Integrand size = 26, antiderivative size = 307 \[ \int \frac {\left (d+c^2 d x^2\right )^3 (a+b \text {arcsinh}(c x))^2}{x^2} \, dx=\frac {122}{25} b^2 c^2 d^3 x+\frac {14}{75} b^2 c^4 d^3 x^3+\frac {2}{125} b^2 c^6 d^3 x^5-\frac {22}{5} b c d^3 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))-\frac {2}{5} b c d^3 \left (1+c^2 x^2\right )^{3/2} (a+b \text {arcsinh}(c x))-\frac {2}{25} b c d^3 \left (1+c^2 x^2\right )^{5/2} (a+b \text {arcsinh}(c x))+\frac {16}{5} c^2 d^3 x (a+b \text {arcsinh}(c x))^2+\frac {8}{5} c^2 d^3 x \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))^2+\frac {6}{5} c^2 d^3 x \left (1+c^2 x^2\right )^2 (a+b \text {arcsinh}(c x))^2-\frac {d^3 \left (1+c^2 x^2\right )^3 (a+b \text {arcsinh}(c x))^2}{x}-4 b c d^3 (a+b \text {arcsinh}(c x)) \text {arctanh}\left (e^{\text {arcsinh}(c x)}\right )-2 b^2 c d^3 \operatorname {PolyLog}\left (2,-e^{\text {arcsinh}(c x)}\right )+2 b^2 c d^3 \operatorname {PolyLog}\left (2,e^{\text {arcsinh}(c x)}\right ) \]
122/25*b^2*c^2*d^3*x+14/75*b^2*c^4*d^3*x^3+2/125*b^2*c^6*d^3*x^5-2/5*b*c*d ^3*(c^2*x^2+1)^(3/2)*(a+b*arcsinh(c*x))-2/25*b*c*d^3*(c^2*x^2+1)^(5/2)*(a+ b*arcsinh(c*x))+16/5*c^2*d^3*x*(a+b*arcsinh(c*x))^2+8/5*c^2*d^3*x*(c^2*x^2 +1)*(a+b*arcsinh(c*x))^2+6/5*c^2*d^3*x*(c^2*x^2+1)^2*(a+b*arcsinh(c*x))^2- d^3*(c^2*x^2+1)^3*(a+b*arcsinh(c*x))^2/x-4*b*c*d^3*(a+b*arcsinh(c*x))*arct anh(c*x+(c^2*x^2+1)^(1/2))-2*b^2*c*d^3*polylog(2,-c*x-(c^2*x^2+1)^(1/2))+2 *b^2*c*d^3*polylog(2,c*x+(c^2*x^2+1)^(1/2))-22/5*b*c*d^3*(a+b*arcsinh(c*x) )*(c^2*x^2+1)^(1/2)
Time = 1.07 (sec) , antiderivative size = 466, normalized size of antiderivative = 1.52 \[ \int \frac {\left (d+c^2 d x^2\right )^3 (a+b \text {arcsinh}(c x))^2}{x^2} \, dx=\frac {1}{720} d^3 \left (-\frac {720 a^2}{x}+2160 a^2 c^2 x+3460 b^2 c^2 x+720 a^2 c^4 x^3+144 a^2 c^6 x^5-\frac {17568}{5} a b c \sqrt {1+c^2 x^2}-\frac {2016}{5} a b c^3 x^2 \sqrt {1+c^2 x^2}-\frac {288}{5} a b c^5 x^4 \sqrt {1+c^2 x^2}-\frac {1440 a b \text {arcsinh}(c x)}{x}+4320 a b c^2 x \text {arcsinh}(c x)+1440 a b c^4 x^3 \text {arcsinh}(c x)+288 a b c^6 x^5 \text {arcsinh}(c x)-3420 b^2 c \sqrt {1+c^2 x^2} \text {arcsinh}(c x)-\frac {720 b^2 \text {arcsinh}(c x)^2}{x}+1890 b^2 c^2 x \text {arcsinh}(c x)^2-1440 a b c \text {arctanh}\left (\sqrt {1+c^2 x^2}\right )+80 b^2 c^2 x \cosh (2 \text {arcsinh}(c x))+360 b^2 c^2 x \text {arcsinh}(c x)^2 \cosh (2 \text {arcsinh}(c x))-90 b^2 c \text {arcsinh}(c x) \cosh (3 \text {arcsinh}(c x))-\frac {18}{5} b^2 c \text {arcsinh}(c x) \cosh (5 \text {arcsinh}(c x))+1440 b^2 c \text {arcsinh}(c x) \log \left (1-e^{-\text {arcsinh}(c x)}\right )-1440 b^2 c \text {arcsinh}(c x) \log \left (1+e^{-\text {arcsinh}(c x)}\right )+1440 b^2 c \operatorname {PolyLog}\left (2,-e^{-\text {arcsinh}(c x)}\right )-1440 b^2 c \operatorname {PolyLog}\left (2,e^{-\text {arcsinh}(c x)}\right )-10 b^2 c \sinh (3 \text {arcsinh}(c x))-45 b^2 c \text {arcsinh}(c x)^2 \sinh (3 \text {arcsinh}(c x))+\frac {18}{25} b^2 c \sinh (5 \text {arcsinh}(c x))+9 b^2 c \text {arcsinh}(c x)^2 \sinh (5 \text {arcsinh}(c x))\right ) \]
(d^3*((-720*a^2)/x + 2160*a^2*c^2*x + 3460*b^2*c^2*x + 720*a^2*c^4*x^3 + 1 44*a^2*c^6*x^5 - (17568*a*b*c*Sqrt[1 + c^2*x^2])/5 - (2016*a*b*c^3*x^2*Sqr t[1 + c^2*x^2])/5 - (288*a*b*c^5*x^4*Sqrt[1 + c^2*x^2])/5 - (1440*a*b*ArcS inh[c*x])/x + 4320*a*b*c^2*x*ArcSinh[c*x] + 1440*a*b*c^4*x^3*ArcSinh[c*x] + 288*a*b*c^6*x^5*ArcSinh[c*x] - 3420*b^2*c*Sqrt[1 + c^2*x^2]*ArcSinh[c*x] - (720*b^2*ArcSinh[c*x]^2)/x + 1890*b^2*c^2*x*ArcSinh[c*x]^2 - 1440*a*b*c *ArcTanh[Sqrt[1 + c^2*x^2]] + 80*b^2*c^2*x*Cosh[2*ArcSinh[c*x]] + 360*b^2* c^2*x*ArcSinh[c*x]^2*Cosh[2*ArcSinh[c*x]] - 90*b^2*c*ArcSinh[c*x]*Cosh[3*A rcSinh[c*x]] - (18*b^2*c*ArcSinh[c*x]*Cosh[5*ArcSinh[c*x]])/5 + 1440*b^2*c *ArcSinh[c*x]*Log[1 - E^(-ArcSinh[c*x])] - 1440*b^2*c*ArcSinh[c*x]*Log[1 + E^(-ArcSinh[c*x])] + 1440*b^2*c*PolyLog[2, -E^(-ArcSinh[c*x])] - 1440*b^2 *c*PolyLog[2, E^(-ArcSinh[c*x])] - 10*b^2*c*Sinh[3*ArcSinh[c*x]] - 45*b^2* c*ArcSinh[c*x]^2*Sinh[3*ArcSinh[c*x]] + (18*b^2*c*Sinh[5*ArcSinh[c*x]])/25 + 9*b^2*c*ArcSinh[c*x]^2*Sinh[5*ArcSinh[c*x]]))/720
Result contains complex when optimal does not.
