Optimal. Leaf size=40 \[ \frac {\sin ^{\frac {3}{2}}(2 a+2 b x)}{6 b}+\frac {E\left (\left .a+b x-\frac {\pi }{4}\right |2\right )}{2 b} \]
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Rubi [A] time = 0.03, antiderivative size = 40, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {4297, 2639} \[ \frac {\sin ^{\frac {3}{2}}(2 a+2 b x)}{6 b}+\frac {E\left (\left .a+b x-\frac {\pi }{4}\right |2\right )}{2 b} \]
Antiderivative was successfully verified.
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Rule 2639
Rule 4297
Rubi steps
\begin {align*} \int \cos ^2(a+b x) \sqrt {\sin (2 a+2 b x)} \, dx &=\frac {\sin ^{\frac {3}{2}}(2 a+2 b x)}{6 b}+\frac {1}{2} \int \sqrt {\sin (2 a+2 b x)} \, dx\\ &=\frac {E\left (\left .a-\frac {\pi }{4}+b x\right |2\right )}{2 b}+\frac {\sin ^{\frac {3}{2}}(2 a+2 b x)}{6 b}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 34, normalized size = 0.85 \[ \frac {\sin ^{\frac {3}{2}}(2 (a+b x))+3 E\left (\left .a+b x-\frac {\pi }{4}\right |2\right )}{6 b} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.54, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\cos \left (b x + a\right )^{2} \sqrt {\sin \left (2 \, b x + 2 \, a\right )}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \cos \left (b x + a\right )^{2} \sqrt {\sin \left (2 \, b x + 2 \, a\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 7.57, size = 24858211, normalized size = 621455.28 \[ \text {output too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \cos \left (b x + a\right )^{2} \sqrt {\sin \left (2 \, b x + 2 \, a\right )}\,{d x} \]
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 \[ \int {\cos \left (a+b\,x\right )}^2\,\sqrt {\sin \left (2\,a+2\,b\,x\right )} \,d x \]
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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