Optimal. Leaf size=124 \[ \frac{(a+b (2 p+3)) \sin (e+f x) \left (a+b \sin ^2(e+f x)\right )^p \left (\frac{b \sin ^2(e+f x)}{a}+1\right )^{-p} \, _2F_1\left (\frac{1}{2},-p;\frac{3}{2};-\frac{b \sin ^2(e+f x)}{a}\right )}{b f (2 p+3)}-\frac{\sin (e+f x) \left (a+b \sin ^2(e+f x)\right )^{p+1}}{b f (2 p+3)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.104095, antiderivative size = 119, normalized size of antiderivative = 0.96, 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 = {3190, 388, 246, 245} \[ \frac{\left (\frac{a}{2 b p+3 b}+1\right ) \sin (e+f x) \left (a+b \sin ^2(e+f x)\right )^p \left (\frac{b \sin ^2(e+f x)}{a}+1\right )^{-p} \, _2F_1\left (\frac{1}{2},-p;\frac{3}{2};-\frac{b \sin ^2(e+f x)}{a}\right )}{f}-\frac{\sin (e+f x) \left (a+b \sin ^2(e+f x)\right )^{p+1}}{b f (2 p+3)} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3190
Rule 388
Rule 246
Rule 245
Rubi steps
\begin{align*} \int \cos ^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 x^2\right )^p \, dx,x,\sin (e+f x)\right )}{f}\\ &=-\frac{\sin (e+f x) \left (a+b \sin ^2(e+f x)\right )^{1+p}}{b f (3+2 p)}+\frac{\left (1+\frac{a}{3 b+2 b p}\right ) \operatorname{Subst}\left (\int \left (a+b x^2\right )^p \, dx,x,\sin (e+f x)\right )}{f}\\ &=-\frac{\sin (e+f x) \left (a+b \sin ^2(e+f x)\right )^{1+p}}{b f (3+2 p)}+\frac{\left (\left (1+\frac{a}{3 b+2 b p}\right ) \left (a+b \sin ^2(e+f x)\right )^p \left (1+\frac{b \sin ^2(e+f x)}{a}\right )^{-p}\right ) \operatorname{Subst}\left (\int \left (1+\frac{b x^2}{a}\right )^p \, dx,x,\sin (e+f x)\right )}{f}\\ &=-\frac{\sin (e+f x) \left (a+b \sin ^2(e+f x)\right )^{1+p}}{b f (3+2 p)}+\frac{\left (1+\frac{a}{3 b+2 b p}\right ) \, _2F_1\left (\frac{1}{2},-p;\frac{3}{2};-\frac{b \sin ^2(e+f x)}{a}\right ) \sin (e+f x) \left (a+b \sin ^2(e+f x)\right )^p \left (1+\frac{b \sin ^2(e+f x)}{a}\right )^{-p}}{f}\\ \end{align*}
Mathematica [A] time = 0.194375, size = 120, normalized size = 0.97 \[ -\frac{\sin (e+f x) \left (a+b \sin ^2(e+f x)\right )^p \left (\frac{b \sin ^2(e+f x)}{a}+1\right )^{-p} \left (\left (a+b \sin ^2(e+f x)\right ) \left (\frac{b \sin ^2(e+f x)}{a}+1\right )^p-(a+b (2 p+3)) \, _2F_1\left (\frac{1}{2},-p;\frac{3}{2};-\frac{b \sin ^2(e+f x)}{a}\right )\right )}{b f (2 p+3)} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 2.313, size = 0, normalized size = 0. \begin{align*} \int \left ( \cos \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} \cos \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 (-b \cos \left (f x + e\right )^{2} + a + b\right )}^{p} \cos \left (f x + e\right )^{3}, 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} \cos \left (f x + e\right )^{3}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]