Integrand size = 26, antiderivative size = 272 \[ \int \frac {\left (d+c^2 d x^2\right )^2 (a+b \text {arcsinh}(c x))^2}{x^3} \, dx=\frac {1}{4} b^2 c^4 d^2 x^2+\frac {1}{2} b c^3 d^2 x \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))-\frac {b c d^2 \left (1+c^2 x^2\right )^{3/2} (a+b \text {arcsinh}(c x))}{x}+\frac {1}{4} c^2 d^2 (a+b \text {arcsinh}(c x))^2+c^2 d^2 \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))^2-\frac {d^2 \left (1+c^2 x^2\right )^2 (a+b \text {arcsinh}(c x))^2}{2 x^2}+\frac {2 c^2 d^2 (a+b \text {arcsinh}(c x))^3}{3 b}+2 c^2 d^2 (a+b \text {arcsinh}(c x))^2 \log \left (1-e^{-2 \text {arcsinh}(c x)}\right )+b^2 c^2 d^2 \log (x)-2 b c^2 d^2 (a+b \text {arcsinh}(c x)) \operatorname {PolyLog}\left (2,e^{-2 \text {arcsinh}(c x)}\right )-b^2 c^2 d^2 \operatorname {PolyLog}\left (3,e^{-2 \text {arcsinh}(c x)}\right ) \]
1/4*b^2*c^4*d^2*x^2-b*c*d^2*(c^2*x^2+1)^(3/2)*(a+b*arcsinh(c*x))/x+1/4*c^2 *d^2*(a+b*arcsinh(c*x))^2+c^2*d^2*(c^2*x^2+1)*(a+b*arcsinh(c*x))^2-1/2*d^2 *(c^2*x^2+1)^2*(a+b*arcsinh(c*x))^2/x^2+2/3*c^2*d^2*(a+b*arcsinh(c*x))^3/b +2*c^2*d^2*(a+b*arcsinh(c*x))^2*ln(1-1/(c*x+(c^2*x^2+1)^(1/2))^2)+b^2*c^2* d^2*ln(x)-2*b*c^2*d^2*(a+b*arcsinh(c*x))*polylog(2,1/(c*x+(c^2*x^2+1)^(1/2 ))^2)-b^2*c^2*d^2*polylog(3,1/(c*x+(c^2*x^2+1)^(1/2))^2)+1/2*b*c^3*d^2*x*( a+b*arcsinh(c*x))*(c^2*x^2+1)^(1/2)
Time = 0.75 (sec) , antiderivative size = 329, normalized size of antiderivative = 1.21 \[ \int \frac {\left (d+c^2 d x^2\right )^2 (a+b \text {arcsinh}(c x))^2}{x^3} \, dx=\frac {1}{2} d^2 \left (-\frac {a^2}{x^2}+a^2 c^4 x^2+2 a b c^4 x^2 \text {arcsinh}(c x)-\frac {2 a b \left (c x \sqrt {1+c^2 x^2}+\text {arcsinh}(c x)\right )}{x^2}+\frac {1}{4} b^2 c^2 \left (1+2 \text {arcsinh}(c x)^2\right ) \cosh (2 \text {arcsinh}(c x))+4 a^2 c^2 \log (x)-\frac {b^2 \left (2 c x \sqrt {1+c^2 x^2} \text {arcsinh}(c x)+\text {arcsinh}(c x)^2-2 c^2 x^2 \log (c x)\right )}{x^2}-a b c^2 \left (c x \sqrt {1+c^2 x^2}+\log \left (-c x+\sqrt {1+c^2 x^2}\right )\right )-4 a b c^2 \left (\text {arcsinh}(c x) \left (\text {arcsinh}(c x)-2 \log \left (1-e^{2 \text {arcsinh}(c x)}\right )\right )-\operatorname {PolyLog}\left (2,e^{2 \text {arcsinh}(c x)}\right )\right )-\frac {2}{3} b^2 c^2 \left (2 \text {arcsinh}(c x)^2 \left (\text {arcsinh}(c x)-3 \log \left (1-e^{2 \text {arcsinh}(c x)}\right )\right )-6 \text {arcsinh}(c x) \operatorname {PolyLog}\left (2,e^{2 \text {arcsinh}(c x)}\right )+3 \operatorname {PolyLog}\left (3,e^{2 \text {arcsinh}(c x)}\right )\right )-\frac {1}{2} b^2 c^2 \text {arcsinh}(c x) \sinh (2 \text {arcsinh}(c x))\right ) \]
(d^2*(-(a^2/x^2) + a^2*c^4*x^2 + 2*a*b*c^4*x^2*ArcSinh[c*x] - (2*a*b*(c*x* Sqrt[1 + c^2*x^2] + ArcSinh[c*x]))/x^2 + (b^2*c^2*(1 + 2*ArcSinh[c*x]^2)*C osh[2*ArcSinh[c*x]])/4 + 4*a^2*c^2*Log[x] - (b^2*(2*c*x*Sqrt[1 + c^2*x^2]* ArcSinh[c*x] + ArcSinh[c*x]^2 - 2*c^2*x^2*Log[c*x]))/x^2 - a*b*c^2*(c*x*Sq rt[1 + c^2*x^2] + Log[-(c*x) + Sqrt[1 + c^2*x^2]]) - 4*a*b*c^2*(ArcSinh[c* x]*(ArcSinh[c*x] - 2*Log[1 - E^(2*ArcSinh[c*x])]) - PolyLog[2, E^(2*ArcSin h[c*x])]) - (2*b^2*c^2*(2*ArcSinh[c*x]^2*(ArcSinh[c*x] - 3*Log[1 - E^(2*Ar cSinh[c*x])]) - 6*ArcSinh[c*x]*PolyLog[2, E^(2*ArcSinh[c*x])] + 3*PolyLog[ 3, E^(2*ArcSinh[c*x])]))/3 - (b^2*c^2*ArcSinh[c*x]*Sinh[2*ArcSinh[c*x]])/2 ))/2
Result contains complex when optimal does not.
