Integrand size = 14, antiderivative size = 204 \[ \int \frac {\left (a+b \text {csch}^{-1}(c x)\right )^3}{x^5} \, dx=\frac {3 b^3 c \sqrt {1+\frac {1}{c^2 x^2}}}{128 x^3}-\frac {45 b^3 c^3 \sqrt {1+\frac {1}{c^2 x^2}}}{256 x}+\frac {45}{256} b^3 c^4 \text {csch}^{-1}(c x)-\frac {3 b^2 \left (a+b \text {csch}^{-1}(c x)\right )}{32 x^4}+\frac {9 b^2 c^2 \left (a+b \text {csch}^{-1}(c x)\right )}{32 x^2}+\frac {3 b c \sqrt {1+\frac {1}{c^2 x^2}} \left (a+b \text {csch}^{-1}(c x)\right )^2}{16 x^3}-\frac {9 b c^3 \sqrt {1+\frac {1}{c^2 x^2}} \left (a+b \text {csch}^{-1}(c x)\right )^2}{32 x}+\frac {3}{32} c^4 \left (a+b \text {csch}^{-1}(c x)\right )^3-\frac {\left (a+b \text {csch}^{-1}(c x)\right )^3}{4 x^4} \]
45/256*b^3*c^4*arccsch(c*x)-3/32*b^2*(a+b*arccsch(c*x))/x^4+9/32*b^2*c^2*( a+b*arccsch(c*x))/x^2+3/32*c^4*(a+b*arccsch(c*x))^3-1/4*(a+b*arccsch(c*x)) ^3/x^4+3/128*b^3*c*(1+1/c^2/x^2)^(1/2)/x^3-45/256*b^3*c^3*(1+1/c^2/x^2)^(1 /2)/x+3/16*b*c*(a+b*arccsch(c*x))^2*(1+1/c^2/x^2)^(1/2)/x^3-9/32*b*c^3*(a+ b*arccsch(c*x))^2*(1+1/c^2/x^2)^(1/2)/x
Time = 0.34 (sec) , antiderivative size = 277, normalized size of antiderivative = 1.36 \[ \int \frac {\left (a+b \text {csch}^{-1}(c x)\right )^3}{x^5} \, dx=\frac {-64 a^3-24 a b^2+48 a^2 b c \sqrt {1+\frac {1}{c^2 x^2}} x+6 b^3 c \sqrt {1+\frac {1}{c^2 x^2}} x+72 a b^2 c^2 x^2-72 a^2 b c^3 \sqrt {1+\frac {1}{c^2 x^2}} x^3-45 b^3 c^3 \sqrt {1+\frac {1}{c^2 x^2}} x^3-24 b \left (8 a^2+b^2 \left (1-3 c^2 x^2\right )+2 a b c \sqrt {1+\frac {1}{c^2 x^2}} x \left (-2+3 c^2 x^2\right )\right ) \text {csch}^{-1}(c x)+24 b^2 \left (b c \sqrt {1+\frac {1}{c^2 x^2}} x \left (2-3 c^2 x^2\right )+a \left (-8+3 c^4 x^4\right )\right ) \text {csch}^{-1}(c x)^2+8 b^3 \left (-8+3 c^4 x^4\right ) \text {csch}^{-1}(c x)^3+9 b \left (8 a^2+5 b^2\right ) c^4 x^4 \text {arcsinh}\left (\frac {1}{c x}\right )}{256 x^4} \]
(-64*a^3 - 24*a*b^2 + 48*a^2*b*c*Sqrt[1 + 1/(c^2*x^2)]*x + 6*b^3*c*Sqrt[1 + 1/(c^2*x^2)]*x + 72*a*b^2*c^2*x^2 - 72*a^2*b*c^3*Sqrt[1 + 1/(c^2*x^2)]*x ^3 - 45*b^3*c^3*Sqrt[1 + 1/(c^2*x^2)]*x^3 - 24*b*(8*a^2 + b^2*(1 - 3*c^2*x ^2) + 2*a*b*c*Sqrt[1 + 1/(c^2*x^2)]*x*(-2 + 3*c^2*x^2))*ArcCsch[c*x] + 24* b^2*(b*c*Sqrt[1 + 1/(c^2*x^2)]*x*(2 - 3*c^2*x^2) + a*(-8 + 3*c^4*x^4))*Arc Csch[c*x]^2 + 8*b^3*(-8 + 3*c^4*x^4)*ArcCsch[c*x]^3 + 9*b*(8*a^2 + 5*b^2)* c^4*x^4*ArcSinh[1/(c*x)])/(256*x^4)
Time = 0.77 (sec) , antiderivative size = 264, normalized size of antiderivative = 1.29, number of steps used = 21, number of rules used = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.429, Rules used = {6840, 5969, 3042, 3792, 25, 3042, 25, 3115, 25, 3042, 25, 3115, 24, 3792, 17, 25, 3042, 25, 3115, 24}
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.
