Optimal. Leaf size=43 \[ \frac{\tanh ^2(a+b x)}{2 b}-\frac{\coth ^2(a+b x)}{2 b}-\frac{2 \log (\tanh (a+b x))}{b} \]
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Rubi [A] time = 0.0443224, antiderivative size = 43, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176, Rules used = {2620, 266, 43} \[ \frac{\tanh ^2(a+b x)}{2 b}-\frac{\coth ^2(a+b x)}{2 b}-\frac{2 \log (\tanh (a+b x))}{b} \]
Antiderivative was successfully verified.
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Rule 2620
Rule 266
Rule 43
Rubi steps
\begin{align*} \int \text{csch}^3(a+b x) \text{sech}^3(a+b x) \, dx &=-\frac{\operatorname{Subst}\left (\int \frac{\left (1+x^2\right )^2}{x^3} \, dx,x,i \tanh (a+b x)\right )}{b}\\ &=-\frac{\operatorname{Subst}\left (\int \frac{(1+x)^2}{x^2} \, dx,x,-\tanh ^2(a+b x)\right )}{2 b}\\ &=-\frac{\operatorname{Subst}\left (\int \left (1+\frac{1}{x^2}+\frac{2}{x}\right ) \, dx,x,-\tanh ^2(a+b x)\right )}{2 b}\\ &=-\frac{\coth ^2(a+b x)}{2 b}-\frac{2 \log (\tanh (a+b x))}{b}+\frac{\tanh ^2(a+b x)}{2 b}\\ \end{align*}
Mathematica [A] time = 0.011343, size = 47, normalized size = 1.09 \[ 8 \left (-\frac{\text{csch}^2(a+b x)}{16 b}-\frac{\text{sech}^2(a+b x)}{16 b}-\frac{\log (\tanh (a+b x))}{4 b}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.023, size = 48, normalized size = 1.1 \begin{align*} -{\frac{1}{2\,b \left ( \sinh \left ( bx+a \right ) \right ) ^{2} \left ( \cosh \left ( bx+a \right ) \right ) ^{2}}}-{\frac{1}{b \left ( \cosh \left ( bx+a \right ) \right ) ^{2}}}-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 = 1.56957, size = 138, normalized size = 3.21 \begin{align*} -\frac{2 \, \log \left (e^{\left (-b x - a\right )} + 1\right )}{b} - \frac{2 \, \log \left (e^{\left (-b x - a\right )} - 1\right )}{b} + \frac{2 \, \log \left (e^{\left (-2 \, b x - 2 \, a\right )} + 1\right )}{b} + \frac{4 \,{\left (e^{\left (-2 \, b x - 2 \, a\right )} + e^{\left (-6 \, b x - 6 \, a\right )}\right )}}{b{\left (2 \, e^{\left (-4 \, b x - 4 \, 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 = 1.84967, size = 2102, normalized size = 48.88 \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}^{3}{\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 = 1.13687, size = 136, normalized size = 3.16 \begin{align*} \frac{\log \left (e^{\left (2 \, b x + 2 \, a\right )} + e^{\left (-2 \, b x - 2 \, a\right )} + 2\right )}{b} - \frac{\log \left (e^{\left (2 \, b x + 2 \, a\right )} + e^{\left (-2 \, b x - 2 \, a\right )} - 2\right )}{b} - \frac{4 \,{\left (e^{\left (2 \, b x + 2 \, a\right )} + e^{\left (-2 \, b x - 2 \, a\right )}\right )}}{{\left ({\left (e^{\left (2 \, b x + 2 \, a\right )} + e^{\left (-2 \, b x - 2 \, a\right )}\right )}^{2} - 4\right )} b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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