Time = 2.61 (sec) , antiderivative size = 375, normalized size of antiderivative = 1.38, number of steps used = 22, number of rules used = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.808, Rules used = {6222, 27, 6222, 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.
| \(\displaystyle \int \frac {\left (c^2 d x^2+d\right )^2 (a+b \text {arcsinh}(c x))^2}{x^3} \, dx\) |
| \(\Big \downarrow \) 6222 |
| \(\displaystyle b c d^2 \int \frac {\left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))}{x^2}dx+2 c^2 d \int \frac {d \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))^2}{x}dx-\frac {d^2 \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))^2}{2 x^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle b c d^2 \int \frac {\left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))}{x^2}dx+2 c^2 d^2 \int \frac {\left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))^2}{x}dx-\frac {d^2 \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))^2}{2 x^2}\) |
| \(\Big \downarrow \) 6222 |
| \(\displaystyle b c d^2 \left (3 c^2 \int \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))dx+b c \int \frac {c^2 x^2+1}{x}dx-\frac {\left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))}{x}\right )+2 c^2 d^2 \int \frac {\left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))^2}{x}dx-\frac {d^2 \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))^2}{2 x^2}\) |
| \(\Big \downarrow \) 244 |
| \(\displaystyle b c d^2 \left (3 c^2 \int \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))dx+b c \int \left (x c^2+\frac {1}{x}\right )dx-\frac {\left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))}{x}\right )+2 c^2 d^2 \int \frac {\left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))^2}{x}dx-\frac {d^2 \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))^2}{2 x^2}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle 2 c^2 d^2 \int \frac {\left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))^2}{x}dx+b c d^2 \left (3 c^2 \int \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))dx-\frac {\left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))}{x}+b c \left (\frac {c^2 x^2}{2}+\log (x)\right )\right )-\frac {d^2 \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))^2}{2 x^2}\) |
| \(\Big \downarrow \) 6200 |
| \(\displaystyle 2 c^2 d^2 \int \frac {\left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))^2}{x}dx+b c d^2 \left (3 c^2 \left (\frac {1}{2} \int \frac {a+b \text {arcsinh}(c x)}{\sqrt {c^2 x^2+1}}dx-\frac {1}{2} b c \int xdx+\frac {1}{2} x \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))\right )-\frac {\left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))}{x}+b c \left (\frac {c^2 x^2}{2}+\log (x)\right )\right )-\frac {d^2 \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))^2}{2 x^2}\) |
| \(\Big \downarrow \) 15 |
| \(\displaystyle 2 c^2 d^2 \int \frac {\left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))^2}{x}dx+b c d^2 \left (3 c^2 \left (\frac {1}{2} \int \frac {a+b \text {arcsinh}(c x)}{\sqrt {c^2 x^2+1}}dx+\frac {1}{2} x \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))-\frac {1}{4} b c x^2\right )-\frac {\left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))}{x}+b c \left (\frac {c^2 x^2}{2}+\log (x)\right )\right )-\frac {d^2 \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))^2}{2 x^2}\) |
| \(\Big \downarrow \) 6198 |
| \(\displaystyle 2 c^2 d^2 \int \frac {\left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))^2}{x}dx-\frac {d^2 \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))^2}{2 x^2}+b c d^2 \left (3 c^2 \left (\frac {1}{2} x \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))+\frac {(a+b \text {arcsinh}(c x))^2}{4 b c}-\frac {1}{4} b c x^2\right )-\frac {\left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))}{x}+b c \left (\frac {c^2 x^2}{2}+\log (x)\right )\right )\) |
| \(\Big \downarrow \) 6223 |
| \(\displaystyle 2 c^2 d^2 \left (-b c \int \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))dx+\int \frac {(a+b \text {arcsinh}(c x))^2}{x}dx+\frac {1}{2} \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))^2\right )-\frac {d^2 \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))^2}{2 x^2}+b c d^2 \left (3 c^2 \left (\frac {1}{2} x \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))+\frac {(a+b \text {arcsinh}(c x))^2}{4 b c}-\frac {1}{4} b c x^2\right )-\frac {\left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))}{x}+b c \left (\frac {c^2 x^2}{2}+\log (x)\right )\right )\) |
