Optimal. Leaf size=53 \[ -\frac{\tanh ^3(a+b x)}{3 b}+\frac{3 \tanh (a+b x)}{b}-\frac{\coth ^3(a+b x)}{3 b}+\frac{3 \coth (a+b x)}{b} \]
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Rubi [A] time = 0.0395316, antiderivative size = 53, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {2620, 270} \[ -\frac{\tanh ^3(a+b x)}{3 b}+\frac{3 \tanh (a+b x)}{b}-\frac{\coth ^3(a+b x)}{3 b}+\frac{3 \coth (a+b x)}{b} \]
Antiderivative was successfully verified.
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Rule 2620
Rule 270
Rubi steps
\begin{align*} \int \text{csch}^4(a+b x) \text{sech}^4(a+b x) \, dx &=-\frac{i \operatorname{Subst}\left (\int \frac{\left (1+x^2\right )^3}{x^4} \, dx,x,i \tanh (a+b x)\right )}{b}\\ &=-\frac{i \operatorname{Subst}\left (\int \left (3+\frac{1}{x^4}+\frac{3}{x^2}+x^2\right ) \, dx,x,i \tanh (a+b x)\right )}{b}\\ &=\frac{3 \coth (a+b x)}{b}-\frac{\coth ^3(a+b x)}{3 b}+\frac{3 \tanh (a+b x)}{b}-\frac{\tanh ^3(a+b x)}{3 b}\\ \end{align*}
Mathematica [A] time = 0.0192102, size = 43, normalized size = 0.81 \[ 16 \left (\frac{\coth (2 (a+b x))}{3 b}-\frac{\coth (2 (a+b x)) \text{csch}^2(2 (a+b x))}{6 b}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.025, size = 62, normalized size = 1.2 \begin{align*}{\frac{1}{b} \left ( -{\frac{1}{3\, \left ( \cosh \left ( bx+a \right ) \right ) ^{3} \left ( \sinh \left ( bx+a \right ) \right ) ^{3}}}+2\,{\frac{1}{ \left ( \cosh \left ( bx+a \right ) \right ) ^{3}\sinh \left ( bx+a \right ) }}+8\, \left ( 2/3+1/3\, \left ({\rm sech} \left (bx+a\right ) \right ) ^{2} \right ) \tanh \left ( bx+a \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.17381, size = 122, normalized size = 2.3 \begin{align*} \frac{32 \, e^{\left (-4 \, b x - 4 \, a\right )}}{b{\left (3 \, e^{\left (-4 \, b x - 4 \, a\right )} - 3 \, e^{\left (-8 \, b x - 8 \, a\right )} + e^{\left (-12 \, b x - 12 \, a\right )} - 1\right )}} - \frac{32}{3 \, b{\left (3 \, e^{\left (-4 \, b x - 4 \, a\right )} - 3 \, e^{\left (-8 \, b x - 8 \, a\right )} + e^{\left (-12 \, b x - 12 \, a\right )} - 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.26766, size = 900, normalized size = 16.98 \begin{align*} -\frac{64 \,{\left (\cosh \left (b x + a\right )^{2} + 4 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + \sinh \left (b x + a\right )^{2}\right )}}{3 \,{\left (b \cosh \left (b x + a\right )^{10} + 120 \, b \cosh \left (b x + a\right )^{3} \sinh \left (b x + a\right )^{7} + 45 \, b \cosh \left (b x + a\right )^{2} \sinh \left (b x + a\right )^{8} + 10 \, b \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{9} + b \sinh \left (b x + a\right )^{10} - 3 \, b \cosh \left (b x + a\right )^{6} + 3 \,{\left (70 \, b \cosh \left (b x + a\right )^{4} - b\right )} \sinh \left (b x + a\right )^{6} + 18 \,{\left (14 \, b \cosh \left (b x + a\right )^{5} - b \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )^{5} + 15 \,{\left (14 \, b \cosh \left (b x + a\right )^{6} - 3 \, b \cosh \left (b x + a\right )^{2}\right )} \sinh \left (b x + a\right )^{4} + 60 \,{\left (2 \, b \cosh \left (b x + a\right )^{7} - b \cosh \left (b x + a\right )^{3}\right )} \sinh \left (b x + a\right )^{3} + 2 \, b \cosh \left (b x + a\right )^{2} +{\left (45 \, b \cosh \left (b x + a\right )^{8} - 45 \, b \cosh \left (b x + a\right )^{4} + 2 \, b\right )} \sinh \left (b x + a\right )^{2} + 2 \,{\left (5 \, b \cosh \left (b x + a\right )^{9} - 9 \, b \cosh \left (b x + a\right )^{5} + 4 \, b \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )\right )}} \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}^{4}{\left (a + b x \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.1614, size = 42, normalized size = 0.79 \begin{align*} -\frac{32 \,{\left (3 \, e^{\left (4 \, b x + 4 \, a\right )} - 1\right )}}{3 \, b{\left (e^{\left (4 \, b x + 4 \, a\right )} - 1\right )}^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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