Optimal. Leaf size=78 \[ -\frac {5}{24} a \sin (x) \cos (x) \sqrt {a \sin ^4(x)}-\frac {1}{6} a \sin ^3(x) \cos (x) \sqrt {a \sin ^4(x)}-\frac {5}{16} a \cot (x) \sqrt {a \sin ^4(x)}+\frac {5}{16} a x \csc ^2(x) \sqrt {a \sin ^4(x)} \]
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Rubi [A] time = 0.03, antiderivative size = 78, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {3207, 2635, 8} \[ -\frac {1}{6} a \sin ^3(x) \cos (x) \sqrt {a \sin ^4(x)}-\frac {5}{24} a \sin (x) \cos (x) \sqrt {a \sin ^4(x)}-\frac {5}{16} a \cot (x) \sqrt {a \sin ^4(x)}+\frac {5}{16} a x \csc ^2(x) \sqrt {a \sin ^4(x)} \]
Antiderivative was successfully verified.
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Rule 8
Rule 2635
Rule 3207
Rubi steps
\begin {align*} \int \left (a \sin ^4(x)\right )^{3/2} \, dx &=\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*}
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Mathematica [A] time = 0.10, size = 38, normalized size = 0.49 \[ -\frac {1}{192} (-60 x+45 \sin (2 x)-9 \sin (4 x)+\sin (6 x)) \csc ^6(x) \left (a \sin ^4(x)\right )^{3/2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.44, size = 56, normalized size = 0.72 \[ -\frac {\sqrt {a \cos \relax (x)^{4} - 2 \, a \cos \relax (x)^{2} + a} {\left (15 \, a x - {\left (8 \, a \cos \relax (x)^{5} - 26 \, a \cos \relax (x)^{3} + 33 \, a \cos \relax (x)\right )} \sin \relax (x)\right )}}{48 \, {\left (\cos \relax (x)^{2} - 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.13, size = 27, normalized size = 0.35 \[ \frac {1}{192} \, a^{\frac {3}{2}} {\left (60 \, x - \sin \left (6 \, x\right ) + 9 \, \sin \left (4 \, x\right ) - 45 \, \sin \left (2 \, x\right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.28, size = 47, normalized size = 0.60 \[ -\frac {\left (a \left (1-\left (\cos ^{2}\relax (x )\right )\right )^{2}\right )^{\frac {3}{2}} \left (8 \left (\cos ^{5}\relax (x )\right ) \sin \relax (x )-26 \left (\cos ^{3}\relax (x )\right ) \sin \relax (x )+33 \sin \relax (x ) \cos \relax (x )-15 x \right )}{48 \sin \relax (x )^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.61, size = 55, normalized size = 0.71 \[ \frac {5}{16} \, a^{\frac {3}{2}} x - \frac {33 \, a^{\frac {3}{2}} \tan \relax (x)^{5} + 40 \, a^{\frac {3}{2}} \tan \relax (x)^{3} + 15 \, a^{\frac {3}{2}} \tan \relax (x)}{48 \, {\left (\tan \relax (x)^{6} + 3 \, \tan \relax (x)^{4} + 3 \, \tan \relax (x)^{2} + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int {\left (a\,{\sin \relax (x)}^4\right )}^{3/2} \,d x \]
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 \[ \int \left (a \sin ^{4}{\relax (x )}\right )^{\frac {3}{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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