Integrand size = 26, antiderivative size = 326 \[ \int \frac {\left (d+c^2 d x^2\right )^3 (a+b \text {arcsinh}(c x))^2}{x^4} \, dx=-\frac {b^2 c^2 d^3}{3 x}+\frac {50}{9} b^2 c^4 d^3 x+\frac {2}{27} b^2 c^6 d^3 x^3-5 b c^3 d^3 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))+\frac {1}{9} b c^3 d^3 \left (1+c^2 x^2\right )^{3/2} (a+b \text {arcsinh}(c x))-\frac {b c d^3 \left (1+c^2 x^2\right )^{5/2} (a+b \text {arcsinh}(c x))}{3 x^2}+\frac {16}{3} c^4 d^3 x (a+b \text {arcsinh}(c x))^2+\frac {8}{3} c^4 d^3 x \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))^2-\frac {2 c^2 d^3 \left (1+c^2 x^2\right )^2 (a+b \text {arcsinh}(c x))^2}{x}-\frac {d^3 \left (1+c^2 x^2\right )^3 (a+b \text {arcsinh}(c x))^2}{3 x^3}-\frac {34}{3} b c^3 d^3 (a+b \text {arcsinh}(c x)) \text {arctanh}\left (e^{\text {arcsinh}(c x)}\right )-\frac {17}{3} b^2 c^3 d^3 \operatorname {PolyLog}\left (2,-e^{\text {arcsinh}(c x)}\right )+\frac {17}{3} b^2 c^3 d^3 \operatorname {PolyLog}\left (2,e^{\text {arcsinh}(c x)}\right ) \] Output:
-1/3*b^2*c^2*d^3/x+50/9*b^2*c^4*d^3*x+2/27*b^2*c^6*d^3*x^3-5*b*c^3*d^3*(c^ 2*x^2+1)^(1/2)*(a+b*arcsinh(c*x))+1/9*b*c^3*d^3*(c^2*x^2+1)^(3/2)*(a+b*arc sinh(c*x))-1/3*b*c*d^3*(c^2*x^2+1)^(5/2)*(a+b*arcsinh(c*x))/x^2+16/3*c^4*d ^3*x*(a+b*arcsinh(c*x))^2+8/3*c^4*d^3*x*(c^2*x^2+1)*(a+b*arcsinh(c*x))^2-2 *c^2*d^3*(c^2*x^2+1)^2*(a+b*arcsinh(c*x))^2/x-1/3*d^3*(c^2*x^2+1)^3*(a+b*a rcsinh(c*x))^2/x^3-34/3*b*c^3*d^3*(a+b*arcsinh(c*x))*arctanh(c*x+(c^2*x^2+ 1)^(1/2))-17/3*b^2*c^3*d^3*polylog(2,-c*x-(c^2*x^2+1)^(1/2))+17/3*b^2*c^3* d^3*polylog(2,c*x+(c^2*x^2+1)^(1/2))
Time = 0.73 (sec) , antiderivative size = 461, normalized size of antiderivative = 1.41 \[ \int \frac {\left (d+c^2 d x^2\right )^3 (a+b \text {arcsinh}(c x))^2}{x^4} \, dx=\frac {d^3 \left (-9 a^2-81 a^2 c^2 x^2-9 b^2 c^2 x^2+81 a^2 c^4 x^4+150 b^2 c^4 x^4+9 a^2 c^6 x^6+2 b^2 c^6 x^6-9 a b c x \sqrt {1+c^2 x^2}-150 a b c^3 x^3 \sqrt {1+c^2 x^2}-6 a b c^5 x^5 \sqrt {1+c^2 x^2}-18 a b \text {arcsinh}(c x)-162 a b c^2 x^2 \text {arcsinh}(c x)+162 a b c^4 x^4 \text {arcsinh}(c x)+18 a b c^6 x^6 \text {arcsinh}(c x)-9 b^2 c x \sqrt {1+c^2 x^2} \text {arcsinh}(c x)-150 b^2 c^3 x^3 \sqrt {1+c^2 x^2} \text {arcsinh}(c x)-6 b^2 c^5 x^5 \sqrt {1+c^2 x^2} \text {arcsinh}(c x)-9 b^2 \text {arcsinh}(c x)^2-81 b^2 c^2 x^2 \text {arcsinh}(c x)^2+81 b^2 c^4 x^4 \text {arcsinh}(c x)^2+9 b^2 c^6 x^6 \text {arcsinh}(c x)^2-153 a b c^3 x^3 \text {arctanh}\left (\sqrt {1+c^2 x^2}\right )+153 b^2 c^3 x^3 \text {arcsinh}(c x) \log \left (1-e^{-\text {arcsinh}(c x)}\right )-153 b^2 c^3 x^3 \text {arcsinh}(c x) \log \left (1+e^{-\text {arcsinh}(c x)}\right )+153 b^2 c^3 x^3 \operatorname {PolyLog}\left (2,-e^{-\text {arcsinh}(c x)}\right )-153 b^2 c^3 x^3 \operatorname {PolyLog}\left (2,e^{-\text {arcsinh}(c x)}\right )\right )}{27 x^3} \] Input:
Integrate[((d + c^2*d*x^2)^3*(a + b*ArcSinh[c*x])^2)/x^4,x]
Output:
(d^3*(-9*a^2 - 81*a^2*c^2*x^2 - 9*b^2*c^2*x^2 + 81*a^2*c^4*x^4 + 150*b^2*c ^4*x^4 + 9*a^2*c^6*x^6 + 2*b^2*c^6*x^6 - 9*a*b*c*x*Sqrt[1 + c^2*x^2] - 150 *a*b*c^3*x^3*Sqrt[1 + c^2*x^2] - 6*a*b*c^5*x^5*Sqrt[1 + c^2*x^2] - 18*a*b* ArcSinh[c*x] - 162*a*b*c^2*x^2*ArcSinh[c*x] + 162*a*b*c^4*x^4*ArcSinh[c*x] + 18*a*b*c^6*x^6*ArcSinh[c*x] - 9*b^2*c*x*Sqrt[1 + c^2*x^2]*ArcSinh[c*x] - 150*b^2*c^3*x^3*Sqrt[1 + c^2*x^2]*ArcSinh[c*x] - 6*b^2*c^5*x^5*Sqrt[1 + c^2*x^2]*ArcSinh[c*x] - 9*b^2*ArcSinh[c*x]^2 - 81*b^2*c^2*x^2*ArcSinh[c*x] ^2 + 81*b^2*c^4*x^4*ArcSinh[c*x]^2 + 9*b^2*c^6*x^6*ArcSinh[c*x]^2 - 153*a* b*c^3*x^3*ArcTanh[Sqrt[1 + c^2*x^2]] + 153*b^2*c^3*x^3*ArcSinh[c*x]*Log[1 - E^(-ArcSinh[c*x])] - 153*b^2*c^3*x^3*ArcSinh[c*x]*Log[1 + E^(-ArcSinh[c* x])] + 153*b^2*c^3*x^3*PolyLog[2, -E^(-ArcSinh[c*x])] - 153*b^2*c^3*x^3*Po lyLog[2, E^(-ArcSinh[c*x])]))/(27*x^3)
Result contains complex when optimal does not.
