Optimal. Leaf size=37 \[ -\frac{\text{csch}^n(a+b x)}{b n}-\frac{\text{csch}^{n+2}(a+b x)}{b (n+2)} \]
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Rubi [A] time = 0.0421384, antiderivative size = 37, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {2621, 14} \[ -\frac{\text{csch}^n(a+b x)}{b n}-\frac{\text{csch}^{n+2}(a+b x)}{b (n+2)} \]
Antiderivative was successfully verified.
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Rule 2621
Rule 14
Rubi steps
\begin{align*} \int \cosh ^3(a+b x) \text{csch}^{3+n}(a+b x) \, dx &=\frac{\operatorname{Subst}\left (\int x^{-1+n} \left (-1-x^2\right ) \, dx,x,\text{csch}(a+b x)\right )}{b}\\ &=\frac{\operatorname{Subst}\left (\int \left (-x^{-1+n}-x^{1+n}\right ) \, dx,x,\text{csch}(a+b x)\right )}{b}\\ &=-\frac{\text{csch}^n(a+b x)}{b n}-\frac{\text{csch}^{2+n}(a+b x)}{b (2+n)}\\ \end{align*}
Mathematica [A] time = 0.0657785, size = 34, normalized size = 0.92 \[ -\frac{\text{csch}^n(a+b x) \left (n \text{csch}^2(a+b x)+n+2\right )}{b n (n+2)} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.152, size = 499, normalized size = 13.5 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.91423, size = 559, normalized size = 15.11 \begin{align*} -\frac{2^{n} n e^{\left (-{\left (b x + a\right )} n - n \log \left (e^{\left (-b x - a\right )} + 1\right ) - n \log \left (-e^{\left (-b x - a\right )} + 1\right )\right )}}{{\left (n^{2} - 2 \,{\left (n^{2} + 2 \, n\right )} e^{\left (-2 \, b x - 2 \, a\right )} +{\left (n^{2} + 2 \, n\right )} e^{\left (-4 \, b x - 4 \, a\right )} + 2 \, n\right )} b} - \frac{{\left (2^{n + 1} n - 2^{n + 2}\right )} e^{\left (-{\left (b x + a\right )} n - 2 \, b x - n \log \left (e^{\left (-b x - a\right )} + 1\right ) - n \log \left (-e^{\left (-b x - a\right )} + 1\right ) - 2 \, a\right )}}{{\left (n^{2} - 2 \,{\left (n^{2} + 2 \, n\right )} e^{\left (-2 \, b x - 2 \, a\right )} +{\left (n^{2} + 2 \, n\right )} e^{\left (-4 \, b x - 4 \, a\right )} + 2 \, n\right )} b} - \frac{{\left (2^{n} n + 2^{n + 1}\right )} e^{\left (-{\left (b x + a\right )} n - 4 \, b x - n \log \left (e^{\left (-b x - a\right )} + 1\right ) - n \log \left (-e^{\left (-b x - a\right )} + 1\right ) - 4 \, a\right )}}{{\left (n^{2} - 2 \,{\left (n^{2} + 2 \, n\right )} e^{\left (-2 \, b x - 2 \, a\right )} +{\left (n^{2} + 2 \, n\right )} e^{\left (-4 \, b x - 4 \, a\right )} + 2 \, n\right )} b} - \frac{2^{n + 1} e^{\left (-{\left (b x + a\right )} n - n \log \left (e^{\left (-b x - a\right )} + 1\right ) - n \log \left (-e^{\left (-b x - a\right )} + 1\right )\right )}}{{\left (n^{2} - 2 \,{\left (n^{2} + 2 \, n\right )} e^{\left (-2 \, b x - 2 \, a\right )} +{\left (n^{2} + 2 \, n\right )} e^{\left (-4 \, b x - 4 \, a\right )} + 2 \, n\right )} b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.98283, size = 603, normalized size = 16.3 \begin{align*} \frac{{\left ({\left (n + 2\right )} \cosh \left (b x + a\right )^{2} +{\left (n + 2\right )} \sinh \left (b x + a\right )^{2} + n - 2\right )} \cosh \left (n \log \left (\frac{2 \,{\left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right )}}{\cosh \left (b x + a\right )^{2} + 2 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + \sinh \left (b x + a\right )^{2} - 1}\right )\right ) +{\left ({\left (n + 2\right )} \cosh \left (b x + a\right )^{2} +{\left (n + 2\right )} \sinh \left (b x + a\right )^{2} + n - 2\right )} \sinh \left (n \log \left (\frac{2 \,{\left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right )}}{\cosh \left (b x + a\right )^{2} + 2 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + \sinh \left (b x + a\right )^{2} - 1}\right )\right )}{b n^{2} -{\left (b n^{2} + 2 \, b n\right )} \cosh \left (b x + a\right )^{2} -{\left (b n^{2} + 2 \, b n\right )} \sinh \left (b x + a\right )^{2} + 2 \, b n} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \operatorname{csch}\left (b x + a\right )^{n} \coth \left (b x + a\right )^{3}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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