Optimal. Leaf size=67 \[ \frac {\sinh (x) \cosh (x)}{a \sqrt {a \cosh ^4(x)}}+\frac {\sinh ^2(x) \tanh ^3(x)}{5 a \sqrt {a \cosh ^4(x)}}-\frac {2 \sinh ^2(x) \tanh (x)}{3 a \sqrt {a \cosh ^4(x)}} \]
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Rubi [A] time = 0.03, 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 = {3207, 3767} \[ \frac {\sinh (x) \cosh (x)}{a \sqrt {a \cosh ^4(x)}}+\frac {\sinh ^2(x) \tanh ^3(x)}{5 a \sqrt {a \cosh ^4(x)}}-\frac {2 \sinh ^2(x) \tanh (x)}{3 a \sqrt {a \cosh ^4(x)}} \]
Antiderivative was successfully verified.
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Rule 3207
Rule 3767
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 ) \operatorname {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 = 0.03, size = 30, normalized size = 0.45 \[ \frac {\sinh (x) \cosh (x) (6 \cosh (2 x)+\cosh (4 x)+8)}{15 \left (a \cosh ^4(x)\right )^{3/2}} \]
Antiderivative was successfully verified.
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fricas [B] time = 1.33, size = 1137, normalized size = 16.97 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 27, normalized size = 0.40 \[ -\frac {16 \, {\left (10 \, e^{\left (4 \, x\right )} + 5 \, e^{\left (2 \, x\right )} + 1\right )}}{15 \, a^{\frac {3}{2}} {\left (e^{\left (2 \, x\right )} + 1\right )}^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.39, size = 80, normalized size = 1.19 \[ \frac {\sqrt {8}\, \sqrt {2}\, \left (2 \left (\cosh ^{2}\left (2 x \right )\right )+6 \cosh \left (2 x \right )+7\right ) \sqrt {a \left (\sinh ^{2}\left (2 x \right )\right )}\, \sqrt {a \left (-1+\cosh \left (2 x \right )\right ) \left (\cosh \left (2 x \right )+1\right )}}{15 a^{2} \left (\cosh \left (2 x \right )+1\right )^{2} \sinh \left (2 x \right ) \sqrt {\left (\cosh \left (2 x \right )+1\right )^{2} a}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.43, size = 165, normalized size = 2.46 \[ \frac {16 \, e^{\left (-2 \, x\right )}}{3 \, {\left (5 \, a^{\frac {3}{2}} e^{\left (-2 \, x\right )} + 10 \, a^{\frac {3}{2}} e^{\left (-4 \, x\right )} + 10 \, a^{\frac {3}{2}} e^{\left (-6 \, x\right )} + 5 \, a^{\frac {3}{2}} e^{\left (-8 \, x\right )} + a^{\frac {3}{2}} e^{\left (-10 \, x\right )} + a^{\frac {3}{2}}\right )}} + \frac {32 \, e^{\left (-4 \, x\right )}}{3 \, {\left (5 \, a^{\frac {3}{2}} e^{\left (-2 \, x\right )} + 10 \, a^{\frac {3}{2}} e^{\left (-4 \, x\right )} + 10 \, a^{\frac {3}{2}} e^{\left (-6 \, x\right )} + 5 \, a^{\frac {3}{2}} e^{\left (-8 \, x\right )} + a^{\frac {3}{2}} e^{\left (-10 \, x\right )} + a^{\frac {3}{2}}\right )}} + \frac {16}{15 \, {\left (5 \, a^{\frac {3}{2}} e^{\left (-2 \, x\right )} + 10 \, a^{\frac {3}{2}} e^{\left (-4 \, x\right )} + 10 \, a^{\frac {3}{2}} e^{\left (-6 \, x\right )} + 5 \, a^{\frac {3}{2}} e^{\left (-8 \, x\right )} + a^{\frac {3}{2}} e^{\left (-10 \, x\right )} + a^{\frac {3}{2}}\right )}} \]
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 \[ -\frac {64\,{\mathrm {e}}^{2\,x}\,\sqrt {a\,{\left (\frac {{\mathrm {e}}^{-x}}{2}+\frac {{\mathrm {e}}^x}{2}\right )}^4}\,\left (5\,{\mathrm {e}}^{2\,x}+10\,{\mathrm {e}}^{4\,x}+1\right )}{15\,a^2\,{\left ({\mathrm {e}}^{2\,x}+1\right )}^7} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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