Optimal. Leaf size=61 \[ \frac {3 \tanh ^{-1}(\sin (x)) \cos (x)}{8 a^2 \sqrt {a \cos ^2(x)}}+\frac {\tan (x)}{4 a \left (a \cos ^2(x)\right )^{3/2}}+\frac {3 \tan (x)}{8 a^2 \sqrt {a \cos ^2(x)}} \]
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Rubi [A]
time = 0.03, antiderivative size = 61, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 4, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {3255, 3283,
3286, 3855} \begin {gather*} \frac {3 \tan (x)}{8 a^2 \sqrt {a \cos ^2(x)}}+\frac {3 \cos (x) \tanh ^{-1}(\sin (x))}{8 a^2 \sqrt {a \cos ^2(x)}}+\frac {\tan (x)}{4 a \left (a \cos ^2(x)\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 3255
Rule 3283
Rule 3286
Rule 3855
Rubi steps
\begin {align*} \int \frac {1}{\left (a-a \sin ^2(x)\right )^{5/2}} \, dx &=\int \frac {1}{\left (a \cos ^2(x)\right )^{5/2}} \, dx\\ &=\frac {\tan (x)}{4 a \left (a \cos ^2(x)\right )^{3/2}}+\frac {3 \int \frac {1}{\left (a \cos ^2(x)\right )^{3/2}} \, dx}{4 a}\\ &=\frac {\tan (x)}{4 a \left (a \cos ^2(x)\right )^{3/2}}+\frac {3 \tan (x)}{8 a^2 \sqrt {a \cos ^2(x)}}+\frac {3 \int \frac {1}{\sqrt {a \cos ^2(x)}} \, dx}{8 a^2}\\ &=\frac {\tan (x)}{4 a \left (a \cos ^2(x)\right )^{3/2}}+\frac {3 \tan (x)}{8 a^2 \sqrt {a \cos ^2(x)}}+\frac {(3 \cos (x)) \int \sec (x) \, dx}{8 a^2 \sqrt {a \cos ^2(x)}}\\ &=\frac {3 \tanh ^{-1}(\sin (x)) \cos (x)}{8 a^2 \sqrt {a \cos ^2(x)}}+\frac {\tan (x)}{4 a \left (a \cos ^2(x)\right )^{3/2}}+\frac {3 \tan (x)}{8 a^2 \sqrt {a \cos ^2(x)}}\\ \end {align*}
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Mathematica [A]
time = 0.12, size = 72, normalized size = 1.18 \begin {gather*} \frac {\cos ^5(x) \left (-6 \log \left (\cos \left (\frac {x}{2}\right )-\sin \left (\frac {x}{2}\right )\right )+6 \log \left (\cos \left (\frac {x}{2}\right )+\sin \left (\frac {x}{2}\right )\right )+\frac {1}{2} \sec ^4(x) (11 \sin (x)+3 \sin (3 x))\right )}{16 \left (a \cos ^2(x)\right )^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 7.03, size = 89, normalized size = 1.46
method | result | size |
default | \(\frac {\sqrt {a \left (\sin ^{2}\left (x \right )\right )}\, \left (3 \ln \left (\frac {2 \sqrt {a}\, \sqrt {a \left (\sin ^{2}\left (x \right )\right )}+2 a}{\cos \left (x \right )}\right ) a \left (\cos ^{4}\left (x \right )\right )+3 \sqrt {a \left (\sin ^{2}\left (x \right )\right )}\, \left (\cos ^{2}\left (x \right )\right ) \sqrt {a}+2 \sqrt {a}\, \sqrt {a \left (\sin ^{2}\left (x \right )\right )}\right )}{8 a^{\frac {7}{2}} \cos \left (x \right )^{3} \sin \left (x \right ) \sqrt {a \left (\cos ^{2}\left (x \right )\right )}}\) | \(89\) |
risch | \(-\frac {i \left (3 \,{\mathrm e}^{6 i x}+11 \,{\mathrm e}^{4 i x}-11 \,{\mathrm e}^{2 i x}-3\right )}{4 a^{2} \left ({\mathrm e}^{2 i x}+1\right )^{3} \sqrt {a \left ({\mathrm e}^{2 i x}+1\right )^{2} {\mathrm e}^{-2 i x}}}-\frac {3 \ln \left ({\mathrm e}^{i x}-i\right ) \cos \left (x \right )}{4 a^{2} \sqrt {a \left ({\mathrm e}^{2 i x}+1\right )^{2} {\mathrm e}^{-2 i x}}}+\frac {3 \ln \left ({\mathrm e}^{i x}+i\right ) \cos \left (x \right )}{4 a^{2} \sqrt {a \left ({\mathrm e}^{2 i x}+1\right )^{2} {\mathrm e}^{-2 i x}}}\) | \(126\) |
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. 933 vs.
