Optimal. Leaf size=77 \[ \frac{\cos ^2(a+b x)^{3/4} (c \sin (a+b x))^{m+1} \, _2F_1\left (\frac{7}{4},\frac{m+1}{2};\frac{m+3}{2};\sin ^2(a+b x)\right )}{b c d (m+1) (d \cos (a+b x))^{3/2}} \]
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Rubi [A] time = 0.0549027, antiderivative size = 77, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.043, Rules used = {2577} \[ \frac{\cos ^2(a+b x)^{3/4} (c \sin (a+b x))^{m+1} \, _2F_1\left (\frac{7}{4},\frac{m+1}{2};\frac{m+3}{2};\sin ^2(a+b x)\right )}{b c d (m+1) (d \cos (a+b x))^{3/2}} \]
Antiderivative was successfully verified.
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Rule 2577
Rubi steps
\begin{align*} \int \frac{(c \sin (a+b x))^m}{(d \cos (a+b x))^{5/2}} \, dx &=\frac{\cos ^2(a+b x)^{3/4} \, _2F_1\left (\frac{7}{4},\frac{1+m}{2};\frac{3+m}{2};\sin ^2(a+b x)\right ) (c \sin (a+b x))^{1+m}}{b c d (1+m) (d \cos (a+b x))^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.0837122, size = 78, normalized size = 1.01 \[ \frac{\cos ^2(a+b x)^{3/4} \tan (a+b x) (c \sin (a+b x))^m \, _2F_1\left (\frac{7}{4},\frac{m+1}{2};\frac{m+3}{2};\sin ^2(a+b x)\right )}{b d^2 (m+1) \sqrt{d \cos (a+b x)}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.069, size = 0, normalized size = 0. \begin{align*} \int{ \left ( c\sin \left ( bx+a \right ) \right ) ^{m} \left ( d\cos \left ( bx+a \right ) \right ) ^{-{\frac{5}{2}}}}\, dx \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 )^{m}}{\left (d \cos \left (b x + a\right )\right )^{\frac{5}{2}}}\,{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 )} \left (c \sin \left (b x + a\right )\right )^{m}}{d^{3} \cos \left (b x + a\right )^{3}}, 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 (c \sin \left (b x + a\right )\right )^{m}}{\left (d \cos \left (b x + a\right )\right )^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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