Optimal. Leaf size=37 \[ -\frac{\text{csch}^3(a+b x)}{3 b}+\frac{\text{csch}(a+b x)}{b}+\frac{\tan ^{-1}(\sinh (a+b x))}{b} \]
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Rubi [A] time = 0.0270247, antiderivative size = 37, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {2621, 302, 207} \[ -\frac{\text{csch}^3(a+b x)}{3 b}+\frac{\text{csch}(a+b x)}{b}+\frac{\tan ^{-1}(\sinh (a+b x))}{b} \]
Antiderivative was successfully verified.
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Rule 2621
Rule 302
Rule 207
Rubi steps
\begin{align*} \int \text{csch}^4(a+b x) \text{sech}(a+b x) \, dx &=\frac{i \operatorname{Subst}\left (\int \frac{x^4}{-1+x^2} \, dx,x,-i \text{csch}(a+b x)\right )}{b}\\ &=\frac{i \operatorname{Subst}\left (\int \left (1+x^2+\frac{1}{-1+x^2}\right ) \, dx,x,-i \text{csch}(a+b x)\right )}{b}\\ &=\frac{\text{csch}(a+b x)}{b}-\frac{\text{csch}^3(a+b x)}{3 b}+\frac{i \operatorname{Subst}\left (\int \frac{1}{-1+x^2} \, dx,x,-i \text{csch}(a+b x)\right )}{b}\\ &=\frac{\tan ^{-1}(\sinh (a+b x))}{b}+\frac{\text{csch}(a+b x)}{b}-\frac{\text{csch}^3(a+b x)}{3 b}\\ \end{align*}
Mathematica [C] time = 0.0169181, size = 33, normalized size = 0.89 \[ -\frac{\text{csch}^3(a+b x) \, _2F_1\left (-\frac{3}{2},1;-\frac{1}{2};-\sinh ^2(a+b x)\right )}{3 b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.017, size = 39, normalized size = 1.1 \begin{align*} -{\frac{1}{3\,b \left ( \sinh \left ( bx+a \right ) \right ) ^{3}}}+{\frac{1}{b\sinh \left ( bx+a \right ) }}+2\,{\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 = 1.78521, size = 122, normalized size = 3.3 \begin{align*} -\frac{2 \, \arctan \left (e^{\left (-b x - a\right )}\right )}{b} - \frac{2 \,{\left (3 \, e^{\left (-b x - a\right )} - 10 \, e^{\left (-3 \, b x - 3 \, a\right )} + 3 \, e^{\left (-5 \, b x - 5 \, a\right )}\right )}}{3 \, b{\left (3 \, e^{\left (-2 \, b x - 2 \, a\right )} - 3 \, 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 = 2.33626, size = 1423, normalized size = 38.46 \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}^{4}{\left (a + b x \right )} \operatorname{sech}{\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 = 1.25549, size = 111, normalized size = 3. \begin{align*} \frac{\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 )}{2 \, b} + \frac{2 \,{\left (3 \,{\left (e^{\left (b x + a\right )} - e^{\left (-b x - a\right )}\right )}^{2} - 4\right )}}{3 \, b{\left (e^{\left (b x + a\right )} - e^{\left (-b x - a\right )}\right )}^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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