Optimal. Leaf size=67 \[ \frac {\cosh (x) \sinh (x)}{a \sqrt {a \cosh ^4(x)}}-\frac {2 \sinh ^2(x) \tanh (x)}{3 a \sqrt {a \cosh ^4(x)}}+\frac {\sinh ^2(x) \tanh ^3(x)}{5 a \sqrt {a \cosh ^4(x)}} \]
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Rubi [A]
time = 0.02, antiderivative size = 67, 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 \sqrt {a \cosh ^4(x)}}+\frac {\sinh ^2(x) \tanh ^3(x)}{5 a \sqrt {a \cosh ^4(x)}}-\frac {2 \sinh ^2(x) \tanh (x)}{3 a \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 )^{3/2}} \, dx &=\frac {\cosh ^2(x) \int \text {sech}^6(x) \, dx}{a \sqrt {a \cosh ^4(x)}}\\ &=\frac {\left (i \cosh ^2(x)\right ) \text {Subst}\left (\int \left (1+2 x^2+x^4\right ) \, dx,x,-i \tanh (x)\right )}{a \sqrt {a \cosh ^4(x)}}\\ &=\frac {\cosh (x) \sinh (x)}{a \sqrt {a \cosh ^4(x)}}-\frac {2 \sinh ^2(x) \tanh (x)}{3 a \sqrt {a \cosh ^4(x)}}+\frac {\sinh ^2(x) \tanh ^3(x)}{5 a \sqrt {a \cosh ^4(x)}}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 30, normalized size = 0.45 \begin {gather*} \frac {\cosh (x) (8+6 \cosh (2 x)+\cosh (4 x)) \sinh (x)}{15 \left (a \cosh ^4(x)\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 1.57, size = 80, normalized size = 1.19
method | result | size |
risch | \(-\frac {16 \,{\mathrm e}^{-2 x} \left (10 \,{\mathrm e}^{4 x}+5 \,{\mathrm e}^{2 x}+1\right )}{15 a \left (1+{\mathrm e}^{2 x}\right )^{3} \sqrt {a \left (1+{\mathrm e}^{2 x}\right )^{4} {\mathrm e}^{-4 x}}}\) | \(48\) |
default | \(\frac {\sqrt {8}\, \sqrt {2}\, \left (2 \left (\cosh ^{2}\left (2 x \right )\right )+6 \cosh \left (2 x \right )+7\right ) \sqrt {a \left (\sinh ^{2}\left (2 x \right )\right )}\, \sqrt {a \left (-1+\cosh \left (2 x \right )\right ) \left (1+\cosh \left (2 x \right )\right )}}{15 a^{2} \left (1+\cosh \left (2 x \right )\right )^{2} \sinh \left (2 x \right ) \sqrt {a \left (1+\cosh \left (2 x \right )\right )^{2}}}\) | \(80\) |
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. 165 vs.
\(2 (57) = 114\).
time = 0.49, size = 165, normalized size = 2.46 \begin {gather*} \frac {16 \, e^{\left (-2 \, x\right )}}{3 \, {\left (5 \, a^{\frac {3}{2}} e^{\left (-2 \, x\right )} + 10 \, a^{\frac {3}{2}} e^{\left (-4 \, x\right )} + 10 \, a^{\frac {3}{2}} e^{\left (-6 \, x\right )} + 5 \, a^{\frac {3}{2}} e^{\left (-8 \, x\right )} + a^{\frac {3}{2}} e^{\left (-10 \, x\right )} + a^{\frac {3}{2}}\right )}} + \frac {32 \, e^{\left (-4 \, x\right )}}{3 \, {\left (5 \, a^{\frac {3}{2}} e^{\left (-2 \, x\right )} + 10 \, a^{\frac {3}{2}} e^{\left (-4 \, x\right )} + 10 \, a^{\frac {3}{2}} e^{\left (-6 \, x\right )} + 5 \, a^{\frac {3}{2}} e^{\left (-8 \, x\right )} + a^{\frac {3}{2}} e^{\left (-10 \, x\right )} + a^{\frac {3}{2}}\right )}} + \frac {16}{15 \, {\left (5 \, a^{\frac {3}{2}} e^{\left (-2 \, x\right )} + 10 \, a^{\frac {3}{2}} e^{\left (-4 \, x\right )} + 10 \, a^{\frac {3}{2}} e^{\left (-6 \, x\right )} + 5 \, a^{\frac {3}{2}} e^{\left (-8 \, x\right )} + a^{\frac {3}{2}} e^{\left (-10 \, x\right )} + a^{\frac {3}{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. 1137 vs.
\(2 (57) = 114\).
time = 0.42, size = 1137, normalized size = 16.97 \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 = 0.42, size = 27, normalized size = 0.40 \begin {gather*} -\frac {16 \, {\left (10 \, e^{\left (4 \, x\right )} + 5 \, e^{\left (2 \, x\right )} + 1\right )}}{15 \, a^{\frac {3}{2}} {\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 = 0.97, size = 48, normalized size = 0.72 \begin {gather*} -\frac {64\,{\mathrm {e}}^{2\,x}\,\sqrt {a\,{\left (\frac {{\mathrm {e}}^{-x}}{2}+\frac {{\mathrm {e}}^x}{2}\right )}^4}\,\left (5\,{\mathrm {e}}^{2\,x}+10\,{\mathrm {e}}^{4\,x}+1\right )}{15\,a^2\,{\left ({\mathrm {e}}^{2\,x}+1\right )}^7} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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