\(2 (49) = 98\).
time = 0.86, size = 933, normalized size = 15.30 \begin {gather*} \frac {4 \, {\left (3 \, \sin \left (7 \, x\right ) + 11 \, \sin \left (5 \, x\right ) - 11 \, \sin \left (3 \, x\right ) - 3 \, \sin \left (x\right )\right )} \cos \left (8 \, x\right ) - 24 \, {\left (2 \, \sin \left (6 \, x\right ) + 3 \, \sin \left (4 \, x\right ) + 2 \, \sin \left (2 \, x\right )\right )} \cos \left (7 \, x\right ) + 16 \, {\left (11 \, \sin \left (5 \, x\right ) - 11 \, \sin \left (3 \, x\right ) - 3 \, \sin \left (x\right )\right )} \cos \left (6 \, x\right ) - 88 \, {\left (3 \, \sin \left (4 \, x\right ) + 2 \, \sin \left (2 \, x\right )\right )} \cos \left (5 \, x\right ) - 24 \, {\left (11 \, \sin \left (3 \, x\right ) + 3 \, \sin \left (x\right )\right )} \cos \left (4 \, x\right ) + 3 \, {\left (2 \, {\left (4 \, \cos \left (6 \, x\right ) + 6 \, \cos \left (4 \, x\right ) + 4 \, \cos \left (2 \, x\right ) + 1\right )} \cos \left (8 \, x\right ) + \cos \left (8 \, x\right )^{2} + 8 \, {\left (6 \, \cos \left (4 \, x\right ) + 4 \, \cos \left (2 \, x\right ) + 1\right )} \cos \left (6 \, x\right ) + 16 \, \cos \left (6 \, x\right )^{2} + 12 \, {\left (4 \, \cos \left (2 \, x\right ) + 1\right )} \cos \left (4 \, x\right ) + 36 \, \cos \left (4 \, x\right )^{2} + 16 \, \cos \left (2 \, x\right )^{2} + 4 \, {\left (2 \, \sin \left (6 \, x\right ) + 3 \, \sin \left (4 \, x\right ) + 2 \, \sin \left (2 \, x\right )\right )} \sin \left (8 \, x\right ) + \sin \left (8 \, x\right )^{2} + 16 \, {\left (3 \, \sin \left (4 \, x\right ) + 2 \, \sin \left (2 \, x\right )\right )} \sin \left (6 \, x\right ) + 16 \, \sin \left (6 \, x\right )^{2} + 36 \, \sin \left (4 \, x\right )^{2} + 48 \, \sin \left (4 \, x\right ) \sin \left (2 \, x\right ) + 16 \, \sin \left (2 \, x\right )^{2} + 8 \, \cos \left (2 \, x\right ) + 1\right )} \log \left (\cos \left (x\right )^{2} + \sin \left (x\right )^{2} + 2 \, \sin \left (x\right ) + 1\right ) - 3 \, {\left (2 \, {\left (4 \, \cos \left (6 \, x\right ) + 6 \, \cos \left (4 \, x\right ) + 4 \, \cos \left (2 \, x\right ) + 1\right )} \cos \left (8 \, x\right ) + \cos \left (8 \, x\right )^{2} + 8 \, {\left (6 \, \cos \left (4 \, x\right ) + 4 \, \cos \left (2 \, x\right ) + 1\right )} \cos \left (6 \, x\right ) + 16 \, \cos \left (6 \, x\right )^{2} + 12 \, {\left (4 \, \cos \left (2 \, x\right ) + 1\right )} \cos \left (4 \, x\right ) + 36 \, \cos \left (4 \, x\right )^{2} + 16 \, \cos \left (2 \, x\right )^{2} + 4 \, {\left (2 \, \sin \left (6 \, x\right ) + 3 \, \sin \left (4 \, x\right ) + 2 \, \sin \left (2 \, x\right )\right )} \sin \left (8 \, x\right ) + \sin \left (8 \, x\right )^{2} + 16 \, {\left (3 \, \sin \left (4 \, x\right ) + 2 \, \sin \left (2 \, x\right )\right )} \sin \left (6 \, x\right ) + 16 \, \sin \left (6 \, x\right )^{2} + 36 \, \sin \left (4 \, x\right )^{2} + 48 \, \sin \left (4 \, x\right ) \sin \left (2 \, x\right ) + 16 \, \sin \left (2 \, x\right )^{2} + 8 \, \cos \left (2 \, x\right ) + 1\right )} \log \left (\cos \left (x\right )^{2} + \sin \left (x\right )^{2} - 2 \, \sin \left (x\right ) + 1\right ) - 4 \, {\left (3 \, \cos \left (7 \, x\right ) + 11 \, \cos \left (5 \, x\right ) - 11 \, \cos \left (3 \, x\right ) - 3 \, \cos \left (x\right )\right )} \sin \left (8 \, x\right ) + 12 \, {\left (4 \, \cos \left (6 \, x\right ) + 6 \, \cos \left (4 \, x\right ) + 4 \, \cos \left (2 \, x\right ) + 1\right )} \sin \left (7 \, x\right ) - 16 \, {\left (11 \, \cos \left (5 \, x\right ) - 11 \, \cos \left (3 \, x\right ) - 3 \, \cos \left (x\right )\right )} \sin \left (6 \, x\right ) + 44 \, {\left (6 \, \cos \left (4 \, x\right ) + 4 \, \cos \left (2 \, x\right ) + 1\right )} \sin \left (5 \, x\right ) + 24 \, {\left (11 \, \cos \left (3 \, x\right ) + 3 \, \cos \left (x\right )\right )} \sin \left (4 \, x\right ) - 44 \, {\left (4 \, \cos \left (2 \, x\right ) + 1\right )} \sin \left (3 \, x\right ) + 176 \, \cos \left (3 \, x\right ) \sin \left (2 \, x\right ) + 48 \, \cos \left (x\right ) \sin \left (2 \, x\right ) - 48 \, \cos \left (2 \, x\right ) \sin \left (x\right ) - 12 \, \sin \left (x\right )}{16 \, {\left (a^{2} \cos \left (8 \, x\right )^{2} + 16 \, a^{2} \cos \left (6 \, x\right )^{2} + 36 \, a^{2} \cos \left (4 \, x\right )^{2} + 16 \, a^{2} \cos \left (2 \, x\right )^{2} + a^{2} \sin \left (8 \, x\right )^{2} + 16 \, a^{2} \sin \left (6 \, x\right )^{2} + 36 \, a^{2} \sin \left (4 \, x\right )^{2} + 48 \, a^{2} \sin \left (4 \, x\right ) \sin \left (2 \, x\right ) + 16 \, a^{2} \sin \left (2 \, x\right )^{2} + 8 \, a^{2} \cos \left (2 \, x\right ) + a^{2} + 2 \, {\left (4 \, a^{2} \cos \left (6 \, x\right ) + 6 \, a^{2} \cos \left (4 \, x\right ) + 4 \, a^{2} \cos \left (2 \, x\right ) + a^{2}\right )} \cos \left (8 \, x\right ) + 8 \, {\left (6 \, a^{2} \cos \left (4 \, x\right ) + 4 \, a^{2} \cos \left (2 \, x\right ) + a^{2}\right )} \cos \left (6 \, x\right ) + 12 \, {\left (4 \, a^{2} \cos \left (2 \, x\right ) + a^{2}\right )} \cos \left (4 \, x\right ) + 4 \, {\left (2 \, a^{2} \sin \left (6 \, x\right ) + 3 \, a^{2} \sin \left (4 \, x\right ) + 2 \, a^{2} \sin \left (2 \, x\right )\right )} \sin \left (8 \, x\right ) + 16 \, {\left (3 \, a^{2} \sin \left (4 \, x\right ) + 2 \, a^{2} \sin \left (2 \, x\right )\right )} \sin \left (6 \, x\right )\right )} \sqrt {a}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.43, size = 49, normalized size = 0.80 \begin {gather*} -\frac {{\left (3 \, \cos \left (x\right )^{4} \log \left (-\frac {\sin \left (x\right ) - 1}{\sin \left (x\right ) + 1}\right ) - 2 \, {\left (3 \, \cos \left (x\right )^{2} + 2\right )} \sin \left (x\right )\right )} \sqrt {a \cos \left (x\right )^{2}}}{16 \, a^{3} \cos \left (x\right )^{5}} \end {gather*}
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 \begin {gather*} \int \frac {1}{\left (- a \sin ^{2}{\left (x \right )} + a\right )^{\frac {5}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 129 vs.
\(2 (49) = 98\).
time = 0.52, size = 129, normalized size = 2.11 \begin {gather*} -\frac {3 \, \log \left ({\left | \frac {1}{\tan \left (\frac {1}{2} \, x\right )} + \tan \left (\frac {1}{2} \, x\right ) + 2 \right |}\right )}{16 \, a^{\frac {5}{2}} \mathrm {sgn}\left (\tan \left (\frac {1}{2} \, x\right )^{4} - 1\right )} + \frac {3 \, \log \left ({\left | \frac {1}{\tan \left (\frac {1}{2} \, x\right )} + \tan \left (\frac {1}{2} \, x\right ) - 2 \right |}\right )}{16 \, a^{\frac {5}{2}} \mathrm {sgn}\left (\tan \left (\frac {1}{2} \, x\right )^{4} - 1\right )} - \frac {5 \, \sqrt {a} {\left (\frac {1}{\tan \left (\frac {1}{2} \, x\right )} + \tan \left (\frac {1}{2} \, x\right )\right )}^{3} - 12 \, \sqrt {a} {\left (\frac {1}{\tan \left (\frac {1}{2} \, x\right )} + \tan \left (\frac {1}{2} \, x\right )\right )}}{4 \, {\left ({\left (\frac {1}{\tan \left (\frac {1}{2} \, x\right )} + \tan \left (\frac {1}{2} \, x\right )\right )}^{2} - 4\right )}^{2} a^{3} \mathrm {sgn}\left (\tan \left (\frac {1}{2} \, x\right )^{4} - 1\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 \frac {1}{{\left (a-a\,{\sin \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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