Optimal. Leaf size=78 \[ -\frac{\sqrt [4]{\sin ^2(a+b x)} (d \cos (a+b x))^{n+1} \, _2F_1\left (\frac{5}{4},\frac{n+1}{2};\frac{n+3}{2};\cos ^2(a+b x)\right )}{b c d (n+1) \sqrt{c \sin (a+b x)}} \]
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Rubi [A] time = 0.0562769, antiderivative size = 78, 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 = {2576} \[ -\frac{\sqrt [4]{\sin ^2(a+b x)} (d \cos (a+b x))^{n+1} \, _2F_1\left (\frac{5}{4},\frac{n+1}{2};\frac{n+3}{2};\cos ^2(a+b x)\right )}{b c d (n+1) \sqrt{c \sin (a+b x)}} \]
Antiderivative was successfully verified.
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Rule 2576
Rubi steps
\begin{align*} \int \frac{(d \cos (a+b x))^n}{(c \sin (a+b x))^{3/2}} \, dx &=-\frac{(d \cos (a+b x))^{1+n} \, _2F_1\left (\frac{5}{4},\frac{1+n}{2};\frac{3+n}{2};\cos ^2(a+b x)\right ) \sqrt [4]{\sin ^2(a+b x)}}{b c d (1+n) \sqrt{c \sin (a+b x)}}\\ \end{align*}
Mathematica [A] time = 0.136956, size = 79, normalized size = 1.01 \[ -\frac{\sqrt [4]{\sin ^2(a+b x)} \cot (a+b x) \sqrt{c \sin (a+b x)} (d \cos (a+b x))^n \, _2F_1\left (\frac{5}{4},\frac{n+1}{2};\frac{n+3}{2};\cos ^2(a+b x)\right )}{b c^2 (n+1)} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.059, size = 0, normalized size = 0. \begin{align*} \int{ \left ( d\cos \left ( bx+a \right ) \right ) ^{n} \left ( c\sin \left ( bx+a \right ) \right ) ^{-{\frac{3}{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 (d \cos \left (b x + a\right )\right )^{n}}{\left (c \sin \left (b x + a\right )\right )^{\frac{3}{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{c \sin \left (b x + a\right )} \left (d \cos \left (b x + a\right )\right )^{n}}{c^{2} \cos \left (b x + a\right )^{2} - c^{2}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (d \cos{\left (a + b x \right )}\right )^{n}}{\left (c \sin{\left (a + b x \right )}\right )^{\frac{3}{2}}}\, dx \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 (d \cos \left (b x + a\right )\right )^{n}}{\left (c \sin \left (b x + a\right )\right )^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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