Time = 3.69 (sec) , antiderivative size = 529, normalized size of antiderivative = 1.62, number of steps used = 21, number of rules used = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.769, Rules used = {6222, 27, 6222, 244, 2009, 6201, 6187, 6213, 24, 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.
| \(\displaystyle \int \frac {\left (c^2 d x^2+d\right )^3 (a+b \text {arcsinh}(c x))^2}{x^4} \, dx\) |
| \(\Big \downarrow \) 6222 |
| \(\displaystyle \frac {2}{3} b c d^3 \int \frac {\left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))}{x^3}dx+2 c^2 d \int \frac {d^2 \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))^2}{x^2}dx-\frac {d^3 \left (c^2 x^2+1\right )^3 (a+b \text {arcsinh}(c x))^2}{3 x^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 2 c^2 d^3 \int \frac {\left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))^2}{x^2}dx+\frac {2}{3} b c d^3 \int \frac {\left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))}{x^3}dx-\frac {d^3 \left (c^2 x^2+1\right )^3 (a+b \text {arcsinh}(c x))^2}{3 x^3}\) |
| \(\Big \downarrow \) 6222 |
| \(\displaystyle \frac {2}{3} b c d^3 \left (\frac {5}{2} c^2 \int \frac {\left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))}{x}dx+\frac {1}{2} b c \int \frac {\left (c^2 x^2+1\right )^2}{x^2}dx-\frac {\left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))}{2 x^2}\right )+2 c^2 d^3 \left (4 c^2 \int \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))^2dx+2 b c \int \frac {\left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))}{x}dx-\frac {\left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))^2}{x}\right )-\frac {d^3 \left (c^2 x^2+1\right )^3 (a+b \text {arcsinh}(c x))^2}{3 x^3}\) |
| \(\Big \downarrow \) 244 |
| \(\displaystyle 2 c^2 d^3 \left (4 c^2 \int \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))^2dx+2 b c \int \frac {\left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))}{x}dx-\frac {\left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))^2}{x}\right )+\frac {2}{3} b c d^3 \left (\frac {5}{2} c^2 \int \frac {\left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))}{x}dx+\frac {1}{2} b c \int \left (x^2 c^4+2 c^2+\frac {1}{x^2}\right )dx-\frac {\left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))}{2 x^2}\right )-\frac {d^3 \left (c^2 x^2+1\right )^3 (a+b \text {arcsinh}(c x))^2}{3 x^3}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle 2 c^2 d^3 \left (4 c^2 \int \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))^2dx+2 b c \int \frac {\left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))}{x}dx-\frac {\left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))^2}{x}\right )+\frac {2}{3} b c d^3 \left (\frac {5}{2} c^2 \int \frac {\left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))}{x}dx-\frac {\left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))}{2 x^2}+\frac {1}{2} b c \left (\frac {c^4 x^3}{3}+2 c^2 x-\frac {1}{x}\right )\right )-\frac {d^3 \left (c^2 x^2+1\right )^3 (a+b \text {arcsinh}(c x))^2}{3 x^3}\) |
| \(\Big \downarrow \) 6201 |
| \(\displaystyle 2 c^2 d^3 \left (4 c^2 \left (-\frac {2}{3} b c \int x \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))dx+\frac {2}{3} \int (a+b \text {arcsinh}(c x))^2dx+\frac {1}{3} x \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))^2\right )+2 b c \int \frac {\left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))}{x}dx-\frac {\left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))^2}{x}\right )+\frac {2}{3} b c d^3 \left (\frac {5}{2} c^2 \int \frac {\left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))}{x}dx-\frac {\left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))}{2 x^2}+\frac {1}{2} b c \left (\frac {c^4 x^3}{3}+2 c^2 x-\frac {1}{x}\right )\right )-\frac {d^3 \left (c^2 x^2+1\right )^3 (a+b \text {arcsinh}(c x))^2}{3 x^3}\) |
| \(\Big \downarrow \) 6187 |
| \(\displaystyle 2 c^2 d^3 \left (4 c^2 \left (\frac {2}{3} \left (x (a+b \text {arcsinh}(c x))^2-2 b c \int \frac {x (a+b \text {arcsinh}(c x))}{\sqrt {c^2 x^2+1}}dx\right )-\frac {2}{3} b c \int x \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))dx+\frac {1}{3} x \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))^2\right )+2 b c \int \frac {\left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))}{x}dx-\frac {\left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))^2}{x}\right )+\frac {2}{3} b c d^3 \left (\frac {5}{2} c^2 \int \frac {\left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))}{x}dx-\frac {\left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))}{2 x^2}+\frac {1}{2} b c \left (\frac {c^4 x^3}{3}+2 c^2 x-\frac {1}{x}\right )\right )-\frac {d^3 \left (c^2 x^2+1\right )^3 (a+b \text {arcsinh}(c x))^2}{3 x^3}\) |
| \(\Big \downarrow \) 6213 |
