\(\int \sin ^3(e+f x) (a+b \sin ^2(e+f x))^p \, dx\) [123]

Optimal result

Integrand size = 23, antiderivative size = 131 \[ \int \sin ^3(e+f x) \left (a+b \sin ^2(e+f x)\right )^p \, dx=-\frac {\cos (e+f x) \left (a+b-b \cos ^2(e+f x)\right )^{1+p}}{b f (3+2 p)}+\frac {(a-2 b (1+p)) \cos (e+f x) \left (a+b-b \cos ^2(e+f x)\right )^p \left (1-\frac {b \cos ^2(e+f x)}{a+b}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-p,\frac {3}{2},\frac {b \cos ^2(e+f x)}{a+b}\right )}{b f (3+2 p)} \] Output:

-cos(f*x+e)*(a+b-b*cos(f*x+e)^2)^(p+1)/b/f/(3+2*p)+(a-2*b*(p+1))*cos(f*x+e 
)*(a+b-b*cos(f*x+e)^2)^p*hypergeom([1/2, -p],[3/2],b*cos(f*x+e)^2/(a+b))/b 
/f/(3+2*p)/((1-b*cos(f*x+e)^2/(a+b))^p)
 

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 6 vs. order 5 in optimal.

Time = 0.79 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.75 \[ \int \sin ^3(e+f x) \left (a+b \sin ^2(e+f x)\right )^p \, dx=\frac {\operatorname {AppellF1}\left (2,\frac {1}{2},-p,3,\sin ^2(e+f x),-\frac {b \sin ^2(e+f x)}{a}\right ) \sqrt {\cos ^2(e+f x)} \sin ^3(e+f x) \left (a+b \sin ^2(e+f x)\right )^p \left (\frac {a+b \sin ^2(e+f x)}{a}\right )^{-p} \tan (e+f x)}{4 f} \] Input:

Integrate[Sin[e + f*x]^3*(a + b*Sin[e + f*x]^2)^p,x]
 

Output:

(AppellF1[2, 1/2, -p, 3, Sin[e + f*x]^2, -((b*Sin[e + f*x]^2)/a)]*Sqrt[Cos 
[e + f*x]^2]*Sin[e + f*x]^3*(a + b*Sin[e + f*x]^2)^p*Tan[e + f*x])/(4*f*(( 
a + b*Sin[e + f*x]^2)/a)^p)
 

Rubi [A] (verified)

Time = 0.30 (sec) , antiderivative size = 130, normalized size of antiderivative = 0.99, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {3042, 3665, 299, 238, 237}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \sin ^3(e+f x) \left (a+b \sin ^2(e+f x)\right )^p \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \sin (e+f x)^3 \left (a+b \sin (e+f x)^2\right )^pdx\)

\(\Big \downarrow \) 3665

\(\displaystyle -\frac {\int \left (1-\cos ^2(e+f x)\right ) \left (-b \cos ^2(e+f x)+a+b\right )^pd\cos (e+f x)}{f}\)

\(\Big \downarrow \) 299

\(\displaystyle -\frac {\frac {\cos (e+f x) \left (a-b \cos ^2(e+f x)+b\right )^{p+1}}{b (2 p+3)}-\frac {(a-2 b (p+1)) \int \left (-b \cos ^2(e+f x)+a+b\right )^pd\cos (e+f x)}{b (2 p+3)}}{f}\)

\(\Big \downarrow \) 238

\(\displaystyle -\frac {\frac {\cos (e+f x) \left (a-b \cos ^2(e+f x)+b\right )^{p+1}}{b (2 p+3)}-\frac {(a-2 b (p+1)) \left (a-b \cos ^2(e+f x)+b\right )^p \left (1-\frac {b \cos ^2(e+f x)}{a+b}\right )^{-p} \int \left (1-\frac {b \cos ^2(e+f x)}{a+b}\right )^pd\cos (e+f x)}{b (2 p+3)}}{f}\)

\(\Big \downarrow \) 237

\(\displaystyle -\frac {\frac {\cos (e+f x) \left (a-b \cos ^2(e+f x)+b\right )^{p+1}}{b (2 p+3)}-\frac {(a-2 b (p+1)) \cos (e+f x) \left (a-b \cos ^2(e+f x)+b\right )^p \left (1-\frac {b \cos ^2(e+f x)}{a+b}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-p,\frac {3}{2},\frac {b \cos ^2(e+f x)}{a+b}\right )}{b (2 p+3)}}{f}\)

Input:

Int[Sin[e + f*x]^3*(a + b*Sin[e + f*x]^2)^p,x]
 

Output:

-(((Cos[e + f*x]*(a + b - b*Cos[e + f*x]^2)^(1 + p))/(b*(3 + 2*p)) - ((a - 
 2*b*(1 + p))*Cos[e + f*x]*(a + b - b*Cos[e + f*x]^2)^p*Hypergeometric2F1[ 
1/2, -p, 3/2, (b*Cos[e + f*x]^2)/(a + b)])/(b*(3 + 2*p)*(1 - (b*Cos[e + f* 
x]^2)/(a + b))^p))/f)
 

