Optimal. Leaf size=117 \[ \frac {\cosh (x) \sinh (x)}{a^2 \sqrt {a \cosh ^4(x)}}-\frac {4 \sinh ^2(x) \tanh (x)}{3 a^2 \sqrt {a \cosh ^4(x)}}+\frac {6 \sinh ^2(x) \tanh ^3(x)}{5 a^2 \sqrt {a \cosh ^4(x)}}-\frac {4 \sinh ^2(x) \tanh ^5(x)}{7 a^2 \sqrt {a \cosh ^4(x)}}+\frac {\sinh ^2(x) \tanh ^7(x)}{9 a^2 \sqrt {a \cosh ^4(x)}} \]
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Rubi [A]
time = 0.02, antiderivative size = 117, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {3286, 3852}
\begin {gather*} \frac {\sinh (x) \cosh (x)}{a^2 \sqrt {a \cosh ^4(x)}}+\frac {\sinh ^2(x) \tanh ^7(x)}{9 a^2 \sqrt {a \cosh ^4(x)}}-\frac {4 \sinh ^2(x) \tanh ^5(x)}{7 a^2 \sqrt {a \cosh ^4(x)}}+\frac {6 \sinh ^2(x) \tanh ^3(x)}{5 a^2 \sqrt {a \cosh ^4(x)}}-\frac {4 \sinh ^2(x) \tanh (x)}{3 a^2 \sqrt {a \cosh ^4(x)}} \end {gather*}
Antiderivative was successfully verified.
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Rule 3286
Rule 3852
Rubi steps
\begin {align*} \int \frac {1}{\left (a \cosh ^4(x)\right )^{5/2}} \, dx &=\frac {\cosh ^2(x) \int \text {sech}^{10}(x) \, dx}{a^2 \sqrt {a \cosh ^4(x)}}\\ &=\frac {\left (i \cosh ^2(x)\right ) \text {Subst}\left (\int \left (1+4 x^2+6 x^4+4 x^6+x^8\right ) \, dx,x,-i \tanh (x)\right )}{a^2 \sqrt {a \cosh ^4(x)}}\\ &=\frac {\cosh (x) \sinh (x)}{a^2 \sqrt {a \cosh ^4(x)}}-\frac {4 \sinh ^2(x) \tanh (x)}{3 a^2 \sqrt {a \cosh ^4(x)}}+\frac {6 \sinh ^2(x) \tanh ^3(x)}{5 a^2 \sqrt {a \cosh ^4(x)}}-\frac {4 \sinh ^2(x) \tanh ^5(x)}{7 a^2 \sqrt {a \cosh ^4(x)}}+\frac {\sinh ^2(x) \tanh ^7(x)}{9 a^2 \sqrt {a \cosh ^4(x)}}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 47, normalized size = 0.40 \begin {gather*} \frac {(128+130 \cosh (2 x)+46 \cosh (4 x)+10 \cosh (6 x)+\cosh (8 x)) \text {sech}^6(x) \tanh (x)}{315 a^2 \sqrt {a \cosh ^4(x)}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 1.58, size = 96, normalized size = 0.82
method | result | size |
risch | \(-\frac {256 \,{\mathrm e}^{-2 x} \left (126 \,{\mathrm e}^{8 x}+84 \,{\mathrm e}^{6 x}+36 \,{\mathrm e}^{4 x}+9 \,{\mathrm e}^{2 x}+1\right )}{315 a^{2} \left (1+{\mathrm e}^{2 x}\right )^{7} \sqrt {a \left (1+{\mathrm e}^{2 x}\right )^{4} {\mathrm e}^{-4 x}}}\) | \(60\) |
default | \(\frac {4 \sqrt {8}\, \sqrt {2}\, \left (8 \left (\cosh ^{4}\left (2 x \right )\right )+40 \left (\cosh ^{3}\left (2 x \right )\right )+84 \left (\cosh ^{2}\left (2 x \right )\right )+100 \cosh \left (2 x \right )+83\right ) \sqrt {a \left (\sinh ^{2}\left (2 x \right )\right )}\, \sqrt {a \left (-1+\cosh \left (2 x \right )\right ) \left (1+\cosh \left (2 x \right )\right )}}{315 a^{3} \left (1+\cosh \left (2 x \right )\right )^{4} \sinh \left (2 x \right ) \sqrt {a \left (1+\cosh \left (2 x \right )\right )^{2}}}\) | \(96\) |
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. 457 vs.
