Optimal. Leaf size=75 \[ \frac{8 (c \sin (a+b x))^{3/2}}{21 b c d^3 (d \cos (a+b x))^{3/2}}+\frac{2 (c \sin (a+b x))^{3/2}}{7 b c d (d \cos (a+b x))^{7/2}} \]
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Rubi [A] time = 0.112531, antiderivative size = 75, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.08, Rules used = {2571, 2563} \[ \frac{8 (c \sin (a+b x))^{3/2}}{21 b c d^3 (d \cos (a+b x))^{3/2}}+\frac{2 (c \sin (a+b x))^{3/2}}{7 b c d (d \cos (a+b x))^{7/2}} \]
Antiderivative was successfully verified.
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Rule 2571
Rule 2563
Rubi steps
\begin{align*} \int \frac{\sqrt{c \sin (a+b x)}}{(d \cos (a+b x))^{9/2}} \, dx &=\frac{2 (c \sin (a+b x))^{3/2}}{7 b c d (d \cos (a+b x))^{7/2}}+\frac{4 \int \frac{\sqrt{c \sin (a+b x)}}{(d \cos (a+b x))^{5/2}} \, dx}{7 d^2}\\ &=\frac{2 (c \sin (a+b x))^{3/2}}{7 b c d (d \cos (a+b x))^{7/2}}+\frac{8 (c \sin (a+b x))^{3/2}}{21 b c d^3 (d \cos (a+b x))^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.242327, size = 57, normalized size = 0.76 \[ \frac{2 (2 \cos (2 (a+b x))+5) \sec ^4(a+b x) (c \sin (a+b x))^{3/2} \sqrt{d \cos (a+b x)}}{21 b c d^5} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.107, size = 50, normalized size = 0.7 \begin{align*}{\frac{ \left ( 8\, \left ( \cos \left ( bx+a \right ) \right ) ^{2}+6 \right ) \cos \left ( bx+a \right ) \sin \left ( bx+a \right ) }{21\,b}\sqrt{c\sin \left ( bx+a \right ) } \left ( d\cos \left ( bx+a \right ) \right ) ^{-{\frac{9}{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{\sqrt{c \sin \left (b x + a\right )}}{\left (d \cos \left (b x + a\right )\right )^{\frac{9}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 3.70182, size = 144, normalized size = 1.92 \begin{align*} \frac{2 \, \sqrt{d \cos \left (b x + a\right )}{\left (4 \, \cos \left (b x + a\right )^{2} + 3\right )} \sqrt{c \sin \left (b x + a\right )} \sin \left (b x + a\right )}{21 \, b d^{5} \cos \left (b x + a\right )^{4}} \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{\sqrt{c \sin \left (b x + a\right )}}{\left (d \cos \left (b x + a\right )\right )^{\frac{9}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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