Optimal. Leaf size=136 \[ -\frac {7 \text {ArcSin}(\cos (a+b x)-\sin (a+b x))}{64 b}+\frac {7 \log \left (\cos (a+b x)+\sin (a+b x)+\sqrt {\sin (2 a+2 b x)}\right )}{64 b}-\frac {7 \cos (a+b x) \sqrt {\sin (2 a+2 b x)}}{32 b}+\frac {7 \sin (a+b x) \sin ^{\frac {3}{2}}(2 a+2 b x)}{48 b}+\frac {\cos (a+b x) \sin ^{\frac {5}{2}}(2 a+2 b x)}{12 b} \]
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Rubi [A]
time = 0.07, antiderivative size = 136, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {4382, 4386,
4387, 4390} \begin {gather*} -\frac {7 \text {ArcSin}(\cos (a+b x)-\sin (a+b x))}{64 b}+\frac {7 \sin (a+b x) \sin ^{\frac {3}{2}}(2 a+2 b x)}{48 b}+\frac {\sin ^{\frac {5}{2}}(2 a+2 b x) \cos (a+b x)}{12 b}-\frac {7 \sqrt {\sin (2 a+2 b x)} \cos (a+b x)}{32 b}+\frac {7 \log \left (\sin (a+b x)+\sqrt {\sin (2 a+2 b x)}+\cos (a+b x)\right )}{64 b} \end {gather*}
Antiderivative was successfully verified.
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Rule 4382
Rule 4386
Rule 4387
Rule 4390
Rubi steps
\begin {align*} \int \cos ^3(a+b x) \sin ^{\frac {3}{2}}(2 a+2 b x) \, dx &=\frac {\cos (a+b x) \sin ^{\frac {5}{2}}(2 a+2 b x)}{12 b}+\frac {7}{12} \int \cos (a+b x) \sin ^{\frac {3}{2}}(2 a+2 b x) \, dx\\ &=\frac {7 \sin (a+b x) \sin ^{\frac {3}{2}}(2 a+2 b x)}{48 b}+\frac {\cos (a+b x) \sin ^{\frac {5}{2}}(2 a+2 b x)}{12 b}+\frac {7}{16} \int \sin (a+b x) \sqrt {\sin (2 a+2 b x)} \, dx\\ &=-\frac {7 \cos (a+b x) \sqrt {\sin (2 a+2 b x)}}{32 b}+\frac {7 \sin (a+b x) \sin ^{\frac {3}{2}}(2 a+2 b x)}{48 b}+\frac {\cos (a+b x) \sin ^{\frac {5}{2}}(2 a+2 b x)}{12 b}+\frac {7}{32} \int \frac {\cos (a+b x)}{\sqrt {\sin (2 a+2 b x)}} \, dx\\ &=-\frac {7 \sin ^{-1}(\cos (a+b x)-\sin (a+b x))}{64 b}+\frac {7 \log \left (\cos (a+b x)+\sin (a+b x)+\sqrt {\sin (2 a+2 b x)}\right )}{64 b}-\frac {7 \cos (a+b x) \sqrt {\sin (2 a+2 b x)}}{32 b}+\frac {7 \sin (a+b x) \sin ^{\frac {3}{2}}(2 a+2 b x)}{48 b}+\frac {\cos (a+b x) \sin ^{\frac {5}{2}}(2 a+2 b x)}{12 b}\\ \end {align*}
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Mathematica [A]
time = 0.37, size = 99, normalized size = 0.73 \begin {gather*} \frac {-7 \text {ArcSin}(\cos (a+b x)-\sin (a+b x))+7 \log \left (\cos (a+b x)+\sin (a+b x)+\sqrt {\sin (2 (a+b x))}\right )-\frac {2}{3} (10 \cos (a+b x)+9 \cos (3 (a+b x))+2 \cos (5 (a+b x))) \sqrt {\sin (2 (a+b x))}}{64 b} \end {gather*}
Antiderivative was successfully verified.
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Maple [F(-1)]
time = 180.00, size = 0, normalized size = 0.00 hanged
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 [B] Leaf count of result is larger than twice the leaf count of optimal. 291 vs.
\(2 (118) = 236\).
time = 1.95, size = 291, normalized size = 2.14 \begin {gather*} -\frac {8 \, \sqrt {2} {\left (32 \, \cos \left (b x + a\right )^{5} - 4 \, \cos \left (b x + a\right )^{3} - 7 \, \cos \left (b x + a\right )\right )} \sqrt {\cos \left (b x + a\right ) \sin \left (b x + a\right )} - 42 \, \arctan \left (-\frac {\sqrt {2} \sqrt {\cos \left (b x + a\right ) \sin \left (b x + a\right )} {\left (\cos \left (b x + a\right ) - \sin \left (b x + a\right )\right )} + \cos \left (b x + a\right ) \sin \left (b x + a\right )}{\cos \left (b x + a\right )^{2} + 2 \, \cos \left (b x + a\right ) \sin \left (b x + a\right ) - 1}\right ) + 42 \, \arctan \left (-\frac {2 \, \sqrt {2} \sqrt {\cos \left (b x + a\right ) \sin \left (b x + a\right )} - \cos \left (b x + a\right ) - \sin \left (b x + a\right )}{\cos \left (b x + a\right ) - \sin \left (b x + a\right )}\right ) + 21 \, \log \left (-32 \, \cos \left (b x + a\right )^{4} + 4 \, \sqrt {2} {\left (4 \, \cos \left (b x + a\right )^{3} - {\left (4 \, \cos \left (b x + a\right )^{2} + 1\right )} \sin \left (b x + a\right ) - 5 \, \cos \left (b x + a\right )\right )} \sqrt {\cos \left (b x + a\right ) \sin \left (b x + a\right )} + 32 \, \cos \left (b x + a\right )^{2} + 16 \, \cos \left (b x + a\right ) \sin \left (b x + a\right ) + 1\right )}{768 \, b} \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 )}^3\,{\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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