Optimal. Leaf size=33 \[ \frac {5}{36} \left (2-3 \sin ^2(x)\right )^{8/5}-\frac {20}{117} \left (2-3 \sin ^2(x)\right )^{13/5} \]
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Rubi [A] time = 0.06, antiderivative size = 33, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {12, 444, 43} \[ \frac {5}{36} \left (2-3 \sin ^2(x)\right )^{8/5}-\frac {20}{117} \left (2-3 \sin ^2(x)\right )^{13/5} \]
Antiderivative was successfully verified.
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Rule 12
Rule 43
Rule 444
Rubi steps
\begin {align*} \int \left (2-3 \sin ^2(x)\right )^{3/5} \sin (4 x) \, dx &=\operatorname {Subst}\left (\int 4 x \left (2-3 x^2\right )^{3/5} \left (1-2 x^2\right ) \, dx,x,\sin (x)\right )\\ &=4 \operatorname {Subst}\left (\int x \left (2-3 x^2\right )^{3/5} \left (1-2 x^2\right ) \, dx,x,\sin (x)\right )\\ &=2 \operatorname {Subst}\left (\int (2-3 x)^{3/5} (1-2 x) \, dx,x,\sin ^2(x)\right )\\ &=2 \operatorname {Subst}\left (\int \left (-\frac {1}{3} (2-3 x)^{3/5}+\frac {2}{3} (2-3 x)^{8/5}\right ) \, dx,x,\sin ^2(x)\right )\\ &=\frac {5}{36} \left (2-3 \sin ^2(x)\right )^{8/5}-\frac {20}{117} \left (2-3 \sin ^2(x)\right )^{13/5}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 29, normalized size = 0.88 \[ -\frac {5 (3 \cos (2 x)+1)^{8/5} (24 \cos (2 x)-5)}{936\ 2^{3/5}} \]
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (2-3 \sin ^2(x)\right )^{3/5} \sin (4 x) \, dx \]
Verification is Not applicable to the result.
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fricas [A] time = 1.50, size = 26, normalized size = 0.79 \[ -\frac {5}{468} \, {\left (144 \, \cos \relax (x)^{4} - 135 \, \cos \relax (x)^{2} + 29\right )} {\left (3 \, \cos \relax (x)^{2} - 1\right )}^{\frac {3}{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.66, size = 35, normalized size = 1.06 \[ -\frac {20}{117} \, {\left (3 \, \sin \relax (x)^{2} - 2\right )}^{2} {\left (-3 \, \sin \relax (x)^{2} + 2\right )}^{\frac {3}{5}} + \frac {5}{36} \, {\left (-3 \, \sin \relax (x)^{2} + 2\right )}^{\frac {8}{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.25, size = 38, normalized size = 1.15
method | result | size |
default | \(\frac {5 \left (3 \left (\cos ^{2}\relax (x )\right )-1\right )^{\frac {8}{5}}}{12}-\frac {20 \left (\frac {1}{2}+\frac {3 \cos \left (2 x \right )}{2}\right )^{\frac {13}{5}}}{117}-\frac {5 \left (\frac {1}{2}+\frac {3 \cos \left (2 x \right )}{2}\right )^{\frac {8}{5}}}{18}\) | \(38\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.44, size = 25, normalized size = 0.76 \[ -\frac {20}{117} \, {\left (-3 \, \sin \relax (x)^{2} + 2\right )}^{\frac {13}{5}} + \frac {5}{36} \, {\left (-3 \, \sin \relax (x)^{2} + 2\right )}^{\frac {8}{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.03 \[ \int \sin \left (4\,x\right )\,{\left (2-3\,{\sin \relax (x)}^2\right )}^{3/5} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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