Optimal. Leaf size=63 \[ \frac{2 b \tanh ^{-1}\left (\frac{\tanh \left (\frac{1}{2} \text{csch}^{-1}(a+b x)\right )+a}{\sqrt{a^2+1}}\right )}{a \sqrt{a^2+1}}-\frac{b \text{csch}^{-1}(a+b x)}{a}-\frac{\text{csch}^{-1}(a+b x)}{x} \]
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Rubi [A] time = 0.101801, antiderivative size = 63, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.6, Rules used = {6322, 5469, 3783, 2660, 618, 206} \[ \frac{2 b \tanh ^{-1}\left (\frac{\tanh \left (\frac{1}{2} \text{csch}^{-1}(a+b x)\right )+a}{\sqrt{a^2+1}}\right )}{a \sqrt{a^2+1}}-\frac{b \text{csch}^{-1}(a+b x)}{a}-\frac{\text{csch}^{-1}(a+b x)}{x} \]
Antiderivative was successfully verified.
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Rule 6322
Rule 5469
Rule 3783
Rule 2660
Rule 618
Rule 206
Rubi steps
\begin{align*} \int \frac{\text{csch}^{-1}(a+b x)}{x^2} \, dx &=-\left (b \operatorname{Subst}\left (\int \frac{x \coth (x) \text{csch}(x)}{(-a+\text{csch}(x))^2} \, dx,x,\text{csch}^{-1}(a+b x)\right )\right )\\ &=-\frac{\text{csch}^{-1}(a+b x)}{x}+b \operatorname{Subst}\left (\int \frac{1}{-a+\text{csch}(x)} \, dx,x,\text{csch}^{-1}(a+b x)\right )\\ &=-\frac{b \text{csch}^{-1}(a+b x)}{a}-\frac{\text{csch}^{-1}(a+b x)}{x}+\frac{b \operatorname{Subst}\left (\int \frac{1}{1-a \sinh (x)} \, dx,x,\text{csch}^{-1}(a+b x)\right )}{a}\\ &=-\frac{b \text{csch}^{-1}(a+b x)}{a}-\frac{\text{csch}^{-1}(a+b x)}{x}+\frac{(2 b) \operatorname{Subst}\left (\int \frac{1}{1-2 a x-x^2} \, dx,x,\tanh \left (\frac{1}{2} \text{csch}^{-1}(a+b x)\right )\right )}{a}\\ &=-\frac{b \text{csch}^{-1}(a+b x)}{a}-\frac{\text{csch}^{-1}(a+b x)}{x}-\frac{(4 b) \operatorname{Subst}\left (\int \frac{1}{4 \left (1+a^2\right )-x^2} \, dx,x,-2 a-2 \tanh \left (\frac{1}{2} \text{csch}^{-1}(a+b x)\right )\right )}{a}\\ &=-\frac{b \text{csch}^{-1}(a+b x)}{a}-\frac{\text{csch}^{-1}(a+b x)}{x}+\frac{2 b \tanh ^{-1}\left (\frac{a+\tanh \left (\frac{1}{2} \text{csch}^{-1}(a+b x)\right )}{\sqrt{1+a^2}}\right )}{a \sqrt{1+a^2}}\\ \end{align*}
Mathematica [B] time = 0.162538, size = 141, normalized size = 2.24 \[ -\frac{b \left (-\log \left (\sqrt{a^2+1} a \sqrt{\frac{a^2+2 a b x+b^2 x^2+1}{(a+b x)^2}}+\sqrt{a^2+1} b x \sqrt{\frac{a^2+2 a b x+b^2 x^2+1}{(a+b x)^2}}+a^2+a b x+1\right )+\sqrt{a^2+1} \sinh ^{-1}\left (\frac{1}{a+b x}\right )+\log (x)\right )}{a \sqrt{a^2+1}}-\frac{\text{csch}^{-1}(a+b x)}{x} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.233, size = 154, normalized size = 2.4 \begin{align*} -{\frac{{\rm arccsch} \left (bx+a\right )}{x}}-{\frac{b}{a \left ( bx+a \right ) }\sqrt{1+ \left ( bx+a \right ) ^{2}}{\it Artanh} \left ({\frac{1}{\sqrt{1+ \left ( bx+a \right ) ^{2}}}} \right ){\frac{1}{\sqrt{{\frac{1+ \left ( bx+a \right ) ^{2}}{ \left ( bx+a \right ) ^{2}}}}}}}+{\frac{b}{a \left ( bx+a \right ) }\sqrt{1+ \left ( bx+a \right ) ^{2}}\ln \left ( 2\,{\frac{\sqrt{{a}^{2}+1}\sqrt{1+ \left ( bx+a \right ) ^{2}}+a \left ( bx+a \right ) +1}{bx}} \right ){\frac{1}{\sqrt{{\frac{1+ \left ( bx+a \right ) ^{2}}{ \left ( bx+a \right ) ^{2}}}}}}{\frac{1}{\sqrt{{a}^{2}+1}}}} \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 \, b{\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 \,{\left (a^{2} + 1\right )}} - \frac{b \log \left (x\right )}{a^{3} + a} - \frac{a^{2} b x \log \left (b^{2} x^{2} + 2 \, a b x + a^{2} + 1\right ) - 2 \,{\left (a^{3} +{\left (a^{2} b + b\right )} x + a\right )} \log \left (b x + a\right ) + 2 \,{\left (a^{3} + a\right )} \log \left (\sqrt{b^{2} x^{2} + 2 \, a b x + a^{2} + 1} + 1\right )}{2 \,{\left (a^{3} + a\right )} x} - \int \frac{b^{2} x + a b}{b^{2} x^{3} + 2 \, a b x^{2} +{\left (a^{2} + 1\right )} x +{\left (b^{2} x^{3} + 2 \, a b x^{2} +{\left (a^{2} + 1\right )} x\right )} \sqrt{b^{2} x^{2} + 2 \, a b x + a^{2} + 1}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.79782, size = 792, normalized size = 12.57 \begin{align*} -\frac{{\left (a^{2} + 1\right )} b x \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 (a^{2} + 1\right )} b x \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 ) - \sqrt{a^{2} + 1} b x \log \left (-\frac{a^{2} b x + a^{3} +{\left (a b x + a^{2} +{\left (a b x + 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}}} + 1\right )} \sqrt{a^{2} + 1} +{\left (a^{3} +{\left (a^{2} + 1\right )} 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}{x}\right ) +{\left (a^{3} + a\right )} \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 )}{{\left (a^{3} + a\right )} x} \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 \frac{\operatorname{acsch}{\left (a + b x \right )}}{x^{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 \frac{\operatorname{arcsch}\left (b x + a\right )}{x^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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