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 = 0.05, antiderivative size = 88, normalized size of antiderivative = 1.00, 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*}
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Mathematica [A] time = 0.09, 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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fricas [F] time = 0.96, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\left (d \cos \left (f x + e\right )\right )^{n} \left (b \sin \left (f x + e\right )\right )^{m}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (d \cos \left (f x + e\right )\right )^{n} \left (b \sin \left (f x + e\right )\right )^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.57, size = 0, normalized size = 0.00 \[ \int \left (d \cos \left (f x +e \right )\right )^{n} \left (b \sin \left (f x +e \right )\right )^{m}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (d \cos \left (f x + e\right )\right )^{n} \left (b \sin \left (f x + e\right )\right )^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int {\left (d\,\cos \left (e+f\,x\right )\right )}^n\,{\left (b\,\sin \left (e+f\,x\right )\right )}^m \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (b \sin {\left (e + f x \right )}\right )^{m} \left (d \cos {\left (e + f x \right )}\right )^{n}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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