Optimal. Leaf size=112 \[ \frac{64 (c \sin (a+b x))^{3/2}}{231 b c d^5 (d \cos (a+b x))^{3/2}}+\frac{16 (c \sin (a+b x))^{3/2}}{77 b c d^3 (d \cos (a+b x))^{7/2}}+\frac{2 (c \sin (a+b x))^{3/2}}{11 b c d (d \cos (a+b x))^{11/2}} \]
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Rubi [A] time = 0.171186, antiderivative size = 112, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.08, Rules used = {2571, 2563} \[ \frac{64 (c \sin (a+b x))^{3/2}}{231 b c d^5 (d \cos (a+b x))^{3/2}}+\frac{16 (c \sin (a+b x))^{3/2}}{77 b c d^3 (d \cos (a+b x))^{7/2}}+\frac{2 (c \sin (a+b x))^{3/2}}{11 b c d (d \cos (a+b x))^{11/2}} \]
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))^{13/2}} \, dx &=\frac{2 (c \sin (a+b x))^{3/2}}{11 b c d (d \cos (a+b x))^{11/2}}+\frac{8 \int \frac{\sqrt{c \sin (a+b x)}}{(d \cos (a+b x))^{9/2}} \, dx}{11 d^2}\\ &=\frac{2 (c \sin (a+b x))^{3/2}}{11 b c d (d \cos (a+b x))^{11/2}}+\frac{16 (c \sin (a+b x))^{3/2}}{77 b c d^3 (d \cos (a+b x))^{7/2}}+\frac{32 \int \frac{\sqrt{c \sin (a+b x)}}{(d \cos (a+b x))^{5/2}} \, dx}{77 d^4}\\ &=\frac{2 (c \sin (a+b x))^{3/2}}{11 b c d (d \cos (a+b x))^{11/2}}+\frac{16 (c \sin (a+b x))^{3/2}}{77 b c d^3 (d \cos (a+b x))^{7/2}}+\frac{64 (c \sin (a+b x))^{3/2}}{231 b c d^5 (d \cos (a+b x))^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.234605, size = 67, normalized size = 0.6 \[ \frac{2 (28 \cos (2 (a+b x))+4 \cos (4 (a+b x))+45) \sec ^6(a+b x) (c \sin (a+b x))^{3/2} \sqrt{d \cos (a+b x)}}{231 b c d^7} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.144, size = 60, normalized size = 0.5 \begin{align*}{\frac{ \left ( 64\, \left ( \cos \left ( bx+a \right ) \right ) ^{4}+48\, \left ( \cos \left ( bx+a \right ) \right ) ^{2}+42 \right ) \cos \left ( bx+a \right ) \sin \left ( bx+a \right ) }{231\,b}\sqrt{c\sin \left ( bx+a \right ) } \left ( d\cos \left ( bx+a \right ) \right ) ^{-{\frac{13}{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{13}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 4.00519, size = 176, normalized size = 1.57 \begin{align*} \frac{2 \,{\left (32 \, \cos \left (b x + a\right )^{4} + 24 \, \cos \left (b x + a\right )^{2} + 21\right )} \sqrt{d \cos \left (b x + a\right )} \sqrt{c \sin \left (b x + a\right )} \sin \left (b x + a\right )}{231 \, b d^{7} \cos \left (b x + a\right )^{6}} \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{13}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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