| \(\displaystyle 2 c^2 d^3 \left (4 c^2 \left (\frac {2}{3} \left (x (a+b \text {arcsinh}(c x))^2-2 b c \left (\frac {\sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{c^2}-\frac {b \int 1dx}{c}\right )\right )-\frac {2}{3} b c \left (\frac {\left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))}{3 c^2}-\frac {b \int \left (c^2 x^2+1\right )dx}{3 c}\right )+\frac {1}{3} x \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))^2\right )+2 b c \int \frac {\left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))}{x}dx-\frac {\left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))^2}{x}\right )+\frac {2}{3} b c d^3 \left (\frac {5}{2} c^2 \int \frac {\left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))}{x}dx-\frac {\left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))}{2 x^2}+\frac {1}{2} b c \left (\frac {c^4 x^3}{3}+2 c^2 x-\frac {1}{x}\right )\right )-\frac {d^3 \left (c^2 x^2+1\right )^3 (a+b \text {arcsinh}(c x))^2}{3 x^3}\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle 2 c^2 d^3 \left (4 c^2 \left (-\frac {2}{3} b c \left (\frac {\left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))}{3 c^2}-\frac {b \int \left (c^2 x^2+1\right )dx}{3 c}\right )+\frac {1}{3} x \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))^2+\frac {2}{3} \left (x (a+b \text {arcsinh}(c x))^2-2 b c \left (\frac {\sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{c^2}-\frac {b x}{c}\right )\right )\right )+2 b c \int \frac {\left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))}{x}dx-\frac {\left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))^2}{x}\right )+\frac {2}{3} b c d^3 \left (\frac {5}{2} c^2 \int \frac {\left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))}{x}dx-\frac {\left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))}{2 x^2}+\frac {1}{2} b c \left (\frac {c^4 x^3}{3}+2 c^2 x-\frac {1}{x}\right )\right )-\frac {d^3 \left (c^2 x^2+1\right )^3 (a+b \text {arcsinh}(c x))^2}{3 x^3}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle 2 c^2 d^3 \left (2 b c \int \frac {\left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))}{x}dx-\frac {\left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))^2}{x}+4 c^2 \left (\frac {1}{3} x \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))^2+\frac {2}{3} \left (x (a+b \text {arcsinh}(c x))^2-2 b c \left (\frac {\sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{c^2}-\frac {b x}{c}\right )\right )-\frac {2}{3} b c \left (\frac {\left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))}{3 c^2}-\frac {b \left (\frac {c^2 x^3}{3}+x\right )}{3 c}\right )\right )\right )+\frac {2}{3} b c d^3 \left (\frac {5}{2} c^2 \int \frac {\left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))}{x}dx-\frac {\left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))}{2 x^2}+\frac {1}{2} b c \left (\frac {c^4 x^3}{3}+2 c^2 x-\frac {1}{x}\right )\right )-\frac {d^3 \left (c^2 x^2+1\right )^3 (a+b \text {arcsinh}(c x))^2}{3 x^3}\) |
| \(\Big \downarrow \) 6223 |
| \(\displaystyle 2 c^2 d^3 \left (2 b c \left (\int \frac {\sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{x}dx-\frac {1}{3} b c \int \left (c^2 x^2+1\right )dx+\frac {1}{3} \left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))\right )-\frac {\left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))^2}{x}+4 c^2 \left (\frac {1}{3} x \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))^2+\frac {2}{3} \left (x (a+b \text {arcsinh}(c x))^2-2 b c \left (\frac {\sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{c^2}-\frac {b x}{c}\right )\right )-\frac {2}{3} b c \left (\frac {\left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))}{3 c^2}-\frac {b \left (\frac {c^2 x^3}{3}+x\right )}{3 c}\right )\right )\right )+\frac {2}{3} b c d^3 \left (\frac {5}{2} c^2 \left (\int \frac {\sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{x}dx-\frac {1}{3} b c \int \left (c^2 x^2+1\right )dx+\frac {1}{3} \left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))\right )-\frac {\left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))}{2 x^2}+\frac {1}{2} b c \left (\frac {c^4 x^3}{3}+2 c^2 x-\frac {1}{x}\right )\right )-\frac {d^3 \left (c^2 x^2+1\right )^3 (a+b \text {arcsinh}(c x))^2}{3 x^3}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle 2 c^2 d^3 \left (2 b c \left (\int \frac {\sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{x}dx+\frac {1}{3} \left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))-\frac {1}{3} b c \left (\frac {c^2 x^3}{3}+x\right )\right )-\frac {\left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))^2}{x}+4 c^2 \left (\frac {1}{3} x \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))^2+\frac {2}{3} \left (x (a+b \text {arcsinh}(c x))^2-2 b c \left (\frac {\sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{c^2}-\frac {b x}{c}\right )\right )-\frac {2}{3} b c \left (\frac {\left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))}{3 c^2}-\frac {b \left (\frac {c^2 x^3}{3}+x\right )}{3 c}\right )\right )\right )+\frac {2}{3} b c d^3 \left (\frac {5}{2} c^2 \left (\int \frac {\sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{x}dx+\frac {1}{3} \left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))-\frac {1}{3} b c \left (\frac {c^2 x^3}{3}+x\right )\right )-\frac {\left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))}{2 x^2}+\frac {1}{2} b c \left (\frac {c^4 x^3}{3}+2 c^2 x-\frac {1}{x}\right )\right )-\frac {d^3 \left (c^2 x^2+1\right )^3 (a+b \text {arcsinh}(c x))^2}{3 x^3}\) |
