Optimal. Leaf size=42 \[ \frac {2 \sin (x) \cos (x)}{\sqrt {a \cos ^3(x)}}-\frac {2 \cos ^{\frac {3}{2}}(x) E\left (\left .\frac {x}{2}\right |2\right )}{\sqrt {a \cos ^3(x)}} \]
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Rubi [A] time = 0.02, antiderivative size = 42, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {3207, 2636, 2639} \[ \frac {2 \sin (x) \cos (x)}{\sqrt {a \cos ^3(x)}}-\frac {2 \cos ^{\frac {3}{2}}(x) E\left (\left .\frac {x}{2}\right |2\right )}{\sqrt {a \cos ^3(x)}} \]
Antiderivative was successfully verified.
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Rule 2636
Rule 2639
Rule 3207
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {a \cos ^3(x)}} \, dx &=\frac {\cos ^{\frac {3}{2}}(x) \int \frac {1}{\cos ^{\frac {3}{2}}(x)} \, dx}{\sqrt {a \cos ^3(x)}}\\ &=\frac {2 \cos (x) \sin (x)}{\sqrt {a \cos ^3(x)}}-\frac {\cos ^{\frac {3}{2}}(x) \int \sqrt {\cos (x)} \, dx}{\sqrt {a \cos ^3(x)}}\\ &=-\frac {2 \cos ^{\frac {3}{2}}(x) E\left (\left .\frac {x}{2}\right |2\right )}{\sqrt {a \cos ^3(x)}}+\frac {2 \cos (x) \sin (x)}{\sqrt {a \cos ^3(x)}}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 31, normalized size = 0.74 \[ \frac {\sin (2 x)-2 \cos ^{\frac {3}{2}}(x) E\left (\left .\frac {x}{2}\right |2\right )}{\sqrt {a \cos ^3(x)}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.63, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {a \cos \relax (x)^{3}}}{a \cos \relax (x)^{3}}, x\right ) \]
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 \[ \int \frac {1}{\sqrt {a \cos \relax (x)^{3}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.31, size = 191, normalized size = 4.55 \[ \frac {2 \left (\cos \relax (x )+1\right )^{2} \left (-1+\cos \relax (x )\right )^{2} \left (i \cos \relax (x ) \sin \relax (x ) \EllipticE \left (\frac {i \left (-1+\cos \relax (x )\right )}{\sin \relax (x )}, i\right ) \sqrt {\frac {1}{\cos \relax (x )+1}}\, \sqrt {\frac {\cos \relax (x )}{\cos \relax (x )+1}}-i \cos \relax (x ) \sin \relax (x ) \EllipticF \left (\frac {i \left (-1+\cos \relax (x )\right )}{\sin \relax (x )}, i\right ) \sqrt {\frac {1}{\cos \relax (x )+1}}\, \sqrt {\frac {\cos \relax (x )}{\cos \relax (x )+1}}+i \sin \relax (x ) \EllipticE \left (\frac {i \left (-1+\cos \relax (x )\right )}{\sin \relax (x )}, i\right ) \sqrt {\frac {1}{\cos \relax (x )+1}}\, \sqrt {\frac {\cos \relax (x )}{\cos \relax (x )+1}}-i \sqrt {\frac {1}{\cos \relax (x )+1}}\, \sqrt {\frac {\cos \relax (x )}{\cos \relax (x )+1}}\, \EllipticF \left (\frac {i \left (-1+\cos \relax (x )\right )}{\sin \relax (x )}, i\right ) \sin \relax (x )-\cos \relax (x )+1\right ) \cos \relax (x )}{\sqrt {a \left (\cos ^{3}\relax (x )\right )}\, \sin \relax (x )^{5}} \]
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 \[ \int \frac {1}{\sqrt {a \cos \relax (x)^{3}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {1}{\sqrt {a\,{\cos \relax (x)}^3}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {a \cos ^{3}{\relax (x )}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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