Optimal. Leaf size=119 \[ -\frac{\sqrt{2} (1-\sin (e+f x)) \cos (e+f x) (c+d \sin (e+f x))^n \left (\frac{c+d \sin (e+f x)}{c+d}\right )^{-n} F_1\left (\frac{3}{2};\frac{1}{2},-n;\frac{5}{2};\frac{1}{2} (1-\sin (e+f x)),\frac{d (1-\sin (e+f x))}{c+d}\right )}{3 a f \sqrt{\sin (e+f x)+1}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.228802, antiderivative size = 119, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.121, Rules used = {2914, 2755, 139, 138} \[ -\frac{\sqrt{2} (1-\sin (e+f x)) \cos (e+f x) (c+d \sin (e+f x))^n \left (\frac{c+d \sin (e+f x)}{c+d}\right )^{-n} F_1\left (\frac{3}{2};\frac{1}{2},-n;\frac{5}{2};\frac{1}{2} (1-\sin (e+f x)),\frac{d (1-\sin (e+f x))}{c+d}\right )}{3 a f \sqrt{\sin (e+f x)+1}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2914
Rule 2755
Rule 139
Rule 138
Rubi steps
\begin{align*} \int \frac{\cos ^2(e+f x) (c+d \sin (e+f x))^n}{a+a \sin (e+f x)} \, dx &=\frac{\int (a-a \sin (e+f x)) (c+d \sin (e+f x))^n \, dx}{a^2}\\ &=\frac{\cos (e+f x) \operatorname{Subst}\left (\int \frac{\sqrt{1-x} (c+d x)^n}{\sqrt{1+x}} \, dx,x,\sin (e+f x)\right )}{a f \sqrt{1-\sin (e+f x)} \sqrt{1+\sin (e+f x)}}\\ &=\frac{\left (\cos (e+f x) (c+d \sin (e+f x))^n \left (-\frac{c+d \sin (e+f x)}{-c-d}\right )^{-n}\right ) \operatorname{Subst}\left (\int \frac{\sqrt{1-x} \left (-\frac{c}{-c-d}-\frac{d x}{-c-d}\right )^n}{\sqrt{1+x}} \, dx,x,\sin (e+f x)\right )}{a f \sqrt{1-\sin (e+f x)} \sqrt{1+\sin (e+f x)}}\\ &=-\frac{\sqrt{2} F_1\left (\frac{3}{2};\frac{1}{2},-n;\frac{5}{2};\frac{1}{2} (1-\sin (e+f x)),\frac{d (1-\sin (e+f x))}{c+d}\right ) \cos (e+f x) (1-\sin (e+f x)) (c+d \sin (e+f x))^n \left (\frac{c+d \sin (e+f x)}{c+d}\right )^{-n}}{3 a f \sqrt{1+\sin (e+f x)}}\\ \end{align*}
Mathematica [A] time = 0.994821, size = 229, normalized size = 1.92 \[ -\frac{\sec (e+f x) \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right )^2 \sqrt{-\frac{d (\sin (e+f x)-1)}{c+d}} (c+d \sin (e+f x))^{n+1} \left ((n+1) (c+d \sin (e+f x)) F_1\left (n+2;\frac{1}{2},\frac{1}{2};n+3;\frac{c+d \sin (e+f x)}{c-d},\frac{c+d \sin (e+f x)}{c+d}\right )-(n+2) (c+d) F_1\left (n+1;\frac{1}{2},\frac{1}{2};n+2;\frac{c+d \sin (e+f x)}{c-d},\frac{c+d \sin (e+f x)}{c+d}\right )\right )}{a d f (n+1) (n+2) (d-c) \sqrt{\frac{d (\sin (e+f x)+1)}{d-c}}} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.414, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( \cos \left ( fx+e \right ) \right ) ^{2} \left ( c+d\sin \left ( fx+e \right ) \right ) ^{n}}{a+a\sin \left ( fx+e \right ) }}\, 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 \frac{{\left (d \sin \left (f x + e\right ) + c\right )}^{n} \cos \left (f x + e\right )^{2}}{a \sin \left (f x + e\right ) + a}\,{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 (\frac{{\left (d \sin \left (f x + e\right ) + c\right )}^{n} \cos \left (f x + e\right )^{2}}{a \sin \left (f x + e\right ) + a}, 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 \frac{{\left (d \sin \left (f x + e\right ) + c\right )}^{n} \cos \left (f x + e\right )^{2}}{a \sin \left (f x + e\right ) + a}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]