Optimal. Leaf size=39 \[ -\frac{\coth ^4(a+b x)}{4 b}+\frac{\coth ^2(a+b x)}{b}+\frac{\log (\tanh (a+b x))}{b} \]
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Rubi [A] time = 0.0320298, antiderivative size = 39, 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 = {2620, 266, 43} \[ -\frac{\coth ^4(a+b x)}{4 b}+\frac{\coth ^2(a+b x)}{b}+\frac{\log (\tanh (a+b x))}{b} \]
Antiderivative was successfully verified.
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Rule 2620
Rule 266
Rule 43
Rubi steps
\begin{align*} \int \text{csch}^5(a+b x) \text{sech}(a+b x) \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\left (1+x^2\right )^2}{x^5} \, dx,x,i \tanh (a+b x)\right )}{b}\\ &=\frac{\operatorname{Subst}\left (\int \frac{(1+x)^2}{x^3} \, dx,x,-\tanh ^2(a+b x)\right )}{2 b}\\ &=\frac{\operatorname{Subst}\left (\int \left (\frac{1}{x^3}+\frac{2}{x^2}+\frac{1}{x}\right ) \, dx,x,-\tanh ^2(a+b x)\right )}{2 b}\\ &=\frac{\coth ^2(a+b x)}{b}-\frac{\coth ^4(a+b x)}{4 b}+\frac{\log (\tanh (a+b x))}{b}\\ \end{align*}
Mathematica [A] time = 0.123431, size = 46, normalized size = 1.18 \[ \frac{-\text{csch}^4(a+b x)+2 \text{csch}^2(a+b x)+4 \log (\sinh (a+b x))-4 \log (\cosh (a+b x))}{4 b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.019, size = 39, normalized size = 1. \begin{align*} -{\frac{1}{4\,b \left ( \sinh \left ( bx+a \right ) \right ) ^{4}}}+{\frac{1}{2\,b \left ( \sinh \left ( bx+a \right ) \right ) ^{2}}}+{\frac{\ln \left ( \tanh \left ( bx+a \right ) \right ) }{b}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.55373, size = 180, normalized size = 4.62 \begin{align*} \frac{\log \left (e^{\left (-b x - a\right )} + 1\right )}{b} + \frac{\log \left (e^{\left (-b x - a\right )} - 1\right )}{b} - \frac{\log \left (e^{\left (-2 \, b x - 2 \, a\right )} + 1\right )}{b} - \frac{2 \,{\left (e^{\left (-2 \, b x - 2 \, a\right )} - 4 \, e^{\left (-4 \, b x - 4 \, a\right )} + e^{\left (-6 \, b x - 6 \, a\right )}\right )}}{b{\left (4 \, e^{\left (-2 \, b x - 2 \, a\right )} - 6 \, e^{\left (-4 \, b x - 4 \, a\right )} + 4 \, e^{\left (-6 \, b x - 6 \, a\right )} - e^{\left (-8 \, b x - 8 \, a\right )} - 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.41554, size = 2952, normalized size = 75.69 \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}^{5}{\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 = 1.20856, size = 171, normalized size = 4.38 \begin{align*} -\frac{\log \left (e^{\left (2 \, b x + 2 \, a\right )} + e^{\left (-2 \, b x - 2 \, a\right )} + 2\right )}{2 \, b} + \frac{\log \left (e^{\left (2 \, b x + 2 \, a\right )} + e^{\left (-2 \, b x - 2 \, a\right )} - 2\right )}{2 \, b} - \frac{3 \,{\left (e^{\left (2 \, b x + 2 \, a\right )} + e^{\left (-2 \, b x - 2 \, a\right )}\right )}^{2} - 20 \, e^{\left (2 \, b x + 2 \, a\right )} - 20 \, e^{\left (-2 \, b x - 2 \, a\right )} + 44}{4 \, b{\left (e^{\left (2 \, b x + 2 \, a\right )} + e^{\left (-2 \, b x - 2 \, a\right )} - 2\right )}^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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