Optimal. Leaf size=50 \[ \frac{a^2-b^2}{2 a^3 (a \cosh (x)+b)^2}+\frac{2 b}{a^3 (a \cosh (x)+b)}+\frac{\log (a \cosh (x)+b)}{a^3} \]
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Rubi [A] time = 0.107477, antiderivative size = 50, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273, Rules used = {4392, 2668, 697} \[ \frac{a^2-b^2}{2 a^3 (a \cosh (x)+b)^2}+\frac{2 b}{a^3 (a \cosh (x)+b)}+\frac{\log (a \cosh (x)+b)}{a^3} \]
Antiderivative was successfully verified.
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Rule 4392
Rule 2668
Rule 697
Rubi steps
\begin{align*} \int \frac{1}{(a \coth (x)+b \text{csch}(x))^3} \, dx &=-\left (i \int \frac{\sinh ^3(x)}{(i b+i a \cosh (x))^3} \, dx\right )\\ &=-\frac{\operatorname{Subst}\left (\int \frac{-a^2-x^2}{(i b+x)^3} \, dx,x,i a \cosh (x)\right )}{a^3}\\ &=-\frac{\operatorname{Subst}\left (\int \left (\frac{-a^2+b^2}{(i b+x)^3}+\frac{2 i b}{(i b+x)^2}-\frac{1}{i b+x}\right ) \, dx,x,i a \cosh (x)\right )}{a^3}\\ &=\frac{a^2-b^2}{2 a^3 (b+a \cosh (x))^2}+\frac{2 b}{a^3 (b+a \cosh (x))}+\frac{\log (b+a \cosh (x))}{a^3}\\ \end{align*}
Mathematica [A] time = 0.107567, size = 77, normalized size = 1.54 \[ \frac{a^2 \cosh (2 x) \log (a \cosh (x)+b)+a^2 \log (a \cosh (x)+b)+a^2+2 b^2 \log (a \cosh (x)+b)+4 a b \cosh (x) (\log (a \cosh (x)+b)+1)+3 b^2}{2 a^3 (a \cosh (x)+b)^2} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.055, size = 144, normalized size = 2.9 \begin{align*} -{\frac{1}{{a}^{3}}\ln \left ( \tanh \left ({\frac{x}{2}} \right ) +1 \right ) }-2\,{\frac{1}{{a}^{2} \left ( a \left ( \tanh \left ( x/2 \right ) \right ) ^{2}- \left ( \tanh \left ( x/2 \right ) \right ) ^{2}b+a+b \right ) }}+2\,{\frac{1}{ \left ( a-b \right ) \left ( a \left ( \tanh \left ( x/2 \right ) \right ) ^{2}- \left ( \tanh \left ( x/2 \right ) \right ) ^{2}b+a+b \right ) ^{2}}}+2\,{\frac{b}{a \left ( a-b \right ) \left ( a \left ( \tanh \left ( x/2 \right ) \right ) ^{2}- \left ( \tanh \left ( x/2 \right ) \right ) ^{2}b+a+b \right ) ^{2}}}+{\frac{1}{{a}^{3}}\ln \left ( a \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{2}- \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{2}b+a+b \right ) }-{\frac{1}{{a}^{3}}\ln \left ( \tanh \left ({\frac{x}{2}} \right ) -1 \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.06751, size = 150, normalized size = 3. \begin{align*} \frac{2 \,{\left (2 \, a b e^{\left (-x\right )} + 2 \, a b e^{\left (-3 \, x\right )} +{\left (a^{2} + 3 \, b^{2}\right )} e^{\left (-2 \, x\right )}\right )}}{4 \, a^{4} b e^{\left (-x\right )} + 4 \, a^{4} b e^{\left (-3 \, x\right )} + a^{5} e^{\left (-4 \, x\right )} + a^{5} + 2 \,{\left (a^{5} + 2 \, a^{3} b^{2}\right )} e^{\left (-2 \, x\right )}} + \frac{x}{a^{3}} + \frac{\log \left (2 \, b e^{\left (-x\right )} + a e^{\left (-2 \, x\right )} + a\right )}{a^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.1715, size = 1362, normalized size = 27.24 \begin{align*} \text{result too large to display} \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{1}{\left (a \coth{\left (x \right )} + b \operatorname{csch}{\left (x \right )}\right )^{3}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.16278, size = 89, normalized size = 1.78 \begin{align*} \frac{\log \left ({\left | a{\left (e^{\left (-x\right )} + e^{x}\right )} + 2 \, b \right |}\right )}{a^{3}} - \frac{3 \, a{\left (e^{\left (-x\right )} + e^{x}\right )}^{2} + 4 \, b{\left (e^{\left (-x\right )} + e^{x}\right )} - 4 \, a}{2 \,{\left (a{\left (e^{\left (-x\right )} + e^{x}\right )} + 2 \, b\right )}^{2} a^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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