Optimal. Leaf size=38 \[ \frac{\tanh ^3(a+b x)}{3 b}-\frac{2 \tanh (a+b x)}{b}-\frac{\coth (a+b x)}{b} \]
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Rubi [A] time = 0.0352183, antiderivative size = 38, 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} \[ \frac{\tanh ^3(a+b x)}{3 b}-\frac{2 \tanh (a+b x)}{b}-\frac{\coth (a+b x)}{b} \]
Antiderivative was successfully verified.
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Rule 2620
Rule 270
Rubi steps
\begin{align*} \int \text{csch}^2(a+b x) \text{sech}^4(a+b x) \, dx &=\frac{i \operatorname{Subst}\left (\int \frac{\left (1+x^2\right )^2}{x^2} \, dx,x,i \tanh (a+b x)\right )}{b}\\ &=\frac{i \operatorname{Subst}\left (\int \left (2+\frac{1}{x^2}+x^2\right ) \, dx,x,i \tanh (a+b x)\right )}{b}\\ &=-\frac{\coth (a+b x)}{b}-\frac{2 \tanh (a+b x)}{b}+\frac{\tanh ^3(a+b x)}{3 b}\\ \end{align*}
Mathematica [A] time = 0.0383215, size = 46, normalized size = 1.21 \[ -\frac{5 \tanh (a+b x)}{3 b}-\frac{\coth (a+b x)}{b}-\frac{\tanh (a+b x) \text{sech}^2(a+b x)}{3 b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.019, size = 44, normalized size = 1.2 \begin{align*}{\frac{1}{b} \left ( -{\frac{1}{ \left ( \cosh \left ( bx+a \right ) \right ) ^{3}\sinh \left ( bx+a \right ) }}-4\, \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 [B] time = 1.1555, size = 127, normalized size = 3.34 \begin{align*} -\frac{32 \, e^{\left (-2 \, b x - 2 \, a\right )}}{3 \, b{\left (2 \, e^{\left (-2 \, b x - 2 \, a\right )} - 2 \, e^{\left (-6 \, b x - 6 \, a\right )} - e^{\left (-8 \, b x - 8 \, a\right )} + 1\right )}} - \frac{16}{3 \, b{\left (2 \, e^{\left (-2 \, b x - 2 \, a\right )} - 2 \, 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 = 2.06823, size = 622, normalized size = 16.37 \begin{align*} -\frac{16 \,{\left (3 \, \cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right )}}{3 \,{\left (b \cosh \left (b x + a\right )^{7} + 7 \, b \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{6} + b \sinh \left (b x + a\right )^{7} + 2 \, b \cosh \left (b x + a\right )^{5} +{\left (21 \, b \cosh \left (b x + a\right )^{2} + 2 \, b\right )} \sinh \left (b x + a\right )^{5} + 5 \,{\left (7 \, b \cosh \left (b x + a\right )^{3} + 2 \, b \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )^{4} + 5 \,{\left (7 \, b \cosh \left (b x + a\right )^{4} + 4 \, b \cosh \left (b x + a\right )^{2}\right )} \sinh \left (b x + a\right )^{3} +{\left (21 \, b \cosh \left (b x + a\right )^{5} + 20 \, b \cosh \left (b x + a\right )^{3}\right )} \sinh \left (b x + a\right )^{2} - 3 \, b \cosh \left (b x + a\right ) +{\left (7 \, b \cosh \left (b x + a\right )^{6} + 10 \, b \cosh \left (b x + a\right )^{4} - b\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}^{2}{\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 = 1.2102, size = 82, normalized size = 2.16 \begin{align*} -\frac{2}{b{\left (e^{\left (2 \, b x + 2 \, a\right )} - 1\right )}} + \frac{2 \,{\left (3 \, e^{\left (4 \, b x + 4 \, a\right )} + 12 \, e^{\left (2 \, b x + 2 \, a\right )} + 5\right )}}{3 \, b{\left (e^{\left (2 \, b x + 2 \, a\right )} + 1\right )}^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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