Optimal. Leaf size=132 \[ \frac {3}{16} a b c^4 \text {csch}^{-1}(c x)+\frac {b c \sqrt {\frac {1}{c^2 x^2}+1} \left (a+b \text {csch}^{-1}(c x)\right )}{8 x^3}-\frac {3 b c^3 \sqrt {\frac {1}{c^2 x^2}+1} \left (a+b \text {csch}^{-1}(c x)\right )}{16 x}-\frac {\left (a+b \text {csch}^{-1}(c x)\right )^2}{4 x^4}+\frac {3}{32} b^2 c^4 \text {csch}^{-1}(c x)^2+\frac {3 b^2 c^2}{32 x^2}-\frac {b^2}{32 x^4} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.12, antiderivative size = 132, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {6286, 5446, 3310} \[ -\frac {3 b c^3 \sqrt {\frac {1}{c^2 x^2}+1} \left (a+b \text {csch}^{-1}(c x)\right )}{16 x}+\frac {b c \sqrt {\frac {1}{c^2 x^2}+1} \left (a+b \text {csch}^{-1}(c x)\right )}{8 x^3}+\frac {3}{16} a b c^4 \text {csch}^{-1}(c x)-\frac {\left (a+b \text {csch}^{-1}(c x)\right )^2}{4 x^4}+\frac {3 b^2 c^2}{32 x^2}+\frac {3}{32} b^2 c^4 \text {csch}^{-1}(c x)^2-\frac {b^2}{32 x^4} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3310
Rule 5446
Rule 6286
Rubi steps
\begin {align*} \int \frac {\left (a+b \text {csch}^{-1}(c x)\right )^2}{x^5} \, dx &=-\left (c^4 \operatorname {Subst}\left (\int (a+b x)^2 \cosh (x) \sinh ^3(x) \, dx,x,\text {csch}^{-1}(c x)\right )\right )\\ &=-\frac {\left (a+b \text {csch}^{-1}(c x)\right )^2}{4 x^4}+\frac {1}{2} \left (b c^4\right ) \operatorname {Subst}\left (\int (a+b x) \sinh ^4(x) \, dx,x,\text {csch}^{-1}(c x)\right )\\ &=-\frac {b^2}{32 x^4}+\frac {b c \sqrt {1+\frac {1}{c^2 x^2}} \left (a+b \text {csch}^{-1}(c x)\right )}{8 x^3}-\frac {\left (a+b \text {csch}^{-1}(c x)\right )^2}{4 x^4}-\frac {1}{8} \left (3 b c^4\right ) \operatorname {Subst}\left (\int (a+b x) \sinh ^2(x) \, dx,x,\text {csch}^{-1}(c x)\right )\\ &=-\frac {b^2}{32 x^4}+\frac {3 b^2 c^2}{32 x^2}+\frac {b c \sqrt {1+\frac {1}{c^2 x^2}} \left (a+b \text {csch}^{-1}(c x)\right )}{8 x^3}-\frac {3 b c^3 \sqrt {1+\frac {1}{c^2 x^2}} \left (a+b \text {csch}^{-1}(c x)\right )}{16 x}-\frac {\left (a+b \text {csch}^{-1}(c x)\right )^2}{4 x^4}+\frac {1}{16} \left (3 b c^4\right ) \operatorname {Subst}\left (\int (a+b x) \, dx,x,\text {csch}^{-1}(c x)\right )\\ &=-\frac {b^2}{32 x^4}+\frac {3 b^2 c^2}{32 x^2}+\frac {3}{16} a b c^4 \text {csch}^{-1}(c x)+\frac {3}{32} b^2 c^4 \text {csch}^{-1}(c x)^2+\frac {b c \sqrt {1+\frac {1}{c^2 x^2}} \left (a+b \text {csch}^{-1}(c x)\right )}{8 x^3}-\frac {3 b c^3 \sqrt {1+\frac {1}{c^2 x^2}} \left (a+b \text {csch}^{-1}(c x)\right )}{16 x}-\frac {\left (a+b \text {csch}^{-1}(c x)\right )^2}{4 x^4}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.18, size = 147, normalized size = 1.11 \[ \frac {-8 a^2+6 a b c^4 x^4 \sinh ^{-1}\left (\frac {1}{c x}\right )+4 a b c x \sqrt {\frac {1}{c^2 x^2}+1}-2 b \text {csch}^{-1}(c x) \left (8 a+b c x \sqrt {\frac {1}{c^2 x^2}+1} \left (3 c^2 x^2-2\right )\right )-6 a b c^3 x^3 \sqrt {\frac {1}{c^2 x^2}+1}+b^2 \left (3 c^4 x^4-8\right ) \text {csch}^{-1}(c x)^2+3 b^2 c^2 x^2-b^2}{32 