Time = 3.17 (sec) , antiderivative size = 451, normalized size of antiderivative = 1.47, number of steps used = 23, number of rules used = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.846, Rules used = {6222, 27, 6201, 6201, 6187, 6213, 24, 210, 2009, 6223, 210, 2009, 6223, 2009, 6221, 24, 6231, 3042, 26, 4670, 2715, 2838}
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.
| \(\displaystyle \int \frac {\left (c^2 d x^2+d\right )^3 (a+b \text {arcsinh}(c x))^2}{x^2} \, dx\) |
| \(\Big \downarrow \) 6222 |
| \(\displaystyle 2 b c d^3 \int \frac {\left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))}{x}dx+6 c^2 d \int d^2 \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))^2dx-\frac {d^3 \left (c^2 x^2+1\right )^3 (a+b \text {arcsinh}(c x))^2}{x}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 2 b c d^3 \int \frac {\left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))}{x}dx+6 c^2 d^3 \int \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))^2dx-\frac {d^3 \left (c^2 x^2+1\right )^3 (a+b \text {arcsinh}(c x))^2}{x}\) |
| \(\Big \downarrow \) 6201 |
| \(\displaystyle 2 b c d^3 \int \frac {\left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))}{x}dx+6 c^2 d^3 \left (-\frac {2}{5} b c \int x \left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))dx+\frac {4}{5} \int \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))^2dx+\frac {1}{5} x \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))^2\right )-\frac {d^3 \left (c^2 x^2+1\right )^3 (a+b \text {arcsinh}(c x))^2}{x}\) |
| \(\Big \downarrow \) 6201 |
| \(\displaystyle 2 b c d^3 \int \frac {\left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))}{x}dx+6 c^2 d^3 \left (-\frac {2}{5} b c \int x \left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))dx+\frac {4}{5} \left (-\frac {2}{3} b c \int x \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))dx+\frac {2}{3} \int (a+b \text {arcsinh}(c x))^2dx+\frac {1}{3} x \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))^2\right )+\frac {1}{5} x \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))^2\right )-\frac {d^3 \left (c^2 x^2+1\right )^3 (a+b \text {arcsinh}(c x))^2}{x}\) |
| \(\Big \downarrow \) 6187 |
| \(\displaystyle 6 c^2 d^3 \left (\frac {4}{5} \left (\frac {2}{3} \left (x (a+b \text {arcsinh}(c x))^2-2 b c \int \frac {x (a+b \text {arcsinh}(c x))}{\sqrt {c^2 x^2+1}}dx\right )-\frac {2}{3} b c \int x \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))dx+\frac {1}{3} x \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))^2\right )-\frac {2}{5} b c \int x \left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))dx+\frac {1}{5} x \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))^2\right )+2 b c d^3 \int \frac {\left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))}{x}dx-\frac {d^3 \left (c^2 x^2+1\right )^3 (a+b \text {arcsinh}(c x))^2}{x}\) |
| \(\Big \downarrow \) 6213 |
| \(\displaystyle 6 c^2 d^3 \left (\frac {4}{5} \left (\frac {2}{3} \left (x (a+b \text {arcsinh}(c x))^2-2 b c \left (\frac {\sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{c^2}-\frac {b \int 1dx}{c}\right )\right )-\frac {2}{3} b c \left (\frac {\left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))}{3 c^2}-\frac {b \int \left (c^2 x^2+1\right )dx}{3 c}\right )+\frac {1}{3} x \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))^2\right )-\frac {2}{5} b c \left (\frac {\left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))}{5 c^2}-\frac {b \int \left (c^2 x^2+1\right )^2dx}{5 c}\right )+\frac {1}{5} x \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))^2\right )+2 b c d^3 \int \frac {\left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))}{x}dx-\frac {d^3 \left (c^2 x^2+1\right )^3 (a+b \text {arcsinh}(c x))^2}{x}\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle 6 c^2 d^3 \left (\frac {4}{5} \left (-\frac {2}{3} b c \left (\frac {\left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))}{3 c^2}-\frac {b \int \left (c^2 x^2+1\right )dx}{3 c}\right )+\frac {1}{3} x \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))^2+\frac {2}{3} \left (x (a+b \text {arcsinh}(c x))^2-2 b c \left (\frac {\sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{c^2}-\frac {b x}{c}\right )\right )\right )-\frac {2}{5} b c \left (\frac {\left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))}{5 c^2}-\frac {b \int \left (c^2 x^2+1\right )^2dx}{5 c}\right )+\frac {1}{5} x \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))^2\right )+2 b c d^3 \int \frac {\left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))}{x}dx-\frac {d^3 \left (c^2 x^2+1\right )^3 (a+b \text {arcsinh}(c x))^2}{x}\) |
| \(\Big \downarrow \) 210 |
