Optimal. Leaf size=61 \[ \frac{(a \sin (c+d x)+a)^m (e \cos (c+d x))^{-2 m} \, _2F_1\left (1,-m;1-m;\frac{1}{2} (1-\sin (c+d x))\right )}{2 d e m} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.067174, antiderivative size = 61, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {2689, 7, 68} \[ \frac{(a \sin (c+d x)+a)^m (e \cos (c+d x))^{-2 m} \, _2F_1\left (1,-m;1-m;\frac{1}{2} (1-\sin (c+d x))\right )}{2 d e m} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2689
Rule 7
Rule 68
Rubi steps
\begin{align*} \int (e \cos (c+d x))^{-1-2 m} (a+a \sin (c+d x))^m \, dx &=\frac{\left (a^2 (e \cos (c+d x))^{-2 m} (a-a \sin (c+d x))^m (a+a \sin (c+d x))^m\right ) \operatorname{Subst}\left (\int (a-a x)^{\frac{1}{2} (-2-2 m)} (a+a x)^{\frac{1}{2} (-2-2 m)+m} \, dx,x,\sin (c+d x)\right )}{d e}\\ &=\frac{\left (a^2 (e \cos (c+d x))^{-2 m} (a-a \sin (c+d x))^m (a+a \sin (c+d x))^m\right ) \operatorname{Subst}\left (\int \frac{(a-a x)^{\frac{1}{2} (-2-2 m)}}{a+a x} \, dx,x,\sin (c+d x)\right )}{d e}\\ &=\frac{(e \cos (c+d x))^{-2 m} \, _2F_1\left (1,-m;1-m;\frac{1}{2} (1-\sin (c+d x))\right ) (a+a \sin (c+d x))^m}{2 d e m}\\ \end{align*}
Mathematica [A] time = 0.0674885, size = 61, normalized size = 1. \[ \frac{(a (\sin (c+d x)+1))^m (e \cos (c+d x))^{-2 m} \, _2F_1\left (1,-m;1-m;\frac{1}{2} (1-\sin (c+d x))\right )}{2 d e m} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.773, size = 0, normalized size = 0. \begin{align*} \int \left ( e\cos \left ( dx+c \right ) \right ) ^{-1-2\,m} \left ( a+a\sin \left ( dx+c \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 (e \cos \left (d x + c\right )\right )^{-2 \, m - 1}{\left (a \sin \left (d x + c\right ) + a\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 (e \cos \left (d x + c\right )\right )^{-2 \, m - 1}{\left (a \sin \left (d x + c\right ) + a\right )}^{m}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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 (e \cos \left (d x + c\right )\right )^{-2 \, m - 1}{\left (a \sin \left (d x + c\right ) + a\right )}^{m}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]