Optimal. Leaf size=61 \[ -\frac {\cot (x)}{4 a \left (a \sin ^2(x)\right )^{3/2}}-\frac {3 \cot (x)}{8 a^2 \sqrt {a \sin ^2(x)}}-\frac {3 \tanh ^{-1}(\cos (x)) \sin (x)}{8 a^2 \sqrt {a \sin ^2(x)}} \]
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Rubi [A]
time = 0.02, antiderivative size = 61, 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 = {3283, 3286,
3855} \begin {gather*} -\frac {3 \cot (x)}{8 a^2 \sqrt {a \sin ^2(x)}}-\frac {3 \sin (x) \tanh ^{-1}(\cos (x))}{8 a^2 \sqrt {a \sin ^2(x)}}-\frac {\cot (x)}{4 a \left (a \sin ^2(x)\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 3283
Rule 3286
Rule 3855
Rubi steps
\begin {align*} \int \frac {1}{\left (a \sin ^2(x)\right )^{5/2}} \, dx &=-\frac {\cot (x)}{4 a \left (a \sin ^2(x)\right )^{3/2}}+\frac {3 \int \frac {1}{\left (a \sin ^2(x)\right )^{3/2}} \, dx}{4 a}\\ &=-\frac {\cot (x)}{4 a \left (a \sin ^2(x)\right )^{3/2}}-\frac {3 \cot (x)}{8 a^2 \sqrt {a \sin ^2(x)}}+\frac {3 \int \frac {1}{\sqrt {a \sin ^2(x)}} \, dx}{8 a^2}\\ &=-\frac {\cot (x)}{4 a \left (a \sin ^2(x)\right )^{3/2}}-\frac {3 \cot (x)}{8 a^2 \sqrt {a \sin ^2(x)}}+\frac {(3 \sin (x)) \int \csc (x) \, dx}{8 a^2 \sqrt {a \sin ^2(x)}}\\ &=-\frac {\cot (x)}{4 a \left (a \sin ^2(x)\right )^{3/2}}-\frac {3 \cot (x)}{8 a^2 \sqrt {a \sin ^2(x)}}-\frac {3 \tanh ^{-1}(\cos (x)) \sin (x)}{8 a^2 \sqrt {a \sin ^2(x)}}\\ \end {align*}
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Mathematica [A]
time = 0.16, size = 77, normalized size = 1.26 \begin {gather*} -\frac {\csc (x) \left (6 \csc ^2\left (\frac {x}{2}\right )+\csc ^4\left (\frac {x}{2}\right )+24 \left (\log \left (\cos \left (\frac {x}{2}\right )\right )-\log \left (\sin \left (\frac {x}{2}\right )\right )\right )-6 \sec ^2\left (\frac {x}{2}\right )-\sec ^4\left (\frac {x}{2}\right )\right ) \sqrt {a \sin ^2(x)}}{64 a^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 7.38, size = 89, normalized size = 1.46
method | result | size |
default | \(-\frac {\sqrt {a \left (\cos ^{2}\left (x \right )\right )}\, \left (3 \ln \left (\frac {2 \sqrt {a}\, \sqrt {a \left (\cos ^{2}\left (x \right )\right )}+2 a}{\sin \left (x \right )}\right ) a \left (\sin ^{4}\left (x \right )\right )+3 \sqrt {a \left (\cos ^{2}\left (x \right )\right )}\, \left (\sin ^{2}\left (x \right )\right ) \sqrt {a}+2 \sqrt {a}\, \sqrt {a \left (\cos ^{2}\left (x \right )\right )}\right )}{8 a^{\frac {7}{2}} \sin \left (x \right )^{3} \cos \left (x \right ) \sqrt {a \left (\sin ^{2}\left (x \right )\right )}}\) | \(89\) |
risch | \(-\frac {i \left (3 \,{\mathrm e}^{6 i x}-11 \,{\mathrm e}^{4 i x}-11 \,{\mathrm e}^{2 i x}+3\right )}{4 a^{2} \left ({\mathrm e}^{2 i x}-1\right )^{3} \sqrt {-a \left ({\mathrm e}^{2 i x}-1\right )^{2} {\mathrm e}^{-2 i x}}}+\frac {3 \ln \left ({\mathrm e}^{i x}-1\right ) \sin \left (x \right )}{4 a^{2} \sqrt {-a \left ({\mathrm e}^{2 i x}-1\right )^{2} {\mathrm e}^{-2 i x}}}-\frac {3 \ln \left ({\mathrm e}^{i x}+1\right ) \sin \left (x \right )}{4 a^{2} \sqrt {-a \left ({\mathrm e}^{2 i x}-1\right )^{2} {\mathrm e}^{-2 i x}}}\) | \(127\) |
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. 931 vs.
