Optimal. Leaf size=49 \[ -\frac{3 \text{sech}(a+b x)}{2 b}+\frac{3 \tanh ^{-1}(\cosh (a+b x))}{2 b}-\frac{\text{csch}^2(a+b x) \text{sech}(a+b x)}{2 b} \]
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Rubi [A] time = 0.0456363, antiderivative size = 49, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.235, Rules used = {2622, 288, 321, 207} \[ -\frac{3 \text{sech}(a+b x)}{2 b}+\frac{3 \tanh ^{-1}(\cosh (a+b x))}{2 b}-\frac{\text{csch}^2(a+b x) \text{sech}(a+b x)}{2 b} \]
Antiderivative was successfully verified.
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Rule 2622
Rule 288
Rule 321
Rule 207
Rubi steps
\begin{align*} \int \text{csch}^3(a+b x) \text{sech}^2(a+b x) \, dx &=-\frac{\operatorname{Subst}\left (\int \frac{x^4}{\left (-1+x^2\right )^2} \, dx,x,\text{sech}(a+b x)\right )}{b}\\ &=-\frac{\text{csch}^2(a+b x) \text{sech}(a+b x)}{2 b}-\frac{3 \operatorname{Subst}\left (\int \frac{x^2}{-1+x^2} \, dx,x,\text{sech}(a+b x)\right )}{2 b}\\ &=-\frac{3 \text{sech}(a+b x)}{2 b}-\frac{\text{csch}^2(a+b x) \text{sech}(a+b x)}{2 b}-\frac{3 \operatorname{Subst}\left (\int \frac{1}{-1+x^2} \, dx,x,\text{sech}(a+b x)\right )}{2 b}\\ &=\frac{3 \tanh ^{-1}(\cosh (a+b x))}{2 b}-\frac{3 \text{sech}(a+b x)}{2 b}-\frac{\text{csch}^2(a+b x) \text{sech}(a+b x)}{2 b}\\ \end{align*}
Mathematica [A] time = 0.0333699, size = 68, normalized size = 1.39 \[ -\frac{\text{csch}^2\left (\frac{1}{2} (a+b x)\right )}{8 b}-\frac{\text{sech}^2\left (\frac{1}{2} (a+b x)\right )}{8 b}-\frac{\text{sech}(a+b x)}{b}-\frac{3 \log \left (\tanh \left (\frac{1}{2} (a+b x)\right )\right )}{2 b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.02, size = 43, normalized size = 0.9 \begin{align*}{\frac{1}{b} \left ( -{\frac{1}{2\,\cosh \left ( bx+a \right ) \left ( \sinh \left ( bx+a \right ) \right ) ^{2}}}-{\frac{3}{2\,\cosh \left ( bx+a \right ) }}+3\,{\it Artanh} \left ({{\rm e}^{bx+a}} \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.08435, size = 143, normalized size = 2.92 \begin{align*} \frac{3 \, \log \left (e^{\left (-b x - a\right )} + 1\right )}{2 \, b} - \frac{3 \, \log \left (e^{\left (-b x - a\right )} - 1\right )}{2 \, b} + \frac{3 \, e^{\left (-b x - a\right )} - 2 \, e^{\left (-3 \, b x - 3 \, a\right )} + 3 \, e^{\left (-5 \, b x - 5 \, a\right )}}{b{\left (e^{\left (-2 \, b x - 2 \, a\right )} + e^{\left (-4 \, b x - 4 \, a\right )} - e^{\left (-6 \, b x - 6 \, a\right )} - 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.46611, size = 1935, normalized size = 39.49 \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}^{2}{\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 = 1.17436, size = 154, normalized size = 3.14 \begin{align*} \frac{3 \, \log \left (e^{\left (b x + a\right )} + e^{\left (-b x - a\right )} + 2\right )}{4 \, b} - \frac{3 \, \log \left (e^{\left (b x + a\right )} + e^{\left (-b x - a\right )} - 2\right )}{4 \, b} - \frac{3 \,{\left (e^{\left (b x + a\right )} + e^{\left (-b x - a\right )}\right )}^{2} - 8}{{\left ({\left (e^{\left (b x + a\right )} + e^{\left (-b x - a\right )}\right )}^{3} - 4 \, e^{\left (b x + a\right )} - 4 \, e^{\left (-b x - a\right )}\right )} b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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