Optimal. Leaf size=23 \[ -\frac{\tanh (a+b x)}{b}-\frac{\coth (a+b x)}{b} \]
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Rubi [A] time = 0.033186, antiderivative size = 23, 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, 14} \[ -\frac{\tanh (a+b x)}{b}-\frac{\coth (a+b x)}{b} \]
Antiderivative was successfully verified.
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Rule 2620
Rule 14
Rubi steps
\begin{align*} \int \text{csch}^2(a+b x) \text{sech}^2(a+b x) \, dx &=\frac{i \operatorname{Subst}\left (\int \frac{1+x^2}{x^2} \, dx,x,i \tanh (a+b x)\right )}{b}\\ &=\frac{i \operatorname{Subst}\left (\int \left (1+\frac{1}{x^2}\right ) \, dx,x,i \tanh (a+b x)\right )}{b}\\ &=-\frac{\coth (a+b x)}{b}-\frac{\tanh (a+b x)}{b}\\ \end{align*}
Mathematica [A] time = 0.0097339, size = 13, normalized size = 0.57 \[ -\frac{2 \coth (2 (a+b x))}{b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0., size = 32, normalized size = 1.4 \begin{align*}{\frac{1}{b} \left ( -{\frac{1}{\cosh \left ( bx+a \right ) \sinh \left ( bx+a \right ) }}-2\,\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 = 1.07723, size = 24, normalized size = 1.04 \begin{align*} \frac{4}{b{\left (e^{\left (-4 \, b x - 4 \, a\right )} - 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.25071, size = 213, normalized size = 9.26 \begin{align*} -\frac{4}{b \cosh \left (b x + a\right )^{4} + 4 \, b \cosh \left (b x + a\right )^{3} \sinh \left (b x + a\right ) + 6 \, b \cosh \left (b x + a\right )^{2} \sinh \left (b x + a\right )^{2} + 4 \, b \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{3} + b \sinh \left (b x + a\right )^{4} - b} \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}^{2}{\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 = 1.17213, size = 24, normalized size = 1.04 \begin{align*} -\frac{4}{b{\left (e^{\left (4 \, b x + 4 \, a\right )} - 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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