Optimal. Leaf size=110 \[ -\frac{\left (1-6 a^2\right ) \tanh ^{-1}\left (\sqrt{\frac{1}{(a+b x)^2}+1}\right )}{6 b^3}+\frac{a^3 \text{csch}^{-1}(a+b x)}{3 b^3}-\frac{5 a (a+b x) \sqrt{\frac{1}{(a+b x)^2}+1}}{6 b^3}+\frac{x (a+b x) \sqrt{\frac{1}{(a+b x)^2}+1}}{6 b^2}+\frac{1}{3} x^3 \text{csch}^{-1}(a+b x) \]
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Rubi [A] time = 0.0952465, antiderivative size = 110, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.6, Rules used = {6322, 5469, 3782, 3770, 3767, 8} \[ -\frac{\left (1-6 a^2\right ) \tanh ^{-1}\left (\sqrt{\frac{1}{(a+b x)^2}+1}\right )}{6 b^3}+\frac{a^3 \text{csch}^{-1}(a+b x)}{3 b^3}-\frac{5 a (a+b x) \sqrt{\frac{1}{(a+b x)^2}+1}}{6 b^3}+\frac{x (a+b x) \sqrt{\frac{1}{(a+b x)^2}+1}}{6 b^2}+\frac{1}{3} x^3 \text{csch}^{-1}(a+b x) \]
Antiderivative was successfully verified.
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Rule 6322
Rule 5469
Rule 3782
Rule 3770
Rule 3767
Rule 8
Rubi steps
\begin{align*} \int x^2 \text{csch}^{-1}(a+b x) \, dx &=-\frac{\operatorname{Subst}\left (\int x \coth (x) \text{csch}(x) (-a+\text{csch}(x))^2 \, dx,x,\text{csch}^{-1}(a+b x)\right )}{b^3}\\ &=\frac{1}{3} x^3 \text{csch}^{-1}(a+b x)-\frac{\operatorname{Subst}\left (\int (-a+\text{csch}(x))^3 \, dx,x,\text{csch}^{-1}(a+b x)\right )}{3 b^3}\\ &=\frac{x (a+b x) \sqrt{1+\frac{1}{(a+b x)^2}}}{6 b^2}+\frac{1}{3} x^3 \text{csch}^{-1}(a+b x)-\frac{\operatorname{Subst}\left (\int \left (-2 a^3-\left (1-6 a^2\right ) \text{csch}(x)-5 a \text{csch}^2(x)\right ) \, dx,x,\text{csch}^{-1}(a+b x)\right )}{6 b^3}\\ &=\frac{x (a+b x) \sqrt{1+\frac{1}{(a+b x)^2}}}{6 b^2}+\frac{a^3 \text{csch}^{-1}(a+b x)}{3 b^3}+\frac{1}{3} x^3 \text{csch}^{-1}(a+b x)+\frac{(5 a) \operatorname{Subst}\left (\int \text{csch}^2(x) \, dx,x,\text{csch}^{-1}(a+b x)\right )}{6 b^3}+\frac{\left (1-6 a^2\right ) \operatorname{Subst}\left (\int \text{csch}(x) \, dx,x,\text{csch}^{-1}(a+b x)\right )}{6 b^3}\\ &=\frac{x (a+b x) \sqrt{1+\frac{1}{(a+b x)^2}}}{6 b^2}+\frac{a^3 \text{csch}^{-1}(a+b x)}{3 b^3}+\frac{1}{3} x^3 \text{csch}^{-1}(a+b x)-\frac{\left (1-6 a^2\right ) \tanh ^{-1}\left (\sqrt{1+\frac{1}{(a+b x)^2}}\right )}{6 b^3}-\frac{(5 i a) \operatorname{Subst}\left (\int 1 \, dx,x,-i (a+b x) \sqrt{1+\frac{1}{(a+b x)^2}}\right )}{6 b^3}\\ &=-\frac{5 a (a+b x) \sqrt{1+\frac{1}{(a+b x)^2}}}{6 b^3}+\frac{x (a+b x) \sqrt{1+\frac{1}{(a+b x)^2}}}{6 b^2}+\frac{a^3 \text{csch}^{-1}(a+b x)}{3 b^3}+\frac{1}{3} x^3 \text{csch}^{-1}(a+b x)-\frac{\left (1-6 a^2\right ) \tanh ^{-1}\left (\sqrt{1+\frac{1}{(a+b x)^2}}\right )}{6 b^3}\\ \end{align*}