| \(\Big \downarrow \) 6221 |
| \(\displaystyle 2 c^2 d^3 \left (2 b c \left (\int \frac {a+b \text {arcsinh}(c x)}{x \sqrt {c^2 x^2+1}}dx-b c \int 1dx+\frac {1}{3} \left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))+\sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))-\frac {1}{3} b c \left (\frac {c^2 x^3}{3}+x\right )\right )-\frac {\left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))^2}{x}+4 c^2 \left (\frac {1}{3} x \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))^2+\frac {2}{3} \left (x (a+b \text {arcsinh}(c x))^2-2 b c \left (\frac {\sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{c^2}-\frac {b x}{c}\right )\right )-\frac {2}{3} b c \left (\frac {\left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))}{3 c^2}-\frac {b \left (\frac {c^2 x^3}{3}+x\right )}{3 c}\right )\right )\right )+\frac {2}{3} b c d^3 \left (\frac {5}{2} c^2 \left (\int \frac {a+b \text {arcsinh}(c x)}{x \sqrt {c^2 x^2+1}}dx-b c \int 1dx+\frac {1}{3} \left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))+\sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))-\frac {1}{3} b c \left (\frac {c^2 x^3}{3}+x\right )\right )-\frac {\left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))}{2 x^2}+\frac {1}{2} b c \left (\frac {c^4 x^3}{3}+2 c^2 x-\frac {1}{x}\right )\right )-\frac {d^3 \left (c^2 x^2+1\right )^3 (a+b \text {arcsinh}(c x))^2}{3 x^3}\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle 2 c^2 d^3 \left (2 b c \left (\int \frac {a+b \text {arcsinh}(c x)}{x \sqrt {c^2 x^2+1}}dx+\frac {1}{3} \left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))+\sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))-\frac {1}{3} b c \left (\frac {c^2 x^3}{3}+x\right )-b c x\right )-\frac {\left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))^2}{x}+4 c^2 \left (\frac {1}{3} x \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))^2+\frac {2}{3} \left (x (a+b \text {arcsinh}(c x))^2-2 b c \left (\frac {\sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{c^2}-\frac {b x}{c}\right )\right )-\frac {2}{3} b c \left (\frac {\left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))}{3 c^2}-\frac {b \left (\frac {c^2 x^3}{3}+x\right )}{3 c}\right )\right )\right )+\frac {2}{3} b c d^3 \left (\frac {5}{2} c^2 \left (\int \frac {a+b \text {arcsinh}(c x)}{x \sqrt {c^2 x^2+1}}dx+\frac {1}{3} \left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))+\sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))-\frac {1}{3} b c \left (\frac {c^2 x^3}{3}+x\right )-b c x\right )-\frac {\left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))}{2 x^2}+\frac {1}{2} b c \left (\frac {c^4 x^3}{3}+2 c^2 x-\frac {1}{x}\right )\right )-\frac {d^3 \left (c^2 x^2+1\right )^3 (a+b \text {arcsinh}(c x))^2}{3 x^3}\) |
| \(\Big \downarrow \) 6231 |
| \(\displaystyle 2 c^2 d^3 \left (2 b c \left (\int \frac {a+b \text {arcsinh}(c x)}{c x}d\text {arcsinh}(c x)+\frac {1}{3} \left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))+\sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))-\frac {1}{3} b c \left (\frac {c^2 x^3}{3}+x\right )-b c x\right )-\frac {\left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))^2}{x}+4 c^2 \left (\frac {1}{3} x \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))^2+\frac {2}{3} \left (x (a+b \text {arcsinh}(c x))^2-2 b c \left (\frac {\sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{c^2}-\frac {b x}{c}\right )\right )-\frac {2}{3} b c \left (\frac {\left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))}{3 c^2}-\frac {b \left (\frac {c^2 x^3}{3}+x\right )}{3 c}\right )\right )\right )+\frac {2}{3} b c d^3 \left (\frac {5}{2} c^2 \left (\int \frac {a+b \text {arcsinh}(c x)}{c x}d\text {arcsinh}(c x)+\frac {1}{3} \left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))+\sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))-\frac {1}{3} b c \left (\frac {c^2 x^3}{3}+x\right )-b c x\right )-\frac {\left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))}{2 x^2}+\frac {1}{2} b c \left (\frac {c^4 x^3}{3}+2 c^2 x-\frac {1}{x}\right )\right )-\frac {d^3 \left (c^2 x^2+1\right )^3 (a+b \text {arcsinh}(c x))^2}{3 x^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle 2 c^2 d^3 \left (2 b c \left (\int i (a+b \text {arcsinh}(c x)) \csc (i \text {arcsinh}(c x))d\text {arcsinh}(c x)+\frac {1}{3} \left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))+\sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))-\frac {1}{3} b c \left (\frac {c^2 x^3}{3}+x\right )-b c x\right )-\frac {\left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))^2}{x}+4 c^2 \left (\frac {1}{3} x \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))^2+\frac {2}{3} \left (x (a+b \text {arcsinh}(c x))^2-2 b c \left (\frac {\sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{c^2}-\frac {b x}{c}\right )\right )-\frac {2}{3} b c \left (\frac {\left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))}{3 c^2}-\frac {b \left (\frac {c^2 x^3}{3}+x\right )}{3 c}\right )\right )\right )+\frac {2}{3} b c d^3 \left (\frac {5}{2} c^2 \left (\int i (a+b \text {arcsinh}(c x)) \csc (i \text {arcsinh}(c x))d\text {arcsinh}(c x)+\frac {1}{3} \left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))+\sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))-\frac {1}{3} b c \left (\frac {c^2 x^3}{3}+x\right )-b c x\right )-\frac {\left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))}{2 x^2}+\frac {1}{2} b c \left (\frac {c^4 x^3}{3}+2 c^2 x-\frac {1}{x}\right )\right )-\frac {d^3 \left (c^2 x^2+1\right )^3 (a+b \text {arcsinh}(c x))^2}{3 x^3}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle 2 c^2 d^3 \left (2 b c \left (i \int (a+b \text {arcsinh}(c x)) \csc (i \text {arcsinh}(c x))d\text {arcsinh}(c x)+\frac {1}{3} \left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))+\sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))-\frac {1}{3} b c \left (\frac {c^2 x^3}{3}+x\right )-b c x\right )-\frac {\left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))^2}{x}+4 c^2 \left (\frac {1}{3} x \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))^2+\frac {2}{3} \left (x (a+b \text {arcsinh}(c x))^2-2 b c \left (\frac {\sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{c^2}-\frac {b x}{c}\right )\right )-\frac {2}{3} b c \left (\frac {\left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))}{3 c^2}-\frac {b \left (\frac {c^2 x^3}{3}+x\right )}{3 c}\right )\right )\right )+\frac {2}{3} b c d^3 \left (\frac {5}{2} c^2 \left (i \int (a+b \text {arcsinh}(c x)) \csc (i \text {arcsinh}(c x))d\text {arcsinh}(c x)+\frac {1}{3} \left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))+\sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))-\frac {1}{3} b c \left (\frac {c^2 x^3}{3}+x\right )-b c x\right )-\frac {\left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))}{2 x^2}+\frac {1}{2} b c \left (\frac {c^4 x^3}{3}+2 c^2 x-\frac {1}{x}\right )\right )-\frac {d^3 \left (c^2 x^2+1\right )^3 (a+b \text {arcsinh}(c x))^2}{3 x^3}\) |
| \(\Big \downarrow \) 4670 |
| \(\displaystyle 2 c^2 d^3 \left (2 b c \left (i \left (i b \int \log \left (1-e^{\text {arcsinh}(c x)}\right )d\text {arcsinh}(c x)-i b \int \log \left (1+e^{\text {arcsinh}(c x)}\right )d\text {arcsinh}(c x)+2 i \text {arctanh}\left (e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))\right )+\frac {1}{3} \left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))+\sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))-\frac {1}{3} b c \left (\frac {c^2 x^3}{3}+x\right )-b c x\right )-\frac {\left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))^2}{x}+4 c^2 \left (\frac {1}{3} x \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))^2+\frac {2}{3} \left (x (a+b \text {arcsinh}(c x))^2-2 b c \left (\frac {\sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{c^2}-\frac {b x}{c}\right )\right )-\frac {2}{3} b c \left (\frac {\left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))}{3 c^2}-\frac {b \left (\frac {c^2 x^3}{3}+x\right )}{3 c}\right )\right )\right )+\frac {2}{3} b c d^3 \left (\frac {5}{2} c^2 \left (i \left (i b \int \log \left (1-e^{\text {arcsinh}(c x)}\right )d\text {arcsinh}(c x)-i b \int \log \left (1+e^{\text {arcsinh}(c x)}\right )d\text {arcsinh}(c x)+2 i \text {arctanh}\left (e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))\right )+\frac {1}{3} \left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))+\sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))-\frac {1}{3} b c \left (\frac {c^2 x^3}{3}+x\right )-b c x\right )-\frac {\left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))}{2 x^2}+\frac {1}{2} b c \left (\frac {c^4 x^3}{3}+2 c^2 x-\frac {1}{x}\right )\right )-\frac {d^3 \left (c^2 x^2+1\right )^3 (a+b \text {arcsinh}(c x))^2}{3 x^3}\) |
| \(\Big \downarrow \) 2715 |
| \(\displaystyle 2 c^2 d^3 \left (2 b c \left (i \left (i b \int e^{-\text {arcsinh}(c x)} \log \left (1-e^{\text {arcsinh}(c x)}\right )de^{\text {arcsinh}(c x)}-i b \int e^{-\text {arcsinh}(c x)} \log \left (1+e^{\text {arcsinh}(c x)}\right )de^{\text {arcsinh}(c x)}+2 i \text {arctanh}\left (e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))\right )+\frac {1}{3} \left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))+\sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))-\frac {1}{3} b c \left (\frac {c^2 x^3}{3}+x\right )-b c x\right )-\frac {\left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))^2}{x}+4 c^2 \left (\frac {1}{3} x \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))^2+\frac {2}{3} \left (x (a+b \text {arcsinh}(c x))^2-2 b c \left (\frac {\sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{c^2}-\frac {b x}{c}\right )\right )-\frac {2}{3} b c \left (\frac {\left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))}{3 c^2}-\frac {b \left (\frac {c^2 x^3}{3}+x\right )}{3 c}\right )\right )\right )+\frac {2}{3} b c d^3 \left (\frac {5}{2} c^2 \left (i \left (i b \int e^{-\text {arcsinh}(c x)} \log \left (1-e^{\text {arcsinh}(c x)}\right )de^{\text {arcsinh}(c x)}-i b \int e^{-\text {arcsinh}(c x)} \log \left (1+e^{\text {arcsinh}(c x)}\right )de^{\text {arcsinh}(c x)}+2 i \text {arctanh}\left (e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))\right )+\frac {1}{3} \left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))+\sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))-\frac {1}{3} b c \left (\frac {c^2 x^3}{3}+x\right )-b c x\right )-\frac {\left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))}{2 x^2}+\frac {1}{2} b c \left (\frac {c^4 x^3}{3}+2 c^2 x-\frac {1}{x}\right )\right )-\frac {d^3 \left (c^2 x^2+1\right )^3 (a+b \text {arcsinh}(c x))^2}{3 x^3}\) |
| \(\Big \downarrow \) 2838 |
| \(\displaystyle 2 c^2 d^3 \left (2 b c \left (i \left (2 i \text {arctanh}\left (e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))+i b \operatorname {PolyLog}\left (2,-e^{\text {arcsinh}(c x)}\right )-i b \operatorname {PolyLog}\left (2,e^{\text {arcsinh}(c x)}\right )\right )+\frac {1}{3} \left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))+\sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))-\frac {1}{3} b c \left (\frac {c^2 x^3}{3}+x\right )-b c x\right )-\frac {\left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))^2}{x}+4 c^2 \left (\frac {1}{3} x \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))^2+\frac {2}{3} \left (x (a+b \text {arcsinh}(c x))^2-2 b c \left (\frac {\sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{c^2}-\frac {b x}{c}\right )\right )-\frac {2}{3} b c \left (\frac {\left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))}{3 c^2}-\frac {b \left (\frac {c^2 x^3}{3}+x\right )}{3 c}\right )\right )\right )+\frac {2}{3} b c d^3 \left (\frac {5}{2} c^2 \left (i \left (2 i \text {arctanh}\left (e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))+i b \operatorname {PolyLog}\left (2,-e^{\text {arcsinh}(c x)}\right )-i b \operatorname {PolyLog}\left (2,e^{\text {arcsinh}(c x)}\right )\right )+\frac {1}{3} \left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))+\sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))-\frac {1}{3} b c \left (\frac {c^2 x^3}{3}+x\right )-b c x\right )-\frac {\left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))}{2 x^2}+\frac {1}{2} b c \left (\frac {c^4 x^3}{3}+2 c^2 x-\frac {1}{x}\right )\right )-\frac {d^3 \left (c^2 x^2+1\right )^3 (a+b \text {arcsinh}(c x))^2}{3 x^3}\) |
Input:
Int[((d + c^2*d*x^2)^3*(a + b*ArcSinh[c*x])^2)/x^4,x]
Output:
-1/3*(d^3*(1 + c^2*x^2)^3*(a + b*ArcSinh[c*x])^2)/x^3 + 2*c^2*d^3*(-(((1 + c^2*x^2)^2*(a + b*ArcSinh[c*x])^2)/x) + 4*c^2*((x*(1 + c^2*x^2)*(a + b*Ar cSinh[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) + 2*b *c*(-(b*c*x) - (b*c*(x + (c^2*x^3)/3))/3 + Sqrt[1 + c^2*x^2]*(a + b*ArcSin h[c*x]) + ((1 + c^2*x^2)^(3/2)*(a + b*ArcSinh[c*x]))/3 + I*((2*I)*(a + b*A rcSinh[c*x])*ArcTanh[E^ArcSinh[c*x]] + I*b*PolyLog[2, -E^ArcSinh[c*x]] - I *b*PolyLog[2, E^ArcSinh[c*x]]))) + (2*b*c*d^3*((b*c*(-x^(-1) + 2*c^2*x + ( c^4*x^3)/3))/2 - ((1 + c^2*x^2)^(5/2)*(a + b*ArcSinh[c*x]))/(2*x^2) + (5*c ^2*(-(b*c*x) - (b*c*(x + (c^2*x^3)/3))/3 + Sqrt[1 + c^2*x^2]*(a + b*ArcSin h[c*x]) + ((1 + c^2*x^2)^(3/2)*(a + b*ArcSinh[c*x]))/3 + I*((2*I)*(a + b*A rcSinh[c*x])*ArcTanh[E^ArcSinh[c*x]] + I*b*PolyLog[2, -E^ArcSinh[c*x]] - I *b*PolyLog[2, E^ArcSinh[c*x]])))/2))/3
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[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 = 1.39 (sec) , antiderivative size = 468, normalized size of antiderivative = 1.44
| method | result | size |
| derivativedivides | \(c^{3} \left (d^{3} a^{2} \left (\frac {x^{3} c^{3}}{3}+3 x c -\frac {3}{x c}-\frac {1}{3 x^{3} c^{3}}\right )+\frac {17 d^{3} b^{2} \operatorname {polylog}\left (2, x c +\sqrt {c^{2} x^{2}+1}\right )}{3}+\frac {50 d^{3} b^{2} x c}{9}-\frac {d^{3} b^{2}}{3 x c}+\frac {2 d^{3} b^{2} x^{3} c^{3}}{27}-\frac {17 d^{3} b^{2} \operatorname {polylog}\left (2, -x c -\sqrt {c^{2} x^{2}+1}\right )}{3}-\frac {3 d^{3} b^{2} \operatorname {arcsinh}\left (x c \right )^{2}}{x c}-\frac {d^{3} b^{2} \operatorname {arcsinh}\left (x c \right )^{2}}{3 x^{3} c^{3}}+\frac {d^{3} b^{2} \operatorname {arcsinh}\left (x c \right )^{2} x^{3} c^{3}}{3}+3 d^{3} b^{2} \operatorname {arcsinh}\left (x c \right )^{2} x c -\frac {50 d^{3} b^{2} \operatorname {arcsinh}\left (x c \right ) \sqrt {c^{2} x^{2}+1}}{9}+\frac {17 d^{3} b^{2} \operatorname {arcsinh}\left (x c \right ) \ln \left (1-x c -\sqrt {c^{2} x^{2}+1}\right )}{3}-\frac {17 d^{3} b^{2} \operatorname {arcsinh}\left (x c \right ) \ln \left (1+x c +\sqrt {c^{2} x^{2}+1}\right )}{3}-\frac {d^{3} b^{2} \operatorname {arcsinh}\left (x c \right ) \sqrt {c^{2} x^{2}+1}}{3 x^{2} c^{2}}-\frac {2 d^{3} b^{2} \operatorname {arcsinh}\left (x c \right ) \sqrt {c^{2} x^{2}+1}\, x^{2} c^{2}}{9}+2 d^{3} a b \left (\frac {\operatorname {arcsinh}\left (x c \right ) x^{3} c^{3}}{3}+3 x c \,\operatorname {arcsinh}\left (x c \right )-\frac {3 \,\operatorname {arcsinh}\left (x c \right )}{x c}-\frac {\operatorname {arcsinh}\left (x c \right )}{3 x^{3} c^{3}}-\frac {x^{2} c^{2} \sqrt {c^{2} x^{2}+1}}{9}-\frac {25 \sqrt {c^{2} x^{2}+1}}{9}-\frac {\sqrt {c^{2} x^{2}+1}}{6 x^{2} c^{2}}-\frac {17 \,\operatorname {arctanh}\left (\frac {1}{\sqrt {c^{2} x^{2}+1}}\right )}{6}\right )\right )\) | \(468\) |