| \(\Big \downarrow \) 6190 |
| \(\displaystyle 2 c^2 d^2 \left (-b c \int \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))dx+\frac {\int -(a+b \text {arcsinh}(c x))^2 \coth \left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c x)}{b}\right )d(a+b \text {arcsinh}(c x))}{b}+\frac {1}{2} \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))^2\right )-\frac {d^2 \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))^2}{2 x^2}+b c d^2 \left (3 c^2 \left (\frac {1}{2} x \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))+\frac {(a+b \text {arcsinh}(c x))^2}{4 b c}-\frac {1}{4} b c x^2\right )-\frac {\left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))}{x}+b c \left (\frac {c^2 x^2}{2}+\log (x)\right )\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle 2 c^2 d^2 \left (-b c \int \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))dx-\frac {\int (a+b \text {arcsinh}(c x))^2 \coth \left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c x)}{b}\right )d(a+b \text {arcsinh}(c x))}{b}+\frac {1}{2} \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))^2\right )-\frac {d^2 \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))^2}{2 x^2}+b c d^2 \left (3 c^2 \left (\frac {1}{2} x \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))+\frac {(a+b \text {arcsinh}(c x))^2}{4 b c}-\frac {1}{4} b c x^2\right )-\frac {\left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))}{x}+b c \left (\frac {c^2 x^2}{2}+\log (x)\right )\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle 2 c^2 d^2 \left (-b c \int \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))dx-\frac {\int -i (a+b \text {arcsinh}(c x))^2 \tan \left (\frac {i a}{b}-\frac {i (a+b \text {arcsinh}(c x))}{b}+\frac {\pi }{2}\right )d(a+b \text {arcsinh}(c x))}{b}+\frac {1}{2} \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))^2\right )-\frac {d^2 \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))^2}{2 x^2}+b c d^2 \left (3 c^2 \left (\frac {1}{2} x \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))+\frac {(a+b \text {arcsinh}(c x))^2}{4 b c}-\frac {1}{4} b c x^2\right )-\frac {\left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))}{x}+b c \left (\frac {c^2 x^2}{2}+\log (x)\right )\right )\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle 2 c^2 d^2 \left (-b c \int \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))dx+\frac {i \int (a+b \text {arcsinh}(c x))^2 \tan \left (\frac {1}{2} \left (\frac {2 i a}{b}+\pi \right )-\frac {i (a+b \text {arcsinh}(c x))}{b}\right )d(a+b \text {arcsinh}(c x))}{b}+\frac {1}{2} \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))^2\right )-\frac {d^2 \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))^2}{2 x^2}+b c d^2 \left (3 c^2 \left (\frac {1}{2} x \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))+\frac {(a+b \text {arcsinh}(c x))^2}{4 b c}-\frac {1}{4} b c x^2\right )-\frac {\left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))}{x}+b c \left (\frac {c^2 x^2}{2}+\log (x)\right )\right )\) |
| \(\Big \downarrow \) 4201 |
| \(\displaystyle 2 c^2 d^2 \left (-b c \int \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))dx+\frac {i \left (2 i \int \frac {e^{\frac {2 a}{b}-\frac {2 (a+b \text {arcsinh}(c x))}{b}-i \pi } (a+b \text {arcsinh}(c x))^2}{1+e^{\frac {2 a}{b}-\frac {2 (a+b \text {arcsinh}(c x))}{b}-i \pi }}d(a+b \text {arcsinh}(c x))-\frac {1}{3} i (a+b \text {arcsinh}(c x))^3\right )}{b}+\frac {1}{2} \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))^2\right )-\frac {d^2 \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))^2}{2 x^2}+b c d^2 \left (3 c^2 \left (\frac {1}{2} x \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))+\frac {(a+b \text {arcsinh}(c x))^2}{4 b c}-\frac {1}{4} b c x^2\right )-\frac {\left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))}{x}+b c \left (\frac {c^2 x^2}{2}+\log (x)\right )\right )\) |
| \(\Big \downarrow \) 2620 |
