Optimal. Leaf size=76 \[ -\frac{c \sin ^2(a+b x)^{3/4} (d \cos (a+b x))^{n+1} \, _2F_1\left (\frac{3}{4},\frac{n+1}{2};\frac{n+3}{2};\cos ^2(a+b x)\right )}{b d (n+1) (c \sin (a+b x))^{3/2}} \]
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Rubi [A] time = 0.0484558, antiderivative size = 76, 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} \[ -\frac{c \sin ^2(a+b x)^{3/4} (d \cos (a+b x))^{n+1} \, _2F_1\left (\frac{3}{4},\frac{n+1}{2};\frac{n+3}{2};\cos ^2(a+b x)\right )}{b d (n+1) (c \sin (a+b x))^{3/2}} \]
Antiderivative was successfully verified.
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Rule 2576
Rubi steps
\begin{align*} \int \frac{(d \cos (a+b x))^n}{\sqrt{c \sin (a+b x)}} \, dx &=-\frac{c (d \cos (a+b x))^{1+n} \, _2F_1\left (\frac{3}{4},\frac{1+n}{2};\frac{3+n}{2};\cos ^2(a+b x)\right ) \sin ^2(a+b x)^{3/4}}{b d (1+n) (c \sin (a+b x))^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.118294, size = 82, normalized size = 1.08 \[ -\frac{\sin (a+b x) \cos (a+b x) (d \cos (a+b x))^n \, _2F_1\left (\frac{3}{4},\frac{n+1}{2};\frac{n+3}{2};\cos ^2(a+b x)\right )}{b (n+1) \sqrt [4]{\sin ^2(a+b x)} \sqrt{c \sin (a+b x)}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.06, size = 0, normalized size = 0. \begin{align*} \int{ \left ( d\cos \left ( bx+a \right ) \right ) ^{n}{\frac{1}{\sqrt{c\sin \left ( bx+a \right ) }}}}\, 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}}{\sqrt{c \sin \left (b x + a\right )}}\,{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 \sin \left (b x + a\right )}, 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}}{\sqrt{c \sin{\left (a + b x \right )}}}\, 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}}{\sqrt{c \sin \left (b x + a\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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