Defintions of rubi rules used

rule 237

Int[((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[a^p*x*Hypergeometric2F1[- 
p, 1/2, 1/2 + 1, (-b)*(x^2/a)], x] /; FreeQ[{a, b, p}, x] &&  !IntegerQ[2*p 
] && GtQ[a, 0]
 

rule 238

Int[((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[a^IntPart[p]*((a + b*x^2) 
^FracPart[p]/(1 + b*(x^2/a))^FracPart[p])   Int[(1 + b*(x^2/a))^p, x], x] / 
; FreeQ[{a, b, p}, x] &&  !IntegerQ[2*p] &&  !GtQ[a, 0]
 

rule 299

Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2), x_Symbol] :> Simp[d*x 
*((a + b*x^2)^(p + 1)/(b*(2*p + 3))), x] - Simp[(a*d - b*c*(2*p + 3))/(b*(2 
*p + 3))   Int[(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - 
 a*d, 0] && NeQ[2*p + 3, 0]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3665

Int[sin[(e_.) + (f_.)*(x_)]^(m_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^ 
(p_.), x_Symbol] :> With[{ff = FreeFactors[Cos[e + f*x], x]}, Simp[-ff/f 
Subst[Int[(1 - ff^2*x^2)^((m - 1)/2)*(a + b - b*ff^2*x^2)^p, x], x, Cos[e + 
 f*x]/ff], x]] /; FreeQ[{a, b, e, f, p}, x] && IntegerQ[(m - 1)/2]
 

Maple [F]

\[\int \sin \left (f x +e \right )^{3} \left (a +b \sin \left (f x +e \right )^{2}\right )^{p}d x\]

Input:

int(sin(f*x+e)^3*(a+b*sin(f*x+e)^2)^p,x)
 

Output:

int(sin(f*x+e)^3*(a+b*sin(f*x+e)^2)^p,x)
 

Fricas [F]

\[ \int \sin ^3(e+f x) \left (a+b \sin ^2(e+f x)\right )^p \, dx=\int { {\left (b \sin \left (f x + e\right )^{2} + a\right )}^{p} \sin \left (f x + e\right )^{3} \,d x } \] Input:

integrate(sin(f*x+e)^3*(a+b*sin(f*x+e)^2)^p,x, algorithm="fricas")
 

Output:

integral(-(cos(f*x + e)^2 - 1)*(-b*cos(f*x + e)^2 + a + b)^p*sin(f*x + e), 
 x)
 

Sympy [F(-1)]

Timed out. \[ \int \sin ^3(e+f x) \left (a+b \sin ^2(e+f x)\right )^p \, dx=\text {Timed out} \] Input:

integrate(sin(f*x+e)**3*(a+b*sin(f*x+e)**2)**p,x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \sin ^3(e+f x) \left (a+b \sin ^2(e+f x)\right )^p \, dx=\int { {\left (b \sin \left (f x + e\right )^{2} + a\right )}^{p} \sin \left (f x + e\right )^{3} \,d x } \] Input:

integrate(sin(f*x+e)^3*(a+b*sin(f*x+e)^2)^p,x, algorithm="maxima")
 

Output:

integrate((b*sin(f*x + e)^2 + a)^p*sin(f*x + e)^3, x)
 

Giac [F]

\[ \int \sin ^3(e+f x) \left (a+b \sin ^2(e+f x)\right )^p \, dx=\int { {\left (b \sin \left (f x + e\right )^{2} + a\right )}^{p} \sin \left (f x + e\right )^{3} \,d x } \] Input:

integrate(sin(f*x+e)^3*(a+b*sin(f*x+e)^2)^p,x, algorithm="giac")
 

Output:

integrate((b*sin(f*x + e)^2 + a)^p*sin(f*x + e)^3, x)
 

Mupad [F(-1)]

Timed out. \[ \int \sin ^3(e+f x) \left (a+b \sin ^2(e+f x)\right )^p \, dx=\int {\sin \left (e+f\,x\right )}^3\,{\left (b\,{\sin \left (e+f\,x\right )}^2+a\right )}^p \,d x \] Input:

int(sin(e + f*x)^3*(a + b*sin(e + f*x)^2)^p,x)
 

Output:

int(sin(e + f*x)^3*(a + b*sin(e + f*x)^2)^p, x)
 

Reduce [F]

\[ \int \sin ^3(e+f x) \left (a+b \sin ^2(e+f x)\right )^p \, dx=\int \left (\sin \left (f x +e \right )^{2} b +a \right )^{p} \sin \left (f x +e \right )^{3}d x \] Input:

int(sin(f*x+e)^3*(a+b*sin(f*x+e)^2)^p,x)
 

Output:

int((sin(e + f*x)**2*b + a)**p*sin(e + f*x)**3,x)