| \(\displaystyle 2 c^2 d^2 \left (-b c \int \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))dx+\frac {i \left (2 i \left (b \int (a+b \text {arcsinh}(c x)) \log \left (1+e^{\frac {2 a}{b}-\frac {2 (a+b \text {arcsinh}(c x))}{b}-i \pi }\right )d(a+b \text {arcsinh}(c x))-\frac {1}{2} b (a+b \text {arcsinh}(c x))^2 \log \left (1+e^{-\frac {2 (a+b \text {arcsinh}(c x))}{b}+\frac {2 a}{b}-i \pi }\right )\right )-\frac {1}{3} i (a+b \text {arcsinh}(c x))^3\right )}{b}+\frac {1}{2} \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))^2\right )-\frac {d^2 \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))^2}{2 x^2}+b c d^2 \left (3 c^2 \left (\frac {1}{2} x \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))+\frac {(a+b \text {arcsinh}(c x))^2}{4 b c}-\frac {1}{4} b c x^2\right )-\frac {\left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))}{x}+b c \left (\frac {c^2 x^2}{2}+\log (x)\right )\right )\) |
| \(\Big \downarrow \) 3011 |
| \(\displaystyle 2 c^2 d^2 \left (-b c \int \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))dx+\frac {i \left (2 i \left (b \left (\frac {1}{2} b (a+b \text {arcsinh}(c x)) \operatorname {PolyLog}\left (2,-e^{\frac {2 a}{b}-\frac {2 (a+b \text {arcsinh}(c x))}{b}-i \pi }\right )-\frac {1}{2} b \int \operatorname {PolyLog}\left (2,-e^{\frac {2 a}{b}-\frac {2 (a+b \text {arcsinh}(c x))}{b}-i \pi }\right )d(a+b \text {arcsinh}(c x))\right )-\frac {1}{2} b (a+b \text {arcsinh}(c x))^2 \log \left (1+e^{-\frac {2 (a+b \text {arcsinh}(c x))}{b}+\frac {2 a}{b}-i \pi }\right )\right )-\frac {1}{3} i (a+b \text {arcsinh}(c x))^3\right )}{b}+\frac {1}{2} \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))^2\right )-\frac {d^2 \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))^2}{2 x^2}+b c d^2 \left (3 c^2 \left (\frac {1}{2} x \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))+\frac {(a+b \text {arcsinh}(c x))^2}{4 b c}-\frac {1}{4} b c x^2\right )-\frac {\left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))}{x}+b c \left (\frac {c^2 x^2}{2}+\log (x)\right )\right )\) |
| \(\Big \downarrow \) 2720 |
| \(\displaystyle 2 c^2 d^2 \left (\frac {i \left (2 i \left (b \left (\frac {1}{4} b^2 \int e^{-\frac {2 a}{b}+\frac {2 (a+b \text {arcsinh}(c x))}{b}+i \pi } \operatorname {PolyLog}(2,-a-b \text {arcsinh}(c x))de^{\frac {2 a}{b}-\frac {2 (a+b \text {arcsinh}(c x))}{b}-i \pi }+\frac {1}{2} b (a+b \text {arcsinh}(c x)) \operatorname {PolyLog}\left (2,-e^{\frac {2 a}{b}-\frac {2 (a+b \text {arcsinh}(c x))}{b}-i \pi }\right )\right )-\frac {1}{2} b (a+b \text {arcsinh}(c x))^2 \log \left (1+e^{-\frac {2 (a+b \text {arcsinh}(c x))}{b}+\frac {2 a}{b}-i \pi }\right )\right )-\frac {1}{3} i (a+b \text {arcsinh}(c x))^3\right )}{b}-b c \int \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))dx+\frac {1}{2} \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))^2\right )-\frac {d^2 \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))^2}{2 x^2}+b c d^2 \left (3 c^2 \left (\frac {1}{2} x \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))+\frac {(a+b \text {arcsinh}(c x))^2}{4 b c}-\frac {1}{4} b c x^2\right )-\frac {\left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))}{x}+b c \left (\frac {c^2 x^2}{2}+\log (x)\right )\right )\) |
| \(\Big \downarrow \) 6200 |
| \(\displaystyle 2 c^2 d^2 \left (\frac {i \left (2 i \left (b \left (\frac {1}{4} b^2 \int e^{-\frac {2 a}{b}+\frac {2 (a+b \text {arcsinh}(c x))}{b}+i \pi } \operatorname {PolyLog}(2,-a-b \text {arcsinh}(c x))de^{\frac {2 a}{b}-\frac {2 (a+b \text {arcsinh}(c x))}{b}-i \pi }+\frac {1}{2} b (a+b \text {arcsinh}(c x)) \operatorname {PolyLog}\left (2,-e^{\frac {2 a}{b}-\frac {2 (a+b \text {arcsinh}(c x))}{b}-i \pi }\right )\right )-\frac {1}{2} b (a+b \text {arcsinh}(c x))^2 \log \left (1+e^{-\frac {2 (a+b \text {arcsinh}(c x))}{b}+\frac {2 a}{b}-i \pi }\right )\right )-\frac {1}{3} i (a+b \text {arcsinh}(c x))^3\right )}{b}-b c \left (\frac {1}{2} \int \frac {a+b \text {arcsinh}(c x)}{\sqrt {c^2 x^2+1}}dx-\frac {1}{2} b c \int xdx+\frac {1}{2} x \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))\right )+\frac {1}{2} \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))^2\right )-\frac {d^2 \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))^2}{2 x^2}+b c d^2 \left (3 c^2 \left (\frac {1}{2} x \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))+\frac {(a+b \text {arcsinh}(c x))^2}{4 b c}-\frac {1}{4} b c x^2\right )-\frac {\left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))}{x}+b c \left (\frac {c^2 x^2}{2}+\log (x)\right )\right )\) |
| \(\Big \downarrow \) 15 |