\(2 (49) = 98\).
time = 0.88, size = 931, normalized size = 15.26 \begin {gather*} -\frac {{\left (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 )} \arctan \left (\sin \left (x\right ), \cos \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 )} \arctan \left (\sin \left (x\right ), \cos \left (x\right ) - 1\right ) + 2 \, {\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 ) + 12 \, {\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 ) + 8 \, {\left (11 \, \sin \left (5 \, x\right ) + 11 \, \sin \left (3 \, x\right ) - 3 \, \sin \left (x\right )\right )} \cos \left (6 \, x\right ) + 44 \, {\left (3 \, \sin \left (4 \, x\right ) - 2 \, \sin \left (2 \, x\right )\right )} \cos \left (5 \, x\right ) - 12 \, {\left (11 \, \sin \left (3 \, x\right ) - 3 \, \sin \left (x\right )\right )} \cos \left (4 \, x\right ) - 2 \, {\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 ) - 6 \, {\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 ) - 8 \, {\left (11 \, \cos \left (5 \, x\right ) + 11 \, \cos \left (3 \, x\right ) - 3 \, \cos \left (x\right )\right )} \sin \left (6 \, x\right ) - 22 \, {\left (6 \, \cos \left (4 \, x\right ) - 4 \, \cos \left (2 \, x\right ) + 1\right )} \sin \left (5 \, x\right ) + 12 \, {\left (11 \, \cos \left (3 \, x\right ) - 3 \, \cos \left (x\right )\right )} \sin \left (4 \, x\right ) + 22 \, {\left (4 \, \cos \left (2 \, x\right ) - 1\right )} \sin \left (3 \, x\right ) - 88 \, \cos \left (3 \, x\right ) \sin \left (2 \, x\right ) + 24 \, \cos \left (x\right ) \sin \left (2 \, x\right ) - 24 \, \cos \left (2 \, x\right ) \sin \left (x\right ) + 6 \, \sin \left (x\right )\right )} \sqrt {-a}}{8 \, {\left (a^{3} \cos \left (8 \, x\right )^{2} + 16 \, a^{3} \cos \left (6 \, x\right )^{2} + 36 \, a^{3} \cos \left (4 \, x\right )^{2} + 16 \, a^{3} \cos \left (2 \, x\right )^{2} + a^{3} \sin \left (8 \, x\right )^{2} + 16 \, a^{3} \sin \left (6 \, x\right )^{2} + 36 \, a^{3} \sin \left (4 \, x\right )^{2} - 48 \, a^{3} \sin \left (4 \, x\right ) \sin \left (2 \, x\right ) + 16 \, a^{3} \sin \left (2 \, x\right )^{2} - 8 \, a^{3} \cos \left (2 \, x\right ) + a^{3} - 2 \, {\left (4 \, a^{3} \cos \left (6 \, x\right ) - 6 \, a^{3} \cos \left (4 \, x\right ) + 4 \, a^{3} \cos \left (2 \, x\right ) - a^{3}\right )} \cos \left (8 \, x\right ) - 8 \, {\left (6 \, a^{3} \cos \left (4 \, x\right ) - 4 \, a^{3} \cos \left (2 \, x\right ) + a^{3}\right )} \cos \left (6 \, x\right ) - 12 \, {\left (4 \, a^{3} \cos \left (2 \, x\right ) - a^{3}\right )} \cos \left (4 \, x\right ) - 4 \, {\left (2 \, a^{3} \sin \left (6 \, x\right ) - 3 \, a^{3} \sin \left (4 \, x\right ) + 2 \, a^{3} \sin \left (2 \, x\right )\right )} \sin \left (8 \, x\right ) - 16 \, {\left (3 \, a^{3} \sin \left (4 \, x\right ) - 2 \, a^{3} \sin \left (2 \, x\right )\right )} \sin \left (6 \, x\right )\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.42, size = 78, normalized size = 1.28 \begin {gather*} \frac {\sqrt {-a \cos \left (x\right )^{2} + a} {\left (6 \, \cos \left (x\right )^{3} + 3 \, {\left (\cos \left (x\right )^{4} - 2 \, \cos \left (x\right )^{2} + 1\right )} \log \left (-\frac {\cos \left (x\right ) - 1}{\cos \left (x\right ) + 1}\right ) - 10 \, \cos \left (x\right )\right )}}{16 \, {\left (a^{3} \cos \left (x\right )^{4} - 2 \, a^{3} \cos \left (x\right )^{2} + a^{3}\right )} \sin \left (x\right )} \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 )}\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. 126 vs.
\(2 (49) = 98\).
time = 0.46, size = 126, normalized size = 2.07 \begin {gather*} \frac {3 \, \log \left (-\frac {\cos \left (x\right ) - 1}{\cos \left (x\right ) + 1}\right )}{16 \, a^{\frac {5}{2}} \mathrm {sgn}\left (\sin \left (x\right )\right )} - \frac {{\left (\sqrt {a} - \frac {8 \, \sqrt {a} {\left (\cos \left (x\right ) - 1\right )}}{\cos \left (x\right ) + 1} + \frac {18 \, \sqrt {a} {\left (\cos \left (x\right ) - 1\right )}^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}}\right )} {\left (\cos \left (x\right ) + 1\right )}^{2}}{64 \, a^{3} {\left (\cos \left (x\right ) - 1\right )}^{2} \mathrm {sgn}\left (\sin \left (x\right )\right )} - \frac {\frac {8 \, a^{\frac {7}{2}} {\left (\cos \left (x\right ) - 1\right )} \mathrm {sgn}\left (\sin \left (x\right )\right )}{\cos \left (x\right ) + 1} - \frac {a^{\frac {7}{2}} {\left (\cos \left (x\right ) - 1\right )}^{2} \mathrm {sgn}\left (\sin \left (x\right )\right )}{{\left (\cos \left (x\right ) + 1\right )}^{2}}}{64 \, a^{6}} \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\,{\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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