| \(\displaystyle 2 b c d^3 \int \frac {\left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))}{x}dx+6 c^2 d^3 \left (\frac {4}{5} \left (-\frac {2}{3} b c \left (\frac {\left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))}{3 c^2}-\frac {b \int \left (c^2 x^2+1\right )dx}{3 c}\right )+\frac {1}{3} x \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))^2+\frac {2}{3} \left (x (a+b \text {arcsinh}(c x))^2-2 b c \left (\frac {\sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{c^2}-\frac {b x}{c}\right )\right )\right )-\frac {2}{5} b c \left (\frac {\left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))}{5 c^2}-\frac {b \int \left (c^4 x^4+2 c^2 x^2+1\right )dx}{5 c}\right )+\frac {1}{5} x \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))^2\right )-\frac {d^3 \left (c^2 x^2+1\right )^3 (a+b \text {arcsinh}(c x))^2}{x}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle 2 b c d^3 \int \frac {\left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))}{x}dx-\frac {d^3 \left (c^2 x^2+1\right )^3 (a+b \text {arcsinh}(c x))^2}{x}+6 c^2 d^3 \left (\frac {1}{5} x \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))^2+\frac {4}{5} \left (\frac {1}{3} x \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))^2+\frac {2}{3} \left (x (a+b \text {arcsinh}(c x))^2-2 b c \left (\frac {\sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{c^2}-\frac {b x}{c}\right )\right )-\frac {2}{3} b c \left (\frac {\left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))}{3 c^2}-\frac {b \left (\frac {c^2 x^3}{3}+x\right )}{3 c}\right )\right )-\frac {2}{5} b c \left (\frac {\left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))}{5 c^2}-\frac {b \left (\frac {c^4 x^5}{5}+\frac {2 c^2 x^3}{3}+x\right )}{5 c}\right )\right )\) |
| \(\Big \downarrow \) 6223 |
| \(\displaystyle 2 b c d^3 \left (\int \frac {\left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))}{x}dx-\frac {1}{5} b c \int \left (c^2 x^2+1\right )^2dx+\frac {1}{5} \left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))\right )-\frac {d^3 \left (c^2 x^2+1\right )^3 (a+b \text {arcsinh}(c x))^2}{x}+6 c^2 d^3 \left (\frac {1}{5} x \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))^2+\frac {4}{5} \left (\frac {1}{3} x \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))^2+\frac {2}{3} \left (x (a+b \text {arcsinh}(c x))^2-2 b c \left (\frac {\sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{c^2}-\frac {b x}{c}\right )\right )-\frac {2}{3} b c \left (\frac {\left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))}{3 c^2}-\frac {b \left (\frac {c^2 x^3}{3}+x\right )}{3 c}\right )\right )-\frac {2}{5} b c \left (\frac {\left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))}{5 c^2}-\frac {b \left (\frac {c^4 x^5}{5}+\frac {2 c^2 x^3}{3}+x\right )}{5 c}\right )\right )\) |
| \(\Big \downarrow \) 210 |
| \(\displaystyle 2 b c d^3 \left (\int \frac {\left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))}{x}dx-\frac {1}{5} b c \int \left (c^4 x^4+2 c^2 x^2+1\right )dx+\frac {1}{5} \left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))\right )-\frac {d^3 \left (c^2 x^2+1\right )^3 (a+b \text {arcsinh}(c x))^2}{x}+6 c^2 d^3 \left (\frac {1}{5} x \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))^2+\frac {4}{5} \left (\frac {1}{3} x \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))^2+\frac {2}{3} \left (x (a+b \text {arcsinh}(c x))^2-2 b c \left (\frac {\sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{c^2}-\frac {b x}{c}\right )\right )-\frac {2}{3} b c \left (\frac {\left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))}{3 c^2}-\frac {b \left (\frac {c^2 x^3}{3}+x\right )}{3 c}\right )\right )-\frac {2}{5} b c \left (\frac {\left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))}{5 c^2}-\frac {b \left (\frac {c^4 x^5}{5}+\frac {2 c^2 x^3}{3}+x\right )}{5 c}\right )\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle 2 b c d^3 \left (\int \frac {\left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))}{x}dx+\frac {1}{5} \left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))-\frac {1}{5} b c \left (\frac {c^4 x^5}{5}+\frac {2 c^2 x^3}{3}+x\right )\right )-\frac {d^3 \left (c^2 x^2+1\right )^3 (a+b \text {arcsinh}(c x))^2}{x}+6 c^2 d^3 \left (\frac {1}{5} x \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))^2+\frac {4}{5} \left (\frac {1}{3} x \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))^2+\frac {2}{3} \left (x (a+b \text {arcsinh}(c x))^2-2 b c \left (\frac {\sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{c^2}-\frac {b x}{c}\right )\right )-\frac {2}{3} b c \left (\frac {\left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))}{3 c^2}-\frac {b \left (\frac {c^2 x^3}{3}+x\right )}{3 c}\right )\right )-\frac {2}{5} b c \left (\frac {\left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))}{5 c^2}-\frac {b \left (\frac {c^4 x^5}{5}+\frac {2 c^2 x^3}{3}+x\right )}{5 c}\right )\right )\) |
| \(\Big \downarrow \) 6223 |
