Optimal. Leaf size=60 \[ -\frac{\cos (e+f x) F_1\left (\frac{1}{2};-n,2;\frac{3}{2};1-\sin (e+f x),\frac{1}{2} (1-\sin (e+f x))\right )}{2 f \sqrt{\sin (e+f x)+1}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0722906, antiderivative size = 60, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {2785, 130, 429} \[ -\frac{\cos (e+f x) F_1\left (\frac{1}{2};-n,2;\frac{3}{2};1-\sin (e+f x),\frac{1}{2} (1-\sin (e+f x))\right )}{2 f \sqrt{\sin (e+f x)+1}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2785
Rule 130
Rule 429
Rubi steps
\begin{align*} \int \frac{\sin ^n(e+f x)}{(1+\sin (e+f x))^{3/2}} \, dx &=-\frac{\cos (e+f x) \operatorname{Subst}\left (\int \frac{(1-x)^n}{(2-x)^2 \sqrt{x}} \, dx,x,1-\sin (e+f x)\right )}{f \sqrt{1-\sin (e+f x)} \sqrt{1+\sin (e+f x)}}\\ &=-\frac{(2 \cos (e+f x)) \operatorname{Subst}\left (\int \frac{\left (1-x^2\right )^n}{\left (2-x^2\right )^2} \, dx,x,\sqrt{1-\sin (e+f x)}\right )}{f \sqrt{1-\sin (e+f x)} \sqrt{1+\sin (e+f x)}}\\ &=-\frac{F_1\left (\frac{1}{2};-n,2;\frac{3}{2};1-\sin (e+f x),\frac{1}{2} (1-\sin (e+f x))\right ) \cos (e+f x)}{2 f \sqrt{1+\sin (e+f x)}}\\ \end{align*}
Mathematica [B] time = 3.65186, size = 263, normalized size = 4.38 \[ \frac{\sec (e+f x) \sin ^n(e+f x) \left (\sqrt{2-2 \sin (e+f x)} (\sin (e+f x)+1)^2 (-\sin (e+f x))^{-n} F_1\left (1;\frac{1}{2},-n;2;\frac{1}{2} (\sin (e+f x)+1),\sin (e+f x)+1\right )-\frac{4 (\sin (e+f x)+1) \sqrt{1-\frac{2}{\sin (e+f x)+1}} \left (1-\frac{1}{\sin (e+f x)+1}\right )^{-n} \left (2 (2 n+1) F_1\left (\frac{1}{2}-n;-\frac{1}{2},-n;\frac{3}{2}-n;\frac{2}{\sin (e+f x)+1},\frac{1}{\sin (e+f x)+1}\right )+(2 n-1) (\sin (e+f x)+1) F_1\left (-n-\frac{1}{2};-\frac{1}{2},-n;\frac{1}{2}-n;\frac{2}{\sin (e+f x)+1},\frac{1}{\sin (e+f x)+1}\right )\right )}{4 n^2-1}\right )}{8 f \sqrt{\sin (e+f x)+1}} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.088, size = 0, normalized size = 0. \begin{align*} \int{ \left ( \sin \left ( fx+e \right ) \right ) ^{n} \left ( 1+\sin \left ( fx+e \right ) \right ) ^{-{\frac{3}{2}}}}\, 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{\sin \left (f x + e\right )^{n}}{{\left (\sin \left (f x + e\right ) + 1\right )}^{\frac{3}{2}}}\,{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{\sin \left (f x + e\right )^{n} \sqrt{\sin \left (f x + e\right ) + 1}}{\cos \left (f x + e\right )^{2} - 2 \, \sin \left (f x + e\right ) - 2}, 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 \frac{\sin ^{n}{\left (e + f x \right )}}{\left (\sin{\left (e + f x \right )} + 1\right )^{\frac{3}{2}}}\, 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 \frac{\sin \left (f x + e\right )^{n}}{{\left (\sin \left (f x + e\right ) + 1\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]