Optimal. Leaf size=85 \[ -\frac{\sin ^{m-1}(e+f x) \sin ^2(e+f x)^{\frac{1-m}{2}} (d \cos (e+f x))^{n+1} \, _2F_1\left (\frac{1-m}{2},\frac{n+1}{2};\frac{n+3}{2};\cos ^2(e+f x)\right )}{d f (n+1)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0445437, antiderivative size = 85, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.053, Rules used = {2576} \[ -\frac{\sin ^{m-1}(e+f x) \sin ^2(e+f x)^{\frac{1-m}{2}} (d \cos (e+f x))^{n+1} \, _2F_1\left (\frac{1-m}{2},\frac{n+1}{2};\frac{n+3}{2};\cos ^2(e+f x)\right )}{d f (n+1)} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2576
Rubi steps
\begin{align*} \int (d \cos (e+f x))^n \sin ^m(e+f x) \, dx &=-\frac{(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 ) \sin ^{-1+m}(e+f x) \sin ^2(e+f x)^{\frac{1-m}{2}}}{d f (1+n)}\\ \end{align*}
Mathematica [A] time = 0.116539, size = 82, normalized size = 0.96 \[ \frac{d \sin ^{m+1}(e+f x) \cos ^2(e+f x)^{\frac{1-n}{2}} (d \cos (e+f x))^{n-1} \, _2F_1\left (\frac{m+1}{2},\frac{1-n}{2};\frac{m+3}{2};\sin ^2(e+f x)\right )}{f (m+1)} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.573, size = 0, normalized size = 0. \begin{align*} \int \left ( d\cos \left ( fx+e \right ) \right ) ^{n} \left ( \sin \left ( fx+e \right ) \right ) ^{m}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (d \cos \left (f x + e\right )\right )^{n} \sin \left (f x + e\right )^{m}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\left (d \cos \left (f x + e\right )\right )^{n} \sin \left (f x + e\right )^{m}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (d \cos{\left (e + f x \right )}\right )^{n} \sin ^{m}{\left (e + f x \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (d \cos \left (f x + e\right )\right )^{n} \sin \left (f x + e\right )^{m}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]