Optimal. Leaf size=88 \[ -\frac{b \sin ^2(e+f x)^{\frac{1-m}{2}} (b \sin (e+f x))^{m-1} (d \cos (e+f x))^{n+1} \, _2F_1\left (\frac{1-m}{2},\frac{n+1}{2};\frac{n+3}{2};\cos ^2(e+f x)\right )}{d f (n+1)} \]
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Rubi [A] time = 0.0480973, antiderivative size = 88, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.048, Rules used = {2576} \[ -\frac{b \sin ^2(e+f x)^{\frac{1-m}{2}} (b \sin (e+f x))^{m-1} (d \cos (e+f x))^{n+1} \, _2F_1\left (\frac{1-m}{2},\frac{n+1}{2};\frac{n+3}{2};\cos ^2(e+f x)\right )}{d f (n+1)} \]
Antiderivative was successfully verified.
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Rule 2576
Rubi steps
\begin{align*} \int (d \cos (e+f x))^n (b \sin (e+f x))^m \, dx &=-\frac{b (d \cos (e+f x))^{1+n} \, _2F_1\left (\frac{1-m}{2},\frac{1+n}{2};\frac{3+n}{2};\cos ^2(e+f x)\right ) (b \sin (e+f x))^{-1+m} \sin ^2(e+f x)^{\frac{1-m}{2}}}{d f (1+n)}\\ \end{align*}
Mathematica [A] time = 0.094801, size = 85, normalized size = 0.97 \[ \frac{\tan (e+f x) \cos ^2(e+f x)^{\frac{1-n}{2}} (b \sin (e+f x))^m (d \cos (e+f x))^n \, _2F_1\left (\frac{m+1}{2},\frac{1-n}{2};\frac{m+3}{2};\sin ^2(e+f x)\right )}{f (m+1)} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.432, size = 0, normalized size = 0. \begin{align*} \int \left ( d\cos \left ( fx+e \right ) \right ) ^{n} \left ( b\sin \left ( fx+e \right ) \right ) ^{m}\, 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 \left (d \cos \left (f x + e\right )\right )^{n} \left (b \sin \left (f x + e\right )\right )^{m}\,{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 (\left (d \cos \left (f x + e\right )\right )^{n} \left (b \sin \left (f x + e\right )\right )^{m}, 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 \left (b \sin{\left (e + f x \right )}\right )^{m} \left (d \cos{\left (e + f x \right )}\right )^{n}\, 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 \left (d \cos \left (f x + e\right )\right )^{n} \left (b \sin \left (f x + e\right )\right )^{m}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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