| \(\displaystyle 2 b c d^3 \left (\int \frac {\sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{x}dx-\frac {1}{3} b c \int \left (c^2 x^2+1\right )dx+\frac {1}{5} \left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))+\frac {1}{3} \left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))-\frac {1}{5} b c \left (\frac {c^4 x^5}{5}+\frac {2 c^2 x^3}{3}+x\right )\right )-\frac {d^3 \left (c^2 x^2+1\right )^3 (a+b \text {arcsinh}(c x))^2}{x}+6 c^2 d^3 \left (\frac {1}{5} x \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))^2+\frac {4}{5} \left (\frac {1}{3} x \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))^2+\frac {2}{3} \left (x (a+b \text {arcsinh}(c x))^2-2 b c \left (\frac {\sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{c^2}-\frac {b x}{c}\right )\right )-\frac {2}{3} b c \left (\frac {\left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))}{3 c^2}-\frac {b \left (\frac {c^2 x^3}{3}+x\right )}{3 c}\right )\right )-\frac {2}{5} b c \left (\frac {\left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))}{5 c^2}-\frac {b \left (\frac {c^4 x^5}{5}+\frac {2 c^2 x^3}{3}+x\right )}{5 c}\right )\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle 2 b c d^3 \left (\int \frac {\sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{x}dx+\frac {1}{5} \left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))+\frac {1}{3} \left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))-\frac {1}{3} b c \left (\frac {c^2 x^3}{3}+x\right )-\frac {1}{5} b c \left (\frac {c^4 x^5}{5}+\frac {2 c^2 x^3}{3}+x\right )\right )-\frac {d^3 \left (c^2 x^2+1\right )^3 (a+b \text {arcsinh}(c x))^2}{x}+6 c^2 d^3 \left (\frac {1}{5} x \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))^2+\frac {4}{5} \left (\frac {1}{3} x \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))^2+\frac {2}{3} \left (x (a+b \text {arcsinh}(c x))^2-2 b c \left (\frac {\sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{c^2}-\frac {b x}{c}\right )\right )-\frac {2}{3} b c \left (\frac {\left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))}{3 c^2}-\frac {b \left (\frac {c^2 x^3}{3}+x\right )}{3 c}\right )\right )-\frac {2}{5} b c \left (\frac {\left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))}{5 c^2}-\frac {b \left (\frac {c^4 x^5}{5}+\frac {2 c^2 x^3}{3}+x\right )}{5 c}\right )\right )\) |
| \(\Big \downarrow \) 6221 |
| \(\displaystyle 2 b c d^3 \left (\int \frac {a+b \text {arcsinh}(c x)}{x \sqrt {c^2 x^2+1}}dx-b c \int 1dx+\frac {1}{5} \left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))+\frac {1}{3} \left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))+\sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))-\frac {1}{3} b c \left (\frac {c^2 x^3}{3}+x\right )-\frac {1}{5} b c \left (\frac {c^4 x^5}{5}+\frac {2 c^2 x^3}{3}+x\right )\right )-\frac {d^3 \left (c^2 x^2+1\right )^3 (a+b \text {arcsinh}(c x))^2}{x}+6 c^2 d^3 \left (\frac {1}{5} x \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))^2+\frac {4}{5} \left (\frac {1}{3} x \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))^2+\frac {2}{3} \left (x (a+b \text {arcsinh}(c x))^2-2 b c \left (\frac {\sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{c^2}-\frac {b x}{c}\right )\right )-\frac {2}{3} b c \left (\frac {\left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))}{3 c^2}-\frac {b \left (\frac {c^2 x^3}{3}+x\right )}{3 c}\right )\right )-\frac {2}{5} b c \left (\frac {\left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))}{5 c^2}-\frac {b \left (\frac {c^4 x^5}{5}+\frac {2 c^2 x^3}{3}+x\right )}{5 c}\right )\right )\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle 2 b c d^3 \left (\int \frac {a+b \text {arcsinh}(c x)}{x \sqrt {c^2 x^2+1}}dx+\frac {1}{5} \left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))+\frac {1}{3} \left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))+\sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))-\frac {1}{3} b c \left (\frac {c^2 x^3}{3}+x\right )-\frac {1}{5} b c \left (\frac {c^4 x^5}{5}+\frac {2 c^2 x^3}{3}+x\right )-b c x\right )-\frac {d^3 \left (c^2 x^2+1\right )^3 (a+b \text {arcsinh}(c x))^2}{x}+6 c^2 d^3 \left (\frac {1}{5} x \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))^2+\frac {4}{5} \left (\frac {1}{3} x \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))^2+\frac {2}{3} \left (x (a+b \text {arcsinh}(c x))^2-2 b c \left (\frac {\sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{c^2}-\frac {b x}{c}\right )\right )-\frac {2}{3} b c \left (\frac {\left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))}{3 c^2}-\frac {b \left (\frac {c^2 x^3}{3}+x\right )}{3 c}\right )\right )-\frac {2}{5} b c \left (\frac {\left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))}{5 c^2}-\frac {b \left (\frac {c^4 x^5}{5}+\frac {2 c^2 x^3}{3}+x\right )}{5 c}\right )\right )\) |
| \(\Big \downarrow \) 6231 |