\(2 (99) = 198\).
time = 0.50, size = 457, normalized size = 3.91 \begin {gather*} \frac {256 \, e^{\left (-2 \, x\right )}}{35 \, {\left (9 \, a^{\frac {5}{2}} e^{\left (-2 \, x\right )} + 36 \, a^{\frac {5}{2}} e^{\left (-4 \, x\right )} + 84 \, a^{\frac {5}{2}} e^{\left (-6 \, x\right )} + 126 \, a^{\frac {5}{2}} e^{\left (-8 \, x\right )} + 126 \, a^{\frac {5}{2}} e^{\left (-10 \, x\right )} + 84 \, a^{\frac {5}{2}} e^{\left (-12 \, x\right )} + 36 \, a^{\frac {5}{2}} e^{\left (-14 \, x\right )} + 9 \, a^{\frac {5}{2}} e^{\left (-16 \, x\right )} + a^{\frac {5}{2}} e^{\left (-18 \, x\right )} + a^{\frac {5}{2}}\right )}} + \frac {1024 \, e^{\left (-4 \, x\right )}}{35 \, {\left (9 \, a^{\frac {5}{2}} e^{\left (-2 \, x\right )} + 36 \, a^{\frac {5}{2}} e^{\left (-4 \, x\right )} + 84 \, a^{\frac {5}{2}} e^{\left (-6 \, x\right )} + 126 \, a^{\frac {5}{2}} e^{\left (-8 \, x\right )} + 126 \, a^{\frac {5}{2}} e^{\left (-10 \, x\right )} + 84 \, a^{\frac {5}{2}} e^{\left (-12 \, x\right )} + 36 \, a^{\frac {5}{2}} e^{\left (-14 \, x\right )} + 9 \, a^{\frac {5}{2}} e^{\left (-16 \, x\right )} + a^{\frac {5}{2}} e^{\left (-18 \, x\right )} + a^{\frac {5}{2}}\right )}} + \frac {1024 \, e^{\left (-6 \, x\right )}}{15 \, {\left (9 \, a^{\frac {5}{2}} e^{\left (-2 \, x\right )} + 36 \, a^{\frac {5}{2}} e^{\left (-4 \, x\right )} + 84 \, a^{\frac {5}{2}} e^{\left (-6 \, x\right )} + 126 \, a^{\frac {5}{2}} e^{\left (-8 \, x\right )} + 126 \, a^{\frac {5}{2}} e^{\left (-10 \, x\right )} + 84 \, a^{\frac {5}{2}} e^{\left (-12 \, x\right )} + 36 \, a^{\frac {5}{2}} e^{\left (-14 \, x\right )} + 9 \, a^{\frac {5}{2}} e^{\left (-16 \, x\right )} + a^{\frac {5}{2}} e^{\left (-18 \, x\right )} + a^{\frac {5}{2}}\right )}} + \frac {512 \, e^{\left (-8 \, x\right )}}{5 \, {\left (9 \, a^{\frac {5}{2}} e^{\left (-2 \, x\right )} + 36 \, a^{\frac {5}{2}} e^{\left (-4 \, x\right )} + 84 \, a^{\frac {5}{2}} e^{\left (-6 \, x\right )} + 126 \, a^{\frac {5}{2}} e^{\left (-8 \, x\right )} + 126 \, a^{\frac {5}{2}} e^{\left (-10 \, x\right )} + 84 \, a^{\frac {5}{2}} e^{\left (-12 \, x\right )} + 36 \, a^{\frac {5}{2}} e^{\left (-14 \, x\right )} + 9 \, a^{\frac {5}{2}} e^{\left (-16 \, x\right )} + a^{\frac {5}{2}} e^{\left (-18 \, x\right )} + a^{\frac {5}{2}}\right )}} + \frac {256}{315 \, {\left (9 \, a^{\frac {5}{2}} e^{\left (-2 \, x\right )} + 36 \, a^{\frac {5}{2}} e^{\left (-4 \, x\right )} + 84 \, a^{\frac {5}{2}} e^{\left (-6 \, x\right )} + 126 \, a^{\frac {5}{2}} e^{\left (-8 \, x\right )} + 126 \, a^{\frac {5}{2}} e^{\left (-10 \, x\right )} + 84 \, a^{\frac {5}{2}} e^{\left (-12 \, x\right )} + 36 \, a^{\frac {5}{2}} e^{\left (-14 \, x\right )} + 9 \, a^{\frac {5}{2}} e^{\left (-16 \, x\right )} + a^{\frac {5}{2}} e^{\left (-18 \, x\right )} + a^{\frac {5}{2}}\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. 3065 vs.
