Optimal. Leaf size=68 \[ x \left (a+b \text {csch}^{-1}(c x)\right )^2+\frac {4 b \left (a+b \text {csch}^{-1}(c x)\right ) \tanh ^{-1}\left (e^{\text {csch}^{-1}(c x)}\right )}{c}+\frac {2 b^2 \text {PolyLog}\left (2,-e^{\text {csch}^{-1}(c x)}\right )}{c}-\frac {2 b^2 \text {PolyLog}\left (2,e^{\text {csch}^{-1}(c x)}\right )}{c} \]
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Rubi [A]
time = 0.05, antiderivative size = 68, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 5, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6415, 5560,
4267, 2317, 2438} \begin {gather*} x \left (a+b \text {csch}^{-1}(c x)\right )^2+\frac {4 b \tanh ^{-1}\left (e^{\text {csch}^{-1}(c x)}\right ) \left (a+b \text {csch}^{-1}(c x)\right )}{c}+\frac {2 b^2 \text {Li}_2\left (-e^{\text {csch}^{-1}(c x)}\right )}{c}-\frac {2 b^2 \text {Li}_2\left (e^{\text {csch}^{-1}(c x)}\right )}{c} \end {gather*}
Antiderivative was successfully verified.
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Rule 2317
Rule 2438
Rule 4267
Rule 5560
Rule 6415
Rubi steps
\begin {align*} \int \left (a+b \text {csch}^{-1}(c x)\right )^2 \, dx &=-\frac {\text {Subst}\left (\int (a+b x)^2 \coth (x) \text {csch}(x) \, dx,x,\text {csch}^{-1}(c x)\right )}{c}\\ &=x \left (a+b \text {csch}^{-1}(c x)\right )^2-\frac {(2 b) \text {Subst}\left (\int (a+b x) \text {csch}(x) \, dx,x,\text {csch}^{-1}(c x)\right )}{c}\\ &=x \left (a+b \text {csch}^{-1}(c x)\right )^2+\frac {4 b \left (a+b \text {csch}^{-1}(c x)\right ) \tanh ^{-1}\left (e^{\text {csch}^{-1}(c x)}\right )}{c}+\frac {\left (2 b^2\right ) \text {Subst}\left (\int \log \left (1-e^x\right ) \, dx,x,\text {csch}^{-1}(c x)\right )}{c}-\frac {\left (2 b^2\right ) \text {Subst}\left (\int \log \left (1+e^x\right ) \, dx,x,\text {csch}^{-1}(c x)\right )}{c}\\ &=x \left (a+b \text {csch}^{-1}(c x)\right )^2+\frac {4 b \left (a+b \text {csch}^{-1}(c x)\right ) \tanh ^{-1}\left (e^{\text {csch}^{-1}(c x)}\right )}{c}+\frac {\left (2 b^2\right ) \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{\text {csch}^{-1}(c x)}\right )}{c}-\frac {\left (2 b^2\right ) \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{\text {csch}^{-1}(c x)}\right )}{c}\\ &=x \left (a+b \text {csch}^{-1}(c x)\right )^2+\frac {4 b \left (a+b \text {csch}^{-1}(c x)\right ) \tanh ^{-1}\left (e^{\text {csch}^{-1}(c x)}\right )}{c}+\frac {2 b^2 \text {Li}_2\left (-e^{\text {csch}^{-1}(c x)}\right )}{c}-\frac {2 b^2 \text {Li}_2\left (e^{\text {csch}^{-1}(c x)}\right )}{c}\\ \end {align*}
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Mathematica [A]
time = 0.16, size = 121, normalized size = 1.78 \begin {gather*} \frac {a^2 c x+2 a b c x \text {csch}^{-1}(c x)+b^2 c x \text {csch}^{-1}(c x)^2-2 b^2 \text {csch}^{-1}(c x) \log \left (1-e^{-\text {csch}^{-1}(c x)}\right )+2 b^2 \text {csch}^{-1}(c x) \log \left (1+e^{-\text {csch}^{-1}(c x)}\right )-2 a b \log \left (\tanh \left (\frac {1}{2} \text {csch}^{-1}(c x)\right )\right )-2 b^2 \text {PolyLog}\left (2,-e^{-\text {csch}^{-1}(c x)}\right )+2 b^2 \text {PolyLog}\left (2,e^{-\text {csch}^{-1}(c x)}\right )}{c} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int \left (a +b \,\mathrm {arccsch}\left (c x \right )\right )^{2}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \left (a + b \operatorname {acsch}{\left (c x \right )}\right )^{2}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int {\left (a+b\,\mathrm {asinh}\left (\frac {1}{c\,x}\right )\right )}^2 \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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