Optimal. Leaf size=221 \[ -\frac{(A-B) \cos (e+f x) (c+d \sin (e+f x))^n \left (\frac{c+d \sin (e+f x)}{c+d}\right )^{-n} F_1\left (\frac{1}{2};\frac{3}{2},-n;\frac{3}{2};\frac{1}{2} (1-\sin (e+f x)),\frac{d (1-\sin (e+f x))}{c+d}\right )}{\sqrt{2} a f \sqrt{\sin (e+f x)+1}}-\frac{\sqrt{2} B \cos (e+f x) (c+d \sin (e+f x))^n \left (\frac{c+d \sin (e+f x)}{c+d}\right )^{-n} F_1\left (\frac{1}{2};\frac{1}{2},-n;\frac{3}{2};\frac{1}{2} (1-\sin (e+f x)),\frac{d (1-\sin (e+f x))}{c+d}\right )}{a f \sqrt{\sin (e+f x)+1}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.304206, antiderivative size = 221, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {2987, 2784, 139, 138, 2665} \[ -\frac{(A-B) \cos (e+f x) (c+d \sin (e+f x))^n \left (\frac{c+d \sin (e+f x)}{c+d}\right )^{-n} F_1\left (\frac{1}{2};\frac{3}{2},-n;\frac{3}{2};\frac{1}{2} (1-\sin (e+f x)),\frac{d (1-\sin (e+f x))}{c+d}\right )}{\sqrt{2} a f \sqrt{\sin (e+f x)+1}}-\frac{\sqrt{2} B \cos (e+f x) (c+d \sin (e+f x))^n \left (\frac{c+d \sin (e+f x)}{c+d}\right )^{-n} F_1\left (\frac{1}{2};\frac{1}{2},-n;\frac{3}{2};\frac{1}{2} (1-\sin (e+f x)),\frac{d (1-\sin (e+f x))}{c+d}\right )}{a f \sqrt{\sin (e+f x)+1}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2987
Rule 2784
Rule 139
Rule 138
Rule 2665
Rubi steps
\begin{align*} \int \frac{(A+B \sin (e+f x)) (c+d \sin (e+f x))^n}{a+a \sin (e+f x)} \, dx &=(A-B) \int \frac{(c+d \sin (e+f x))^n}{a+a \sin (e+f x)} \, dx+\frac{B \int (c+d \sin (e+f x))^n \, dx}{a}\\ &=\frac{((A-B) \cos (e+f x)) \operatorname{Subst}\left (\int \frac{(c+d x)^n}{\sqrt{1-x} (1+x)^{3/2}} \, dx,x,\sin (e+f x)\right )}{a f \sqrt{1-\sin (e+f x)} \sqrt{1+\sin (e+f x)}}+\frac{(B \cos (e+f x)) \operatorname{Subst}\left (\int \frac{(c+d x)^n}{\sqrt{1-x} \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 ((A-B) \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{\left (-\frac{c}{-c-d}-\frac{d x}{-c-d}\right )^n}{\sqrt{1-x} (1+x)^{3/2}} \, dx,x,\sin (e+f x)\right )}{a f \sqrt{1-\sin (e+f x)} \sqrt{1+\sin (e+f x)}}+\frac{\left (B \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{\left (-\frac{c}{-c-d}-\frac{d x}{-c-d}\right )^n}{\sqrt{1-x} \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} B F_1\left (\frac{1}{2};\frac{1}{2},-n;\frac{3}{2};\frac{1}{2} (1-\sin (e+f x)),\frac{d (1-\sin (e+f x))}{c+d}\right ) \cos (e+f x) (c+d \sin (e+f x))^n \left (\frac{c+d \sin (e+f x)}{c+d}\right )^{-n}}{a f \sqrt{1+\sin (e+f x)}}-\frac{(A-B) F_1\left (\frac{1}{2};\frac{3}{2},-n;\frac{3}{2};\frac{1}{2} (1-\sin (e+f x)),\frac{d (1-\sin (e+f x))}{c+d}\right ) \cos (e+f x) (c+d \sin (e+f x))^n \left (\frac{c+d \sin (e+f x)}{c+d}\right )^{-n}}{\sqrt{2} a f \sqrt{1+\sin (e+f x)}}\\ \end{align*}
Mathematica [F] time = 4.92391, size = 0, normalized size = 0. \[ \int \frac{(A+B \sin (e+f x)) (c+d \sin (e+f x))^n}{a+a \sin (e+f x)} \, dx \]
Verification is Not applicable to the result.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.299, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( A+B\sin \left ( fx+e \right ) \right ) \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 (B \sin \left (f x + e\right ) + A\right )}{\left (d \sin \left (f x + e\right ) + c\right )}^{n}}{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 (B \sin \left (f x + e\right ) + A\right )}{\left (d \sin \left (f x + e\right ) + c\right )}^{n}}{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 (B \sin \left (f x + e\right ) + A\right )}{\left (d \sin \left (f x + e\right ) + c\right )}^{n}}{a \sin \left (f x + e\right ) + a}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]