| default | \(c^{3} \left (d^{3} a^{2} \left (\frac {x^{3} c^{3}}{3}+3 x c -\frac {3}{x c}-\frac {1}{3 x^{3} c^{3}}\right )+\frac {17 d^{3} b^{2} \operatorname {polylog}\left (2, x c +\sqrt {c^{2} x^{2}+1}\right )}{3}+\frac {50 d^{3} b^{2} x c}{9}-\frac {d^{3} b^{2}}{3 x c}+\frac {2 d^{3} b^{2} x^{3} c^{3}}{27}-\frac {17 d^{3} b^{2} \operatorname {polylog}\left (2, -x c -\sqrt {c^{2} x^{2}+1}\right )}{3}-\frac {3 d^{3} b^{2} \operatorname {arcsinh}\left (x c \right )^{2}}{x c}-\frac {d^{3} b^{2} \operatorname {arcsinh}\left (x c \right )^{2}}{3 x^{3} c^{3}}+\frac {d^{3} b^{2} \operatorname {arcsinh}\left (x c \right )^{2} x^{3} c^{3}}{3}+3 d^{3} b^{2} \operatorname {arcsinh}\left (x c \right )^{2} x c -\frac {50 d^{3} b^{2} \operatorname {arcsinh}\left (x c \right ) \sqrt {c^{2} x^{2}+1}}{9}+\frac {17 d^{3} b^{2} \operatorname {arcsinh}\left (x c \right ) \ln \left (1-x c -\sqrt {c^{2} x^{2}+1}\right )}{3}-\frac {17 d^{3} b^{2} \operatorname {arcsinh}\left (x c \right ) \ln \left (1+x c +\sqrt {c^{2} x^{2}+1}\right )}{3}-\frac {d^{3} b^{2} \operatorname {arcsinh}\left (x c \right ) \sqrt {c^{2} x^{2}+1}}{3 x^{2} c^{2}}-\frac {2 d^{3} b^{2} \operatorname {arcsinh}\left (x c \right ) \sqrt {c^{2} x^{2}+1}\, x^{2} c^{2}}{9}+2 d^{3} a b \left (\frac {\operatorname {arcsinh}\left (x c \right ) x^{3} c^{3}}{3}+3 x c \,\operatorname {arcsinh}\left (x c \right )-\frac {3 \,\operatorname {arcsinh}\left (x c \right )}{x c}-\frac {\operatorname {arcsinh}\left (x c \right )}{3 x^{3} c^{3}}-\frac {x^{2} c^{2} \sqrt {c^{2} x^{2}+1}}{9}-\frac {25 \sqrt {c^{2} x^{2}+1}}{9}-\frac {\sqrt {c^{2} x^{2}+1}}{6 x^{2} c^{2}}-\frac {17 \,\operatorname {arctanh}\left (\frac {1}{\sqrt {c^{2} x^{2}+1}}\right )}{6}\right )\right )\) | \(468\) |
| parts | \(d^{3} a^{2} \left (\frac {x^{3} c^{6}}{3}+3 c^{4} x -\frac {1}{3 x^{3}}-\frac {3 c^{2}}{x}\right )+\frac {50 b^{2} c^{4} d^{3} x}{9}+\frac {17 d^{3} b^{2} c^{3} \operatorname {arcsinh}\left (x c \right ) \ln \left (1-x c -\sqrt {c^{2} x^{2}+1}\right )}{3}-\frac {17 d^{3} b^{2} c^{3} \operatorname {arcsinh}\left (x c \right ) \ln \left (1+x c +\sqrt {c^{2} x^{2}+1}\right )}{3}-\frac {d^{3} b^{2} c \,\operatorname {arcsinh}\left (x c \right ) \sqrt {c^{2} x^{2}+1}}{3 x^{2}}-\frac {2 d^{3} b^{2} c^{5} \sqrt {c^{2} x^{2}+1}\, \operatorname {arcsinh}\left (x c \right ) x^{2}}{9}-\frac {17 b^{2} c^{3} d^{3} \operatorname {polylog}\left (2, -x c -\sqrt {c^{2} x^{2}+1}\right )}{3}+\frac {17 b^{2} c^{3} d^{3} \operatorname {polylog}\left (2, x c +\sqrt {c^{2} x^{2}+1}\right )}{3}+\frac {2 b^{2} c^{6} d^{3} x^{3}}{27}-\frac {b^{2} c^{2} d^{3}}{3 x}-\frac {50 d^{3} b^{2} c^{3} \sqrt {c^{2} x^{2}+1}\, \operatorname {arcsinh}\left (x c \right )}{9}-\frac {3 d^{3} b^{2} c^{2} \operatorname {arcsinh}\left (x c \right )^{2}}{x}-\frac {d^{3} b^{2} \operatorname {arcsinh}\left (x c \right )^{2}}{3 x^{3}}+\frac {d^{3} b^{2} c^{6} \operatorname {arcsinh}\left (x c \right )^{2} x^{3}}{3}+3 d^{3} b^{2} c^{4} \operatorname {arcsinh}\left (x c \right )^{2} x +2 d^{3} a b \,c^{3} \left (\frac {\operatorname {arcsinh}\left (x c \right ) x^{3} c^{3}}{3}+3 x c \,\operatorname {arcsinh}\left (x c \right )-\frac {3 \,\operatorname {arcsinh}\left (x c \right )}{x c}-\frac {\operatorname {arcsinh}\left (x c \right )}{3 x^{3} c^{3}}-\frac {x^{2} c^{2} \sqrt {c^{2} x^{2}+1}}{9}-\frac {25 \sqrt {c^{2} x^{2}+1}}{9}-\frac {\sqrt {c^{2} x^{2}+1}}{6 x^{2} c^{2}}-\frac {17 \,\operatorname {arctanh}\left (\frac {1}{\sqrt {c^{2} x^{2}+1}}\right )}{6}\right )\) | \(480\) |
Input:
int((c^2*d*x^2+d)^3*(a+b*arcsinh(x*c))^2/x^4,x,method=_RETURNVERBOSE)
Output:
c^3*(d^3*a^2*(1/3*x^3*c^3+3*x*c-3/x/c-1/3/x^3/c^3)+17/3*d^3*b^2*polylog(2, x*c+(c^2*x^2+1)^(1/2))+50/9*d^3*b^2*x*c-1/3*d^3*b^2/x/c+2/27*d^3*b^2*x^3*c ^3-17/3*d^3*b^2*polylog(2,-x*c-(c^2*x^2+1)^(1/2))-3*d^3*b^2*arcsinh(x*c)^2 /x/c-1/3*d^3*b^2/x^3/c^3*arcsinh(x*c)^2+1/3*d^3*b^2*arcsinh(x*c)^2*x^3*c^3 +3*d^3*b^2*arcsinh(x*c)^2*x*c-50/9*d^3*b^2*arcsinh(x*c)*(c^2*x^2+1)^(1/2)+ 17/3*d^3*b^2*arcsinh(x*c)*ln(1-x*c-(c^2*x^2+1)^(1/2))-17/3*d^3*b^2*arcsinh (x*c)*ln(1+x*c+(c^2*x^2+1)^(1/2))-1/3*d^3*b^2/x^2/c^2*arcsinh(x*c)*(c^2*x^ 2+1)^(1/2)-2/9*d^3*b^2*arcsinh(x*c)*(c^2*x^2+1)^(1/2)*x^2*c^2+2*d^3*a*b*(1 /3*arcsinh(x*c)*x^3*c^3+3*x*c*arcsinh(x*c)-3*arcsinh(x*c)/x/c-1/3*arcsinh( x*c)/x^3/c^3-1/9*x^2*c^2*(c^2*x^2+1)^(1/2)-25/9*(c^2*x^2+1)^(1/2)-1/6/x^2/ c^2*(c^2*x^2+1)^(1/2)-17/6*arctanh(1/(c^2*x^2+1)^(1/2))))
\[ \int \frac {\left (d+c^2 d x^2\right )^3 (a+b \text {arcsinh}(c x))^2}{x^4} \, dx=\int { \frac {{\left (c^{2} d x^{2} + d\right )}^{3} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2}}{x^{4}} \,d x } \] Input:
