Optimal. Leaf size=54 \[ \frac {b x \sqrt {\frac {1}{c^2 x^2}+1} \left (a+b \text {csch}^{-1}(c x)\right )}{c}+\frac {1}{2} x^2 \left (a+b \text {csch}^{-1}(c x)\right )^2+\frac {b^2 \log (x)}{c^2} \]
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Rubi [A] time = 0.08, antiderivative size = 54, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {6286, 5452, 4184, 3475} \[ \frac {b x \sqrt {\frac {1}{c^2 x^2}+1} \left (a+b \text {csch}^{-1}(c x)\right )}{c}+\frac {1}{2} x^2 \left (a+b \text {csch}^{-1}(c x)\right )^2+\frac {b^2 \log (x)}{c^2} \]
Antiderivative was successfully verified.
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Rule 3475
Rule 4184
Rule 5452
Rule 6286
Rubi steps
\begin {align*} \int x \left (a+b \text {csch}^{-1}(c x)\right )^2 \, dx &=-\frac {\operatorname {Subst}\left (\int (a+b x)^2 \coth (x) \text {csch}^2(x) \, dx,x,\text {csch}^{-1}(c x)\right )}{c^2}\\ &=\frac {1}{2} x^2 \left (a+b \text {csch}^{-1}(c x)\right )^2-\frac {b \operatorname {Subst}\left (\int (a+b x) \text {csch}^2(x) \, dx,x,\text {csch}^{-1}(c x)\right )}{c^2}\\ &=\frac {b \sqrt {1+\frac {1}{c^2 x^2}} x \left (a+b \text {csch}^{-1}(c x)\right )}{c}+\frac {1}{2} x^2 \left (a+b \text {csch}^{-1}(c x)\right )^2-\frac {b^2 \operatorname {Subst}\left (\int \coth (x) \, dx,x,\text {csch}^{-1}(c x)\right )}{c^2}\\ &=\frac {b \sqrt {1+\frac {1}{c^2 x^2}} x \left (a+b \text {csch}^{-1}(c x)\right )}{c}+\frac {1}{2} x^2 \left (a+b \text {csch}^{-1}(c x)\right )^2+\frac {b^2 \log (x)}{c^2}\\ \end {align*}
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Mathematica [A] time = 0.14, size = 87, normalized size = 1.61 \[ \frac {a c x \left (a c x+2 b \sqrt {\frac {1}{c^2 x^2}+1}\right )+2 b c x \text {csch}^{-1}(c x) \left (a c x+b \sqrt {\frac {1}{c^2 x^2}+1}\right )+b^2 c^2 x^2 \text {csch}^{-1}(c x)^2+2 b^2 \log (c x)}{2 c^2} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.78, size = 234, normalized size = 4.33 \[ \frac {b^{2} c^{2} x^{2} \log \left (\frac {c x \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}} + 1}{c x}\right )^{2} + a^{2} c^{2} x^{2} + 2 \, a b c^{2} \log \left (c x \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}} - c x + 1\right ) - 2 \, a b c^{2} \log \left (c x \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}} - c x - 1\right ) + 2 \, a b c x \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}} + 2 \, b^{2} \log \relax (x) + 2 \, {\left (a b c^{2} x^{2} + b^{2} c x \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}} - a b c^{2}\right )} \log \left (\frac {c x \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}} + 1}{c x}\right )}{2 \, c^{2}} \]
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 \[ \int {\left (b \operatorname {arcsch}\left (c x\right ) + a\right )}^{2} x\,{d x} \]
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 \[ \int x \left (a +b \,\mathrm {arccsch}\left (c x \right )\right )^{2}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.36, size = 82, normalized size = 1.52 \[ \frac {1}{2} \, b^{2} x^{2} \operatorname {arcsch}\left (c x\right )^{2} + \frac {1}{2} \, a^{2} x^{2} + {\left (x^{2} \operatorname {arcsch}\left (c x\right ) + \frac {x \sqrt {\frac {1}{c^{2} x^{2}} + 1}}{c}\right )} a b + {\left (\frac {x \sqrt {\frac {1}{c^{2} x^{2}} + 1} \operatorname {arcsch}\left (c x\right )}{c} + \frac {\log \relax (x)}{c^{2}}\right )} b^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int x\,{\left (a+b\,\mathrm {asinh}\left (\frac {1}{c\,x}\right )\right )}^2 \,d x \]
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 \[ \int x \left (a + b \operatorname {acsch}{\left (c x \right )}\right )^{2}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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