Optimal. Leaf size=33 \[ -\frac {1}{2} \coth ^2(x)+\frac {\coth ^3(x)}{3}+\frac {\coth ^4(x)}{4}-\frac {\coth ^5(x)}{5} \]
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Rubi [A]
time = 0.04, antiderivative size = 33, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {3597, 862, 76}
\begin {gather*} -\frac {1}{5} \coth ^5(x)+\frac {\coth ^4(x)}{4}+\frac {\coth ^3(x)}{3}-\frac {\coth ^2(x)}{2} \end {gather*}
Antiderivative was successfully verified.
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Rule 76
Rule 862
Rule 3597
Rubi steps
\begin {align*} \int \frac {\text {csch}^6(x)}{1+\tanh (x)} \, dx &=\text {Subst}\left (\int \frac {\left (-1+x^2\right )^2}{x^6 (1+x)} \, dx,x,\tanh (x)\right )\\ &=\text {Subst}\left (\int \frac {(-1+x)^2 (1+x)}{x^6} \, dx,x,\tanh (x)\right )\\ &=\text {Subst}\left (\int \left (\frac {1}{x^6}-\frac {1}{x^5}-\frac {1}{x^4}+\frac {1}{x^3}\right ) \, dx,x,\tanh (x)\right )\\ &=-\frac {1}{2} \coth ^2(x)+\frac {\coth ^3(x)}{3}+\frac {\coth ^4(x)}{4}-\frac {\coth ^5(x)}{5}\\ \end {align*}
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Mathematica [A]
time = 0.04, size = 27, normalized size = 0.82 \begin {gather*} \frac {1}{120} \text {csch}^5(x) (-20 \cosh (x)-5 \cosh (3 x)+\cosh (5 x)+30 \sinh (x)) \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(79\) vs.
\(2(25)=50\).
time = 0.62, size = 80, normalized size = 2.42
method | result | size |
risch | \(-\frac {4 \left (20 \,{\mathrm e}^{4 x}+5 \,{\mathrm e}^{2 x}-1\right )}{15 \left ({\mathrm e}^{2 x}-1\right )^{5}}\) | \(25\) |
default | \(-\frac {\left (\tanh ^{5}\left (\frac {x}{2}\right )\right )}{160}+\frac {\left (\tanh ^{4}\left (\frac {x}{2}\right )\right )}{64}+\frac {\left (\tanh ^{3}\left (\frac {x}{2}\right )\right )}{96}-\frac {\left (\tanh ^{2}\left (\frac {x}{2}\right )\right )}{16}+\frac {\tanh \left (\frac {x}{2}\right )}{16}-\frac {1}{160 \tanh \left (\frac {x}{2}\right )^{5}}+\frac {1}{64 \tanh \left (\frac {x}{2}\right )^{4}}+\frac {1}{96 \tanh \left (\frac {x}{2}\right )^{3}}+\frac {1}{16 \tanh \left (\frac {x}{2}\right )}-\frac {1}{16 \tanh \left (\frac {x}{2}\right )^{2}}\) | \(80\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 149 vs.
\(2 (25) = 50\).
time = 0.26, size = 149, normalized size = 4.52 \begin {gather*} \frac {4 \, e^{\left (-2 \, x\right )}}{3 \, {\left (5 \, e^{\left (-2 \, x\right )} - 10 \, e^{\left (-4 \, x\right )} + 10 \, e^{\left (-6 \, x\right )} - 5 \, e^{\left (-8 \, x\right )} + e^{\left (-10 \, x\right )} - 1\right )}} - \frac {8 \, e^{\left (-4 \, x\right )}}{3 \, {\left (5 \, e^{\left (-2 \, x\right )} - 10 \, e^{\left (-4 \, x\right )} + 10 \, e^{\left (-6 \, x\right )} - 5 \, e^{\left (-8 \, x\right )} + e^{\left (-10 \, x\right )} - 1\right )}} + \frac {8 \, e^{\left (-6 \, x\right )}}{5 \, e^{\left (-2 \, x\right )} - 10 \, e^{\left (-4 \, x\right )} + 10 \, e^{\left (-6 \, x\right )} - 5 \, e^{\left (-8 \, x\right )} + e^{\left (-10 \, x\right )} - 1} - \frac {4}{15 \, {\left (5 \, e^{\left (-2 \, x\right )} - 10 \, e^{\left (-4 \, x\right )} + 10 \, e^{\left (-6 \, x\right )} - 5 \, e^{\left (-8 \, x\right )} + e^{\left (-10 \, x\right )} - 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 185 vs.
\(2 (25) = 50\).
time = 0.33, size = 185, normalized size = 5.61 \begin {gather*} -\frac {4 \, {\left (19 \, \cosh \left (x\right )^{2} + 42 \, \cosh \left (x\right ) \sinh \left (x\right ) + 19 \, \sinh \left (x\right )^{2} + 5\right )}}{15 \, {\left (\cosh \left (x\right )^{8} + 8 \, \cosh \left (x\right ) \sinh \left (x\right )^{7} + \sinh \left (x\right )^{8} + {\left (28 \, \cosh \left (x\right )^{2} - 5\right )} \sinh \left (x\right )^{6} - 5 \, \cosh \left (x\right )^{6} + 2 \, {\left (28 \, \cosh \left (x\right )^{3} - 15 \, \cosh \left (x\right )\right )} \sinh \left (x\right )^{5} + 5 \, {\left (14 \, \cosh \left (x\right )^{4} - 15 \, \cosh \left (x\right )^{2} + 2\right )} \sinh \left (x\right )^{4} + 10 \, \cosh \left (x\right )^{4} + 4 \, {\left (14 \, \cosh \left (x\right )^{5} - 25 \, \cosh \left (x\right )^{3} + 10 \, \cosh \left (x\right )\right )} \sinh \left (x\right )^{3} + {\left (28 \, \cosh \left (x\right )^{6} - 75 \, \cosh \left (x\right )^{4} + 60 \, \cosh \left (x\right )^{2} - 11\right )} \sinh \left (x\right )^{2} - 11 \, \cosh \left (x\right )^{2} + 2 \, {\left (4 \, \cosh \left (x\right )^{7} - 15 \, \cosh \left (x\right )^{5} + 20 \, \cosh \left (x\right )^{3} - 9 \, \cosh \left (x\right )\right )} \sinh \left (x\right ) + 5\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\operatorname {csch}^{6}{\left (x \right )}}{\tanh {\left (x \right )} + 1}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.41, size = 24, normalized size = 0.73 \begin {gather*} -\frac {4 \, {\left (20 \, e^{\left (4 \, x\right )} + 5 \, e^{\left (2 \, x\right )} - 1\right )}}{15 \, {\left (e^{\left (2 \, x\right )} - 1\right )}^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.10, size = 24, normalized size = 0.73 \begin {gather*} -\frac {4\,\left (5\,{\mathrm {e}}^{2\,x}+20\,{\mathrm {e}}^{4\,x}-1\right )}{15\,{\left ({\mathrm {e}}^{2\,x}-1\right )}^5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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