integrate((c^2*d*x^2+d)^3*(a+b*arcsinh(c*x))^2/x^4,x, algorithm="fricas")
Output:
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^4, x)
\[ \int \frac {\left (d+c^2 d x^2\right )^3 (a+b \text {arcsinh}(c x))^2}{x^4} \, dx=d^{3} \left (\int 3 a^{2} c^{4}\, dx + \int \frac {a^{2}}{x^{4}}\, dx + \int \frac {3 a^{2} c^{2}}{x^{2}}\, dx + \int a^{2} c^{6} x^{2}\, dx + \int 3 b^{2} c^{4} \operatorname {asinh}^{2}{\left (c x \right )}\, dx + \int \frac {b^{2} \operatorname {asinh}^{2}{\left (c x \right )}}{x^{4}}\, dx + \int 6 a b c^{4} \operatorname {asinh}{\left (c x \right )}\, dx + \int \frac {2 a b \operatorname {asinh}{\left (c x \right )}}{x^{4}}\, dx + \int \frac {3 b^{2} c^{2} \operatorname {asinh}^{2}{\left (c x \right )}}{x^{2}}\, dx + \int b^{2} c^{6} x^{2} \operatorname {asinh}^{2}{\left (c x \right )}\, dx + \int \frac {6 a b c^{2} \operatorname {asinh}{\left (c x \right )}}{x^{2}}\, dx + \int 2 a b c^{6} x^{2} \operatorname {asinh}{\left (c x \right )}\, dx\right ) \] Input:
integrate((c**2*d*x**2+d)**3*(a+b*asinh(c*x))**2/x**4,x)
Output:
d**3*(Integral(3*a**2*c**4, x) + Integral(a**2/x**4, x) + Integral(3*a**2* c**2/x**2, x) + Integral(a**2*c**6*x**2, x) + Integral(3*b**2*c**4*asinh(c *x)**2, x) + Integral(b**2*asinh(c*x)**2/x**4, x) + Integral(6*a*b*c**4*as inh(c*x), x) + Integral(2*a*b*asinh(c*x)/x**4, x) + Integral(3*b**2*c**2*a sinh(c*x)**2/x**2, x) + Integral(b**2*c**6*x**2*asinh(c*x)**2, x) + Integr al(6*a*b*c**2*asinh(c*x)/x**2, x) + Integral(2*a*b*c**6*x**2*asinh(c*x), x ))
\[ \int \frac {\left (d+c^2 d x^2\right )^3 (a+b \text {arcsinh}(c x))^2}{x^4} \, dx=\int { \frac {{\left (c^{2} d x^{2} + d\right )}^{3} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2}}{x^{4}} \,d x } \] Input:
integrate((c^2*d*x^2+d)^3*(a+b*arcsinh(c*x))^2/x^4,x, algorithm="maxima")
Output:
1/3*a^2*c^6*d^3*x^3 + 2/9*(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^6*d^3 + 3*b^2*c^4*d^3*x*arcsinh(c*x)^ 2 + 6*b^2*c^4*d^3*(x - sqrt(c^2*x^2 + 1)*arcsinh(c*x)/c) + 3*a^2*c^4*d^3*x + 6*(c*x*arcsinh(c*x) - sqrt(c^2*x^2 + 1))*a*b*c^3*d^3 - 6*(c*arcsinh(1/( c*abs(x))) + arcsinh(c*x)/x)*a*b*c^2*d^3 + 1/3*((c^2*arcsinh(1/(c*abs(x))) - sqrt(c^2*x^2 + 1)/x^2)*c - 2*arcsinh(c*x)/x^3)*a*b*d^3 - 3*a^2*c^2*d^3/ x - 1/3*a^2*d^3/x^3 + 1/3*(b^2*c^6*d^3*x^6 - 9*b^2*c^2*d^3*x^2 - b^2*d^3)* log(c*x + sqrt(c^2*x^2 + 1))^2/x^3 - integrate(2/3*(b^2*c^9*d^3*x^8 + b^2* c^7*d^3*x^6 - 9*b^2*c^5*d^3*x^4 - 10*b^2*c^3*d^3*x^2 - b^2*c*d^3 + (b^2*c^ 8*d^3*x^7 - 9*b^2*c^4*d^3*x^3 - 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^6 + c*x^4 + (c^2*x^5 + x^3)*sqrt(c^2*x^2 + 1)) , x)
Exception generated. \[ \int \frac {\left (d+c^2 d x^2\right )^3 (a+b \text {arcsinh}(c x))^2}{x^4} \, dx=\text {Exception raised: TypeError} \] Input:
integrate((c^2*d*x^2+d)^3*(a+b*arcsinh(c*x))^2/x^4,x, algorithm="giac")
Output:
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. \[ \int \frac {\left (d+c^2 d x^2\right )^3 (a+b \text {arcsinh}(c x))^2}{x^4} \, dx=\int \frac {{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2\,{\left (d\,c^2\,x^2+d\right )}^3}{x^4} \,d x \] Input:
int(((a + b*asinh(c*x))^2*(d + c^2*d*x^2)^3)/x^4,x)
Output:
int(((a + b*asinh(c*x))^2*(d + c^2*d*x^2)^3)/x^4, x)
\[ \int \frac {\left (d+c^2 d x^2\right )^3 (a+b \text {arcsinh}(c x))^2}{x^4} \, dx=\frac {d^{3} \left (27 \mathit {asinh} \left (c x \right )^{2} b^{2} c^{4} x^{4}-54 \sqrt {c^{2} x^{2}+1}\, \mathit {asinh} \left (c x \right ) b^{2} c^{3} x^{3}+6 \mathit {asinh} \left (c x \right ) a b \,c^{6} x^{6}+54 \mathit {asinh} \left (c x \right ) a b \,c^{4} x^{4}-54 \mathit {asinh} \left (c x \right ) a b \,c^{2} x^{2}-6 \mathit {asinh} \left (c x \right ) a b -2 \sqrt {c^{2} x^{2}+1}\, a b \,c^{5} x^{5}-50 \sqrt {c^{2} x^{2}+1}\, a b \,c^{3} x^{3}-3 \sqrt {c^{2} x^{2}+1}\, a b c x +9 \left (\int \frac {\mathit {asinh} \left (c x \right )^{2}}{x^{4}}d x \right ) b^{2} x^{3}+27 \left (\int \frac {\mathit {asinh} \left (c x \right )^{2}}{x^{2}}d x \right ) b^{2} c^{2} x^{3}+9 \left (\int \mathit {asinh} \left (c x \right )^{2} x^{2}d x \right ) b^{2} c^{6} x^{3}+51 \,\mathrm {log}\left (\sqrt {c^{2} x^{2}+1}+c x -1\right ) a b \,c^{3} x^{3}-51 \,\mathrm {log}\left (\sqrt {c^{2} x^{2}+1}+c x +1\right ) a b \,c^{3} x^{3}+3 a^{2} c^{6} x^{6}+27 a^{2} c^{4} x^{4}-27 a^{2} c^{2} x^{2}-3 a^{2}+54 b^{2} c^{4} x^{4}\right )}{9 x^{3}} \] Input:
int((c^2*d*x^2+d)^3*(a+b*asinh(c*x))^2/x^4,x)
Output:
(d**3*(27*asinh(c*x)**2*b**2*c**4*x**4 - 54*sqrt(c**2*x**2 + 1)*asinh(c*x) *b**2*c**3*x**3 + 6*asinh(c*x)*a*b*c**6*x**6 + 54*asinh(c*x)*a*b*c**4*x**4 - 54*asinh(c*x)*a*b*c**2*x**2 - 6*asinh(c*x)*a*b - 2*sqrt(c**2*x**2 + 1)* a*b*c**5*x**5 - 50*sqrt(c**2*x**2 + 1)*a*b*c**3*x**3 - 3*sqrt(c**2*x**2 + 1)*a*b*c*x + 9*int(asinh(c*x)**2/x**4,x)*b**2*x**3 + 27*int(asinh(c*x)**2/ x**2,x)*b**2*c**2*x**3 + 9*int(asinh(c*x)**2*x**2,x)*b**2*c**6*x**3 + 51*l og(sqrt(c**2*x**2 + 1) + c*x - 1)*a*b*c**3*x**3 - 51*log(sqrt(c**2*x**2 + 1) + c*x + 1)*a*b*c**3*x**3 + 3*a**2*c**6*x**6 + 27*a**2*c**4*x**4 - 27*a* *2*c**2*x**2 - 3*a**2 + 54*b**2*c**4*x**4))/(9*x**3)