Optimal. Leaf size=53 \[ \frac {8}{15} a^2 \sqrt {a \cos ^2(x)} \tan (x)+\frac {4}{15} a \left (a \cos ^2(x)\right )^{3/2} \tan (x)+\frac {1}{5} \left (a \cos ^2(x)\right )^{5/2} \tan (x) \]
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Rubi [A]
time = 0.03, antiderivative size = 53, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {3282, 3286,
2717} \begin {gather*} \frac {8}{15} a^2 \tan (x) \sqrt {a \cos ^2(x)}+\frac {1}{5} \tan (x) \left (a \cos ^2(x)\right )^{5/2}+\frac {4}{15} a \tan (x) \left (a \cos ^2(x)\right )^{3/2} \end {gather*}
Antiderivative was successfully verified.
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Rule 2717
Rule 3282
Rule 3286
Rubi steps
\begin {align*} \int \left (a \cos ^2(x)\right )^{5/2} \, dx &=\frac {1}{5} \left (a \cos ^2(x)\right )^{5/2} \tan (x)+\frac {1}{5} (4 a) \int \left (a \cos ^2(x)\right )^{3/2} \, dx\\ &=\frac {4}{15} a \left (a \cos ^2(x)\right )^{3/2} \tan (x)+\frac {1}{5} \left (a \cos ^2(x)\right )^{5/2} \tan (x)+\frac {1}{15} \left (8 a^2\right ) \int \sqrt {a \cos ^2(x)} \, dx\\ &=\frac {4}{15} a \left (a \cos ^2(x)\right )^{3/2} \tan (x)+\frac {1}{5} \left (a \cos ^2(x)\right )^{5/2} \tan (x)+\frac {1}{15} \left (8 a^2 \sqrt {a \cos ^2(x)} \sec (x)\right ) \int \cos (x) \, dx\\ &=\frac {8}{15} a^2 \sqrt {a \cos ^2(x)} \tan (x)+\frac {4}{15} a \left (a \cos ^2(x)\right )^{3/2} \tan (x)+\frac {1}{5} \left (a \cos ^2(x)\right )^{5/2} \tan (x)\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 36, normalized size = 0.68 \begin {gather*} \frac {1}{240} a^2 \sqrt {a \cos ^2(x)} \sec (x) (150 \sin (x)+25 \sin (3 x)+3 \sin (5 x)) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.05, size = 32, normalized size = 0.60
method | result | size |
default | \(\frac {a^{3} \cos \left (x \right ) \sin \left (x \right ) \left (3 \left (\cos ^{4}\left (x \right )\right )+4 \left (\cos ^{2}\left (x \right )\right )+8\right )}{15 \sqrt {a \left (\cos ^{2}\left (x \right )\right )}}\) | \(32\) |
risch | \(-\frac {i a^{2} {\mathrm e}^{6 i x} \sqrt {a \left ({\mathrm e}^{2 i x}+1\right )^{2} {\mathrm e}^{-2 i x}}}{160 \left ({\mathrm e}^{2 i x}+1\right )}-\frac {5 i a^{2} {\mathrm e}^{2 i x} \sqrt {a \left ({\mathrm e}^{2 i x}+1\right )^{2} {\mathrm e}^{-2 i x}}}{16 \left ({\mathrm e}^{2 i x}+1\right )}+\frac {5 i a^{2} \sqrt {a \left ({\mathrm e}^{2 i x}+1\right )^{2} {\mathrm e}^{-2 i x}}}{16 \left ({\mathrm e}^{2 i x}+1\right )}+\frac {5 i a^{2} {\mathrm e}^{-2 i x} \sqrt {a \left ({\mathrm e}^{2 i x}+1\right )^{2} {\mathrm e}^{-2 i x}}}{96 \left ({\mathrm e}^{2 i x}+1\right )}-\frac {11 i a^{2} \sqrt {a \left ({\mathrm e}^{2 i x}+1\right )^{2} {\mathrm e}^{-2 i x}}\, \cos \left (4 x \right )}{240 \left ({\mathrm e}^{2 i x}+1\right )}+\frac {7 a^{2} \sqrt {a \left ({\mathrm e}^{2 i x}+1\right )^{2} {\mathrm e}^{-2 i x}}\, \sin \left (4 x \right )}{120 \left ({\mathrm e}^{2 i x}+1\right )}\) | \(222\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.52, size = 31, normalized size = 0.58 \begin {gather*} \frac {1}{240} \, {\left (3 \, a^{2} \sin \left (5 \, x\right ) + 25 \, a^{2} \sin \left (3 \, x\right ) + 150 \, a^{2} \sin \left (x\right )\right )} \sqrt {a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 40, normalized size = 0.75 \begin {gather*} \frac {{\left (3 \, a^{2} \cos \left (x\right )^{4} + 4 \, a^{2} \cos \left (x\right )^{2} + 8 \, a^{2}\right )} \sqrt {a \cos \left (x\right )^{2}} \sin \left (x\right )}{15 \, \cos \left (x\right )} \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 = 0.43, size = 34, normalized size = 0.64 \begin {gather*} \frac {1}{15} \, {\left (3 \, a^{2} \sin \left (x\right )^{5} - 10 \, a^{2} \sin \left (x\right )^{3} + 15 \, a^{2} \sin \left (x\right )\right )} \sqrt {a} \mathrm {sgn}\left (\cos \left (x\right )\right ) \end {gather*}
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 \begin {gather*} \int {\left (a\,{\cos \left (x\right )}^2\right )}^{5/2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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