Mathematica [A] time = 0.176214, size = 129, normalized size = 1.17 \[ \frac{\left (-5 a^2-4 a b x+b^2 x^2\right ) \sqrt{\frac{a^2+2 a b x+b^2 x^2+1}{(a+b x)^2}}+\left (6 a^2-1\right ) \log \left ((a+b x) \left (\sqrt{\frac{a^2+2 a b x+b^2 x^2+1}{(a+b x)^2}}+1\right )\right )+2 a^3 \sinh ^{-1}\left (\frac{1}{a+b x}\right )+2 b^3 x^3 \text{csch}^{-1}(a+b x)}{6 b^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.219, size = 170, normalized size = 1.6 \begin{align*}{\frac{1}{{b}^{3}} \left ({\frac{{\rm arccsch} \left (bx+a\right ) \left ( bx+a \right ) ^{3}}{3}}-{\rm arccsch} \left (bx+a\right ) \left ( bx+a \right ) ^{2}a+{\rm arccsch} \left (bx+a\right ) \left ( bx+a \right ){a}^{2}-{\frac{{\rm arccsch} \left (bx+a\right ){a}^{3}}{3}}+{\frac{1}{6\,bx+6\,a}\sqrt{1+ \left ( bx+a \right ) ^{2}} \left ( 2\,{a}^{3}{\it Artanh} \left ({\frac{1}{\sqrt{1+ \left ( bx+a \right ) ^{2}}}} \right ) +6\,{\it Arcsinh} \left ( bx+a \right ){a}^{2}+ \left ( bx+a \right ) \sqrt{1+ \left ( bx+a \right ) ^{2}}-6\,a\sqrt{1+ \left ( bx+a \right ) ^{2}}-{\it Arcsinh} \left ( bx+a \right ) \right ){\frac{1}{\sqrt{{\frac{1+ \left ( bx+a \right ) ^{2}}{ \left ( bx+a \right ) ^{2}}}}}}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{{\left (3 i \, a^{2} - i\right )}{\left (\log \left (\frac{i \,{\left (b^{2} x + a b\right )}}{b} + 1\right ) - \log \left (-\frac{i \,{\left (b^{2} x + a b\right )}}{b} + 1\right )\right )}}{6 \, b^{3}} + \frac{2 \, b^{3} x^{3} \log \left (\sqrt{b^{2} x^{2} + 2 \, a b x + a^{2} + 1} + 1\right ) + 2 \, b x +{\left (a^{3} - 3 \, a\right )} \log \left (b^{2} x^{2} + 2 \, a b x + a^{2} + 1\right ) - 2 \,{\left (b^{3} x^{3} + a^{3}\right )} \log \left (b x + a\right )}{6 \, b^{3}} + \int \frac{b^{2} x^{4} + a b x^{3}}{3 \,{\left (b^{2} x^{2} + 2 \, a b x + a^{2} +{\left (b^{2} x^{2} + 2 \, a b x + a^{2} + 1\right )}^{\frac{3}{2}} + 1\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 3.31905, size = 686, normalized size = 6.24 \begin{align*} \frac{2 \, b^{3} x^{3} \log \left (\frac{{\left (b x + a\right )} \sqrt{\frac{b^{2} x^{2} + 2 \, a b x + a^{2} + 1}{b^{2} x^{2} + 2 \, a b x + a^{2}}} + 1}{b x + a}\right ) + 2 \, a^{3} \log \left (-b x +{\left (b x + a\right )} \sqrt{\frac{b^{2} x^{2} + 2 \, a b x + a^{2} + 1}{b^{2} x^{2} + 2 \, a b x + a^{2}}} - a + 1\right ) - 2 \, a^{3} \log \left (-b x +{\left (b x + a\right )} \sqrt{\frac{b^{2} x^{2} + 2 \, a b x + a^{2} + 1}{b^{2} x^{2} + 2 \, a b x + a^{2}}} - a - 1\right ) -{\left (6 \, a^{2} - 1\right )} \log \left (-b x +{\left (b x + a\right )} \sqrt{\frac{b^{2} x^{2} + 2 \, a b x + a^{2} + 1}{b^{2} x^{2} + 2 \, a b x + a^{2}}} - a\right ) +{\left (b^{2} x^{2} - 4 \, a b x - 5 \, a^{2}\right )} \sqrt{\frac{b^{2} x^{2} + 2 \, a b x + a^{2} + 1}{b^{2} x^{2} + 2 \, a b x + a^{2}}}}{6 \, b^{3}} \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^{2} \operatorname{acsch}{\left (a + b x \right )}\, 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 x^{2} \operatorname{arcsch}\left (b x + a\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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