Optimal. Leaf size=114 \[ \frac{b^2 \text{csch}^{-1}(a+b x)}{2 a^2}-\frac{\left (2 a^2+1\right ) b^2 \tanh ^{-1}\left (\frac{\tanh \left (\frac{1}{2} \text{csch}^{-1}(a+b x)\right )+a}{\sqrt{a^2+1}}\right )}{a^2 \left (a^2+1\right )^{3/2}}+\frac{b (a+b x) \sqrt{\frac{1}{(a+b x)^2}+1}}{2 a \left (a^2+1\right ) x}-\frac{\text{csch}^{-1}(a+b x)}{2 x^2} \]
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Rubi [A] time = 0.209962, antiderivative size = 114, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.8, Rules used = {6322, 5469, 3785, 3919, 3831, 2660, 618, 206} \[ \frac{b^2 \text{csch}^{-1}(a+b x)}{2 a^2}-\frac{\left (2 a^2+1\right ) b^2 \tanh ^{-1}\left (\frac{\tanh \left (\frac{1}{2} \text{csch}^{-1}(a+b x)\right )+a}{\sqrt{a^2+1}}\right )}{a^2 \left (a^2+1\right )^{3/2}}+\frac{b (a+b x) \sqrt{\frac{1}{(a+b x)^2}+1}}{2 a \left (a^2+1\right ) x}-\frac{\text{csch}^{-1}(a+b x)}{2 x^2} \]
Antiderivative was successfully verified.
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Rule 6322
Rule 5469
Rule 3785
Rule 3919
Rule 3831
Rule 2660
Rule 618
Rule 206
Rubi steps
\begin{align*} \int \frac{\text{csch}^{-1}(a+b x)}{x^3} \, dx &=-\left (b^2 \operatorname{Subst}\left (\int \frac{x \coth (x) \text{csch}(x)}{(-a+\text{csch}(x))^3} \, dx,x,\text{csch}^{-1}(a+b x)\right )\right )\\ &=-\frac{\text{csch}^{-1}(a+b x)}{2 x^2}+\frac{1}{2} b^2 \operatorname{Subst}\left (\int \frac{1}{(-a+\text{csch}(x))^2} \, dx,x,\text{csch}^{-1}(a+b x)\right )\\ &=\frac{b (a+b x) \sqrt{1+\frac{1}{(a+b x)^2}}}{2 a \left (1+a^2\right ) x}-\frac{\text{csch}^{-1}(a+b x)}{2 x^2}+\frac{b^2 \operatorname{Subst}\left (\int \frac{-1-a^2-a \text{csch}(x)}{-a+\text{csch}(x)} \, dx,x,\text{csch}^{-1}(a+b x)\right )}{2 a \left (1+a^2\right )}\\ &=\frac{b (a+b x) \sqrt{1+\frac{1}{(a+b x)^2}}}{2 a \left (1+a^2\right ) x}+\frac{b^2 \text{csch}^{-1}(a+b x)}{2 a^2}-\frac{\text{csch}^{-1}(a+b x)}{2 x^2}-\frac{\left (\left (1+2 a^2\right ) b^2\right ) \operatorname{Subst}\left (\int \frac{\text{csch}(x)}{-a+\text{csch}(x)} \, dx,x,\text{csch}^{-1}(a+b x)\right )}{2 a^2 \left (1+a^2\right )}\\ &=\frac{b (a+b x) \sqrt{1+\frac{1}{(a+b x)^2}}}{2 a \left (1+a^2\right ) x}+\frac{b^2 \text{csch}^{-1}(a+b x)}{2 a^2}-\frac{\text{csch}^{-1}(a+b x)}{2 x^2}-\frac{\left (\left (1+2 a^2\right ) b^2\right ) \operatorname{Subst}\left (\int \frac{1}{1-a \sinh (x)} \, dx,x,\text{csch}^{-1}(a+b x)\right )}{2 a^2 \left (1+a^2\right )}\\ &=\frac{b (a+b x) \sqrt{1+\frac{1}{(a+b x)^2}}}{2 a \left (1+a^2\right ) x}+\frac{b^2 \text{csch}^{-1}(a+b x)}{2 a^2}-\frac{\text{csch}^{-1}(a+b x)}{2 x^2}-\frac{\left (\left (1+2 a^2\right ) b^2\right ) \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^2 \left (1+a^2\right )}\\ &=\frac{b (a+b x) \sqrt{1+\frac{1}{(a+b x)^2}}}{2 a \left (1+a^2\right ) x}+\frac{b^2 \text{csch}^{-1}(a+b x)}{2 a^2}-\frac{\text{csch}^{-1}(a+b x)}{2 x^2}+\frac{\left (2 \left (1+2 a^2\right ) b^2\right ) \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^2 \left (1+a^2\right )}\\ &=\frac{b (a+b x) \sqrt{1+\frac{1}{(a+b x)^2}}}{2 a \left (1+a^2\right ) x}+\frac{b^2 \text{csch}^{-1}(a+b x)}{2 a^2}-\frac{\text{csch}^{-1}(a+b x)}{2 x^2}-\frac{\left (1+2 a^2\right ) b^2 \tanh ^{-1}\left (\frac{a+\tanh \left (\frac{1}{2} \text{csch}^{-1}(a+b x)\right )}{\sqrt{1+a^2}}\right )}{a^2 \left (1+a^2\right )^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.456366, size = 220, normalized size = 1.93 \[ \frac{1}{2} \left (\frac{b (a+b x) \sqrt{\frac{a^2+2 a b x+b^2 x^2+1}{(a+b x)^2}}}{a \left (a^2+1\right ) x}-\frac{\left (2 a^2+1\right ) b^2 \log \left (\sqrt{a^2+1} a \sqrt{\frac{a^2+2 a b x+b^2 x^2+1}{(a+b x)^2}}+\sqrt{a^2+1} b x \sqrt{\frac{a^2+2 a b x+b^2 x^2+1}{(a+b x)^2}}+a^2+a b x+1\right )}{a^2 \left (a^2+1\right )^{3/2}}+\frac{\left (2 a^2+1\right ) b^2 \log (x)}{a^2 \left (a^2+1\right )^{3/2}}+\frac{b^2 \sinh ^{-1}\left (\frac{1}{a+b x}\right )}{a^2}-\frac{\text{csch}^{-1}(a+b x)}{x^2}\right ) \]
Antiderivative was successfully verified.
