Optimal. Leaf size=50 \[ -\frac {16}{15} \coth (x) \sqrt {\cosh (x) \coth (x)}+\frac {2}{5} \cosh ^2(x) \coth (x) \sqrt {\cosh (x) \coth (x)}+\frac {64}{15} \sqrt {\cosh (x) \coth (x)} \tanh (x) \]
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Rubi [A]
time = 0.09, antiderivative size = 50, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.556, Rules used = {4483, 4485,
2678, 2674, 2669} \begin {gather*} \frac {2}{5} \cosh ^2(x) \coth (x) \sqrt {\cosh (x) \coth (x)}-\frac {16}{15} \coth (x) \sqrt {\cosh (x) \coth (x)}+\frac {64}{15} \tanh (x) \sqrt {\cosh (x) \coth (x)} \end {gather*}
Antiderivative was successfully verified.
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Rule 2669
Rule 2674
Rule 2678
Rule 4483
Rule 4485
Rubi steps
\begin {align*} \int (\cosh (x) \coth (x))^{5/2} \, dx &=-\frac {\sqrt {\cosh (x) \coth (x)} \int (-i \cosh (x) \coth (x))^{5/2} \, dx}{\sqrt {-i \cosh (x) \coth (x)}}\\ &=-\frac {\sqrt {\cosh (x) \coth (x)} \int \cosh ^{\frac {5}{2}}(x) (-i \coth (x))^{5/2} \, dx}{\sqrt {\cosh (x)} \sqrt {-i \coth (x)}}\\ &=\frac {2}{5} \cosh ^2(x) \coth (x) \sqrt {\cosh (x) \coth (x)}-\frac {\left (8 \sqrt {\cosh (x) \coth (x)}\right ) \int \sqrt {\cosh (x)} (-i \coth (x))^{5/2} \, dx}{5 \sqrt {\cosh (x)} \sqrt {-i \coth (x)}}\\ &=-\frac {16}{15} \coth (x) \sqrt {\cosh (x) \coth (x)}+\frac {2}{5} \cosh ^2(x) \coth (x) \sqrt {\cosh (x) \coth (x)}+\frac {\left (32 \sqrt {\cosh (x) \coth (x)}\right ) \int \sqrt {\cosh (x)} \sqrt {-i \coth (x)} \, dx}{15 \sqrt {\cosh (x)} \sqrt {-i \coth (x)}}\\ &=-\frac {16}{15} \coth (x) \sqrt {\cosh (x) \coth (x)}+\frac {2}{5} \cosh ^2(x) \coth (x) \sqrt {\cosh (x) \coth (x)}+\frac {64}{15} \sqrt {\cosh (x) \coth (x)} \tanh (x)\\ \end {align*}
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Mathematica [A]
time = 0.23, size = 44, normalized size = 0.88 \begin {gather*} \frac {1}{15} \sqrt {\cosh (x) \coth (x)} \left (-10 \coth (x)+6 \cosh (x) \sinh (x)+57 \text {csch}(x) \text {sech}(x) \left (-\sinh ^2(x)\right )^{3/4}+64 \tanh (x)\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 1.90, size = 0, normalized size = 0.00 \[\int \left (\cosh \left (x \right ) \coth \left (x \right )\right )^{\frac {5}{2}}\, dx\]
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. 163 vs.
\(2 (38) = 76\).