\(2 (99) = 198\).
time = 0.42, size = 3065, normalized size = 26.20 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.41, size = 39, normalized size = 0.33 \begin {gather*} -\frac {256 \, {\left (126 \, e^{\left (8 \, x\right )} + 84 \, e^{\left (6 \, x\right )} + 36 \, e^{\left (4 \, x\right )} + 9 \, e^{\left (2 \, x\right )} + 1\right )}}{315 \, a^{\frac {5}{2}} {\left (e^{\left (2 \, x\right )} + 1\right )}^{9}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.99, size = 256, normalized size = 2.19 \begin {gather*} \frac {4096\,{\mathrm {e}}^{4\,x}\,\sqrt {a\,{\left (\frac {{\mathrm {e}}^{-x}}{2}+\frac {{\mathrm {e}}^x}{2}\right )}^4}}{3\,a^3\,{\left ({\mathrm {e}}^{2\,x}+1\right )}^6\,\left ({\mathrm {e}}^{2\,x}+2\,{\mathrm {e}}^{4\,x}+{\mathrm {e}}^{6\,x}\right )}-\frac {2048\,{\mathrm {e}}^{4\,x}\,\sqrt {a\,{\left (\frac {{\mathrm {e}}^{-x}}{2}+\frac {{\mathrm {e}}^x}{2}\right )}^4}}{5\,a^3\,{\left ({\mathrm {e}}^{2\,x}+1\right )}^5\,\left ({\mathrm {e}}^{2\,x}+2\,{\mathrm {e}}^{4\,x}+{\mathrm {e}}^{6\,x}\right )}-\frac {12288\,{\mathrm {e}}^{4\,x}\,\sqrt {a\,{\left (\frac {{\mathrm {e}}^{-x}}{2}+\frac {{\mathrm {e}}^x}{2}\right )}^4}}{7\,a^3\,{\left ({\mathrm {e}}^{2\,x}+1\right )}^7\,\left ({\mathrm {e}}^{2\,x}+2\,{\mathrm {e}}^{4\,x}+{\mathrm {e}}^{6\,x}\right )}+\frac {1024\,{\mathrm {e}}^{4\,x}\,\sqrt {a\,{\left (\frac {{\mathrm {e}}^{-x}}{2}+\frac {{\mathrm {e}}^x}{2}\right )}^4}}{a^3\,{\left ({\mathrm {e}}^{2\,x}+1\right )}^8\,\left ({\mathrm {e}}^{2\,x}+2\,{\mathrm {e}}^{4\,x}+{\mathrm {e}}^{6\,x}\right )}-\frac {2048\,{\mathrm {e}}^{4\,x}\,\sqrt {a\,{\left (\frac {{\mathrm {e}}^{-x}}{2}+\frac {{\mathrm {e}}^x}{2}\right )}^4}}{9\,a^3\,{\left ({\mathrm {e}}^{2\,x}+1\right )}^9\,\left ({\mathrm {e}}^{2\,x}+2\,{\mathrm {e}}^{4\,x}+{\mathrm {e}}^{6\,x}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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