x^4} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.83, size = 202, normalized size = 1.53 \[ \frac {3 \, b^{2} c^{2} x^{2} + {\left (3 \, b^{2} c^{4} x^{4} - 8 \, b^{2}\right )} \log \left (\frac {c x \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}} + 1}{c x}\right )^{2} - 8 \, a^{2} - b^{2} + 2 \, {\left (3 \, a b c^{4} x^{4} - 8 \, a b - {\left (3 \, b^{2} c^{3} x^{3} - 2 \, b^{2} c x\right )} \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}}\right )} \log \left (\frac {c x \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}} + 1}{c x}\right ) - 2 \, {\left (3 \, a b c^{3} x^{3} - 2 \, a b c x\right )} \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}}}{32 \, x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b \operatorname {arcsch}\left (c x\right ) + a\right )}^{2}}{x^{5}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.05, size = 0, normalized size = 0.00 \[ \int \frac {\left (a +b \,\mathrm {arccsch}\left (c x \right )\right )^{2}}{x^{5}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {1}{32} \, a b {\left (\frac {3 \, c^{5} \log \left (c x \sqrt {\frac {1}{c^{2} x^{2}} + 1} + 1\right ) - 3 \, c^{5} \log \left (c x \sqrt {\frac {1}{c^{2} x^{2}} + 1} - 1\right ) - \frac {2 \, {\left (3 \, c^{8} x^{3} {\left (\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {3}{2}} - 5 \, c^{6} x \sqrt {\frac {1}{c^{2} x^{2}} + 1}\right )}}{c^{4} x^{4} {\left (\frac {1}{c^{2} x^{2}} + 1\right )}^{2} - 2 \, c^{2} x^{2} {\left (\frac {1}{c^{2} x^{2}} + 1\right )} + 1}}{c} - \frac {16 \, \operatorname {arcsch}\left (c x\right )}{x^{4}}\right )} - \frac {1}{4} \, b^{2} {\left (\frac {\log \left (\sqrt {c^{2} x^{2} + 1} + 1\right )^{2}}{x^{4}} + 4 \, \int -\frac {2 \, c^{2} x^{2} \log \relax (c)^{2} + 2 \, {\left (c^{2} x^{2} + 1\right )} \log \relax (x)^{2} + 2 \, \log \relax (c)^{2} + 4 \, {\left (c^{2} x^{2} \log \relax (c) + \log \relax (c)\right )} \log \relax (x) - {\left (4 \, c^{2} x^{2} \log \relax (c) + 4 \, {\left (c^{2} x^{2} + 1\right )} \log \relax (x) + {\left (c^{2} x^{2} {\left (4 \, \log \relax (c) - 1\right )} + 4 \, {\left (c^{2} x^{2} + 1\right )} \log \relax (x) + 4 \, \log \relax (c)\right )} \sqrt {c^{2} x^{2} + 1} + 4 \, \log \relax (c)\right )} \log \left (\sqrt {c^{2} x^{2} + 1} + 1\right ) + 2 \, {\left (c^{2} x^{2} \log \relax (c)^{2} + {\left (c^{2} x^{2} + 1\right )} \log \relax (x)^{2} + \log \relax (c)^{2} + 2 \, {\left (c^{2} x^{2} \log \relax (c) + \log \relax (c)\right )} \log \relax (x)\right )} \sqrt {c^{2} x^{2} + 1}}{2 \, {\left (c^{2} x^{7} + x^{5} + {\left (c^{2} x^{7} + x^{5}\right )} \sqrt {c^{2} x^{2} + 1}\right )}}\,{d x}\right )} - \frac {a^{2}}{4 \, x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (a+b\,\mathrm {asinh}\left (\frac {1}{c\,x}\right )\right )}^2}{x^5} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (a + b \operatorname {acsch}{\left (c x \right )}\right )^{2}}{x^{5}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________