Optimal. Leaf size=166 \[ \frac {4 b^2 c^2 \left (a+b \text {csch}^{-1}(c x)\right )}{3 x}-\frac {2 b^2 \left (a+b \text {csch}^{-1}(c x)\right )}{9 x^3}+\frac {b c \sqrt {\frac {1}{c^2 x^2}+1} \left (a+b \text {csch}^{-1}(c x)\right )^2}{3 x^2}-\frac {2}{3} b c^3 \sqrt {\frac {1}{c^2 x^2}+1} \left (a+b \text {csch}^{-1}(c x)\right )^2-\frac {\left (a+b \text {csch}^{-1}(c x)\right )^3}{3 x^3}+\frac {2}{27} b^3 c^3 \left (\frac {1}{c^2 x^2}+1\right )^{3/2}-\frac {14}{9} b^3 c^3 \sqrt {\frac {1}{c^2 x^2}+1} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.17, antiderivative size = 166, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {6286, 5446, 3311, 3296, 2638, 2633} \[ \frac {4 b^2 c^2 \left (a+b \text {csch}^{-1}(c x)\right )}{3 x}-\frac {2 b^2 \left (a+b \text {csch}^{-1}(c x)\right )}{9 x^3}-\frac {2}{3} b c^3 \sqrt {\frac {1}{c^2 x^2}+1} \left (a+b \text {csch}^{-1}(c x)\right )^2+\frac {b c \sqrt {\frac {1}{c^2 x^2}+1} \left (a+b \text {csch}^{-1}(c x)\right )^2}{3 x^2}-\frac {\left (a+b \text {csch}^{-1}(c x)\right )^3}{3 x^3}+\frac {2}{27} b^3 c^3 \left (\frac {1}{c^2 x^2}+1\right )^{3/2}-\frac {14}{9} b^3 c^3 \sqrt {\frac {1}{c^2 x^2}+1} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2633
Rule 2638
Rule 3296
Rule 3311
Rule 5446
Rule 6286
Rubi steps
\begin {align*} \int \frac {\left (a+b \text {csch}^{-1}(c x)\right )^3}{x^4} \, dx &=-\left (c^3 \operatorname {Subst}\left (\int (a+b x)^3 \cosh (x) \sinh ^2(x) \, dx,x,\text {csch}^{-1}(c x)\right )\right )\\ &=-\frac {\left (a+b \text {csch}^{-1}(c x)\right )^3}{3 x^3}+\left (b c^3\right ) \operatorname {Subst}\left (\int (a+b x)^2 \sinh ^3(x) \, dx,x,\text {csch}^{-1}(c x)\right )\\ &=-\frac {2 b^2 \left (a+b \text {csch}^{-1}(c x)\right )}{9 x^3}+\frac {b c \sqrt {1+\frac {1}{c^2 x^2}} \left (a+b \text {csch}^{-1}(c x)\right )^2}{3 x^2}-\frac {\left (a+b \text {csch}^{-1}(c x)\right )^3}{3 x^3}-\frac {1}{3} \left (2 b c^3\right ) \operatorname {Subst}\left (\int (a+b x)^2 \sinh (x) \, dx,x,\text {csch}^{-1}(c x)\right )+\frac {1}{9} \left (2 b^3 c^3\right ) \operatorname {Subst}\left (\int \sinh ^3(x) \, dx,x,\text {csch}^{-1}(c x)\right )\\ &=-\frac {2 b^2 \left (a+b \text {csch}^{-1}(c x)\right )}{9 x^3}-\frac {2}{3} b c^3 \sqrt {1+\frac {1}{c^2 x^2}} \left (a+b \text {csch}^{-1}(c x)\right )^2+\frac {b c \sqrt {1+\frac {1}{c^2 x^2}} \left (a+b \text {csch}^{-1}(c x)\right )^2}{3 x^2}-\frac {\left (a+b \text {csch}^{-1}(c x)\right )^3}{3 x^3}+\frac {1}{3} \left (4 b^2 c^3\right ) \operatorname {Subst}\left (\int (a+b x) \cosh (x) \, dx,x,\text {csch}^{-1}(c x)\right )-\frac {1}{9} \left (2 b^3 c^3\right ) \operatorname {Subst}\left (\int \left (1-x^2\right ) \, dx,x,\sqrt {1+\frac {1}{c^2 x^2}}\right )\\ &=-\frac {2}{9} b^3 c^3 \sqrt {1+\frac {1}{c^2 x^2}}+\frac {2}{27} b^3 c^3 \left (1+\frac {1}{c^2 x^2}\right )^{3/2}-\frac {2 b^2 \left (a+b \text {csch}^{-1}(c x)\right )}{9 x^3}+\frac {4 b^2 c^2 \left (a+b \text {csch}^{-1}(c x)\right )}{3 x}-\frac {2}{3} b c^3 \sqrt {1+\frac {1}{c^2 x^2}} \left (a+b \text {csch}^{-1}(c x)\right )^2+\frac {b c \sqrt {1+\frac {1}{c^2 x^2}} \left (a+b \text {csch}^{-1}(c x)\right )^2}{3 x^2}-\frac {\left (a+b \text {csch}^{-1}(c x)\right )^3}{3 x^3}-\frac {1}{3} \left (4 b^3 c^3\right ) \operatorname {Subst}\left (\int \sinh (x) \, dx,x,\text {csch}^{-1}(c x)\right )\\ &=-\frac {14}{9} b^3 c^3 \sqrt {1+\frac {1}{c^2 x^2}}+\frac {2}{27} b^3 c^3 \left (1+\frac {1}{c^2 x^2}\right )^{3/2}-\frac {2 b^2 \left (a+b \text {csch}^{-1}(c x)\right )}{9 x^3}+\frac {4 b^2 c^2 \left (a+b \text {csch}^{-1}(c x)\right )}{3 x}-\frac {2}{3} b c^3 \sqrt {1+\frac {1}{c^2 x^2}} \left (a+b \text {csch}^{-1}(c x)\right )^2+\frac {b c \sqrt {1+\frac {1}{c^2 x^2}} \left (a+b \text {csch}^{-1}(c x)\right )^2}{3 x^2}-\frac {\left (a+b \text {csch}^{-1}(c x)\right )^3}{3 x^3}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.31, size = 200, normalized size = 1.20 \[ \frac {-9 a^3+3 b \text {csch}^{-1}(c x) \left (-9 a^2+6 a b c x \sqrt {\frac {1}{c^2 x^2}+1} \left (1-2 c^2 x^2\right )+2 b^2 \left (6 c^2 x^2-1\right )\right )+9 a^2 b c x \sqrt {\frac {1}{c^2 x^2}+1} \left (1-2 c^2 x^2\right )+6 a b^2 \left (6 c^2 x^2-1\right )-9 b^2 \text {csch}^{-1}(c x)^2 \left (3 a+b c x \sqrt {\frac {1}{c^2 x^2}+1} \left (2 c^2 x^2-1\right )\right )+2 b^3 c x \sqrt {\frac {1}{c^2 x^2}+1} \left (1-20 c^2 x^2\right )-9 b^3 \text {csch}^{-1}(c x)^3}{27 x^3} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 0.77, size = 301, normalized size = 1.81 \[ \frac {36 \, a b^{2} c^{2} x^{2} - 9 \, b^{3} \log \left (\frac {c x \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}} + 1}{c x}\right )^{3} - 9 \, a^{3} - 6 \, a b^{2} - 9 \, {\left (3 \, a b^{2} + {\left (2 \, b^{3} c^{3} x^{3} - b^{3} c x\right )} \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}}\right )} \log \left (\frac {c x \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}} + 1}{c x}\right )^{2} + 3 \, {\left (12 \, b^{3} c^{2} x^{2} - 9 \, a^{2} b - 2 \, b^{3} - 6 \, {\left (2 \, a b^{2} c^{3} x^{3} - a b^{2} c x\right )} \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}}\right )} \log \left (\frac {c x \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}} + 1}{c x}\right ) - {\left (2 \, {\left (9 \, a^{2} b + 20 \, b^{3}\right )} c^{3} x^{3} - {\left (9 \, a^{2} b + 2 \, b^{3}\right )} c x\right )} \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}}}{27 \, x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b \operatorname {arcsch}\left (c x\right ) + a\right )}^{3}}{x^{4}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.05, size = 0, normalized size = 0.00 \[ \int \frac {\left (a +b \,\mathrm {arccsch}\left (c x \right )\right )^{3}}{x^{4}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {1}{3} \, a^{2} b {\left (\frac {c^{4} {\left (\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {3}{2}} - 3 \, c^{4} \sqrt {\frac {1}{c^{2} x^{2}} + 1}}{c} - \frac {3 \, \operatorname {arcsch}\left (c x\right )}{x^{3}}\right )} - \frac {b^{3} \log \left (\sqrt {c^{2} x^{2} + 1} + 1\right )^{3}}{3 \, x^{3}} - \frac {a^{3}}{3 \, x^{3}} - \int \frac {b^{3} \log \relax (c)^{3} - 3 \, a b^{2} \log \relax (c)^{2} + {\left (b^{3} c^{2} x^{2} + b^{3}\right )} \log \relax (x)^{3} + {\left (b^{3} c^{2} \log \relax (c)^{3} - 3 \, a b^{2} c^{2} \log \relax (c)^{2}\right )} x^{2} + 3 \, {\left (b^{3} \log \relax (c) - a b^{2} + {\left (b^{3} c^{2} \log \relax (c) - a b^{2} c^{2}\right )} x^{2}\right )} \log \relax (x)^{2} + {\left (3 \, b^{3} \log \relax (c) - 3 \, a b^{2} + 3 \, {\left (b^{3} c^{2} \log \relax (c) - a b^{2} c^{2}\right )} x^{2} + 3 \, {\left (b^{3} c^{2} x^{2} + b^{3}\right )} \log \relax (x) + \sqrt {c^{2} x^{2} + 1} {\left (3 \, b^{3} \log \relax (c) - 3 \, a b^{2} + {\left (b^{3} c^{2} {\left (3 \, \log \relax (c) - 1\right )} - 3 \, a b^{2} c^{2}\right )} x^{2} + 3 \, {\left (b^{3} c^{2} x^{2} + b^{3}\right )} \log \relax (x)\right )}\right )} \log \left (\sqrt {c^{2} x^{2} + 1} + 1\right )^{2} + 3 \, {\left (b^{3} \log \relax (c)^{2} - 2 \, a b^{2} \log \relax (c) + {\left (b^{3} c^{2} \log \relax (c)^{2} - 2 \, a b^{2} c^{2} \log \relax (c)\right )} x^{2}\right )} \log \relax (x) - 3 \, {\left (b^{3} \log \relax (c)^{2} - 2 \, a b^{2} \log \relax (c) + {\left (b^{3} c^{2} \log \relax (c)^{2} - 2 \, a b^{2} c^{2} \log \relax (c)\right )} x^{2} + {\left (b^{3} c^{2} x^{2} + b^{3}\right )} \log \relax (x)^{2} + 2 \, {\left (b^{3} \log \relax (c) - a b^{2} + {\left (b^{3} c^{2} \log \relax (c) - a b^{2} c^{2}\right )} x^{2}\right )} \log \relax (x) + {\left (b^{3} \log \relax (c)^{2} - 2 \, a b^{2} \log \relax (c) + {\left (b^{3} c^{2} \log \relax (c)^{2} - 2 \, a b^{2} c^{2} \log \relax (c)\right )} x^{2} + {\left (b^{3} c^{2} x^{2} + b^{3}\right )} \log \relax (x)^{2} + 2 \, {\left (b^{3} \log \relax (c) - a b^{2} + {\left (b^{3} c^{2} \log \relax (c) - a b^{2} c^{2}\right )} x^{2}\right )} \log \relax (x)\right )} \sqrt {c^{2} x^{2} + 1}\right )} \log \left (\sqrt {c^{2} x^{2} + 1} + 1\right ) + {\left (b^{3} \log \relax (c)^{3} - 3 \, a b^{2} \log \relax (c)^{2} + {\left (b^{3} c^{2} x^{2} + b^{3}\right )} \log \relax (x)^{3} + {\left (b^{3} c^{2} \log \relax (c)^{3} - 3 \, a b^{2} c^{2} \log \relax (c)^{2}\right )} x^{2} + 3 \, {\left (b^{3} \log \relax (c) - a b^{2} + {\left (b^{3} c^{2} \log \relax (c) - a b^{2} c^{2}\right )} x^{2}\right )} \log \relax (x)^{2} + 3 \, {\left (b^{3} \log \relax (c)^{2} - 2 \, a b^{2} \log \relax (c) + {\left (b^{3} c^{2} \log \relax (c)^{2} - 2 \, a b^{2} c^{2} \log \relax (c)\right )} x^{2}\right )} \log \relax (x)\right )} \sqrt {c^{2} x^{2} + 1}}{c^{2} x^{6} + x^{4} + {\left (c^{2} x^{6} + x^{4}\right )} \sqrt {c^{2} x^{2} + 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (a+b\,\mathrm {asinh}\left (\frac {1}{c\,x}\right )\right )}^3}{x^4} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (a + b \operatorname {acsch}{\left (c x \right )}\right )^{3}}{x^{4}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________