| \(\displaystyle 2 b c d^3 \left (\int \frac {a+b \text {arcsinh}(c x)}{c x}d\text {arcsinh}(c x)+\frac {1}{5} \left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))+\frac {1}{3} \left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))+\sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))-\frac {1}{3} b c \left (\frac {c^2 x^3}{3}+x\right )-\frac {1}{5} b c \left (\frac {c^4 x^5}{5}+\frac {2 c^2 x^3}{3}+x\right )-b c x\right )-\frac {d^3 \left (c^2 x^2+1\right )^3 (a+b \text {arcsinh}(c x))^2}{x}+6 c^2 d^3 \left (\frac {1}{5} x \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))^2+\frac {4}{5} \left (\frac {1}{3} x \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))^2+\frac {2}{3} \left (x (a+b \text {arcsinh}(c x))^2-2 b c \left (\frac {\sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{c^2}-\frac {b x}{c}\right )\right )-\frac {2}{3} b c \left (\frac {\left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))}{3 c^2}-\frac {b \left (\frac {c^2 x^3}{3}+x\right )}{3 c}\right )\right )-\frac {2}{5} b c \left (\frac {\left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))}{5 c^2}-\frac {b \left (\frac {c^4 x^5}{5}+\frac {2 c^2 x^3}{3}+x\right )}{5 c}\right )\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle 2 b c d^3 \left (\int i (a+b \text {arcsinh}(c x)) \csc (i \text {arcsinh}(c x))d\text {arcsinh}(c x)+\frac {1}{5} \left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))+\frac {1}{3} \left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))+\sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))-\frac {1}{3} b c \left (\frac {c^2 x^3}{3}+x\right )-\frac {1}{5} b c \left (\frac {c^4 x^5}{5}+\frac {2 c^2 x^3}{3}+x\right )-b c x\right )-\frac {d^3 \left (c^2 x^2+1\right )^3 (a+b \text {arcsinh}(c x))^2}{x}+6 c^2 d^3 \left (\frac {1}{5} x \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))^2+\frac {4}{5} \left (\frac {1}{3} x \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))^2+\frac {2}{3} \left (x (a+b \text {arcsinh}(c x))^2-2 b c \left (\frac {\sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{c^2}-\frac {b x}{c}\right )\right )-\frac {2}{3} b c \left (\frac {\left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))}{3 c^2}-\frac {b \left (\frac {c^2 x^3}{3}+x\right )}{3 c}\right )\right )-\frac {2}{5} b c \left (\frac {\left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))}{5 c^2}-\frac {b \left (\frac {c^4 x^5}{5}+\frac {2 c^2 x^3}{3}+x\right )}{5 c}\right )\right )\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle 2 b c d^3 \left (i \int (a+b \text {arcsinh}(c x)) \csc (i \text {arcsinh}(c x))d\text {arcsinh}(c x)+\frac {1}{5} \left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))+\frac {1}{3} \left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))+\sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))-\frac {1}{3} b c \left (\frac {c^2 x^3}{3}+x\right )-\frac {1}{5} b c \left (\frac {c^4 x^5}{5}+\frac {2 c^2 x^3}{3}+x\right )-b c x\right )-\frac {d^3 \left (c^2 x^2+1\right )^3 (a+b \text {arcsinh}(c x))^2}{x}+6 c^2 d^3 \left (\frac {1}{5} x \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))^2+\frac {4}{5} \left (\frac {1}{3} x \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))^2+\frac {2}{3} \left (x (a+b \text {arcsinh}(c x))^2-2 b c \left (\frac {\sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{c^2}-\frac {b x}{c}\right )\right )-\frac {2}{3} b c \left (\frac {\left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))}{3 c^2}-\frac {b \left (\frac {c^2 x^3}{3}+x\right )}{3 c}\right )\right )-\frac {2}{5} b c \left (\frac {\left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))}{5 c^2}-\frac {b \left (\frac {c^4 x^5}{5}+\frac {2 c^2 x^3}{3}+x\right )}{5 c}\right )\right )\) |
| \(\Big \downarrow \) 4670 |
| \(\displaystyle 2 b c d^3 \left (i \left (i b \int \log \left (1-e^{\text {arcsinh}(c x)}\right )d\text {arcsinh}(c x)-i b \int \log \left (1+e^{\text {arcsinh}(c x)}\right )d\text {arcsinh}(c x)+2 i \text {arctanh}\left (e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))\right )+\frac {1}{5} \left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))+\frac {1}{3} \left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))+\sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))-\frac {1}{3} b c \left (\frac {c^2 x^3}{3}+x\right )-\frac {1}{5} b c \left (\frac {c^4 x^5}{5}+\frac {2 c^2 x^3}{3}+x\right )-b c x\right )-\frac {d^3 \left (c^2 x^2+1\right )^3 (a+b \text {arcsinh}(c x))^2}{x}+6 c^2 d^3 \left (\frac {1}{5} x \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))^2+\frac {4}{5} \left (\frac {1}{3} x \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))^2+\frac {2}{3} \left (x (a+b \text {arcsinh}(c x))^2-2 b c \left (\frac {\sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{c^2}-\frac {b x}{c}\right )\right )-\frac {2}{3} b c \left (\frac {\left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))}{3 c^2}-\frac {b \left (\frac {c^2 x^3}{3}+x\right )}{3 c}\right )\right )-\frac {2}{5} b c \left (\frac {\left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))}{5 c^2}-\frac {b \left (\frac {c^4 x^5}{5}+\frac {2 c^2 x^3}{3}+x\right )}{5 c}\right )\right )\) |
| \(\Big \downarrow \) 2715 |
