Optimal. Leaf size=69 \[ \frac {F\left (\left .a-\frac {\pi }{4}+b x\right |2\right )}{6 b}-\frac {\cos (2 a+2 b x) \sqrt {\sin (2 a+2 b x)}}{6 b}+\frac {\sin ^{\frac {5}{2}}(2 a+2 b x)}{10 b} \]
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Rubi [A]
time = 0.04, antiderivative size = 69, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {4382, 2715,
2720} \begin {gather*} \frac {\sin ^{\frac {5}{2}}(2 a+2 b x)}{10 b}+\frac {F\left (\left .a+b x-\frac {\pi }{4}\right |2\right )}{6 b}-\frac {\sqrt {\sin (2 a+2 b x)} \cos (2 a+2 b x)}{6 b} \end {gather*}
Antiderivative was successfully verified.
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Rule 2715
Rule 2720
Rule 4382
Rubi steps
\begin {align*} \int \cos ^2(a+b x) \sin ^{\frac {3}{2}}(2 a+2 b x) \, dx &=\frac {\sin ^{\frac {5}{2}}(2 a+2 b x)}{10 b}+\frac {1}{2} \int \sin ^{\frac {3}{2}}(2 a+2 b x) \, dx\\ &=-\frac {\cos (2 a+2 b x) \sqrt {\sin (2 a+2 b x)}}{6 b}+\frac {\sin ^{\frac {5}{2}}(2 a+2 b x)}{10 b}+\frac {1}{6} \int \frac {1}{\sqrt {\sin (2 a+2 b x)}} \, dx\\ &=\frac {F\left (\left .a-\frac {\pi }{4}+b x\right |2\right )}{6 b}-\frac {\cos (2 a+2 b x) \sqrt {\sin (2 a+2 b x)}}{6 b}+\frac {\sin ^{\frac {5}{2}}(2 a+2 b x)}{10 b}\\ \end {align*}
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Mathematica [A]
time = 0.33, size = 76, normalized size = 1.10 \begin {gather*} \frac {20 F\left (\left .a-\frac {\pi }{4}+b x\right |2\right ) \sqrt {\sin (2 (a+b x))}+9 \sin (2 (a+b x))-10 \sin (4 (a+b x))-3 \sin (6 (a+b x))}{120 b \sqrt {\sin (2 (a+b x))}} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] result has leaf size over 500,000. Avoiding possible recursion issues.
time = 71.64, size = 189673602, normalized size = 2748892.78
method | result | size |
default | \(\text {Expression too large to display}\) | \(189673602\) |
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int {\cos \left (a+b\,x\right )}^2\,{\sin \left (2\,a+2\,b\,x\right )}^{3/2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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