Optimal. Leaf size=17 \[ \frac {\coth ^3(x)}{3}-\frac {\coth ^5(x)}{5} \]
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Rubi [A]
time = 0.02, antiderivative size = 17, normalized size of antiderivative = 1.00, 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 = {2687, 14}
\begin {gather*} \frac {\coth ^3(x)}{3}-\frac {\coth ^5(x)}{5} \end {gather*}
Antiderivative was successfully verified.
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Rule 14
Rule 2687
Rubi steps
\begin {align*} \int \coth ^2(x) \text {csch}^4(x) \, dx &=i \text {Subst}\left (\int x^2 \left (1+x^2\right ) \, dx,x,i \coth (x)\right )\\ &=i \text {Subst}\left (\int \left (x^2+x^4\right ) \, dx,x,i \coth (x)\right )\\ &=\frac {\coth ^3(x)}{3}-\frac {\coth ^5(x)}{5}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 27, normalized size = 1.59 \begin {gather*} \frac {2 \coth (x)}{15}-\frac {1}{15} \coth (x) \text {csch}^2(x)-\frac {1}{5} \coth (x) \text {csch}^4(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. \(30\) vs.
\(2(13)=26\).
time = 0.53, size = 31, normalized size = 1.82
method | result | size |
risch | \(-\frac {4 \left (15 \,{\mathrm e}^{6 x}+5 \,{\mathrm e}^{4 x}+5 \,{\mathrm e}^{2 x}-1\right )}{15 \left ({\mathrm e}^{2 x}-1\right )^{5}}\) | \(31\) |
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.
\(2 (13) = 26\).
time = 0.27, size = 149, normalized size = 8.76 \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 {4 \, 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 {4 \, 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. 164 vs.
\(2 (13) = 26\).
time = 0.35, size = 164, normalized size = 9.65 \begin {gather*} -\frac {8 \, {\left (7 \, \cosh \left (x\right )^{3} + 24 \, \cosh \left (x\right )^{2} \sinh \left (x\right ) + 21 \, \cosh \left (x\right ) \sinh \left (x\right )^{2} + 8 \, \sinh \left (x\right )^{3} + 5 \, \cosh \left (x\right )\right )}}{15 \, {\left (\cosh \left (x\right )^{7} + 7 \, \cosh \left (x\right ) \sinh \left (x\right )^{6} + \sinh \left (x\right )^{7} + {\left (21 \, \cosh \left (x\right )^{2} - 5\right )} \sinh \left (x\right )^{5} - 5 \, \cosh \left (x\right )^{5} + 5 \, {\left (7 \, \cosh \left (x\right )^{3} - 5 \, \cosh \left (x\right )\right )} \sinh \left (x\right )^{4} + {\left (35 \, \cosh \left (x\right )^{4} - 50 \, \cosh \left (x\right )^{2} + 11\right )} \sinh \left (x\right )^{3} + 9 \, \cosh \left (x\right )^{3} + {\left (21 \, \cosh \left (x\right )^{5} - 50 \, \cosh \left (x\right )^{3} + 27 \, \cosh \left (x\right )\right )} \sinh \left (x\right )^{2} + {\left (7 \, \cosh \left (x\right )^{6} - 25 \, \cosh \left (x\right )^{4} + 33 \, \cosh \left (x\right )^{2} - 15\right )} \sinh \left (x\right ) - 5 \, \cosh \left (x\right )\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 \coth ^{2}{\left (x \right )} \operatorname {csch}^{4}{\left (x \right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 30 vs.
\(2 (13) = 26\).
time = 0.40, size = 30, normalized size = 1.76 \begin {gather*} -\frac {4 \, {\left (15 \, e^{\left (6 \, x\right )} + 5 \, 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 = 1.46, size = 144, normalized size = 8.47 \begin {gather*} -\frac {\frac {8\,{\mathrm {e}}^{2\,x}}{5}+\frac {16\,{\mathrm {e}}^{4\,x}}{5}+\frac {8\,{\mathrm {e}}^{6\,x}}{5}}{5\,{\mathrm {e}}^{2\,x}-10\,{\mathrm {e}}^{4\,x}+10\,{\mathrm {e}}^{6\,x}-5\,{\mathrm {e}}^{8\,x}+{\mathrm {e}}^{10\,x}-1}-\frac {\frac {4\,{\mathrm {e}}^{2\,x}}{5}+\frac {8}{15}}{3\,{\mathrm {e}}^{2\,x}-3\,{\mathrm {e}}^{4\,x}+{\mathrm {e}}^{6\,x}-1}-\frac {2}{5\,\left ({\mathrm {e}}^{4\,x}-2\,{\mathrm {e}}^{2\,x}+1\right )}-\frac {\frac {8\,{\mathrm {e}}^{2\,x}}{5}+\frac {6\,{\mathrm {e}}^{4\,x}}{5}+\frac {2}{5}}{6\,{\mathrm {e}}^{4\,x}-4\,{\mathrm {e}}^{2\,x}-4\,{\mathrm {e}}^{6\,x}+{\mathrm {e}}^{8\,x}+1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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