Optimal. Leaf size=98 \[ \frac{2 c \sqrt{c \sin (a+b x)}}{3 b d (d \cos (a+b x))^{3/2}}-\frac{c^2 \sqrt{\sin (2 a+2 b x)} F\left (\left .a+b x-\frac{\pi }{4}\right |2\right )}{3 b d^2 \sqrt{c \sin (a+b x)} \sqrt{d \cos (a+b x)}} \]
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Rubi [A] time = 0.122679, antiderivative size = 98, 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, 2573, 2641} \[ \frac{2 c \sqrt{c \sin (a+b x)}}{3 b d (d \cos (a+b x))^{3/2}}-\frac{c^2 \sqrt{\sin (2 a+2 b x)} F\left (\left .a+b x-\frac{\pi }{4}\right |2\right )}{3 b d^2 \sqrt{c \sin (a+b x)} \sqrt{d \cos (a+b x)}} \]
Antiderivative was successfully verified.
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Rule 2566
Rule 2573
Rule 2641
Rubi steps
\begin{align*} \int \frac{(c \sin (a+b x))^{3/2}}{(d \cos (a+b x))^{5/2}} \, dx &=\frac{2 c \sqrt{c \sin (a+b x)}}{3 b d (d \cos (a+b x))^{3/2}}-\frac{c^2 \int \frac{1}{\sqrt{d \cos (a+b x)} \sqrt{c \sin (a+b x)}} \, dx}{3 d^2}\\ &=\frac{2 c \sqrt{c \sin (a+b x)}}{3 b d (d \cos (a+b x))^{3/2}}-\frac{\left (c^2 \sqrt{\sin (2 a+2 b x)}\right ) \int \frac{1}{\sqrt{\sin (2 a+2 b x)}} \, dx}{3 d^2 \sqrt{d \cos (a+b x)} \sqrt{c \sin (a+b x)}}\\ &=\frac{2 c \sqrt{c \sin (a+b x)}}{3 b d (d \cos (a+b x))^{3/2}}-\frac{c^2 F\left (\left .a-\frac{\pi }{4}+b x\right |2\right ) \sqrt{\sin (2 a+2 b x)}}{3 b d^2 \sqrt{d \cos (a+b x)} \sqrt{c \sin (a+b x)}}\\ \end{align*}
Mathematica [C] time = 0.16754, size = 67, normalized size = 0.68 \[ \frac{2 \cos ^2(a+b x)^{3/4} (c \sin (a+b x))^{5/2} \, _2F_1\left (\frac{5}{4},\frac{7}{4};\frac{9}{4};\sin ^2(a+b x)\right )}{5 b c d (d \cos (a+b x))^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.093, size = 186, normalized size = 1.9 \begin{align*}{\frac{\cos \left ( bx+a \right ) \sqrt{2}}{3\,b \left ( -1+\cos \left ( bx+a \right ) \right ) \sin \left ( bx+a \right ) } \left ({\it EllipticF} \left ( \sqrt{{\frac{1-\cos \left ( bx+a \right ) +\sin \left ( bx+a \right ) }{\sin \left ( bx+a \right ) }}},{\frac{\sqrt{2}}{2}} \right ) \sqrt{{\frac{1-\cos \left ( bx+a \right ) +\sin \left ( bx+a \right ) }{\sin \left ( bx+a \right ) }}}\sqrt{{\frac{-1+\cos \left ( bx+a \right ) +\sin \left ( bx+a \right ) }{\sin \left ( bx+a \right ) }}}\sqrt{{\frac{-1+\cos \left ( bx+a \right ) }{\sin \left ( bx+a \right ) }}}\sin \left ( bx+a \right ) \cos \left ( bx+a \right ) +\cos \left ( bx+a \right ) \sqrt{2}-\sqrt{2} \right ) \left ( c\sin \left ( bx+a \right ) \right ) ^{{\frac{3}{2}}} \left ( d\cos \left ( bx+a \right ) \right ) ^{-{\frac{5}{2}}}} \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{3}{2}}}{\left (d \cos \left (b x + a\right )\right )^{\frac{5}{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{\sqrt{d \cos \left (b x + a\right )} \sqrt{c \sin \left (b x + a\right )} c \sin \left (b x + a\right )}{d^{3} \cos \left (b x + a\right )^{3}}, 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{3}{2}}}{\left (d \cos \left (b x + a\right )\right )^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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