Optimal. Leaf size=117 \[ \frac {154 \sin (x) \cos (x)}{195 a^2 \sqrt {a \cos ^3(x)}}+\frac {154 \tan (x)}{585 a^2 \sqrt {a \cos ^3(x)}}-\frac {154 \cos ^{\frac {3}{2}}(x) E\left (\left .\frac {x}{2}\right |2\right )}{195 a^2 \sqrt {a \cos ^3(x)}}+\frac {2 \tan (x) \sec ^4(x)}{13 a^2 \sqrt {a \cos ^3(x)}}+\frac {22 \tan (x) \sec ^2(x)}{117 a^2 \sqrt {a \cos ^3(x)}} \]
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Rubi [A] time = 0.05, antiderivative size = 117, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 3, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {3207, 2636, 2639} \[ \frac {154 \sin (x) \cos (x)}{195 a^2 \sqrt {a \cos ^3(x)}}+\frac {154 \tan (x)}{585 a^2 \sqrt {a \cos ^3(x)}}-\frac {154 \cos ^{\frac {3}{2}}(x) E\left (\left .\frac {x}{2}\right |2\right )}{195 a^2 \sqrt {a \cos ^3(x)}}+\frac {2 \tan (x) \sec ^4(x)}{13 a^2 \sqrt {a \cos ^3(x)}}+\frac {22 \tan (x) \sec ^2(x)}{117 a^2 \sqrt {a \cos ^3(x)}} \]
Antiderivative was successfully verified.
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Rule 2636
Rule 2639
Rule 3207
Rubi steps
\begin {align*} \int \frac {1}{\left (a \cos ^3(x)\right )^{5/2}} \, dx &=\frac {\cos ^{\frac {3}{2}}(x) \int \frac {1}{\cos ^{\frac {15}{2}}(x)} \, dx}{a^2 \sqrt {a \cos ^3(x)}}\\ &=\frac {2 \sec ^4(x) \tan (x)}{13 a^2 \sqrt {a \cos ^3(x)}}+\frac {\left (11 \cos ^{\frac {3}{2}}(x)\right ) \int \frac {1}{\cos ^{\frac {11}{2}}(x)} \, dx}{13 a^2 \sqrt {a \cos ^3(x)}}\\ &=\frac {22 \sec ^2(x) \tan (x)}{117 a^2 \sqrt {a \cos ^3(x)}}+\frac {2 \sec ^4(x) \tan (x)}{13 a^2 \sqrt {a \cos ^3(x)}}+\frac {\left (77 \cos ^{\frac {3}{2}}(x)\right ) \int \frac {1}{\cos ^{\frac {7}{2}}(x)} \, dx}{117 a^2 \sqrt {a \cos ^3(x)}}\\ &=\frac {154 \tan (x)}{585 a^2 \sqrt {a \cos ^3(x)}}+\frac {22 \sec ^2(x) \tan (x)}{117 a^2 \sqrt {a \cos ^3(x)}}+\frac {2 \sec ^4(x) \tan (x)}{13 a^2 \sqrt {a \cos ^3(x)}}+\frac {\left (77 \cos ^{\frac {3}{2}}(x)\right ) \int \frac {1}{\cos ^{\frac {3}{2}}(x)} \, dx}{195 a^2 \sqrt {a \cos ^3(x)}}\\ &=\frac {154 \cos (x) \sin (x)}{195 a^2 \sqrt {a \cos ^3(x)}}+\frac {154 \tan (x)}{585 a^2 \sqrt {a \cos ^3(x)}}+\frac {22 \sec ^2(x) \tan (x)}{117 a^2 \sqrt {a \cos ^3(x)}}+\frac {2 \sec ^4(x) \tan (x)}{13 a^2 \sqrt {a \cos ^3(x)}}-\frac {\left (77 \cos ^{\frac {3}{2}}(x)\right ) \int \sqrt {\cos (x)} \, dx}{195 a^2 \sqrt {a \cos ^3(x)}}\\ &=-\frac {154 \cos ^{\frac {3}{2}}(x) E\left (\left .\frac {x}{2}\right |2\right )}{195 a^2 \sqrt {a \cos ^3(x)}}+\frac {154 \cos (x) \sin (x)}{195 a^2 \sqrt {a \cos ^3(x)}}+\frac {154 \tan (x)}{585 a^2 \sqrt {a \cos ^3(x)}}+\frac {22 \sec ^2(x) \tan (x)}{117 a^2 \sqrt {a \cos ^3(x)}}+\frac {2 \sec ^4(x) \tan (x)}{13 a^2 \sqrt {a \cos ^3(x)}}\\ \end {align*}
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Mathematica [A] time = 0.10, size = 57, normalized size = 0.49 \[ \frac {-462 \cos ^{\frac {3}{2}}(x) E\left (\left .\frac {x}{2}\right |2\right )+462 \sin (x) \cos (x)+2 \tan (x) \left (45 \sec ^4(x)+55 \sec ^2(x)+77\right )}{585 a^2 \sqrt {a \cos ^3(x)}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.92, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {a \cos \relax (x)^{3}}}{a^{3} \cos \relax (x)^{9}}, x\right ) \]
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 \[ \int \frac {1}{\left (a \cos \relax (x)^{3}\right )^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.38, size = 223, normalized size = 1.91 \[ -\frac {2 \left (\cos \relax (x )+1\right )^{2} \left (-1+\cos \relax (x )\right )^{2} \left (231 i \left (\cos ^{7}\relax (x )\right ) \sin \relax (x ) \sqrt {\frac {1}{\cos \relax (x )+1}}\, \sqrt {\frac {\cos \relax (x )}{\cos \relax (x )+1}}\, \EllipticF \left (\frac {i \left (-1+\cos \relax (x )\right )}{\sin \relax (x )}, i\right )-231 i \left (\cos ^{7}\relax (x )\right ) \sin \relax (x ) \sqrt {\frac {1}{\cos \relax (x )+1}}\, \sqrt {\frac {\cos \relax (x )}{\cos \relax (x )+1}}\, \EllipticE \left (\frac {i \left (-1+\cos \relax (x )\right )}{\sin \relax (x )}, i\right )+231 i \left (\cos ^{6}\relax (x )\right ) \sin \relax (x ) \sqrt {\frac {1}{\cos \relax (x )+1}}\, \sqrt {\frac {\cos \relax (x )}{\cos \relax (x )+1}}\, \EllipticF \left (\frac {i \left (-1+\cos \relax (x )\right )}{\sin \relax (x )}, i\right )-231 i \left (\cos ^{6}\relax (x )\right ) \sin \relax (x ) \sqrt {\frac {1}{\cos \relax (x )+1}}\, \sqrt {\frac {\cos \relax (x )}{\cos \relax (x )+1}}\, \EllipticE \left (\frac {i \left (-1+\cos \relax (x )\right )}{\sin \relax (x )}, i\right )+231 \left (\cos ^{7}\relax (x )\right )-154 \left (\cos ^{6}\relax (x )\right )-22 \left (\cos ^{4}\relax (x )\right )-10 \left (\cos ^{2}\relax (x )\right )-45\right ) \cos \relax (x )}{585 \sin \relax (x )^{5} \left (a \left (\cos ^{3}\relax (x )\right )\right )^{\frac {5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (a \cos \relax (x)^{3}\right )^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {1}{{\left (a\,{\cos \relax (x)}^3\right )}^{5/2}} \,d x \]
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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