| \(\displaystyle 2 b c d^3 \left (i \left (i b \int e^{-\text {arcsinh}(c x)} \log \left (1-e^{\text {arcsinh}(c x)}\right )de^{\text {arcsinh}(c x)}-i b \int e^{-\text {arcsinh}(c x)} \log \left (1+e^{\text {arcsinh}(c x)}\right )de^{\text {arcsinh}(c x)}+2 i \text {arctanh}\left (e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))\right )+\frac {1}{5} \left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))+\frac {1}{3} \left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))+\sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))-\frac {1}{3} b c \left (\frac {c^2 x^3}{3}+x\right )-\frac {1}{5} b c \left (\frac {c^4 x^5}{5}+\frac {2 c^2 x^3}{3}+x\right )-b c x\right )-\frac {d^3 \left (c^2 x^2+1\right )^3 (a+b \text {arcsinh}(c x))^2}{x}+6 c^2 d^3 \left (\frac {1}{5} x \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))^2+\frac {4}{5} \left (\frac {1}{3} x \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))^2+\frac {2}{3} \left (x (a+b \text {arcsinh}(c x))^2-2 b c \left (\frac {\sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{c^2}-\frac {b x}{c}\right )\right )-\frac {2}{3} b c \left (\frac {\left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))}{3 c^2}-\frac {b \left (\frac {c^2 x^3}{3}+x\right )}{3 c}\right )\right )-\frac {2}{5} b c \left (\frac {\left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))}{5 c^2}-\frac {b \left (\frac {c^4 x^5}{5}+\frac {2 c^2 x^3}{3}+x\right )}{5 c}\right )\right )\) |
| \(\Big \downarrow \) 2838 |
| \(\displaystyle 2 b c d^3 \left (i \left (2 i \text {arctanh}\left (e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))+i b \operatorname {PolyLog}\left (2,-e^{\text {arcsinh}(c x)}\right )-i b \operatorname {PolyLog}\left (2,e^{\text {arcsinh}(c x)}\right )\right )+\frac {1}{5} \left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))+\frac {1}{3} \left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))+\sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))-\frac {1}{3} b c \left (\frac {c^2 x^3}{3}+x\right )-\frac {1}{5} b c \left (\frac {c^4 x^5}{5}+\frac {2 c^2 x^3}{3}+x\right )-b c x\right )-\frac {d^3 \left (c^2 x^2+1\right )^3 (a+b \text {arcsinh}(c x))^2}{x}+6 c^2 d^3 \left (\frac {1}{5} x \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))^2+\frac {4}{5} \left (\frac {1}{3} x \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))^2+\frac {2}{3} \left (x (a+b \text {arcsinh}(c x))^2-2 b c \left (\frac {\sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{c^2}-\frac {b x}{c}\right )\right )-\frac {2}{3} b c \left (\frac {\left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))}{3 c^2}-\frac {b \left (\frac {c^2 x^3}{3}+x\right )}{3 c}\right )\right )-\frac {2}{5} b c \left (\frac {\left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))}{5 c^2}-\frac {b \left (\frac {c^4 x^5}{5}+\frac {2 c^2 x^3}{3}+x\right )}{5 c}\right )\right )\) |
-((d^3*(1 + c^2*x^2)^3*(a + b*ArcSinh[c*x])^2)/x) + 6*c^2*d^3*((x*(1 + c^2 *x^2)^2*(a + b*ArcSinh[c*x])^2)/5 - (2*b*c*(-1/5*(b*(x + (2*c^2*x^3)/3 + ( c^4*x^5)/5))/c + ((1 + c^2*x^2)^(5/2)*(a + b*ArcSinh[c*x]))/(5*c^2)))/5 + (4*((x*(1 + c^2*x^2)*(a + b*ArcSinh[c*x])^2)/3 - (2*b*c*(-1/3*(b*(x + (c^2 *x^3)/3))/c + ((1 + c^2*x^2)^(3/2)*(a + b*ArcSinh[c*x]))/(3*c^2)))/3 + (2* (x*(a + b*ArcSinh[c*x])^2 - 2*b*c*(-((b*x)/c) + (Sqrt[1 + c^2*x^2]*(a + b* ArcSinh[c*x]))/c^2)))/3))/5) + 2*b*c*d^3*(-(b*c*x) - (b*c*(x + (c^2*x^3)/3 ))/3 - (b*c*(x + (2*c^2*x^3)/3 + (c^4*x^5)/5))/5 + Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]) + ((1 + c^2*x^2)^(3/2)*(a + b*ArcSinh[c*x]))/3 + ((1 + c^2 *x^2)^(5/2)*(a + b*ArcSinh[c*x]))/5 + I*((2*I)*(a + b*ArcSinh[c*x])*ArcTan h[E^ArcSinh[c*x]] + I*b*PolyLog[2, -E^ArcSinh[c*x]] - I*b*PolyLog[2, E^Arc Sinh[c*x]]))
3.3.22.3.1 Defintions of rubi rules used
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[((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Int[ExpandIntegrand[(a + b*x^2 )^p, x], x] /; FreeQ[{a, b}, x] && IGtQ[p, 0]
Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Simp[1/(d*e*n*Log[F]) Subst[Int[Log[a + b*x]/x, x], x, (F^(e*(c + d*x) ))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]
Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2 , (-c)*e*x^n]/n, x] /; FreeQ[{c, d, e, n}, x] && EqQ[c*d, 1]
Int[csc[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x _Symbol] :> Simp[-2*(c + d*x)^m*(ArcTanh[E^((-I)*e + f*fz*x)]/(f*fz*I)), x] + (-Simp[d*(m/(f*fz*I)) Int[(c + d*x)^(m - 1)*Log[1 - E^((-I)*e + f*fz*x )], x], x] + Simp[d*(m/(f*fz*I)) Int[(c + d*x)^(m - 1)*Log[1 + E^((-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_Symbol] :> Simp[x*(a + b*A rcSinh[c*x])^n, x] - Simp[b*c*n Int[x*((a + b*ArcSinh[c*x])^(n - 1)/Sqrt[ 1 + c^2*x^2]), x], x] /; FreeQ[{a, b, c}, x] && 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_.)*(x_)*((d_) + (e_.)*(x_)^2)^(p _.), x_Symbol] :> Simp[(d + e*x^2)^(p + 1)*((a + b*ArcSinh[c*x])^n/(2*e*(p + 1))), x] - Simp[b*(n/(2*c*(p + 1)))*Simp[(d + e*x^2)^p/(1 + c^2*x^2)^p] Int[(1 + c^2*x^2)^(p + 1/2)*(a + b*ArcSinh[c*x])^(n - 1), x], x] /; FreeQ[ {a, b, c, d, e, p}, x] && EqQ[e, c^2*d] && GtQ[n, 0] && NeQ[p, -1]
Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[(f*x)^(m + 1)*Sqrt[d + e*x^2]*((a + b*Arc Sinh[c*x])^n/(f*(m + 2))), x] + (Simp[(1/(m + 2))*Simp[Sqrt[d + e*x^2]/Sqrt [1 + c^2*x^2]] Int[(f*x)^m*((a + b*ArcSinh[c*x])^n/Sqrt[1 + c^2*x^2]), x] , x] - Simp[b*c*(n/(f*(m + 2)))*Simp[Sqrt[d + e*x^2]/Sqrt[1 + c^2*x^2]] I nt[(f*x)^(m + 1)*(a + b*ArcSinh[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d , e, f, m}, x] && EqQ[e, c^2*d] && IGtQ[n, 0] && (IGtQ[m, -2] || EqQ[n, 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 + 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[(((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_))/Sqrt[(d_) + (e_.) *(x_)^2], x_Symbol] :> Simp[(1/c^(m + 1))*Simp[Sqrt[1 + c^2*x^2]/Sqrt[d + e *x^2]] Subst[Int[(a + b*x)^n*Sinh[x]^m, x], x, ArcSinh[c*x]], x] /; FreeQ [{a, b, c, d, e}, x] && EqQ[e, c^2*d] && IGtQ[n, 0] && IntegerQ[m]
Time = 0.28 (sec) , antiderivative size = 463, normalized size of antiderivative = 1.51
| method | result | size |
| derivativedivides | \(c \left (d^{3} a^{2} \left (\frac {c^{5} x^{5}}{5}+c^{3} x^{3}+3 c x -\frac {1}{c x}\right )+\frac {14 d^{3} b^{2} c^{3} x^{3}}{75}+\frac {122 d^{3} b^{2} c x}{25}+2 d^{3} b^{2} \operatorname {polylog}\left (2, c x +\sqrt {c^{2} x^{2}+1}\right )-2 d^{3} b^{2} \operatorname {polylog}\left (2, -c x -\sqrt {c^{2} x^{2}+1}\right )-\frac {2 d^{3} b^{2} \operatorname {arcsinh}\left (c x \right ) \sqrt {c^{2} x^{2}+1}\, c^{4} x^{4}}{25}-\frac {14 d^{3} b^{2} \operatorname {arcsinh}\left (c x \right ) \sqrt {c^{2} x^{2}+1}\, c^{2} x^{2}}{25}+2 d^{3} b^{2} \operatorname {arcsinh}\left (c x \right ) \ln \left (1-c x -\sqrt {c^{2} x^{2}+1}\right )+\frac {2 d^{3} b^{2} c^{5} x^{5}}{125}-2 d^{3} b^{2} \operatorname {arcsinh}\left (c x \right ) \ln \left (1+c x +\sqrt {c^{2} x^{2}+1}\right )-\frac {d^{3} b^{2} \operatorname {arcsinh}\left (c x \right )^{2}}{c x}+\frac {d^{3} b^{2} \operatorname {arcsinh}\left (c x \right )^{2} c^{5} x^{5}}{5}+d^{3} b^{2} \operatorname {arcsinh}\left (c x \right )^{2} c^{3} x^{3}+3 d^{3} b^{2} \operatorname {arcsinh}\left (c x \right )^{2} c x -\frac {122 d^{3} b^{2} \operatorname {arcsinh}\left (c x \right ) \sqrt {c^{2} x^{2}+1}}{25}+2 d^{3} a b \left (\frac {\operatorname {arcsinh}\left (c x \right ) c^{5} x^{5}}{5}+\operatorname {arcsinh}\left (c x \right ) c^{3} x^{3}+3 \,\operatorname {arcsinh}\left (c x \right ) c x -\frac {\operatorname {arcsinh}\left (c x \right )}{c x}-\frac {c^{4} x^{4} \sqrt {c^{2} x^{2}+1}}{25}-\frac {7 c^{2} x^{2} \sqrt {c^{2} x^{2}+1}}{25}-\frac {61 \sqrt {c^{2} x^{2}+1}}{25}-\operatorname {arctanh}\left (\frac {1}{\sqrt {c^{2} x^{2}+1}}\right )\right )\right )\) | \(463\) |
| default | \(c \left (d^{3} a^{2} \left (\frac {c^{5} x^{5}}{5}+c^{3} x^{3}+3 c x -\frac {1}{c x}\right )+\frac {14 d^{3} b^{2} c^{3} x^{3}}{75}+\frac {122 d^{3} b^{2} c x}{25}+2 d^{3} b^{2} \operatorname {polylog}\left (2, c x +\sqrt {c^{2} x^{2}+1}\right )-2 d^{3} b^{2} \operatorname {polylog}\left (2, -c x -\sqrt {c^{2} x^{2}+1}\right )-\frac {2 d^{3} b^{2} \operatorname {arcsinh}\left (c x \right ) \sqrt {c^{2} x^{2}+1}\, c^{4} x^{4}}{25}-\frac {14 d^{3} b^{2} \operatorname {arcsinh}\left (c x \right ) \sqrt {c^{2} x^{2}+1}\, c^{2} x^{2}}{25}+2 d^{3} b^{2} \operatorname {arcsinh}\left (c x \right ) \ln \left (1-c x -\sqrt {c^{2} x^{2}+1}\right )+\frac {2 d^{3} b^{2} c^{5} x^{5}}{125}-2 d^{3} b^{2} \operatorname {arcsinh}\left (c x \right ) \ln \left (1+c x +\sqrt {c^{2} x^{2}+1}\right )-\frac {d^{3} b^{2} \operatorname {arcsinh}\left (c x \right )^{2}}{c x}+\frac {d^{3} b^{2} \operatorname {arcsinh}\left (c x \right )^{2} c^{5} x^{5}}{5}+d^{3} b^{2} \operatorname {arcsinh}\left (c x \right )^{2} c^{3} x^{3}+3 d^{3} b^{2} \operatorname {arcsinh}\left (c x \right )^{2} c x -\frac {122 d^{3} b^{2} \operatorname {arcsinh}\left (c x \right ) \sqrt {c^{2} x^{2}+1}}{25}+2 d^{3} a b \left (\frac {\operatorname {arcsinh}\left (c x \right ) c^{5} x^{5}}{5}+\operatorname {arcsinh}\left (c x \right ) c^{3} x^{3}+3 \,\operatorname {arcsinh}\left (c x \right ) c x -\frac {\operatorname {arcsinh}\left (c x \right )}{c x}-\frac {c^{4} x^{4} \sqrt {c^{2} x^{2}+1}}{25}-\frac {7 c^{2} x^{2} \sqrt {c^{2} x^{2}+1}}{25}-\frac {61 \sqrt {c^{2} x^{2}+1}}{25}-\operatorname {arctanh}\left (\frac {1}{\sqrt {c^{2} x^{2}+1}}\right )\right )\right )\) | \(463\) |
