Optimal. Leaf size=92 \[ \frac{d^2 \sqrt{\sin (2 a+2 b x)} F\left (\left .a+b x-\frac{\pi }{4}\right |2\right )}{2 b \sqrt{c \sin (a+b x)} \sqrt{d \cos (a+b x)}}+\frac{d \sqrt{c \sin (a+b x)} \sqrt{d \cos (a+b x)}}{b c} \]
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Rubi [A] time = 0.112895, antiderivative size = 92, 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 = {2569, 2573, 2641} \[ \frac{d^2 \sqrt{\sin (2 a+2 b x)} F\left (\left .a+b x-\frac{\pi }{4}\right |2\right )}{2 b \sqrt{c \sin (a+b x)} \sqrt{d \cos (a+b x)}}+\frac{d \sqrt{c \sin (a+b x)} \sqrt{d \cos (a+b x)}}{b c} \]
Antiderivative was successfully verified.
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Rule 2569
Rule 2573
Rule 2641
Rubi steps
\begin{align*} \int \frac{(d \cos (a+b x))^{3/2}}{\sqrt{c \sin (a+b x)}} \, dx &=\frac{d \sqrt{d \cos (a+b x)} \sqrt{c \sin (a+b x)}}{b c}+\frac{1}{2} d^2 \int \frac{1}{\sqrt{d \cos (a+b x)} \sqrt{c \sin (a+b x)}} \, dx\\ &=\frac{d \sqrt{d \cos (a+b x)} \sqrt{c \sin (a+b x)}}{b c}+\frac{\left (d^2 \sqrt{\sin (2 a+2 b x)}\right ) \int \frac{1}{\sqrt{\sin (2 a+2 b x)}} \, dx}{2 \sqrt{d \cos (a+b x)} \sqrt{c \sin (a+b x)}}\\ &=\frac{d \sqrt{d \cos (a+b x)} \sqrt{c \sin (a+b x)}}{b c}+\frac{d^2 F\left (\left .a-\frac{\pi }{4}+b x\right |2\right ) \sqrt{\sin (2 a+2 b x)}}{2 b \sqrt{d \cos (a+b x)} \sqrt{c \sin (a+b x)}}\\ \end{align*}
Mathematica [C] time = 0.107491, size = 68, normalized size = 0.74 \[ \frac{2 d^2 \cos ^2(a+b x)^{3/4} \tan (a+b x) \, _2F_1\left (-\frac{1}{4},\frac{1}{4};\frac{5}{4};\sin ^2(a+b x)\right )}{b \sqrt{c \sin (a+b x)} \sqrt{d \cos (a+b x)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.099, size = 188, normalized size = 2. \begin{align*} -{\frac{\sqrt{2}\sin \left ( bx+a \right ) }{2\,b \left ( -1+\cos \left ( bx+a \right ) \right ) \left ( \cos \left ( bx+a \right ) \right ) ^{2}} \left ( \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 ) }}}\sqrt{{\frac{-1+\cos \left ( bx+a \right ) }{\sin \left ( bx+a \right ) }}}{\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 ) - \left ( \cos \left ( bx+a \right ) \right ) ^{2}\sqrt{2}+\cos \left ( bx+a \right ) \sqrt{2} \right ) \left ( d\cos \left ( bx+a \right ) \right ) ^{{\frac{3}{2}}}{\frac{1}{\sqrt{c\sin \left ( bx+a \right ) }}}} \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 (d \cos \left (b x + a\right )\right )^{\frac{3}{2}}}{\sqrt{c \sin \left (b x + a\right )}}\,{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 )} d \cos \left (b x + a\right )}{c \sin \left (b x + a\right )}, 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 (d \cos \left (b x + a\right )\right )^{\frac{3}{2}}}{\sqrt{c \sin \left (b x + a\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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