Optimal. Leaf size=49 \[ -\frac{3 \text{csch}(a+b x)}{2 b}-\frac{3 \tan ^{-1}(\sinh (a+b x))}{2 b}+\frac{\text{csch}(a+b x) \text{sech}^2(a+b x)}{2 b} \]
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Rubi [A] time = 0.0406723, antiderivative size = 49, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.235, Rules used = {2621, 288, 321, 207} \[ -\frac{3 \text{csch}(a+b x)}{2 b}-\frac{3 \tan ^{-1}(\sinh (a+b x))}{2 b}+\frac{\text{csch}(a+b x) \text{sech}^2(a+b x)}{2 b} \]
Antiderivative was successfully verified.
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Rule 2621
Rule 288
Rule 321
Rule 207
Rubi steps
\begin{align*} \int \text{csch}^2(a+b x) \text{sech}^3(a+b x) \, dx &=-\frac{i \operatorname{Subst}\left (\int \frac{x^4}{\left (-1+x^2\right )^2} \, dx,x,-i \text{csch}(a+b x)\right )}{b}\\ &=\frac{\text{csch}(a+b x) \text{sech}^2(a+b x)}{2 b}-\frac{(3 i) \operatorname{Subst}\left (\int \frac{x^2}{-1+x^2} \, dx,x,-i \text{csch}(a+b x)\right )}{2 b}\\ &=-\frac{3 \text{csch}(a+b x)}{2 b}+\frac{\text{csch}(a+b x) \text{sech}^2(a+b x)}{2 b}-\frac{(3 i) \operatorname{Subst}\left (\int \frac{1}{-1+x^2} \, dx,x,-i \text{csch}(a+b x)\right )}{2 b}\\ &=-\frac{3 \tan ^{-1}(\sinh (a+b x))}{2 b}-\frac{3 \text{csch}(a+b x)}{2 b}+\frac{\text{csch}(a+b x) \text{sech}^2(a+b x)}{2 b}\\ \end{align*}
Mathematica [C] time = 0.0151644, size = 29, normalized size = 0.59 \[ -\frac{\text{csch}(a+b x) \, _2F_1\left (-\frac{1}{2},2;\frac{1}{2};-\sinh ^2(a+b x)\right )}{b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.019, size = 52, normalized size = 1.1 \begin{align*} -{\frac{1}{b\sinh \left ( bx+a \right ) \left ( \cosh \left ( bx+a \right ) \right ) ^{2}}}-{\frac{3\,{\rm sech} \left (bx+a\right )\tanh \left ( bx+a \right ) }{2\,b}}-3\,{\frac{\arctan \left ({{\rm e}^{bx+a}} \right ) }{b}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.52873, size = 122, normalized size = 2.49 \begin{align*} \frac{3 \, \arctan \left (e^{\left (-b x - a\right )}\right )}{b} - \frac{3 \, e^{\left (-b x - a\right )} + 2 \, e^{\left (-3 \, b x - 3 \, a\right )} + 3 \, e^{\left (-5 \, b x - 5 \, a\right )}}{b{\left (e^{\left (-2 \, b x - 2 \, a\right )} - e^{\left (-4 \, b x - 4 \, a\right )} - e^{\left (-6 \, b x - 6 \, a\right )} + 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.08012, size = 1401, normalized size = 28.59 \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 \operatorname{csch}^{2}{\left (a + b x \right )} \operatorname{sech}^{3}{\left (a + b x \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.22911, size = 140, normalized size = 2.86 \begin{align*} -\frac{3 \,{\left (\pi + 2 \, \arctan \left (\frac{1}{2} \,{\left (e^{\left (2 \, b x + 2 \, a\right )} - 1\right )} e^{\left (-b x - a\right )}\right )\right )}}{4 \, b} - \frac{3 \,{\left (e^{\left (b x + a\right )} - e^{\left (-b x - a\right )}\right )}^{2} + 8}{{\left ({\left (e^{\left (b x + a\right )} - e^{\left (-b x - a\right )}\right )}^{3} + 4 \, e^{\left (b x + a\right )} - 4 \, e^{\left (-b x - a\right )}\right )} b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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