| parts | \(d^{3} a^{2} \left (\frac {c^{6} x^{5}}{5}+c^{4} x^{3}+3 c^{2} x -\frac {1}{x}\right )-2 d^{3} b^{2} c \,\operatorname {arcsinh}\left (c x \right ) \ln \left (1+c x +\sqrt {c^{2} x^{2}+1}\right )+2 d^{3} b^{2} c \,\operatorname {arcsinh}\left (c x \right ) \ln \left (1-c x -\sqrt {c^{2} x^{2}+1}\right )-\frac {122 d^{3} b^{2} \operatorname {arcsinh}\left (c x \right ) \sqrt {c^{2} x^{2}+1}\, c}{25}-\frac {d^{3} b^{2} \operatorname {arcsinh}\left (c x \right )^{2}}{x}+\frac {d^{3} b^{2} \operatorname {arcsinh}\left (c x \right )^{2} c^{6} x^{5}}{5}+d^{3} b^{2} \operatorname {arcsinh}\left (c x \right )^{2} c^{4} x^{3}+3 d^{3} b^{2} \operatorname {arcsinh}\left (c x \right )^{2} c^{2} x +\frac {2 b^{2} c^{6} d^{3} x^{5}}{125}+\frac {14 b^{2} c^{4} d^{3} x^{3}}{75}+\frac {122 b^{2} c^{2} d^{3} x}{25}-2 b^{2} c \,d^{3} \operatorname {polylog}\left (2, -c x -\sqrt {c^{2} x^{2}+1}\right )+2 b^{2} c \,d^{3} \operatorname {polylog}\left (2, c x +\sqrt {c^{2} x^{2}+1}\right )-\frac {2 d^{3} b^{2} \operatorname {arcsinh}\left (c x \right ) \sqrt {c^{2} x^{2}+1}\, c^{5} x^{4}}{25}-\frac {14 d^{3} b^{2} \operatorname {arcsinh}\left (c x \right ) \sqrt {c^{2} x^{2}+1}\, c^{3} x^{2}}{25}+2 d^{3} a b c \left (\frac {\operatorname {arcsinh}\left (c x \right ) c^{5} x^{5}}{5}+\operatorname {arcsinh}\left (c x \right ) c^{3} x^{3}+3 \,\operatorname {arcsinh}\left (c x \right ) c x -\frac {\operatorname {arcsinh}\left (c x \right )}{c x}-\frac {c^{4} x^{4} \sqrt {c^{2} x^{2}+1}}{25}-\frac {7 c^{2} x^{2} \sqrt {c^{2} x^{2}+1}}{25}-\frac {61 \sqrt {c^{2} x^{2}+1}}{25}-\operatorname {arctanh}\left (\frac {1}{\sqrt {c^{2} x^{2}+1}}\right )\right )\) | \(467\) |
c*(d^3*a^2*(1/5*c^5*x^5+c^3*x^3+3*c*x-1/c/x)+14/75*d^3*b^2*c^3*x^3+122/25* d^3*b^2*c*x+2*d^3*b^2*polylog(2,c*x+(c^2*x^2+1)^(1/2))-2*d^3*b^2*polylog(2 ,-c*x-(c^2*x^2+1)^(1/2))-2/25*d^3*b^2*arcsinh(c*x)*(c^2*x^2+1)^(1/2)*c^4*x ^4-14/25*d^3*b^2*arcsinh(c*x)*(c^2*x^2+1)^(1/2)*c^2*x^2+2*d^3*b^2*arcsinh( c*x)*ln(1-c*x-(c^2*x^2+1)^(1/2))+2/125*d^3*b^2*c^5*x^5-2*d^3*b^2*arcsinh(c *x)*ln(1+c*x+(c^2*x^2+1)^(1/2))-d^3*b^2*arcsinh(c*x)^2/c/x+1/5*d^3*b^2*arc sinh(c*x)^2*c^5*x^5+d^3*b^2*arcsinh(c*x)^2*c^3*x^3+3*d^3*b^2*arcsinh(c*x)^ 2*c*x-122/25*d^3*b^2*arcsinh(c*x)*(c^2*x^2+1)^(1/2)+2*d^3*a*b*(1/5*arcsinh (c*x)*c^5*x^5+arcsinh(c*x)*c^3*x^3+3*arcsinh(c*x)*c*x-arcsinh(c*x)/c/x-1/2 5*c^4*x^4*(c^2*x^2+1)^(1/2)-7/25*c^2*x^2*(c^2*x^2+1)^(1/2)-61/25*(c^2*x^2+ 1)^(1/2)-arctanh(1/(c^2*x^2+1)^(1/2))))
\[ \int \frac {\left (d+c^2 d x^2\right )^3 (a+b \text {arcsinh}(c x))^2}{x^2} \, dx=\int { \frac {{\left (c^{2} d x^{2} + d\right )}^{3} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2}}{x^{2}} \,d x } \]
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^2, x)
\[ \int \frac {\left (d+c^2 d x^2\right )^3 (a+b \text {arcsinh}(c x))^2}{x^2} \, dx=d^{3} \left (\int 3 a^{2} c^{2}\, dx + \int \frac {a^{2}}{x^{2}}\, dx + \int 3 a^{2} c^{4} x^{2}\, dx + \int a^{2} c^{6} x^{4}\, dx + \int 3 b^{2} c^{2} \operatorname {asinh}^{2}{\left (c x \right )}\, dx + \int \frac {b^{2} \operatorname {asinh}^{2}{\left (c x \right )}}{x^{2}}\, dx + \int 6 a b c^{2} \operatorname {asinh}{\left (c x \right )}\, dx + \int \frac {2 a b \operatorname {asinh}{\left (c x \right )}}{x^{2}}\, dx + \int 3 b^{2} c^{4} x^{2} \operatorname {asinh}^{2}{\left (c x \right )}\, dx + \int b^{2} c^{6} x^{4} \operatorname {asinh}^{2}{\left (c x \right )}\, dx + \int 6 a b c^{4} x^{2} \operatorname {asinh}{\left (c x \right )}\, dx + \int 2 a b c^{6} x^{4} \operatorname {asinh}{\left (c x \right )}\, dx\right ) \]
d**3*(Integral(3*a**2*c**2, x) + Integral(a**2/x**2, x) + Integral(3*a**2* c**4*x**2, x) + Integral(a**2*c**6*x**4, x) + Integral(3*b**2*c**2*asinh(c *x)**2, x) + Integral(b**2*asinh(c*x)**2/x**2, x) + Integral(6*a*b*c**2*as inh(c*x), x) + Integral(2*a*b*asinh(c*x)/x**2, x) + Integral(3*b**2*c**4*x **2*asinh(c*x)**2, x) + Integral(b**2*c**6*x**4*asinh(c*x)**2, x) + Integr al(6*a*b*c**4*x**2*asinh(c*x), x) + Integral(2*a*b*c**6*x**4*asinh(c*x), x ))
\[ \int \frac {\left (d+c^2 d x^2\right )^3 (a+b \text {arcsinh}(c x))^2}{x^2} \, dx=\int { \frac {{\left (c^{2} d x^{2} + d\right )}^{3} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2}}{x^{2}} \,d x } \]
1/5*a^2*c^6*d^3*x^5 + 2/75*(15*x^5*arcsinh(c*x) - (3*sqrt(c^2*x^2 + 1)*x^4 /c^2 - 4*sqrt(c^2*x^2 + 1)*x^2/c^4 + 8*sqrt(c^2*x^2 + 1)/c^6)*c)*a*b*c^6*d ^3 + a^2*c^4*d^3*x^3 + 2/3*(3*x^3*arcsinh(c*x) - c*(sqrt(c^2*x^2 + 1)*x^2/ c^2 - 2*sqrt(c^2*x^2 + 1)/c^4))*a*b*c^4*d^3 + 3*b^2*c^2*d^3*x*arcsinh(c*x) ^2 + 6*b^2*c^2*d^3*(x - sqrt(c^2*x^2 + 1)*arcsinh(c*x)/c) + 3*a^2*c^2*d^3* x + 6*(c*x*arcsinh(c*x) - sqrt(c^2*x^2 + 1))*a*b*c*d^3 - 2*(c*arcsinh(1/(c *abs(x))) + arcsinh(c*x)/x)*a*b*d^3 - a^2*d^3/x + 1/5*(b^2*c^6*d^3*x^6 + 5 *b^2*c^4*d^3*x^4 - 5*b^2*d^3)*log(c*x + sqrt(c^2*x^2 + 1))^2/x - integrate (2/5*(b^2*c^9*d^3*x^8 + 6*b^2*c^7*d^3*x^6 + 5*b^2*c^5*d^3*x^4 - 5*b^2*c^3* d^3*x^2 - 5*b^2*c*d^3 + (b^2*c^8*d^3*x^7 + 5*b^2*c^6*d^3*x^5 - 5*b^2*c^2*d ^3*x)*sqrt(c^2*x^2 + 1))*log(c*x + sqrt(c^2*x^2 + 1))/(c^3*x^4 + c*x^2 + ( c^2*x^3 + x)*sqrt(c^2*x^2 + 1)), x)
Exception generated. \[ \int \frac {\left (d+c^2 d x^2\right )^3 (a+b \text {arcsinh}(c x))^2}{x^2} \, dx=\text {Exception raised: TypeError} \]
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. \[ \int \frac {\left (d+c^2 d x^2\right )^3 (a+b \text {arcsinh}(c x))^2}{x^2} \, dx=\int \frac {{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2\,{\left (d\,c^2\,x^2+d\right )}^3}{x^2} \,d x \]