Optimal. Leaf size=131 \[ \frac{(a-2 b (p+1)) \cos (e+f x) \left (a-b \cos ^2(e+f x)+b\right )^p \left (1-\frac{b \cos ^2(e+f x)}{a+b}\right )^{-p} \, _2F_1\left (\frac{1}{2},-p;\frac{3}{2};\frac{b \cos ^2(e+f x)}{a+b}\right )}{b f (2 p+3)}-\frac{\cos (e+f x) \left (a-b \cos ^2(e+f x)+b\right )^{p+1}}{b f (2 p+3)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.109882, antiderivative size = 131, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.174, Rules used = {3186, 388, 246, 245} \[ \frac{(a-2 b (p+1)) \cos (e+f x) \left (a-b \cos ^2(e+f x)+b\right )^p \left (1-\frac{b \cos ^2(e+f x)}{a+b}\right )^{-p} \, _2F_1\left (\frac{1}{2},-p;\frac{3}{2};\frac{b \cos ^2(e+f x)}{a+b}\right )}{b f (2 p+3)}-\frac{\cos (e+f x) \left (a-b \cos ^2(e+f x)+b\right )^{p+1}}{b f (2 p+3)} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3186
Rule 388
Rule 246
Rule 245
Rubi steps
\begin{align*} \int \sin ^3(e+f x) \left (a+b \sin ^2(e+f x)\right )^p \, dx &=-\frac{\operatorname{Subst}\left (\int \left (1-x^2\right ) \left (a+b-b x^2\right )^p \, dx,x,\cos (e+f x)\right )}{f}\\ &=-\frac{\cos (e+f x) \left (a+b-b \cos ^2(e+f x)\right )^{1+p}}{b f (3+2 p)}+\frac{(a-2 b (1+p)) \operatorname{Subst}\left (\int \left (a+b-b x^2\right )^p \, dx,x,\cos (e+f x)\right )}{b f (3+2 p)}\\ &=-\frac{\cos (e+f x) \left (a+b-b \cos ^2(e+f x)\right )^{1+p}}{b f (3+2 p)}+\frac{\left ((a-2 b (1+p)) \left (a+b-b \cos ^2(e+f x)\right )^p \left (1-\frac{b \cos ^2(e+f x)}{a+b}\right )^{-p}\right ) \operatorname{Subst}\left (\int \left (1-\frac{b x^2}{a+b}\right )^p \, dx,x,\cos (e+f x)\right )}{b f (3+2 p)}\\ &=-\frac{\cos (e+f x) \left (a+b-b \cos ^2(e+f x)\right )^{1+p}}{b f (3+2 p)}+\frac{(a-2 b (1+p)) \cos (e+f x) \left (a+b-b \cos ^2(e+f x)\right )^p \left (1-\frac{b \cos ^2(e+f x)}{a+b}\right )^{-p} \, _2F_1\left (\frac{1}{2},-p;\frac{3}{2};\frac{b \cos ^2(e+f x)}{a+b}\right )}{b f (3+2 p)}\\ \end{align*}
Mathematica [C] time = 0.37294, size = 98, normalized size = 0.75 \[ \frac{\sin ^3(e+f x) \sqrt{\cos ^2(e+f x)} \tan (e+f x) \left (a+b \sin ^2(e+f x)\right )^p \left (\frac{a+b \sin ^2(e+f x)}{a}\right )^{-p} F_1\left (2;\frac{1}{2},-p;3;\sin ^2(e+f x),-\frac{b \sin ^2(e+f x)}{a}\right )}{4 f} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 2.264, size = 0, normalized size = 0. \begin{align*} \int \left ( \sin \left ( fx+e \right ) \right ) ^{3} \left ( a+b \left ( \sin \left ( fx+e \right ) \right ) ^{2} \right ) ^{p}\, 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 (b \sin \left (f x + e\right )^{2} + a\right )}^{p} \sin \left (f x + e\right )^{3}\,{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 (\cos \left (f x + e\right )^{2} - 1\right )}{\left (-b \cos \left (f x + e\right )^{2} + a + b\right )}^{p} \sin \left (f x + e\right ), 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 (b \sin \left (f x + e\right )^{2} + a\right )}^{p} \sin \left (f x + e\right )^{3}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]