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Maple [B] time = 0.231, size = 453, normalized size = 4. \begin{align*} -{\frac{{\rm arccsch} \left (bx+a\right )}{2\,{x}^{2}}}+{\frac{{b}^{2}}{ \left ( 2\,bx+2\,a \right ) \left ({a}^{2}+1 \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{{a}^{2}{b}^{2}}{bx+a}\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}}}}}} \left ({a}^{2}+1 \right ) ^{-{\frac{5}{2}}}}+{\frac{{b}^{2}}{ \left ( 2\,bx+2\,a \right ){a}^{2} \left ({a}^{2}+1 \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 \left ( 1+ \left ( bx+a \right ) ^{2} \right ) }{ \left ( 2\,bx+2\,a \right ) a \left ({a}^{2}+1 \right ) x}{\frac{1}{\sqrt{{\frac{1+ \left ( bx+a \right ) ^{2}}{ \left ( bx+a \right ) ^{2}}}}}}}-{\frac{3\,{b}^{2}}{2\,bx+2\,a}\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}}}}}} \left ({a}^{2}+1 \right ) ^{-{\frac{5}{2}}}}-{\frac{{b}^{2}}{ \left ( 2\,bx+2\,a \right ){a}^{2}}\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}}}}}} \left ({a}^{2}+1 \right ) ^{-{\frac{5}{2}}}} \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 b^{2}{\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^{4} + 2 \, a^{2} + 1\right )}} + \frac{{\left (3 \, a^{2} b^{2} + b^{2}\right )} \log \left (x\right )}{2 \,{\left (a^{6} + 2 \, a^{4} + a^{2}\right )}} + \frac{{\left (a^{4} b^{2} - a^{2} b^{2}\right )} x^{2} \log \left (b^{2} x^{2} + 2 \, a b x + a^{2} + 1\right ) + 2 \,{\left (a^{3} b + a b\right )} x + 2 \,{\left (a^{6} + 2 \, a^{4} -{\left (a^{4} b^{2} + 2 \, a^{2} b^{2} + b^{2}\right )} x^{2} + a^{2}\right )} \log \left (b x + a\right ) - 2 \,{\left (a^{6} + 2 \, a^{4} + a^{2}\right )} \log \left (\sqrt{b^{2} x^{2} + 2 \, a b x + a^{2} + 1} + 1\right )}{4 \,{\left (a^{6} + 2 \, a^{4} + a^{2}\right )} x^{2}} - \int \frac{b^{2} x + a b}{2 \,{\left (b^{2} x^{4} + 2 \, a b x^{3} +{\left (a^{2} + 1\right )} x^{2} +{\left (b^{2} x^{4} + 2 \, a b x^{3} +{\left (a^{2} + 1\right )} x^{2}\right )} \sqrt{b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.9718, size = 1045, normalized size = 9.17 \begin{align*} \frac{{\left (2 \, a^{2} + 1\right )} \sqrt{a^{2} + 1} b^{2} x^{2} \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^{4} + 2 \, a^{2} + 1\right )} b^{2} x^{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 ) -{\left (a^{4} + 2 \, a^{2} + 1\right )} b^{2} x^{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 ) +{\left (a^{3} + a\right )} b^{2} x^{2} -{\left (a^{6} + 2 \, a^{4} + a^{2}\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 ({\left (a^{3} + a\right )} b^{2} x^{2} +{\left (a^{4} + a^{2}\right )} b x\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 \,{\left (a^{6} + 2 \, a^{4} + a^{2}\right )} x^{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 \frac{\operatorname{acsch}{\left (a + b x \right )}}{x^{3}}\, 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^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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