Optimal. Leaf size=72 \[ \frac {b \sin (c+d x) \sqrt {b \cos (c+d x)}}{d \sqrt {\cos (c+d x)}}-\frac {b \sin ^3(c+d x) \sqrt {b \cos (c+d x)}}{3 d \sqrt {\cos (c+d x)}} \]
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Rubi [A] time = 0.02, antiderivative size = 72, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {17, 2633} \[ \frac {b \sin (c+d x) \sqrt {b \cos (c+d x)}}{d \sqrt {\cos (c+d x)}}-\frac {b \sin ^3(c+d x) \sqrt {b \cos (c+d x)}}{3 d \sqrt {\cos (c+d x)}} \]
Antiderivative was successfully verified.
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Rule 17
Rule 2633
Rubi steps
\begin {align*} \int \cos ^{\frac {3}{2}}(c+d x) (b \cos (c+d x))^{3/2} \, dx &=\frac {\left (b \sqrt {b \cos (c+d x)}\right ) \int \cos ^3(c+d x) \, dx}{\sqrt {\cos (c+d x)}}\\ &=-\frac {\left (b \sqrt {b \cos (c+d x)}\right ) \operatorname {Subst}\left (\int \left (1-x^2\right ) \, dx,x,-\sin (c+d x)\right )}{d \sqrt {\cos (c+d x)}}\\ &=\frac {b \sqrt {b \cos (c+d x)} \sin (c+d x)}{d \sqrt {\cos (c+d x)}}-\frac {b \sqrt {b \cos (c+d x)} \sin ^3(c+d x)}{3 d \sqrt {\cos (c+d x)}}\\ \end {align*}
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Mathematica [A] time = 0.11, size = 45, normalized size = 0.62 \[ \frac {\sin (c+d x) (\cos (2 (c+d x))+5) (b \cos (c+d x))^{3/2}}{6 d \cos ^{\frac {3}{2}}(c+d x)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.55, size = 43, normalized size = 0.60 \[ \frac {{\left (b \cos \left (d x + c\right )^{2} + 2 \, b\right )} \sqrt {b \cos \left (d x + c\right )} \sin \left (d x + c\right )}{3 \, d \sqrt {\cos \left (d x + c\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [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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maple [A] time = 0.08, size = 40, normalized size = 0.56 \[ \frac {\left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right ) \left (b \cos \left (d x +c \right )\right )^{\frac {3}{2}}}{3 d \cos \left (d x +c \right )^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.25, size = 45, normalized size = 0.62 \[ \frac {{\left (b \sin \left (3 \, d x + 3 \, c\right ) + 9 \, b \sin \left (\frac {1}{3} \, \arctan \left (\sin \left (3 \, d x + 3 \, c\right ), \cos \left (3 \, d x + 3 \, c\right )\right )\right )\right )} \sqrt {b}}{12 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.70, size = 58, normalized size = 0.81 \[ \frac {b\,\sqrt {\cos \left (c+d\,x\right )}\,\sqrt {b\,\cos \left (c+d\,x\right )}\,\left (10\,\sin \left (2\,c+2\,d\,x\right )+\sin \left (4\,c+4\,d\,x\right )\right )}{12\,d\,\left (\cos \left (2\,c+2\,d\,x\right )+1\right )} \]
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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