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 = 0.0760874, antiderivative size = 54, normalized size of antiderivative = 1., 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 6286
Rule 5452
Rule 4184
Rule 3475
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*}
Mathematica [A] time = 0.137106, 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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Maple [F] time = 0.19, size = 0, normalized size = 0. \begin{align*} \int x \left ( a+b{\rm arccsch} \left (cx\right ) \right ) ^{2}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.00935, size = 111, normalized size = 2.06 \begin{align*} \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 \left (x\right )}{c^{2}}\right )} b^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.45099, size = 524, normalized size = 9.7 \begin{align*} \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 \left (x\right ) + 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}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int x \left (a + b \operatorname{acsch}{\left (c x \right )}\right )^{2}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b \operatorname{arcsch}\left (c x\right ) + a\right )}^{2} x\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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