Optimal. Leaf size=18 \[ \frac {1}{7} \left (a^2-4 \cos ^2(x)\right )^{7/4} \]
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Rubi [A] time = 0.05, antiderivative size = 19, normalized size of antiderivative = 1.06, number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {12, 261} \[ \frac {1}{7} \left (a^2+4 \sin ^2(x)-4\right )^{7/4} \]
Antiderivative was successfully verified.
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Rule 12
Rule 261
Rubi steps
\begin {align*} \int \left (a^2-4 \cos ^2(x)\right )^{3/4} \sin (2 x) \, dx &=\operatorname {Subst}\left (\int 2 x \left (-4+a^2+4 x^2\right )^{3/4} \, dx,x,\sin (x)\right )\\ &=2 \operatorname {Subst}\left (\int x \left (-4+a^2+4 x^2\right )^{3/4} \, dx,x,\sin (x)\right )\\ &=\frac {1}{7} \left (-4+a^2+4 \sin ^2(x)\right )^{7/4}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 19, normalized size = 1.06 \[ \frac {1}{7} \left (a^2+4 \sin ^2(x)-4\right )^{7/4} \]
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (a^2-4 \cos ^2(x)\right )^{3/4} \sin (2 x) \, dx \]
Verification is Not applicable to the result.
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fricas [A] time = 1.06, size = 14, normalized size = 0.78 \[ \frac {1}{7} \, {\left (a^{2} - 4 \, \cos \relax (x)^{2}\right )}^{\frac {7}{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.00, size = 14, normalized size = 0.78 \[ \frac {1}{7} \, {\left (a^{2} - 4 \, \cos \relax (x)^{2}\right )}^{\frac {7}{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 15, normalized size = 0.83
method | result | size |
derivativedivides | \(\frac {\left (a^{2}-4 \left (\cos ^{2}\relax (x )\right )\right )^{\frac {7}{4}}}{7}\) | \(15\) |
default | \(\frac {\left (a^{2}-4 \left (\cos ^{2}\relax (x )\right )\right )^{\frac {7}{4}}}{7}\) | \(15\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.43, size = 14, normalized size = 0.78 \[ \frac {1}{7} \, {\left (a^{2} - 4 \, \cos \relax (x)^{2}\right )}^{\frac {7}{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.31, size = 14, normalized size = 0.78 \[ \frac {{\left (a^2-4\,{\cos \relax (x)}^2\right )}^{7/4}}{7} \]
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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