Optimal. Leaf size=141 \[ -\frac{64 c (c \sin (a+b x))^{3/2}}{1155 b d^7 (d \cos (a+b x))^{3/2}}-\frac{16 c (c \sin (a+b x))^{3/2}}{385 b d^5 (d \cos (a+b x))^{7/2}}-\frac{2 c (c \sin (a+b x))^{3/2}}{55 b d^3 (d \cos (a+b x))^{11/2}}+\frac{2 c (c \sin (a+b x))^{3/2}}{15 b d (d \cos (a+b x))^{15/2}} \]
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Rubi [A] time = 0.234812, antiderivative size = 141, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.12, Rules used = {2566, 2571, 2563} \[ -\frac{64 c (c \sin (a+b x))^{3/2}}{1155 b d^7 (d \cos (a+b x))^{3/2}}-\frac{16 c (c \sin (a+b x))^{3/2}}{385 b d^5 (d \cos (a+b x))^{7/2}}-\frac{2 c (c \sin (a+b x))^{3/2}}{55 b d^3 (d \cos (a+b x))^{11/2}}+\frac{2 c (c \sin (a+b x))^{3/2}}{15 b d (d \cos (a+b x))^{15/2}} \]
Antiderivative was successfully verified.
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Rule 2566
Rule 2571
Rule 2563
Rubi steps
\begin{align*} \int \frac{(c \sin (a+b x))^{5/2}}{(d \cos (a+b x))^{17/2}} \, dx &=\frac{2 c (c \sin (a+b x))^{3/2}}{15 b d (d \cos (a+b x))^{15/2}}-\frac{c^2 \int \frac{\sqrt{c \sin (a+b x)}}{(d \cos (a+b x))^{13/2}} \, dx}{5 d^2}\\ &=\frac{2 c (c \sin (a+b x))^{3/2}}{15 b d (d \cos (a+b x))^{15/2}}-\frac{2 c (c \sin (a+b x))^{3/2}}{55 b d^3 (d \cos (a+b x))^{11/2}}-\frac{\left (8 c^2\right ) \int \frac{\sqrt{c \sin (a+b x)}}{(d \cos (a+b x))^{9/2}} \, dx}{55 d^4}\\ &=\frac{2 c (c \sin (a+b x))^{3/2}}{15 b d (d \cos (a+b x))^{15/2}}-\frac{2 c (c \sin (a+b x))^{3/2}}{55 b d^3 (d \cos (a+b x))^{11/2}}-\frac{16 c (c \sin (a+b x))^{3/2}}{385 b d^5 (d \cos (a+b x))^{7/2}}-\frac{\left (32 c^2\right ) \int \frac{\sqrt{c \sin (a+b x)}}{(d \cos (a+b x))^{5/2}} \, dx}{385 d^6}\\ &=\frac{2 c (c \sin (a+b x))^{3/2}}{15 b d (d \cos (a+b x))^{15/2}}-\frac{2 c (c \sin (a+b x))^{3/2}}{55 b d^3 (d \cos (a+b x))^{11/2}}-\frac{16 c (c \sin (a+b x))^{3/2}}{385 b d^5 (d \cos (a+b x))^{7/2}}-\frac{64 c (c \sin (a+b x))^{3/2}}{1155 b d^7 (d \cos (a+b x))^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.47729, size = 67, normalized size = 0.48 \[ \frac{2 (44 \cos (2 (a+b x))+4 \cos (4 (a+b x))+117) \sec ^8(a+b x) (c \sin (a+b x))^{7/2} \sqrt{d \cos (a+b x)}}{1155 b c d^9} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.116, size = 60, normalized size = 0.4 \begin{align*}{\frac{ \left ( 64\, \left ( \cos \left ( bx+a \right ) \right ) ^{4}+112\, \left ( \cos \left ( bx+a \right ) \right ) ^{2}+154 \right ) \cos \left ( bx+a \right ) \sin \left ( bx+a \right ) }{1155\,b} \left ( c\sin \left ( bx+a \right ) \right ) ^{{\frac{5}{2}}} \left ( d\cos \left ( bx+a \right ) \right ) ^{-{\frac{17}{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{5}{2}}}{\left (d \cos \left (b x + a\right )\right )^{\frac{17}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 5.81589, size = 227, normalized size = 1.61 \begin{align*} -\frac{2 \,{\left (32 \, c^{2} \cos \left (b x + a\right )^{6} + 24 \, c^{2} \cos \left (b x + a\right )^{4} + 21 \, c^{2} \cos \left (b x + a\right )^{2} - 77 \, c^{2}\right )} \sqrt{d \cos \left (b x + a\right )} \sqrt{c \sin \left (b x + a\right )} \sin \left (b x + a\right )}{1155 \, b d^{9} \cos \left (b x + a\right )^{8}} \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(-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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