Optimal. Leaf size=97 \[ \frac{2 \sqrt{\sin (2 a+2 b x)} F\left (\left .a+b x-\frac{\pi }{4}\right |2\right )}{3 b d^2 \sqrt{c \sin (a+b x)} \sqrt{d \cos (a+b x)}}+\frac{2 \sqrt{c \sin (a+b x)}}{3 b c d (d \cos (a+b x))^{3/2}} \]
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Rubi [A] time = 0.113712, antiderivative size = 97, 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 = {2571, 2573, 2641} \[ \frac{2 \sqrt{\sin (2 a+2 b x)} F\left (\left .a+b x-\frac{\pi }{4}\right |2\right )}{3 b d^2 \sqrt{c \sin (a+b x)} \sqrt{d \cos (a+b x)}}+\frac{2 \sqrt{c \sin (a+b x)}}{3 b c d (d \cos (a+b x))^{3/2}} \]
Antiderivative was successfully verified.
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Rule 2571
Rule 2573
Rule 2641
Rubi steps
\begin{align*} \int \frac{1}{(d \cos (a+b x))^{5/2} \sqrt{c \sin (a+b x)}} \, dx &=\frac{2 \sqrt{c \sin (a+b x)}}{3 b c d (d \cos (a+b x))^{3/2}}+\frac{2 \int \frac{1}{\sqrt{d \cos (a+b x)} \sqrt{c \sin (a+b x)}} \, dx}{3 d^2}\\ &=\frac{2 \sqrt{c \sin (a+b x)}}{3 b c d (d \cos (a+b x))^{3/2}}+\frac{\left (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 \sqrt{c \sin (a+b x)}}{3 b c d (d \cos (a+b x))^{3/2}}+\frac{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 = 0.106237, size = 65, normalized size = 0.67 \[ \frac{2 \cos ^2(a+b x)^{3/4} \sqrt{c \sin (a+b x)} \, _2F_1\left (\frac{1}{4},\frac{7}{4};\frac{5}{4};\sin ^2(a+b x)\right )}{b c d (d \cos (a+b x))^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.112, size = 184, normalized size = 1.9 \begin{align*} -{\frac{\sqrt{2}\sin \left ( bx+a \right ) \cos \left ( bx+a \right ) }{3\,b \left ( -1+\cos \left ( bx+a \right ) \right ) } \left ( 2\,{\it EllipticF} \left ( \sqrt{{\frac{1-\cos \left ( bx+a \right ) +\sin \left ( bx+a \right ) }{\sin \left ( bx+a \right ) }}},1/2\,\sqrt{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 ( d\cos \left ( bx+a \right ) \right ) ^{-{\frac{5}{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{1}{\left (d \cos \left (b x + a\right )\right )^{\frac{5}{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 )}}{c d^{3} \cos \left (b x + a\right )^{3} \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{1}{\left (d \cos \left (b x + a\right )\right )^{\frac{5}{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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