| \(\displaystyle 2 c^2 d^2 \left (\frac {i \left (2 i \left (b \left (\frac {1}{4} b^2 \int e^{-\frac {2 a}{b}+\frac {2 (a+b \text {arcsinh}(c x))}{b}+i \pi } \operatorname {PolyLog}(2,-a-b \text {arcsinh}(c x))de^{\frac {2 a}{b}-\frac {2 (a+b \text {arcsinh}(c x))}{b}-i \pi }+\frac {1}{2} b (a+b \text {arcsinh}(c x)) \operatorname {PolyLog}\left (2,-e^{\frac {2 a}{b}-\frac {2 (a+b \text {arcsinh}(c x))}{b}-i \pi }\right )\right )-\frac {1}{2} b (a+b \text {arcsinh}(c x))^2 \log \left (1+e^{-\frac {2 (a+b \text {arcsinh}(c x))}{b}+\frac {2 a}{b}-i \pi }\right )\right )-\frac {1}{3} i (a+b \text {arcsinh}(c x))^3\right )}{b}-b c \left (\frac {1}{2} \int \frac {a+b \text {arcsinh}(c x)}{\sqrt {c^2 x^2+1}}dx+\frac {1}{2} x \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))-\frac {1}{4} b c x^2\right )+\frac {1}{2} \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))^2\right )-\frac {d^2 \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))^2}{2 x^2}+b c d^2 \left (3 c^2 \left (\frac {1}{2} x \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))+\frac {(a+b \text {arcsinh}(c x))^2}{4 b c}-\frac {1}{4} b c x^2\right )-\frac {\left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))}{x}+b c \left (\frac {c^2 x^2}{2}+\log (x)\right )\right )\) |
| \(\Big \downarrow \) 6198 |
| \(\displaystyle 2 c^2 d^2 \left (\frac {i \left (2 i \left (b \left (\frac {1}{4} b^2 \int e^{-\frac {2 a}{b}+\frac {2 (a+b \text {arcsinh}(c x))}{b}+i \pi } \operatorname {PolyLog}(2,-a-b \text {arcsinh}(c x))de^{\frac {2 a}{b}-\frac {2 (a+b \text {arcsinh}(c x))}{b}-i \pi }+\frac {1}{2} b (a+b \text {arcsinh}(c x)) \operatorname {PolyLog}\left (2,-e^{\frac {2 a}{b}-\frac {2 (a+b \text {arcsinh}(c x))}{b}-i \pi }\right )\right )-\frac {1}{2} b (a+b \text {arcsinh}(c x))^2 \log \left (1+e^{-\frac {2 (a+b \text {arcsinh}(c x))}{b}+\frac {2 a}{b}-i \pi }\right )\right )-\frac {1}{3} i (a+b \text {arcsinh}(c x))^3\right )}{b}+\frac {1}{2} \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))^2-b c \left (\frac {1}{2} x \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))+\frac {(a+b \text {arcsinh}(c x))^2}{4 b c}-\frac {1}{4} b c x^2\right )\right )-\frac {d^2 \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))^2}{2 x^2}+b c d^2 \left (3 c^2 \left (\frac {1}{2} x \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))+\frac {(a+b \text {arcsinh}(c x))^2}{4 b c}-\frac {1}{4} b c x^2\right )-\frac {\left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))}{x}+b c \left (\frac {c^2 x^2}{2}+\log (x)\right )\right )\) |
| \(\Big \downarrow \) 7143 |
| \(\displaystyle 2 c^2 d^2 \left (\frac {i \left (2 i \left (b \left (\frac {1}{4} b^2 \operatorname {PolyLog}(3,-a-b \text {arcsinh}(c x))+\frac {1}{2} b (a+b \text {arcsinh}(c x)) \operatorname {PolyLog}\left (2,-e^{\frac {2 a}{b}-\frac {2 (a+b \text {arcsinh}(c x))}{b}-i \pi }\right )\right )-\frac {1}{2} b (a+b \text {arcsinh}(c x))^2 \log \left (1+e^{-\frac {2 (a+b \text {arcsinh}(c x))}{b}+\frac {2 a}{b}-i \pi }\right )\right )-\frac {1}{3} i (a+b \text {arcsinh}(c x))^3\right )}{b}+\frac {1}{2} \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))^2-b c \left (\frac {1}{2} x \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))+\frac {(a+b \text {arcsinh}(c x))^2}{4 b c}-\frac {1}{4} b c x^2\right )\right )-\frac {d^2 \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))^2}{2 x^2}+b c d^2 \left (3 c^2 \left (\frac {1}{2} x \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))+\frac {(a+b \text {arcsinh}(c x))^2}{4 b c}-\frac {1}{4} b c x^2\right )-\frac {\left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))}{x}+b c \left (\frac {c^2 x^2}{2}+\log (x)\right )\right )\) |
-1/2*(d^2*(1 + c^2*x^2)^2*(a + b*ArcSinh[c*x])^2)/x^2 + b*c*d^2*(-(((1 + c ^2*x^2)^(3/2)*(a + b*ArcSinh[c*x]))/x) + 3*c^2*(-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*((c^2*x^2)/2 + Log[x])) + 2*c^2*d^2*(((1 + c^2*x^2)*(a + b*ArcSinh[c*x]) ^2)/2 - 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)) + (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*ArcSinh[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)
3.3.14.3.1 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[((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_.)*((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 + 1))), x] + (-Simp[2*e*(p/(f^2*(m + 1))) Int[(f*x)^(m + 2)*(d + e*x^2)^(p - 1)*(a + b*ArcSinh[c*x])^n, x], x] - Simp[b*c*(n/(f*( m + 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}, x] && EqQ[e, c^2*d] && GtQ[n, 0] && GtQ[p, 0] && LtQ[m, -1]
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. \(583\) vs. \(2(285)=570\).