time = 0.49, size = 163, normalized size = 3.26 \begin {gather*} \frac {\sqrt {2} e^{\left (\frac {5}{2} \, x\right )}}{20 \, {\left (e^{\left (-x\right )} + 1\right )}^{\frac {5}{2}} {\left (-e^{\left (-x\right )} + 1\right )}^{\frac {5}{2}}} + \frac {7 \, \sqrt {2} e^{\left (\frac {1}{2} \, x\right )}}{4 \, {\left (e^{\left (-x\right )} + 1\right )}^{\frac {5}{2}} {\left (-e^{\left (-x\right )} + 1\right )}^{\frac {5}{2}}} - \frac {41 \, \sqrt {2} e^{\left (-\frac {3}{2} \, x\right )}}{6 \, {\left (e^{\left (-x\right )} + 1\right )}^{\frac {5}{2}} {\left (-e^{\left (-x\right )} + 1\right )}^{\frac {5}{2}}} + \frac {41 \, \sqrt {2} e^{\left (-\frac {7}{2} \, x\right )}}{6 \, {\left (e^{\left (-x\right )} + 1\right )}^{\frac {5}{2}} {\left (-e^{\left (-x\right )} + 1\right )}^{\frac {5}{2}}} - \frac {7 \, \sqrt {2} e^{\left (-\frac {11}{2} \, x\right )}}{4 \, {\left (e^{\left (-x\right )} + 1\right )}^{\frac {5}{2}} {\left (-e^{\left (-x\right )} + 1\right )}^{\frac {5}{2}}} - \frac {\sqrt {2} e^{\left (-\frac {15}{2} \, x\right )}}{20 \, {\left (e^{\left (-x\right )} + 1\right )}^{\frac {5}{2}} {\left (-e^{\left (-x\right )} + 1\right )}^{\frac {5}{2}}} \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. 259 vs.
\(2 (38) = 76\).
time = 0.56, size = 259, normalized size = 5.18 \begin {gather*} \frac {\sqrt {\frac {1}{2}} {\left (3 \, \cosh \left (x\right )^{8} + 24 \, \cosh \left (x\right ) \sinh \left (x\right )^{7} + 3 \, \sinh \left (x\right )^{8} + 12 \, {\left (7 \, \cosh \left (x\right )^{2} + 9\right )} \sinh \left (x\right )^{6} + 108 \, \cosh \left (x\right )^{6} + 24 \, {\left (7 \, \cosh \left (x\right )^{3} + 27 \, \cosh \left (x\right )\right )} \sinh \left (x\right )^{5} + 2 \, {\left (105 \, \cosh \left (x\right )^{4} + 810 \, \cosh \left (x\right )^{2} - 151\right )} \sinh \left (x\right )^{4} - 302 \, \cosh \left (x\right )^{4} + 8 \, {\left (21 \, \cosh \left (x\right )^{5} + 270 \, \cosh \left (x\right )^{3} - 151 \, \cosh \left (x\right )\right )} \sinh \left (x\right )^{3} + 12 \, {\left (7 \, \cosh \left (x\right )^{6} + 135 \, \cosh \left (x\right )^{4} - 151 \, \cosh \left (x\right )^{2} + 9\right )} \sinh \left (x\right )^{2} + 108 \, \cosh \left (x\right )^{2} + 8 \, {\left (3 \, \cosh \left (x\right )^{7} + 81 \, \cosh \left (x\right )^{5} - 151 \, \cosh \left (x\right )^{3} + 27 \, \cosh \left (x\right )\right )} \sinh \left (x\right ) + 3\right )}}{30 \, {\left (\cosh \left (x\right )^{4} + 4 \, \cosh \left (x\right ) \sinh \left (x\right )^{3} + \sinh \left (x\right )^{4} + {\left (6 \, \cosh \left (x\right )^{2} - 1\right )} \sinh \left (x\right )^{2} - \cosh \left (x\right )^{2} + 2 \, {\left (2 \, \cosh \left (x\right )^{3} - \cosh \left (x\right )\right )} \sinh \left (x\right )\right )} \sqrt {\cosh \left (x\right )^{3} + 3 \, \cosh \left (x\right ) \sinh \left (x\right )^{2} + \sinh \left (x\right )^{3} + {\left (3 \, \cosh \left (x\right )^{2} - 1\right )} \sinh \left (x\right ) - \cosh \left (x\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int {\left (\mathrm {cosh}\left (x\right )\,\mathrm {coth}\left (x\right )\right )}^{5/2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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