Optimal. Leaf size=26 \[ \frac{\coth ^{n+1}(x)}{n+1}-\frac{\coth ^{n+3}(x)}{n+3} \]
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Rubi [A] time = 0.0351811, antiderivative size = 26, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {2607, 14} \[ \frac{\coth ^{n+1}(x)}{n+1}-\frac{\coth ^{n+3}(x)}{n+3} \]
Antiderivative was successfully verified.
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Rule 2607
Rule 14
Rubi steps
\begin{align*} \int \coth ^n(x) \text{csch}^4(x) \, dx &=-\left (i \operatorname{Subst}\left (\int (-i x)^n \left (1+x^2\right ) \, dx,x,i \coth (x)\right )\right )\\ &=-\left (i \operatorname{Subst}\left (\int \left ((-i x)^n-(-i x)^{2+n}\right ) \, dx,x,i \coth (x)\right )\right )\\ &=\frac{\coth ^{1+n}(x)}{1+n}-\frac{\coth ^{3+n}(x)}{3+n}\\ \end{align*}
Mathematica [A] time = 0.0860355, size = 30, normalized size = 1.15 \[ \frac{\text{csch}^2(x) (-n+\cosh (2 x)-2) \coth ^{n+1}(x)}{(n+1) (n+3)} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.208, size = 371, normalized size = 14.3 \begin{align*} -2\,{\frac{-{{\rm e}^{6\,x}}+2\,n{{\rm e}^{4\,x}}+3\,{{\rm e}^{4\,x}}+2\,n{{\rm e}^{2\,x}}+3\,{{\rm e}^{2\,x}}-1}{ \left ( n+1 \right ) \left ( n+3 \right ) \left ({{\rm e}^{2\,x}}-1 \right ) ^{3}}{{\rm e}^{-1/2\,n \left ( i\pi \, \left ({\it csgn} \left ({\frac{i \left ({{\rm e}^{2\,x}}+1 \right ) }{{{\rm e}^{x}}+1}} \right ) \right ) ^{3}-i\pi \, \left ({\it csgn} \left ({\frac{i \left ({{\rm e}^{2\,x}}+1 \right ) }{{{\rm e}^{x}}+1}} \right ) \right ) ^{2}{\it csgn} \left ({\frac{i}{{{\rm e}^{x}}+1}} \right ) -i\pi \, \left ({\it csgn} \left ({\frac{i \left ({{\rm e}^{2\,x}}+1 \right ) }{{{\rm e}^{x}}+1}} \right ) \right ) ^{2}{\it csgn} \left ( i \left ({{\rm e}^{2\,x}}+1 \right ) \right ) +i\pi \,{\it csgn} \left ({\frac{i \left ({{\rm e}^{2\,x}}+1 \right ) }{{{\rm e}^{x}}+1}} \right ){\it csgn} \left ({\frac{i}{{{\rm e}^{x}}+1}} \right ){\it csgn} \left ( i \left ({{\rm e}^{2\,x}}+1 \right ) \right ) -i\pi \,{\it csgn} \left ({\frac{i \left ({{\rm e}^{2\,x}}+1 \right ) }{{{\rm e}^{x}}+1}} \right ) \left ({\it csgn} \left ({\frac{i \left ({{\rm e}^{2\,x}}+1 \right ) }{ \left ({{\rm e}^{x}}-1 \right ) \left ({{\rm e}^{x}}+1 \right ) }} \right ) \right ) ^{2}+i\pi \,{\it csgn} \left ({\frac{i \left ({{\rm e}^{2\,x}}+1 \right ) }{{{\rm e}^{x}}+1}} \right ){\it csgn} \left ({\frac{i \left ({{\rm e}^{2\,x}}+1 \right ) }{ \left ({{\rm e}^{x}}-1 \right ) \left ({{\rm e}^{x}}+1 \right ) }} \right ){\it csgn} \left ({\frac{i}{{{\rm e}^{x}}-1}} \right ) +i\pi \, \left ({\it csgn} \left ({\frac{i \left ({{\rm e}^{2\,x}}+1 \right ) }{ \left ({{\rm e}^{x}}-1 \right ) \left ({{\rm e}^{x}}+1 \right ) }} \right ) \right ) ^{3}-i\pi \, \left ({\it csgn} \left ({\frac{i \left ({{\rm e}^{2\,x}}+1 \right ) }{ \left ({{\rm e}^{x}}-1 \right ) \left ({{\rm e}^{x}}+1 \right ) }} \right ) \right ) ^{2}{\it csgn} \left ({\frac{i}{{{\rm e}^{x}}-1}} \right ) +2\,\ln \left ({{\rm e}^{x}}-1 \right ) +2\,\ln \left ({{\rm e}^{x}}+1 \right ) -2\,\ln \left ({{\rm e}^{2\,x}}+1 \right ) \right ) }}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.66412, size = 497, normalized size = 19.12 \begin{align*} -\frac{2 \,{\left (2 \, n + 3\right )} e^{\left (-n \log \left (e^{\left (-x\right )} + 1\right ) - n \log \left (-e^{\left (-x\right )} + 1\right ) + n \log \left (e^{\left (-2 \, x\right )} + 1\right ) - 2 \, x\right )}}{n^{2} - 3 \,{\left (n^{2} + 4 \, n + 3\right )} e^{\left (-2 \, x\right )} + 3 \,{\left (n^{2} + 4 \, n + 3\right )} e^{\left (-4 \, x\right )} -{\left (n^{2} + 4 \, n + 3\right )} e^{\left (-6 \, x\right )} + 4 \, n + 3} - \frac{2 \,{\left (2 \, n + 3\right )} e^{\left (-n \log \left (e^{\left (-x\right )} + 1\right ) - n \log \left (-e^{\left (-x\right )} + 1\right ) + n \log \left (e^{\left (-2 \, x\right )} + 1\right ) - 4 \, x\right )}}{n^{2} - 3 \,{\left (n^{2} + 4 \, n + 3\right )} e^{\left (-2 \, x\right )} + 3 \,{\left (n^{2} + 4 \, n + 3\right )} e^{\left (-4 \, x\right )} -{\left (n^{2} + 4 \, n + 3\right )} e^{\left (-6 \, x\right )} + 4 \, n + 3} + \frac{2 \, e^{\left (-n \log \left (e^{\left (-x\right )} + 1\right ) - n \log \left (-e^{\left (-x\right )} + 1\right ) + n \log \left (e^{\left (-2 \, x\right )} + 1\right ) - 6 \, x\right )}}{n^{2} - 3 \,{\left (n^{2} + 4 \, n + 3\right )} e^{\left (-2 \, x\right )} + 3 \,{\left (n^{2} + 4 \, n + 3\right )} e^{\left (-4 \, x\right )} -{\left (n^{2} + 4 \, n + 3\right )} e^{\left (-6 \, x\right )} + 4 \, n + 3} + \frac{2 \, e^{\left (-n \log \left (e^{\left (-x\right )} + 1\right ) - n \log \left (-e^{\left (-x\right )} + 1\right ) + n \log \left (e^{\left (-2 \, x\right )} + 1\right )\right )}}{n^{2} - 3 \,{\left (n^{2} + 4 \, n + 3\right )} e^{\left (-2 \, x\right )} + 3 \,{\left (n^{2} + 4 \, n + 3\right )} e^{\left (-4 \, x\right )} -{\left (n^{2} + 4 \, n + 3\right )} e^{\left (-6 \, x\right )} + 4 \, n + 3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.00791, size = 347, normalized size = 13.35 \begin{align*} \frac{2 \,{\left ({\left (\cosh \left (x\right )^{3} + 3 \, \cosh \left (x\right ) \sinh \left (x\right )^{2} -{\left (2 \, n + 3\right )} \cosh \left (x\right )\right )} \cosh \left (n \log \left (\frac{\cosh \left (x\right )}{\sinh \left (x\right )}\right )\right ) +{\left (\cosh \left (x\right )^{3} + 3 \, \cosh \left (x\right ) \sinh \left (x\right )^{2} -{\left (2 \, n + 3\right )} \cosh \left (x\right )\right )} \sinh \left (n \log \left (\frac{\cosh \left (x\right )}{\sinh \left (x\right )}\right )\right )\right )}}{{\left (n^{2} + 4 \, n + 3\right )} \sinh \left (x\right )^{3} + 3 \,{\left ({\left (n^{2} + 4 \, n + 3\right )} \cosh \left (x\right )^{2} - n^{2} - 4 \, n - 3\right )} \sinh \left (x\right )} \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 \coth \left (x\right )^{n} \operatorname{csch}\left (x\right )^{4}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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