Time = 0.28 (sec) , antiderivative size = 584, normalized size of antiderivative = 2.15
| method | result | size |
| derivativedivides | \(c^{2} \left (d^{2} a^{2} \left (\frac {c^{2} x^{2}}{2}+2 \ln \left (c x \right )-\frac {1}{2 c^{2} x^{2}}\right )+d^{2} b^{2} \left (-\frac {2 \operatorname {arcsinh}\left (c x \right )^{3}}{3}+\frac {\left (2 \operatorname {arcsinh}\left (c x \right )^{2}-2 \,\operatorname {arcsinh}\left (c x \right )+1\right ) \left (2 c^{2} x^{2}+1+2 c x \sqrt {c^{2} x^{2}+1}\right )}{16}+\frac {\left (-2 c x \sqrt {c^{2} x^{2}+1}+2 c^{2} x^{2}+1\right ) \left (2 \operatorname {arcsinh}\left (c x \right )^{2}+2 \,\operatorname {arcsinh}\left (c x \right )+1\right )}{16}-\frac {\operatorname {arcsinh}\left (c x \right ) \left (-2 c^{2} x^{2}+2 c x \sqrt {c^{2} x^{2}+1}+\operatorname {arcsinh}\left (c x \right )\right )}{2 c^{2} x^{2}}+\ln \left (1+c x +\sqrt {c^{2} x^{2}+1}\right )-2 \ln \left (c x +\sqrt {c^{2} x^{2}+1}\right )+\ln \left (c x +\sqrt {c^{2} x^{2}+1}-1\right )+2 \operatorname {arcsinh}\left (c x \right )^{2} \ln \left (1+c x +\sqrt {c^{2} x^{2}+1}\right )+4 \,\operatorname {arcsinh}\left (c x \right ) \operatorname {polylog}\left (2, -c x -\sqrt {c^{2} x^{2}+1}\right )-4 \operatorname {polylog}\left (3, -c x -\sqrt {c^{2} x^{2}+1}\right )+2 \operatorname {arcsinh}\left (c x \right )^{2} \ln \left (1-c x -\sqrt {c^{2} x^{2}+1}\right )+4 \,\operatorname {arcsinh}\left (c x \right ) \operatorname {polylog}\left (2, c x +\sqrt {c^{2} x^{2}+1}\right )-4 \operatorname {polylog}\left (3, c x +\sqrt {c^{2} x^{2}+1}\right )\right )+2 d^{2} a b \left (-\operatorname {arcsinh}\left (c x \right )^{2}+\frac {\left (-1+2 \,\operatorname {arcsinh}\left (c x \right )\right ) \left (2 c^{2} x^{2}+1+2 c x \sqrt {c^{2} x^{2}+1}\right )}{16}+\frac {\left (-2 c x \sqrt {c^{2} x^{2}+1}+2 c^{2} x^{2}+1\right ) \left (1+2 \,\operatorname {arcsinh}\left (c x \right )\right )}{16}-\frac {c x \sqrt {c^{2} x^{2}+1}-c^{2} x^{2}+\operatorname {arcsinh}\left (c x \right )}{2 c^{2} x^{2}}+2 \,\operatorname {arcsinh}\left (c x \right ) \ln \left (1+c x +\sqrt {c^{2} x^{2}+1}\right )+2 \operatorname {polylog}\left (2, -c x -\sqrt {c^{2} x^{2}+1}\right )+2 \,\operatorname {arcsinh}\left (c x \right ) \ln \left (1-c x -\sqrt {c^{2} x^{2}+1}\right )+2 \operatorname {polylog}\left (2, c x +\sqrt {c^{2} x^{2}+1}\right )\right )\right )\) | \(584\) |
| default | \(c^{2} \left (d^{2} a^{2} \left (\frac {c^{2} x^{2}}{2}+2 \ln \left (c x \right )-\frac {1}{2 c^{2} x^{2}}\right )+d^{2} b^{2} \left (-\frac {2 \operatorname {arcsinh}\left (c x \right )^{3}}{3}+\frac {\left (2 \operatorname {arcsinh}\left (c x \right )^{2}-2 \,\operatorname {arcsinh}\left (c x \right )+1\right ) \left (2 c^{2} x^{2}+1+2 c x \sqrt {c^{2} x^{2}+1}\right )}{16}+\frac {\left (-2 c x \sqrt {c^{2} x^{2}+1}+2 c^{2} x^{2}+1\right ) \left (2 \operatorname {arcsinh}\left (c x \right )^{2}+2 \,\operatorname {arcsinh}\left (c x \right )+1\right )}{16}-\frac {\operatorname {arcsinh}\left (c x \right ) \left (-2 c^{2} x^{2}+2 c x \sqrt {c^{2} x^{2}+1}+\operatorname {arcsinh}\left (c x \right )\right )}{2 c^{2} x^{2}}+\ln \left (1+c x +\sqrt {c^{2} x^{2}+1}\right )-2 \ln \left (c x +\sqrt {c^{2} x^{2}+1}\right )+\ln \left (c x +\sqrt {c^{2} x^{2}+1}-1\right )+2 \operatorname {arcsinh}\left (c x \right )^{2} \ln \left (1+c x +\sqrt {c^{2} x^{2}+1}\right )+4 \,\operatorname {arcsinh}\left (c x \right ) \operatorname {polylog}\left (2, -c x -\sqrt {c^{2} x^{2}+1}\right )-4 \operatorname {polylog}\left (3, -c x -\sqrt {c^{2} x^{2}+1}\right )+2 \operatorname {arcsinh}\left (c x \right )^{2} \ln \left (1-c x -\sqrt {c^{2} x^{2}+1}\right )+4 \,\operatorname {arcsinh}\left (c x \right ) \operatorname {polylog}\left (2, c x +\sqrt {c^{2} x^{2}+1}\right )-4 \operatorname {polylog}\left (3, c x +\sqrt {c^{2} x^{2}+1}\right )\right )+2 d^{2} a b \left (-\operatorname {arcsinh}\left (c x \right )^{2}+\frac {\left (-1+2 \,\operatorname {arcsinh}\left (c x \right )\right ) \left (2 c^{2} x^{2}+1+2 c x \sqrt {c^{2} x^{2}+1}\right )}{16}+\frac {\left (-2 c x \sqrt {c^{2} x^{2}+1}+2 c^{2} x^{2}+1\right ) \left (1+2 \,\operatorname {arcsinh}\left (c x \right )\right )}{16}-\frac {c x \sqrt {c^{2} x^{2}+1}-c^{2} x^{2}+\operatorname {arcsinh}\left (c x \right )}{2 c^{2} x^{2}}+2 \,\operatorname {arcsinh}\left (c x \right ) \ln \left (1+c x +\sqrt {c^{2} x^{2}+1}\right )+2 \operatorname {polylog}\left (2, -c x -\sqrt {c^{2} x^{2}+1}\right )+2 \,\operatorname {arcsinh}\left (c x \right ) \ln \left (1-c x -\sqrt {c^{2} x^{2}+1}\right )+2 \operatorname {polylog}\left (2, c x +\sqrt {c^{2} x^{2}+1}\right )\right )\right )\) | \(584\) |
| parts | \(d^{2} a^{2} \left (\frac {c^{4} x^{2}}{2}-\frac {1}{2 x^{2}}+2 c^{2} \ln \left (x \right )\right )+d^{2} b^{2} c^{2} \left (-\frac {2 \operatorname {arcsinh}\left (c x \right )^{3}}{3}+\frac {\left (2 \operatorname {arcsinh}\left (c x \right )^{2}-2 \,\operatorname {arcsinh}\left (c x \right )+1\right ) \left (2 c^{2} x^{2}+1+2 c x \sqrt {c^{2} x^{2}+1}\right )}{16}+\frac {\left (-2 c x \sqrt {c^{2} x^{2}+1}+2 c^{2} x^{2}+1\right ) \left (2 \operatorname {arcsinh}\left (c x \right )^{2}+2 \,\operatorname {arcsinh}\left (c x \right )+1\right )}{16}-\frac {\operatorname {arcsinh}\left (c x \right ) \left (-2 c^{2} x^{2}+2 c x \sqrt {c^{2} x^{2}+1}+\operatorname {arcsinh}\left (c x \right )\right )}{2 c^{2} x^{2}}+\ln \left (1+c x +\sqrt {c^{2} x^{2}+1}\right )-2 \ln \left (c x +\sqrt {c^{2} x^{2}+1}\right )+\ln \left (c x +\sqrt {c^{2} x^{2}+1}-1\right )+2 \operatorname {arcsinh}\left (c x \right )^{2} \ln \left (1+c x +\sqrt {c^{2} x^{2}+1}\right )+4 \,\operatorname {arcsinh}\left (c x \right ) \operatorname {polylog}\left (2, -c x -\sqrt {c^{2} x^{2}+1}\right )-4 \operatorname {polylog}\left (3, -c x -\sqrt {c^{2} x^{2}+1}\right )+2 \operatorname {arcsinh}\left (c x \right )^{2} \ln \left (1-c x -\sqrt {c^{2} x^{2}+1}\right )+4 \,\operatorname {arcsinh}\left (c x \right ) \operatorname {polylog}\left (2, c x +\sqrt {c^{2} x^{2}+1}\right )-4 \operatorname {polylog}\left (3, c x +\sqrt {c^{2} x^{2}+1}\right )\right )+2 d^{2} a b \,c^{2} \left (-\operatorname {arcsinh}\left (c x \right )^{2}+\frac {\left (-1+2 \,\operatorname {arcsinh}\left (c x \right )\right ) \left (2 c^{2} x^{2}+1+2 c x \sqrt {c^{2} x^{2}+1}\right )}{16}+\frac {\left (-2 c x \sqrt {c^{2} x^{2}+1}+2 c^{2} x^{2}+1\right ) \left (1+2 \,\operatorname {arcsinh}\left (c x \right )\right )}{16}-\frac {c x \sqrt {c^{2} x^{2}+1}-c^{2} x^{2}+\operatorname {arcsinh}\left (c x \right )}{2 c^{2} x^{2}}+2 \,\operatorname {arcsinh}\left (c x \right ) \ln \left (1+c x +\sqrt {c^{2} x^{2}+1}\right )+2 \operatorname {polylog}\left (2, -c x -\sqrt {c^{2} x^{2}+1}\right )+2 \,\operatorname {arcsinh}\left (c x \right ) \ln \left (1-c x -\sqrt {c^{2} x^{2}+1}\right )+2 \operatorname {polylog}\left (2, c x +\sqrt {c^{2} x^{2}+1}\right )\right )\) | \(584\) |
