Optimal. Leaf size=96 \[ -\frac{2 (4 n+5) \cos (e+f x) \, _2F_1\left (\frac{1}{2},-n;\frac{3}{2};1-\sin (e+f x)\right )}{f (2 n+3) \sqrt{\sin (e+f x)+1}}-\frac{2 \cos (e+f x) \sin ^{n+1}(e+f x)}{f (2 n+3) \sqrt{\sin (e+f x)+1}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.11055, antiderivative size = 96, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.19, Rules used = {2763, 21, 2776, 65} \[ -\frac{2 (4 n+5) \cos (e+f x) \, _2F_1\left (\frac{1}{2},-n;\frac{3}{2};1-\sin (e+f x)\right )}{f (2 n+3) \sqrt{\sin (e+f x)+1}}-\frac{2 \cos (e+f x) \sin ^{n+1}(e+f x)}{f (2 n+3) \sqrt{\sin (e+f x)+1}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2763
Rule 21
Rule 2776
Rule 65
Rubi steps
\begin{align*} \int \sin ^n(e+f x) (1+\sin (e+f x))^{3/2} \, dx &=-\frac{2 \cos (e+f x) \sin ^{1+n}(e+f x)}{f (3+2 n) \sqrt{1+\sin (e+f x)}}+\frac{2 \int \frac{\sin ^n(e+f x) \left (\frac{1}{2} (5+4 n)+\frac{1}{2} (5+4 n) \sin (e+f x)\right )}{\sqrt{1+\sin (e+f x)}} \, dx}{3+2 n}\\ &=-\frac{2 \cos (e+f x) \sin ^{1+n}(e+f x)}{f (3+2 n) \sqrt{1+\sin (e+f x)}}+\frac{(5+4 n) \int \sin ^n(e+f x) \sqrt{1+\sin (e+f x)} \, dx}{3+2 n}\\ &=-\frac{2 \cos (e+f x) \sin ^{1+n}(e+f x)}{f (3+2 n) \sqrt{1+\sin (e+f x)}}+\frac{((5+4 n) \cos (e+f x)) \operatorname{Subst}\left (\int \frac{x^n}{\sqrt{1-x}} \, dx,x,\sin (e+f x)\right )}{f (3+2 n) \sqrt{1-\sin (e+f x)} \sqrt{1+\sin (e+f x)}}\\ &=-\frac{2 (5+4 n) \cos (e+f x) \, _2F_1\left (\frac{1}{2},-n;\frac{3}{2};1-\sin (e+f x)\right )}{f (3+2 n) \sqrt{1+\sin (e+f x)}}-\frac{2 \cos (e+f x) \sin ^{1+n}(e+f x)}{f (3+2 n) \sqrt{1+\sin (e+f x)}}\\ \end{align*}
Mathematica [C] time = 23.2207, size = 5109, normalized size = 53.22 \[ \text{Result too large to show} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.118, 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 \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 (\sin \left (f x + e\right )^{n}{\left (\sin \left (f x + e\right ) + 1\right )}^{\frac{3}{2}}, 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 \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]