Optimal. Leaf size=94 \[ \frac{2 c (c \sin (a+b x))^{3/2}}{b d \sqrt{d \cos (a+b x)}}-\frac{3 c^2 E\left (\left .a+b x-\frac{\pi }{4}\right |2\right ) \sqrt{c \sin (a+b x)} \sqrt{d \cos (a+b x)}}{b d^2 \sqrt{\sin (2 a+2 b x)}} \]
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Rubi [A] time = 0.12023, antiderivative size = 94, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.12, Rules used = {2566, 2572, 2639} \[ \frac{2 c (c \sin (a+b x))^{3/2}}{b d \sqrt{d \cos (a+b x)}}-\frac{3 c^2 E\left (\left .a+b x-\frac{\pi }{4}\right |2\right ) \sqrt{c \sin (a+b x)} \sqrt{d \cos (a+b x)}}{b d^2 \sqrt{\sin (2 a+2 b x)}} \]
Antiderivative was successfully verified.
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Rule 2566
Rule 2572
Rule 2639
Rubi steps
\begin{align*} \int \frac{(c \sin (a+b x))^{5/2}}{(d \cos (a+b x))^{3/2}} \, dx &=\frac{2 c (c \sin (a+b x))^{3/2}}{b d \sqrt{d \cos (a+b x)}}-\frac{\left (3 c^2\right ) \int \sqrt{d \cos (a+b x)} \sqrt{c \sin (a+b x)} \, dx}{d^2}\\ &=\frac{2 c (c \sin (a+b x))^{3/2}}{b d \sqrt{d \cos (a+b x)}}-\frac{\left (3 c^2 \sqrt{d \cos (a+b x)} \sqrt{c \sin (a+b x)}\right ) \int \sqrt{\sin (2 a+2 b x)} \, dx}{d^2 \sqrt{\sin (2 a+2 b x)}}\\ &=\frac{2 c (c \sin (a+b x))^{3/2}}{b d \sqrt{d \cos (a+b x)}}-\frac{3 c^2 \sqrt{d \cos (a+b x)} E\left (\left .a-\frac{\pi }{4}+b x\right |2\right ) \sqrt{c \sin (a+b x)}}{b d^2 \sqrt{\sin (2 a+2 b x)}}\\ \end{align*}
Mathematica [C] time = 0.120092, size = 67, normalized size = 0.71 \[ \frac{2 \sqrt [4]{\cos ^2(a+b x)} (c \sin (a+b x))^{7/2} \, _2F_1\left (\frac{5}{4},\frac{7}{4};\frac{11}{4};\sin ^2(a+b x)\right )}{7 b c d \sqrt{d \cos (a+b x)}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.093, size = 508, normalized size = 5.4 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (c \sin \left (b x + a\right )\right )^{\frac{5}{2}}}{\left (d \cos \left (b x + a\right )\right )^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{{\left (c^{2} \cos \left (b x + a\right )^{2} - c^{2}\right )} \sqrt{d \cos \left (b x + a\right )} \sqrt{c \sin \left (b x + a\right )}}{d^{2} \cos \left (b x + a\right )^{2}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (c \sin \left (b x + a\right )\right )^{\frac{5}{2}}}{\left (d \cos \left (b x + a\right )\right )^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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