c^2*(d^2*a^2*(1/2*c^2*x^2+2*ln(c*x)-1/2/c^2/x^2)+d^2*b^2*(-2/3*arcsinh(c*x )^3+1/16*(2*arcsinh(c*x)^2-2*arcsinh(c*x)+1)*(2*c^2*x^2+1+2*c*x*(c^2*x^2+1 )^(1/2))+1/16*(-2*c*x*(c^2*x^2+1)^(1/2)+2*c^2*x^2+1)*(2*arcsinh(c*x)^2+2*a rcsinh(c*x)+1)-1/2*arcsinh(c*x)*(-2*c^2*x^2+2*c*x*(c^2*x^2+1)^(1/2)+arcsin h(c*x))/c^2/x^2+ln(1+c*x+(c^2*x^2+1)^(1/2))-2*ln(c*x+(c^2*x^2+1)^(1/2))+ln (c*x+(c^2*x^2+1)^(1/2)-1)+2*arcsinh(c*x)^2*ln(1+c*x+(c^2*x^2+1)^(1/2))+4*a rcsinh(c*x)*polylog(2,-c*x-(c^2*x^2+1)^(1/2))-4*polylog(3,-c*x-(c^2*x^2+1) ^(1/2))+2*arcsinh(c*x)^2*ln(1-c*x-(c^2*x^2+1)^(1/2))+4*arcsinh(c*x)*polylo g(2,c*x+(c^2*x^2+1)^(1/2))-4*polylog(3,c*x+(c^2*x^2+1)^(1/2)))+2*d^2*a*b*( -arcsinh(c*x)^2+1/16*(-1+2*arcsinh(c*x))*(2*c^2*x^2+1+2*c*x*(c^2*x^2+1)^(1 /2))+1/16*(-2*c*x*(c^2*x^2+1)^(1/2)+2*c^2*x^2+1)*(1+2*arcsinh(c*x))-1/2*(c *x*(c^2*x^2+1)^(1/2)-c^2*x^2+arcsinh(c*x))/c^2/x^2+2*arcsinh(c*x)*ln(1+c*x +(c^2*x^2+1)^(1/2))+2*polylog(2,-c*x-(c^2*x^2+1)^(1/2))+2*arcsinh(c*x)*ln( 1-c*x-(c^2*x^2+1)^(1/2))+2*polylog(2,c*x+(c^2*x^2+1)^(1/2))))
\[ \int \frac {\left (d+c^2 d x^2\right )^2 (a+b \text {arcsinh}(c x))^2}{x^3} \, dx=\int { \frac {{\left (c^{2} d x^{2} + d\right )}^{2} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2}}{x^{3}} \,d x } \]
integral((a^2*c^4*d^2*x^4 + 2*a^2*c^2*d^2*x^2 + a^2*d^2 + (b^2*c^4*d^2*x^4 + 2*b^2*c^2*d^2*x^2 + b^2*d^2)*arcsinh(c*x)^2 + 2*(a*b*c^4*d^2*x^4 + 2*a* b*c^2*d^2*x^2 + a*b*d^2)*arcsinh(c*x))/x^3, x)
\[ \int \frac {\left (d+c^2 d x^2\right )^2 (a+b \text {arcsinh}(c x))^2}{x^3} \, dx=d^{2} \left (\int \frac {a^{2}}{x^{3}}\, dx + \int \frac {2 a^{2} c^{2}}{x}\, dx + \int a^{2} c^{4} x\, dx + \int \frac {b^{2} \operatorname {asinh}^{2}{\left (c x \right )}}{x^{3}}\, dx + \int \frac {2 a b \operatorname {asinh}{\left (c x \right )}}{x^{3}}\, dx + \int \frac {2 b^{2} c^{2} \operatorname {asinh}^{2}{\left (c x \right )}}{x}\, dx + \int b^{2} c^{4} x \operatorname {asinh}^{2}{\left (c x \right )}\, dx + \int \frac {4 a b c^{2} \operatorname {asinh}{\left (c x \right )}}{x}\, dx + \int 2 a b c^{4} x \operatorname {asinh}{\left (c x \right )}\, dx\right ) \]
d**2*(Integral(a**2/x**3, x) + Integral(2*a**2*c**2/x, x) + Integral(a**2* c**4*x, x) + Integral(b**2*asinh(c*x)**2/x**3, x) + Integral(2*a*b*asinh(c *x)/x**3, x) + Integral(2*b**2*c**2*asinh(c*x)**2/x, x) + Integral(b**2*c* *4*x*asinh(c*x)**2, x) + Integral(4*a*b*c**2*asinh(c*x)/x, x) + Integral(2 *a*b*c**4*x*asinh(c*x), x))
\[ \int \frac {\left (d+c^2 d x^2\right )^2 (a+b \text {arcsinh}(c x))^2}{x^3} \, dx=\int { \frac {{\left (c^{2} d x^{2} + d\right )}^{2} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2}}{x^{3}} \,d x } \]
1/2*a^2*c^4*d^2*x^2 + 2*a^2*c^2*d^2*log(x) - a*b*d^2*(sqrt(c^2*x^2 + 1)*c/ x + arcsinh(c*x)/x^2) - 1/2*a^2*d^2/x^2 + integrate(b^2*c^4*d^2*x*log(c*x + sqrt(c^2*x^2 + 1))^2 + 2*a*b*c^4*d^2*x*log(c*x + sqrt(c^2*x^2 + 1)) + 2* b^2*c^2*d^2*log(c*x + sqrt(c^2*x^2 + 1))^2/x + 4*a*b*c^2*d^2*log(c*x + sqr t(c^2*x^2 + 1))/x + b^2*d^2*log(c*x + sqrt(c^2*x^2 + 1))^2/x^3, x)
Exception generated. \[ \int \frac {\left (d+c^2 d x^2\right )^2 (a+b \text {arcsinh}(c x))^2}{x^3} \, dx=\text {Exception raised: TypeError} \]
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. \[ \int \frac {\left (d+c^2 d x^2\right )^2 (a+b \text {arcsinh}(c x))^2}{x^3} \, dx=\int \frac {{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2\,{\left (d\,c^2\,x^2+d\right )}^2}{x^3} \,d x \]