Optimal. Leaf size=87 \[ \frac {b c \sqrt {\frac {1}{c^2 x^2}+1} \left (a+b \text {csch}^{-1}(c x)\right )}{2 x}-\frac {1}{2} a b c^2 \text {csch}^{-1}(c x)-\frac {\left (a+b \text {csch}^{-1}(c x)\right )^2}{2 x^2}-\frac {1}{4} b^2 c^2 \text {csch}^{-1}(c x)^2-\frac {b^2}{4 x^2} \]
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Rubi [A] time = 0.08, antiderivative size = 87, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {6286, 5446, 3310} \[ \frac {b c \sqrt {\frac {1}{c^2 x^2}+1} \left (a+b \text {csch}^{-1}(c x)\right )}{2 x}-\frac {1}{2} a b c^2 \text {csch}^{-1}(c x)-\frac {\left (a+b \text {csch}^{-1}(c x)\right )^2}{2 x^2}-\frac {1}{4} b^2 c^2 \text {csch}^{-1}(c x)^2-\frac {b^2}{4 x^2} \]
Antiderivative was successfully verified.
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Rule 3310
Rule 5446
Rule 6286
Rubi steps
\begin {align*} \int \frac {\left (a+b \text {csch}^{-1}(c x)\right )^2}{x^3} \, dx &=-\left (c^2 \operatorname {Subst}\left (\int (a+b x)^2 \cosh (x) \sinh (x) \, dx,x,\text {csch}^{-1}(c x)\right )\right )\\ &=-\frac {\left (a+b \text {csch}^{-1}(c x)\right )^2}{2 x^2}+\left (b c^2\right ) \operatorname {Subst}\left (\int (a+b x) \sinh ^2(x) \, dx,x,\text {csch}^{-1}(c x)\right )\\ &=-\frac {b^2}{4 x^2}+\frac {b c \sqrt {1+\frac {1}{c^2 x^2}} \left (a+b \text {csch}^{-1}(c x)\right )}{2 x}-\frac {\left (a+b \text {csch}^{-1}(c x)\right )^2}{2 x^2}-\frac {1}{2} \left (b c^2\right ) \operatorname {Subst}\left (\int (a+b x) \, dx,x,\text {csch}^{-1}(c x)\right )\\ &=-\frac {b^2}{4 x^2}-\frac {1}{2} a b c^2 \text {csch}^{-1}(c x)-\frac {1}{4} b^2 c^2 \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 )}{2 x}-\frac {\left (a+b \text {csch}^{-1}(c x)\right )^2}{2 x^2}\\ \end {align*}
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Mathematica [A] time = 0.14, size = 100, normalized size = 1.15 \[ -\frac {2 a^2-2 a b c x \sqrt {\frac {1}{c^2 x^2}+1}+2 a b c^2 x^2 \sinh ^{-1}\left (\frac {1}{c x}\right )-2 b \text {csch}^{-1}(c x) \left (b c x \sqrt {\frac {1}{c^2 x^2}+1}-2 a\right )+b^2 \left (c^2 x^2+2\right ) \text {csch}^{-1}(c x)^2+b^2}{4 x^2} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.67, size = 163, normalized size = 1.87 \[ \frac {2 \, a b c x \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}} - {\left (b^{2} c^{2} x^{2} + 2 \, b^{2}\right )} \log \left (\frac {c x \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}} + 1}{c x}\right )^{2} - 2 \, a^{2} - b^{2} - 2 \, {\left (a b c^{2} x^{2} - b^{2} c x \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}} + 2 \, a b\right )} \log \left (\frac {c x \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}} + 1}{c x}\right )}{4 \, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b \operatorname {arcsch}\left (c x\right ) + a\right )}^{2}}{x^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.06, size = 0, normalized size = 0.00 \[ \int \frac {\left (a +b \,\mathrm {arccsch}\left (c x \right )\right )^{2}}{x^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {1}{4} \, a b {\left (\frac {\frac {2 \, c^{4} x \sqrt {\frac {1}{c^{2} x^{2}} + 1}}{c^{2} x^{2} {\left (\frac {1}{c^{2} x^{2}} + 1\right )} - 1} - c^{3} \log \left (c x \sqrt {\frac {1}{c^{2} x^{2}} + 1} + 1\right ) + c^{3} \log \left (c x \sqrt {\frac {1}{c^{2} x^{2}} + 1} - 1\right )}{c} - \frac {4 \, \operatorname {arcsch}\left (c x\right )}{x^{2}}\right )} - \frac {1}{2} \, b^{2} {\left (\frac {\log \left (\sqrt {c^{2} x^{2} + 1} + 1\right )^{2}}{x^{2}} + 2 \, \int -\frac {c^{2} x^{2} \log \relax (c)^{2} + {\left (c^{2} x^{2} + 1\right )} \log \relax (x)^{2} + \log \relax (c)^{2} + 2 \, {\left (c^{2} x^{2} \log \relax (c) + \log \relax (c)\right )} \log \relax (x) - {\left (2 \, c^{2} x^{2} \log \relax (c) + 2 \, {\left (c^{2} x^{2} + 1\right )} \log \relax (x) + {\left (c^{2} x^{2} {\left (2 \, \log \relax (c) - 1\right )} + 2 \, {\left (c^{2} x^{2} + 1\right )} \log \relax (x) + 2 \, \log \relax (c)\right )} \sqrt {c^{2} x^{2} + 1} + 2 \, \log \relax (c)\right )} \log \left (\sqrt {c^{2} x^{2} + 1} + 1\right ) + {\left (c^{2} x^{2} \log \relax (c)^{2} + {\left (c^{2} x^{2} + 1\right )} \log \relax (x)^{2} + \log \relax (c)^{2} + 2 \, {\left (c^{2} x^{2} \log \relax (c) + \log \relax (c)\right )} \log \relax (x)\right )} \sqrt {c^{2} x^{2} + 1}}{c^{2} x^{5} + x^{3} + {\left (c^{2} x^{5} + x^{3}\right )} \sqrt {c^{2} x^{2} + 1}}\,{d x}\right )} - \frac {a^{2}}{2 \, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (a+b\,\mathrm {asinh}\left (\frac {1}{c\,x}\right )\right )}^2}{x^3} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (a + b \operatorname {acsch}{\left (c x \right )}\right )^{2}}{x^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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