| \(\displaystyle \int \frac {\left (a+b \text {csch}^{-1}(c x)\right )^3}{x^5} \, dx\) |
| \(\Big \downarrow \) 6840 |
| \(\displaystyle -c^4 \int \frac {\sqrt {1+\frac {1}{c^2 x^2}} \left (a+b \text {csch}^{-1}(c x)\right )^3}{c^3 x^3}d\text {csch}^{-1}(c x)\) |
| \(\Big \downarrow \) 5969 |
| \(\displaystyle -c^4 \left (\frac {\left (a+b \text {csch}^{-1}(c x)\right )^3}{4 c^4 x^4}-\frac {3}{4} b \int \frac {\left (a+b \text {csch}^{-1}(c x)\right )^2}{c^4 x^4}d\text {csch}^{-1}(c x)\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -c^4 \left (\frac {\left (a+b \text {csch}^{-1}(c x)\right )^3}{4 c^4 x^4}-\frac {3}{4} b \int \left (a+b \text {csch}^{-1}(c x)\right )^2 \sin \left (i \text {csch}^{-1}(c x)\right )^4d\text {csch}^{-1}(c x)\right )\) |
| \(\Big \downarrow \) 3792 |
| \(\displaystyle -c^4 \left (\frac {\left (a+b \text {csch}^{-1}(c x)\right )^3}{4 c^4 x^4}-\frac {3}{4} b \left (\frac {3}{4} \int -\frac {\left (a+b \text {csch}^{-1}(c x)\right )^2}{c^2 x^2}d\text {csch}^{-1}(c x)+\frac {1}{8} b^2 \int \frac {1}{c^4 x^4}d\text {csch}^{-1}(c x)-\frac {b \left (a+b \text {csch}^{-1}(c x)\right )}{8 c^4 x^4}+\frac {\sqrt {\frac {1}{c^2 x^2}+1} \left (a+b \text {csch}^{-1}(c x)\right )^2}{4 c^3 x^3}\right )\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -c^4 \left (\frac {\left (a+b \text {csch}^{-1}(c x)\right )^3}{4 c^4 x^4}-\frac {3}{4} b \left (-\frac {3}{4} \int \frac {\left (a+b \text {csch}^{-1}(c x)\right )^2}{c^2 x^2}d\text {csch}^{-1}(c x)+\frac {1}{8} b^2 \int \frac {1}{c^4 x^4}d\text {csch}^{-1}(c x)-\frac {b \left (a+b \text {csch}^{-1}(c x)\right )}{8 c^4 x^4}+\frac {\sqrt {\frac {1}{c^2 x^2}+1} \left (a+b \text {csch}^{-1}(c x)\right )^2}{4 c^3 x^3}\right )\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -c^4 \left (\frac {\left (a+b \text {csch}^{-1}(c x)\right )^3}{4 c^4 x^4}-\frac {3}{4} b \left (-\frac {3}{4} \int -\left (a+b \text {csch}^{-1}(c x)\right )^2 \sin \left (i \text {csch}^{-1}(c x)\right )^2d\text {csch}^{-1}(c x)+\frac {1}{8} b^2 \int \sin \left (i \text {csch}^{-1}(c x)\right )^4d\text {csch}^{-1}(c x)-\frac {b \left (a+b \text {csch}^{-1}(c x)\right )}{8 c^4 x^4}+\frac {\sqrt {\frac {1}{c^2 x^2}+1} \left (a+b \text {csch}^{-1}(c x)\right )^2}{4 c^3 x^3}\right )\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -c^4 \left (\frac {\left (a+b \text {csch}^{-1}(c x)\right )^3}{4 c^4 x^4}-\frac {3}{4} b \left (\frac {3}{4} \int \left (a+b \text {csch}^{-1}(c x)\right )^2 \sin \left (i \text {csch}^{-1}(c x)\right )^2d\text {csch}^{-1}(c x)+\frac {1}{8} b^2 \int \sin \left (i \text {csch}^{-1}(c x)\right )^4d\text {csch}^{-1}(c x)-\frac {b \left (a+b \text {csch}^{-1}(c x)\right )}{8 c^4 x^4}+\frac {\sqrt {\frac {1}{c^2 x^2}+1} \left (a+b \text {csch}^{-1}(c x)\right )^2}{4 c^3 x^3}\right )\right )\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle -c^4 \left (\frac {\left (a+b \text {csch}^{-1}(c x)\right )^3}{4 c^4 x^4}-\frac {3}{4} b \left (\frac {3}{4} \int \left (a+b \text {csch}^{-1}(c x)\right )^2 \sin \left (i \text {csch}^{-1}(c x)\right )^2d\text {csch}^{-1}(c x)+\frac {1}{8} b^2 \left (\frac {3}{4} \int -\frac {1}{c^2 x^2}d\text {csch}^{-1}(c x)+\frac {\sqrt {\frac {1}{c^2 x^2}+1}}{4 c^3 x^3}\right )-\frac {b \left (a+b \text {csch}^{-1}(c x)\right )}{8 c^4 x^4}+\frac {\sqrt {\frac {1}{c^2 x^2}+1} \left (a+b \text {csch}^{-1}(c x)\right )^2}{4 c^3 x^3}\right )\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -c^4 \left (\frac {\left (a+b \text {csch}^{-1}(c x)\right )^3}{4 c^4 x^4}-\frac {3}{4} b \left (\frac {3}{4} \int \left (a+b \text {csch}^{-1}(c x)\right )^2 \sin \left (i \text {csch}^{-1}(c x)\right )^2d\text {csch}^{-1}(c x)+\frac {1}{8} b^2 \left (\frac {\sqrt {\frac {1}{c^2 x^2}+1}}{4 c^3 x^3}-\frac {3}{4} \int \frac {1}{c^2 x^2}d\text {csch}^{-1}(c x)\right )-\frac {b \left (a+b \text {csch}^{-1}(c x)\right )}{8 c^4 x^4}+\frac {\sqrt {\frac {1}{c^2 x^2}+1} \left (a+b \text {csch}^{-1}(c x)\right )^2}{4 c^3 x^3}\right )\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -c^4 \left (\frac {\left (a+b \text {csch}^{-1}(c x)\right )^3}{4 c^4 x^4}-\frac {3}{4} b \left (\frac {3}{4} \int \left (a+b \text {csch}^{-1}(c x)\right )^2 \sin \left (i \text {csch}^{-1}(c x)\right )^2d\text {csch}^{-1}(c x)+\frac {1}{8} b^2 \left (\frac {\sqrt {\frac {1}{c^2 x^2}+1}}{4 c^3 x^3}-\frac {3}{4} \int -\sin \left (i \text {csch}^{-1}(c x)\right )^2d\text {csch}^{-1}(c x)\right )-\frac {b \left (a+b \text {csch}^{-1}(c x)\right )}{8 c^4 x^4}+\frac {\sqrt {\frac {1}{c^2 x^2}+1} \left (a+b \text {csch}^{-1}(c x)\right )^2}{4 c^3 x^3}\right )\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -c^4 \left (\frac {\left (a+b \text {csch}^{-1}(c x)\right )^3}{4 c^4 x^4}-\frac {3}{4} b \left (\frac {3}{4} \int \left (a+b \text {csch}^{-1}(c x)\right )^2 \sin \left (i \text {csch}^{-1}(c x)\right )^2d\text {csch}^{-1}(c x)+\frac {1}{8} b^2 \left (\frac {\sqrt {\frac {1}{c^2 x^2}+1}}{4 c^3 x^3}+\frac {3}{4} \int \sin \left (i \text {csch}^{-1}(c x)\right )^2d\text {csch}^{-1}(c x)\right )-\frac {b \left (a+b \text {csch}^{-1}(c x)\right )}{8 c^4 x^4}+\frac {\sqrt {\frac {1}{c^2 x^2}+1} \left (a+b \text {csch}^{-1}(c x)\right )^2}{4 c^3 x^3}\right )\right )\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle -c^4 \left (\frac {\left (a+b \text {csch}^{-1}(c x)\right )^3}{4 c^4 x^4}-\frac {3}{4} b \left (\frac {3}{4} \int \left (a+b \text {csch}^{-1}(c x)\right )^2 \sin \left (i \text {csch}^{-1}(c x)\right )^2d\text {csch}^{-1}(c x)+\frac {1}{8} b^2 \left (\frac {3}{4} \left (\frac {1}{2} \int 1d\text {csch}^{-1}(c x)-\frac {\sqrt {\frac {1}{c^2 x^2}+1}}{2 c x}\right )+\frac {\sqrt {\frac {1}{c^2 x^2}+1}}{4 c^3 x^3}\right )-\frac {b \left (a+b \text {csch}^{-1}(c x)\right )}{8 c^4 x^4}+\frac {\sqrt {\frac {1}{c^2 x^2}+1} \left (a+b \text {csch}^{-1}(c x)\right )^2}{4 c^3 x^3}\right )\right )\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle -c^4 \left (\frac {\left (a+b \text {csch}^{-1}(c x)\right )^3}{4 c^4 x^4}-\frac {3}{4} b \left (\frac {3}{4} \int \left (a+b \text {csch}^{-1}(c x)\right )^2 \sin \left (i \text {csch}^{-1}(c x)\right )^2d\text {csch}^{-1}(c x)-\frac {b \left (a+b \text {csch}^{-1}(c x)\right )}{8 c^4 x^4}+\frac {\sqrt {\frac {1}{c^2 x^2}+1} \left (a+b \text {csch}^{-1}(c x)\right )^2}{4 c^3 x^3}+\frac {1}{8} b^2 \left (\frac {3}{4} \left (\frac {1}{2} \text {csch}^{-1}(c x)-\frac {\sqrt {\frac {1}{c^2 x^2}+1}}{2 c x}\right )+\frac {\sqrt {\frac {1}{c^2 x^2}+1}}{4 c^3 x^3}\right )\right )\right )\) |
| \(\Big \downarrow \) 3792 |
| \(\displaystyle -c^4 \left (\frac {\left (a+b \text {csch}^{-1}(c x)\right )^3}{4 c^4 x^4}-\frac {3}{4} b \left (\frac {3}{4} \left (\frac {1}{2} \int \left (a+b \text {csch}^{-1}(c x)\right )^2d\text {csch}^{-1}(c x)+\frac {1}{2} b^2 \int -\frac {1}{c^2 x^2}d\text {csch}^{-1}(c x)+\frac {b \left (a+b \text {csch}^{-1}(c x)\right )}{2 c^2 x^2}-\frac {\sqrt {\frac {1}{c^2 x^2}+1} \left (a+b \text {csch}^{-1}(c x)\right )^2}{2 c x}\right )-\frac {b \left (a+b \text {csch}^{-1}(c x)\right )}{8 c^4 x^4}+\frac {\sqrt {\frac {1}{c^2 x^2}+1} \left (a+b \text {csch}^{-1}(c x)\right )^2}{4 c^3 x^3}+\frac {1}{8} b^2 \left (\frac {3}{4} \left (\frac {1}{2} \text {csch}^{-1}(c x)-\frac {\sqrt {\frac {1}{c^2 x^2}+1}}{2 c x}\right )+\frac {\sqrt {\frac {1}{c^2 x^2}+1}}{4 c^3 x^3}\right )\right )\right )\) |
| \(\Big \downarrow \) 17 |
| \(\displaystyle -c^4 \left (\frac {\left (a+b \text {csch}^{-1}(c x)\right )^3}{4 c^4 x^4}-\frac {3}{4} b \left (\frac {3}{4} \left (\frac {1}{2} b^2 \int -\frac {1}{c^2 x^2}d\text {csch}^{-1}(c x)-\frac {\sqrt {\frac {1}{c^2 x^2}+1} \left (a+b \text {csch}^{-1}(c x)\right )^2}{2 c x}+\frac {b \left (a+b \text {csch}^{-1}(c x)\right )}{2 c^2 x^2}+\frac {\left (a+b \text {csch}^{-1}(c x)\right )^3}{6 b}\right )-\frac {b \left (a+b \text {csch}^{-1}(c