{x} \,d x } \]
1/6*a^2*c^6*d^3*x^6 + 3/4*a^2*c^4*d^3*x^4 + 3/2*a^2*c^2*d^3*x^2 + a^2*d^3* log(x) + integrate(b^2*c^6*d^3*x^5*log(c*x + sqrt(c^2*x^2 + 1))^2 + 2*a*b* c^6*d^3*x^5*log(c*x + sqrt(c^2*x^2 + 1)) + 3*b^2*c^4*d^3*x^3*log(c*x + sqr t(c^2*x^2 + 1))^2 + 6*a*b*c^4*d^3*x^3*log(c*x + sqrt(c^2*x^2 + 1)) + 3*b^2 *c^2*d^3*x*log(c*x + sqrt(c^2*x^2 + 1))^2 + 6*a*b*c^2*d^3*x*log(c*x + sqrt (c^2*x^2 + 1)) + b^2*d^3*log(c*x + sqrt(c^2*x^2 + 1))^2/x + 2*a*b*d^3*log( c*x + sqrt(c^2*x^2 + 1))/x, x)
Exception generated. \[ \int \frac {\left (d+c^2 d x^2\right )^3 (a+b \text {arcsinh}(c x))^2}{x} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value
Timed out. \[ \int \frac {\left (d+c^2 d x^2\right )^3 (a+b \text {arcsinh}(c x))^2}{x} \, dx=\int \frac {{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2\,{\left (d\,c^2\,x^2+d\right )}^3}{x} \,d x \]