Optimal. Leaf size=75 \[ -\frac{a^2 \text{csch}^{-1}(a+b x)}{2 b^2}+\frac{(a+b x) \sqrt{\frac{1}{(a+b x)^2}+1}}{2 b^2}-\frac{a \tanh ^{-1}\left (\sqrt{\frac{1}{(a+b x)^2}+1}\right )}{b^2}+\frac{1}{2} x^2 \text{csch}^{-1}(a+b x) \]
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Rubi [A] time = 0.0605459, antiderivative size = 75, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.75, Rules used = {6322, 5469, 3773, 3770, 3767, 8} \[ -\frac{a^2 \text{csch}^{-1}(a+b x)}{2 b^2}+\frac{(a+b x) \sqrt{\frac{1}{(a+b x)^2}+1}}{2 b^2}-\frac{a \tanh ^{-1}\left (\sqrt{\frac{1}{(a+b x)^2}+1}\right )}{b^2}+\frac{1}{2} x^2 \text{csch}^{-1}(a+b x) \]
Antiderivative was successfully verified.
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Rule 6322
Rule 5469
Rule 3773
Rule 3770
Rule 3767
Rule 8
Rubi steps
\begin{align*} \int x \text{csch}^{-1}(a+b x) \, dx &=-\frac{\operatorname{Subst}\left (\int x \coth (x) \text{csch}(x) (-a+\text{csch}(x)) \, dx,x,\text{csch}^{-1}(a+b x)\right )}{b^2}\\ &=\frac{1}{2} x^2 \text{csch}^{-1}(a+b x)-\frac{\operatorname{Subst}\left (\int (-a+\text{csch}(x))^2 \, dx,x,\text{csch}^{-1}(a+b x)\right )}{2 b^2}\\ &=-\frac{a^2 \text{csch}^{-1}(a+b x)}{2 b^2}+\frac{1}{2} x^2 \text{csch}^{-1}(a+b x)-\frac{\operatorname{Subst}\left (\int \text{csch}^2(x) \, dx,x,\text{csch}^{-1}(a+b x)\right )}{2 b^2}+\frac{a \operatorname{Subst}\left (\int \text{csch}(x) \, dx,x,\text{csch}^{-1}(a+b x)\right )}{b^2}\\ &=-\frac{a^2 \text{csch}^{-1}(a+b x)}{2 b^2}+\frac{1}{2} x^2 \text{csch}^{-1}(a+b x)-\frac{a \tanh ^{-1}\left (\sqrt{1+\frac{1}{(a+b x)^2}}\right )}{b^2}+\frac{i \operatorname{Subst}\left (\int 1 \, dx,x,-i (a+b x) \sqrt{1+\frac{1}{(a+b x)^2}}\right )}{2 b^2}\\ &=\frac{(a+b x) \sqrt{1+\frac{1}{(a+b x)^2}}}{2 b^2}-\frac{a^2 \text{csch}^{-1}(a+b x)}{2 b^2}+\frac{1}{2} x^2 \text{csch}^{-1}(a+b x)-\frac{a \tanh ^{-1}\left (\sqrt{1+\frac{1}{(a+b x)^2}}\right )}{b^2}\\ \end{align*}
Mathematica [A] time = 0.085407, size = 110, normalized size = 1.47 \[ \frac{(a+b x) \sqrt{\frac{a^2+2 a b x+b^2 x^2+1}{(a+b x)^2}}-2 a \log \left ((a+b x) \left (\sqrt{\frac{a^2+2 a b x+b^2 x^2+1}{(a+b x)^2}}+1\right )\right )+a^2 \left (-\sinh ^{-1}\left (\frac{1}{a+b x}\right )\right )+b^2 x^2 \text{csch}^{-1}(a+b x)}{2 b^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.234, size = 97, normalized size = 1.3 \begin{align*}{\frac{1}{{b}^{2}} \left ({\frac{{\rm arccsch} \left (bx+a\right ) \left ( bx+a \right ) ^{2}}{2}}-{\rm arccsch} \left (bx+a\right )a \left ( bx+a \right ) -{\frac{1}{2\,bx+2\,a}\sqrt{1+ \left ( bx+a \right ) ^{2}} \left ( 2\,a{\it Arcsinh} \left ( bx+a \right ) -\sqrt{1+ \left ( bx+a \right ) ^{2}} \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{i \, a{\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 )}}{2 \, b^{2}} + \frac{2 \, b^{2} x^{2} \log \left (\sqrt{b^{2} x^{2} + 2 \, a b x + a^{2} + 1} + 1\right ) -{\left (a^{2} - 1\right )} \log \left (b^{2} x^{2} + 2 \, a b x + a^{2} + 1\right ) - 2 \,{\left (b^{2} x^{2} - a^{2}\right )} \log \left (b x + a\right )}{4 \, b^{2}} + \int \frac{b^{2} x^{3} + a b x^{2}}{2 \,{\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 = 3.15985, size = 643, normalized size = 8.57 \begin{align*} \frac{b^{2} x^{2} \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 ) - a^{2} \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 ) + a^{2} \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 \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 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}}}}{2 \, b^{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 \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 \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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