x)\right )}{8 c^4 x^4}+\frac {\sqrt {\frac {1}{c^2 x^2}+1} \left (a+b \text {csch}^{-1}(c x)\right )^2}{4 c^3 x^3}+\frac {1}{8} b^2 \left (\frac {3}{4} \left (\frac {1}{2} \text {csch}^{-1}(c x)-\frac {\sqrt {\frac {1}{c^2 x^2}+1}}{2 c x}\right )+\frac {\sqrt {\frac {1}{c^2 x^2}+1}}{4 c^3 x^3}\right )\right )\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -c^4 \left (\frac {\left (a+b \text {csch}^{-1}(c x)\right )^3}{4 c^4 x^4}-\frac {3}{4} b \left (\frac {3}{4} \left (-\frac {1}{2} b^2 \int \frac {1}{c^2 x^2}d\text {csch}^{-1}(c x)-\frac {\sqrt {\frac {1}{c^2 x^2}+1} \left (a+b \text {csch}^{-1}(c x)\right )^2}{2 c x}+\frac {b \left (a+b \text {csch}^{-1}(c x)\right )}{2 c^2 x^2}+\frac {\left (a+b \text {csch}^{-1}(c x)\right )^3}{6 b}\right )-\frac {b \left (a+b \text {csch}^{-1}(c x)\right )}{8 c^4 x^4}+\frac {\sqrt {\frac {1}{c^2 x^2}+1} \left (a+b \text {csch}^{-1}(c x)\right )^2}{4 c^3 x^3}+\frac {1}{8} b^2 \left (\frac {3}{4} \left (\frac {1}{2} \text {csch}^{-1}(c x)-\frac {\sqrt {\frac {1}{c^2 x^2}+1}}{2 c x}\right )+\frac {\sqrt {\frac {1}{c^2 x^2}+1}}{4 c^3 x^3}\right )\right )\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -c^4 \left (\frac {\left (a+b \text {csch}^{-1}(c x)\right )^3}{4 c^4 x^4}-\frac {3}{4} b \left (\frac {3}{4} \left (-\frac {1}{2} b^2 \int -\sin \left (i \text {csch}^{-1}(c x)\right )^2d\text {csch}^{-1}(c x)-\frac {\sqrt {\frac {1}{c^2 x^2}+1} \left (a+b \text {csch}^{-1}(c x)\right )^2}{2 c x}+\frac {b \left (a+b \text {csch}^{-1}(c x)\right )}{2 c^2 x^2}+\frac {\left (a+b \text {csch}^{-1}(c x)\right )^3}{6 b}\right )-\frac {b \left (a+b \text {csch}^{-1}(c x)\right )}{8 c^4 x^4}+\frac {\sqrt {\frac {1}{c^2 x^2}+1} \left (a+b \text {csch}^{-1}(c x)\right )^2}{4 c^3 x^3}+\frac {1}{8} b^2 \left (\frac {3}{4} \left (\frac {1}{2} \text {csch}^{-1}(c x)-\frac {\sqrt {\frac {1}{c^2 x^2}+1}}{2 c x}\right )+\frac {\sqrt {\frac {1}{c^2 x^2}+1}}{4 c^3 x^3}\right )\right )\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -c^4 \left (\frac {\left (a+b \text {csch}^{-1}(c x)\right )^3}{4 c^4 x^4}-\frac {3}{4} b \left (\frac {3}{4} \left (\frac {1}{2} b^2 \int \sin \left (i \text {csch}^{-1}(c x)\right )^2d\text {csch}^{-1}(c x)-\frac {\sqrt {\frac {1}{c^2 x^2}+1} \left (a+b \text {csch}^{-1}(c x)\right )^2}{2 c x}+\frac {b \left (a+b \text {csch}^{-1}(c x)\right )}{2 c^2 x^2}+\frac {\left (a+b \text {csch}^{-1}(c x)\right )^3}{6 b}\right )-\frac {b \left (a+b \text {csch}^{-1}(c x)\right )}{8 c^4 x^4}+\frac {\sqrt {\frac {1}{c^2 x^2}+1} \left (a+b \text {csch}^{-1}(c x)\right )^2}{4 c^3 x^3}+\frac {1}{8} b^2 \left (\frac {3}{4} \left (\frac {1}{2} \text {csch}^{-1}(c x)-\frac {\sqrt {\frac {1}{c^2 x^2}+1}}{2 c x}\right )+\frac {\sqrt {\frac {1}{c^2 x^2}+1}}{4 c^3 x^3}\right )\right )\right )\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle -c^4 \left (\frac {\left (a+b \text {csch}^{-1}(c x)\right )^3}{4 c^4 x^4}-\frac {3}{4} b \left (\frac {3}{4} \left (\frac {1}{2} b^2 \left (\frac {1}{2} \int 1d\text {csch}^{-1}(c x)-\frac {\sqrt {\frac {1}{c^2 x^2}+1}}{2 c x}\right )-\frac {\sqrt {\frac {1}{c^2 x^2}+1} \left (a+b \text {csch}^{-1}(c x)\right )^2}{2 c x}+\frac {b \left (a+b \text {csch}^{-1}(c x)\right )}{2 c^2 x^2}+\frac {\left (a+b \text {csch}^{-1}(c x)\right )^3}{6 b}\right )-\frac {b \left (a+b \text {csch}^{-1}(c x)\right )}{8 c^4 x^4}+\frac {\sqrt {\frac {1}{c^2 x^2}+1} \left (a+b \text {csch}^{-1}(c x)\right )^2}{4 c^3 x^3}+\frac {1}{8} b^2 \left (\frac {3}{4} \left (\frac {1}{2} \text {csch}^{-1}(c x)-\frac {\sqrt {\frac {1}{c^2 x^2}+1}}{2 c x}\right )+\frac {\sqrt {\frac {1}{c^2 x^2}+1}}{4 c^3 x^3}\right )\right )\right )\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle -c^4 \left (\frac {\left (a+b \text {csch}^{-1}(c x)\right )^3}{4 c^4 x^4}-\frac {3}{4} b \left (\frac {3}{4} \left (-\frac {\sqrt {\frac {1}{c^2 x^2}+1} \left (a+b \text {csch}^{-1}(c x)\right )^2}{2 c x}+\frac {b \left (a+b \text {csch}^{-1}(c x)\right )}{2 c^2 x^2}+\frac {\left (a+b \text {csch}^{-1}(c x)\right )^3}{6 b}+\frac {1}{2} b^2 \left (\frac {1}{2} \text {csch}^{-1}(c x)-\frac {\sqrt {\frac {1}{c^2 x^2}+1}}{2 c x}\right )\right )-\frac {b \left (a+b \text {csch}^{-1}(c x)\right )}{8 c^4 x^4}+\frac {\sqrt {\frac {1}{c^2 x^2}+1} \left (a+b \text {csch}^{-1}(c x)\right )^2}{4 c^3 x^3}+\frac {1}{8} b^2 \left (\frac {3}{4} \left (\frac {1}{2} \text {csch}^{-1}(c x)-\frac {\sqrt {\frac {1}{c^2 x^2}+1}}{2 c x}\right )+\frac {\sqrt {\frac {1}{c^2 x^2}+1}}{4 c^3 x^3}\right )\right )\right )\) |
-(c^4*((a + b*ArcCsch[c*x])^3/(4*c^4*x^4) - (3*b*((b^2*(Sqrt[1 + 1/(c^2*x^ 2)]/(4*c^3*x^3) + (3*(-1/2*Sqrt[1 + 1/(c^2*x^2)]/(c*x) + ArcCsch[c*x]/2))/ 4))/8 - (b*(a + b*ArcCsch[c*x]))/(8*c^4*x^4) + (Sqrt[1 + 1/(c^2*x^2)]*(a + b*ArcCsch[c*x])^2)/(4*c^3*x^3) + (3*((b^2*(-1/2*Sqrt[1 + 1/(c^2*x^2)]/(c* x) + ArcCsch[c*x]/2))/2 + (b*(a + b*ArcCsch[c*x]))/(2*c^2*x^2) - (Sqrt[1 + 1/(c^2*x^2)]*(a + b*ArcCsch[c*x])^2)/(2*c*x) + (a + b*ArcCsch[c*x])^3/(6* b)))/4))/4))
3.1.32.3.1 Defintions of rubi rules used
Int[(c_.)*((a_.) + (b_.)*(x_))^(m_.), x_Symbol] :> Simp[c*((a + b*x)^(m + 1 )/(b*(m + 1))), x] /; FreeQ[{a, b, c, m}, x] && NeQ[m, -1]
Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d* x]*((b*Sin[c + d*x])^(n - 1)/(d*n)), x] + Simp[b^2*((n - 1)/n) Int[(b*Sin [c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && IntegerQ[ 2*n]
Int[((c_.) + (d_.)*(x_))^(m_)*((b_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbo l] :> Simp[d*m*(c + d*x)^(m - 1)*((b*Sin[e + f*x])^n/(f^2*n^2)), x] + (-Sim p[b*(c + d*x)^m*Cos[e + f*x]*((b*Sin[e + f*x])^(n - 1)/(f*n)), x] + Simp[b^ 2*((n - 1)/n) Int[(c + d*x)^m*(b*Sin[e + f*x])^(n - 2), x], x] - Simp[d^2 *m*((m - 1)/(f^2*n^2)) Int[(c + d*x)^(m - 2)*(b*Sin[e + f*x])^n, x], x]) /; FreeQ[{b, c, d, e, f}, x] && GtQ[n, 1] && GtQ[m, 1]
Int[Cosh[(a_.) + (b_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.)*Sinh[(a_.) + (b_.)* (x_)]^(n_.), x_Symbol] :> Simp[(c + d*x)^m*(Sinh[a + b*x]^(n + 1)/(b*(n + 1 ))), x] - Simp[d*(m/(b*(n + 1))) Int[(c + d*x)^(m - 1)*Sinh[a + b*x]^(n + 1), x], x] /; FreeQ[{a, b, c, d, n}, x] && IGtQ[m, 0] && NeQ[n, -1]
Int[((a_.) + ArcCsch[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Simp[ -(c^(m + 1))^(-1) Subst[Int[(a + b*x)^n*Csch[x]^(m + 1)*Coth[x], x], x, A rcCsch[c*x]], x] /; FreeQ[{a, b, c}, x] && IntegerQ[n] && IntegerQ[m] && (G tQ[n, 0] || LtQ[m, -1])
\[\int \frac {\left (a +b \,\operatorname {arccsch}\left (c x \right )\right )^{3}}{x^{5}}d x\]
Time = 0.28 (sec) , antiderivative size = 346, normalized size of antiderivative = 1.70 \[ \int \frac {\left (a+b \text {csch}^{-1}(c x)\right )^3}{x^5} \, dx=\frac {72 \, a b^{2} c^{2} x^{2} + 8 \, {\left (3 \, b^{3} c^{4} x^{4} - 8 \, b^{3}\right )} \log \left (\frac {c x \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}} + 1}{c x}\right )^{3} - 64 \, a^{3} - 24 \, a b^{2} + 24 \, {\left (3 \, a b^{2} c^{4} x^{4} - 8 \, a b^{2} - {\left (3 \, b^{3} c^{3} x^{3} - 2 \, b^{3} c x\right )} \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}}\right )} \log \left (\frac {c x \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}} + 1}{c x}\right )^{2} + 3 \, {\left (3 \, {\left (8 \, a^{2} b + 5 \, b^{3}\right )} c^{4} x^{4} + 24 \, b^{3} c^{2} x^{2} - 64 \, a^{2} b - 8 \, b^{3} - 16 \, {\left (3 \, a b^{2} c^{3} x^{3} - 2 \, a b^{2} c x\right )} \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}}\right )} \log \left (\frac {c x \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}} + 1}{c x}\right ) - 3 \, {\left (3 \, {\left (8 \, a^{2} b + 5 \, b^{3}\right )} c^{3} x^{3} - 2 \, {\left (8 \, a^{2} b + b^{3}\right )} c x\right )} \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}}}{256 \, x^{4}} \]
