Integrand size = 10, antiderivative size = 78 \[ \int \left (a \sin ^4(x)\right )^{3/2} \, dx=-\frac {5}{16} a \cot (x) \sqrt {a \sin ^4(x)}+\frac {5}{16} a x \csc ^2(x) \sqrt {a \sin ^4(x)}-\frac {5}{24} a \cos (x) \sin (x) \sqrt {a \sin ^4(x)}-\frac {1}{6} a \cos (x) \sin ^3(x) \sqrt {a \sin ^4(x)} \]
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Time = 0.02 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {3286, 2715, 8} \[ \int \left (a \sin ^4(x)\right )^{3/2} \, dx=-\frac {5}{24} a \sin (x) \cos (x) \sqrt {a \sin ^4(x)}-\frac {1}{6} a \sin ^3(x) \cos (x) \sqrt {a \sin ^4(x)}-\frac {5}{16} a \cot (x) \sqrt {a \sin ^4(x)}+\frac {5}{16} a x \csc ^2(x) \sqrt {a \sin ^4(x)} \]
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Rule 8
Rule 2715
Rule 3286
Rubi steps \begin{align*} \text {integral}& = \left (a \csc ^2(x) \sqrt {a \sin ^4(x)}\right ) \int \sin ^6(x) \, dx \\ & = -\frac {1}{6} a \cos (x) \sin ^3(x) \sqrt {a \sin ^4(x)}+\frac {1}{6} \left (5 a \csc ^2(x) \sqrt {a \sin ^4(x)}\right ) \int \sin ^4(x) \, dx \\ & = -\frac {5}{24} a \cos (x) \sin (x) \sqrt {a \sin ^4(x)}-\frac {1}{6} a \cos (x) \sin ^3(x) \sqrt {a \sin ^4(x)}+\frac {1}{8} \left (5 a \csc ^2(x) \sqrt {a \sin ^4(x)}\right ) \int \sin ^2(x) \, dx \\ & = -\frac {5}{16} a \cot (x) \sqrt {a \sin ^4(x)}-\frac {5}{24} a \cos (x) \sin (x) \sqrt {a \sin ^4(x)}-\frac {1}{6} a \cos (x) \sin ^3(x) \sqrt {a \sin ^4(x)}+\frac {1}{16} \left (5 a \csc ^2(x) \sqrt {a \sin ^4(x)}\right ) \int 1 \, dx \\ & = -\frac {5}{16} a \cot (x) \sqrt {a \sin ^4(x)}+\frac {5}{16} a x \csc ^2(x) \sqrt {a \sin ^4(x)}-\frac {5}{24} a \cos (x) \sin (x) \sqrt {a \sin ^4(x)}-\frac {1}{6} a \cos (x) \sin ^3(x) \sqrt {a \sin ^4(x)} \\ \end{align*}
Time = 0.08 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.49 \[ \int \left (a \sin ^4(x)\right )^{3/2} \, dx=-\frac {1}{192} \csc ^6(x) \left (a \sin ^4(x)\right )^{3/2} (-60 x+45 \sin (2 x)-9 \sin (4 x)+\sin (6 x)) \]
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Time = 0.58 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.55
method | result | size |
default | \(-\frac {a \sqrt {a \left (\sin ^{4}\left (x \right )\right )}\, \left (8 \left (\cos ^{4}\left (x \right )\right ) \cot \left (x \right )-26 \left (\cos ^{2}\left (x \right )\right ) \cot \left (x \right )+33 \cot \left (x \right )-15 \left (\csc ^{2}\left (x \right )\right ) x \right ) \sqrt {16}}{192}\) | \(43\) |
risch | \(-\frac {5 a \,{\mathrm e}^{2 i x} \sqrt {a \left ({\mathrm e}^{2 i x}-1\right )^{4} {\mathrm e}^{-4 i x}}\, x}{16 \left ({\mathrm e}^{2 i x}-1\right )^{2}}-\frac {i a \,{\mathrm e}^{8 i x} \sqrt {a \left ({\mathrm e}^{2 i x}-1\right )^{4} {\mathrm e}^{-4 i x}}}{384 \left ({\mathrm e}^{2 i x}-1\right )^{2}}+\frac {3 i a \,{\mathrm e}^{6 i x} \sqrt {a \left ({\mathrm e}^{2 i x}-1\right )^{4} {\mathrm e}^{-4 i x}}}{128 \left ({\mathrm e}^{2 i x}-1\right )^{2}}-\frac {15 i a \,{\mathrm e}^{4 i x} \sqrt {a \left ({\mathrm e}^{2 i x}-1\right )^{4} {\mathrm e}^{-4 i x}}}{128 \left ({\mathrm e}^{2 i x}-1\right )^{2}}+\frac {15 i a \sqrt {a \left ({\mathrm e}^{2 i x}-1\right )^{4} {\mathrm e}^{-4 i x}}}{128 \left ({\mathrm e}^{2 i x}-1\right )^{2}}-\frac {3 i a \,{\mathrm e}^{-2 i x} \sqrt {a \left ({\mathrm e}^{2 i x}-1\right )^{4} {\mathrm e}^{-4 i x}}}{128 \left ({\mathrm e}^{2 i x}-1\right )^{2}}+\frac {i a \,{\mathrm e}^{-4 i x} \sqrt {a \left ({\mathrm e}^{2 i x}-1\right )^{4} {\mathrm e}^{-4 i x}}}{384 \left ({\mathrm e}^{2 i x}-1\right )^{2}}\) | \(249\) |
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Time = 0.27 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.72 \[ \int \left (a \sin ^4(x)\right )^{3/2} \, dx=-\frac {\sqrt {a \cos \left (x\right )^{4} - 2 \, a \cos \left (x\right )^{2} + a} {\left (15 \, a x - {\left (8 \, a \cos \left (x\right )^{5} - 26 \, a \cos \left (x\right )^{3} + 33 \, a \cos \left (x\right )\right )} \sin \left (x\right )\right )}}{48 \, {\left (\cos \left (x\right )^{2} - 1\right )}} \]
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\[ \int \left (a \sin ^4(x)\right )^{3/2} \, dx=\int \left (a \sin ^{4}{\left (x \right )}\right )^{\frac {3}{2}}\, dx \]
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Time = 0.32 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.71 \[ \int \left (a \sin ^4(x)\right )^{3/2} \, dx=\frac {5}{16} \, a^{\frac {3}{2}} x - \frac {33 \, a^{\frac {3}{2}} \tan \left (x\right )^{5} + 40 \, a^{\frac {3}{2}} \tan \left (x\right )^{3} + 15 \, a^{\frac {3}{2}} \tan \left (x\right )}{48 \, {\left (\tan \left (x\right )^{6} + 3 \, \tan \left (x\right )^{4} + 3 \, \tan \left (x\right )^{2} + 1\right )}} \]
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Time = 0.31 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.35 \[ \int \left (a \sin ^4(x)\right )^{3/2} \, dx=\frac {1}{192} \, a^{\frac {3}{2}} {\left (60 \, x - \sin \left (6 \, x\right ) + 9 \, \sin \left (4 \, x\right ) - 45 \, \sin \left (2 \, x\right )\right )} \]
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Timed out. \[ \int \left (a \sin ^4(x)\right )^{3/2} \, dx=\int {\left (a\,{\sin \left (x\right )}^4\right )}^{3/2} \,d x \]
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