1/256*(72*a*b^2*c^2*x^2 + 8*(3*b^3*c^4*x^4 - 8*b^3)*log((c*x*sqrt((c^2*x^2 + 1)/(c^2*x^2)) + 1)/(c*x))^3 - 64*a^3 - 24*a*b^2 + 24*(3*a*b^2*c^4*x^4 - 8*a*b^2 - (3*b^3*c^3*x^3 - 2*b^3*c*x)*sqrt((c^2*x^2 + 1)/(c^2*x^2)))*log( (c*x*sqrt((c^2*x^2 + 1)/(c^2*x^2)) + 1)/(c*x))^2 + 3*(3*(8*a^2*b + 5*b^3)* c^4*x^4 + 24*b^3*c^2*x^2 - 64*a^2*b - 8*b^3 - 16*(3*a*b^2*c^3*x^3 - 2*a*b^ 2*c*x)*sqrt((c^2*x^2 + 1)/(c^2*x^2)))*log((c*x*sqrt((c^2*x^2 + 1)/(c^2*x^2 )) + 1)/(c*x)) - 3*(3*(8*a^2*b + 5*b^3)*c^3*x^3 - 2*(8*a^2*b + b^3)*c*x)*s qrt((c^2*x^2 + 1)/(c^2*x^2)))/x^4
\[ \int \frac {\left (a+b \text {csch}^{-1}(c x)\right )^3}{x^5} \, dx=\int \frac {\left (a + b \operatorname {acsch}{\left (c x \right )}\right )^{3}}{x^{5}}\, dx \]
\[ \int \frac {\left (a+b \text {csch}^{-1}(c x)\right )^3}{x^5} \, dx=\int { \frac {{\left (b \operatorname {arcsch}\left (c x\right ) + a\right )}^{3}}{x^{5}} \,d x } \]
3/64*a^2*b*((3*c^5*log(c*x*sqrt(1/(c^2*x^2) + 1) + 1) - 3*c^5*log(c*x*sqrt (1/(c^2*x^2) + 1) - 1) - 2*(3*c^8*x^3*(1/(c^2*x^2) + 1)^(3/2) - 5*c^6*x*sq rt(1/(c^2*x^2) + 1))/(c^4*x^4*(1/(c^2*x^2) + 1)^2 - 2*c^2*x^2*(1/(c^2*x^2) + 1) + 1))/c - 16*arccsch(c*x)/x^4) - 1/4*b^3*log(sqrt(c^2*x^2 + 1) + 1)^ 3/x^4 - 1/4*a^3/x^4 - integrate(1/4*(4*b^3*log(c)^3 - 12*a*b^2*log(c)^2 + 4*(b^3*c^2*x^2 + b^3)*log(x)^3 + 4*(b^3*c^2*log(c)^3 - 3*a*b^2*c^2*log(c)^ 2)*x^2 + 12*(b^3*log(c) - a*b^2 + (b^3*c^2*log(c) - a*b^2*c^2)*x^2)*log(x) ^2 + 3*(4*b^3*log(c) - 4*a*b^2 + 4*(b^3*c^2*log(c) - a*b^2*c^2)*x^2 + 4*(b ^3*c^2*x^2 + b^3)*log(x) + sqrt(c^2*x^2 + 1)*(4*b^3*log(c) - 4*a*b^2 + (b^ 3*c^2*(4*log(c) - 1) - 4*a*b^2*c^2)*x^2 + 4*(b^3*c^2*x^2 + b^3)*log(x)))*l og(sqrt(c^2*x^2 + 1) + 1)^2 + 12*(b^3*log(c)^2 - 2*a*b^2*log(c) + (b^3*c^2 *log(c)^2 - 2*a*b^2*c^2*log(c))*x^2)*log(x) - 12*(b^3*log(c)^2 - 2*a*b^2*l og(c) + (b^3*c^2*log(c)^2 - 2*a*b^2*c^2*log(c))*x^2 + (b^3*c^2*x^2 + b^3)* log(x)^2 + 2*(b^3*log(c) - a*b^2 + (b^3*c^2*log(c) - a*b^2*c^2)*x^2)*log(x ) + (b^3*log(c)^2 - 2*a*b^2*log(c) + (b^3*c^2*log(c)^2 - 2*a*b^2*c^2*log(c ))*x^2 + (b^3*c^2*x^2 + b^3)*log(x)^2 + 2*(b^3*log(c) - a*b^2 + (b^3*c^2*l og(c) - a*b^2*c^2)*x^2)*log(x))*sqrt(c^2*x^2 + 1))*log(sqrt(c^2*x^2 + 1) + 1) + 4*(b^3*log(c)^3 - 3*a*b^2*log(c)^2 + (b^3*c^2*x^2 + b^3)*log(x)^3 + (b^3*c^2*log(c)^3 - 3*a*b^2*c^2*log(c)^2)*x^2 + 3*(b^3*log(c) - a*b^2 + (b ^3*c^2*log(c) - a*b^2*c^2)*x^2)*log(x)^2 + 3*(b^3*log(c)^2 - 2*a*b^2*lo...
\[ \int \frac {\left (a+b \text {csch}^{-1}(c x)\right )^3}{x^5} \, dx=\int { \frac {{\left (b \operatorname {arcsch}\left (c x\right ) + a\right )}^{3}}{x^{5}} \,d x } \]
Timed out. \[ \int \frac {\left (a+b \text {csch}^{-1}(c x)\right )^3}{x^5} \, dx=\int \frac {{\left (a+b\,\mathrm {asinh}\left (\frac {1}{c\,x}\right )\right )}^3}{x^5} \,d x \]