\(\int \cos (e+f x) (a+b \sin ^n(e+f x))^p \, dx\) [364]

Optimal result

Integrand size = 21, antiderivative size = 69 \[ \int \cos (e+f x) \left (a+b \sin ^n(e+f x)\right )^p \, dx=\frac {\operatorname {Hypergeometric2F1}\left (\frac {1}{n},-p,1+\frac {1}{n},-\frac {b \sin ^n(e+f x)}{a}\right ) \sin (e+f x) \left (a+b \sin ^n(e+f x)\right )^p \left (1+\frac {b \sin ^n(e+f x)}{a}\right )^{-p}}{f} \] Output:

hypergeom([-p, 1/n],[1+1/n],-b*sin(f*x+e)^n/a)*sin(f*x+e)*(a+b*sin(f*x+e)^ 
n)^p/f/((1+b*sin(f*x+e)^n/a)^p)
 

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.00 \[ \int \cos (e+f x) \left (a+b \sin ^n(e+f x)\right )^p \, dx=\frac {\operatorname {Hypergeometric2F1}\left (\frac {1}{n},-p,1+\frac {1}{n},-\frac {b \sin ^n(e+f x)}{a}\right ) \sin (e+f x) \left (a+b \sin ^n(e+f x)\right )^p \left (1+\frac {b \sin ^n(e+f x)}{a}\right )^{-p}}{f} \] Input:

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

Output:

(Hypergeometric2F1[n^(-1), -p, 1 + n^(-1), -((b*Sin[e + f*x]^n)/a)]*Sin[e 
+ f*x]*(a + b*Sin[e + f*x]^n)^p)/(f*(1 + (b*Sin[e + f*x]^n)/a)^p)
 

Rubi [A] (verified)

Time = 0.24 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {3042, 3702, 779, 778}

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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3702

\(\displaystyle \frac {\int \left (b \sin ^n(e+f x)+a\right )^pd\sin (e+f x)}{f}\)

\(\Big \downarrow \) 779

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

\(\Big \downarrow \) 778

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

Input:

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

Output:

(Hypergeometric2F1[n^(-1), -p, 1 + n^(-1), -((b*Sin[e + f*x]^n)/a)]*Sin[e 
+ f*x]*(a + b*Sin[e + f*x]^n)^p)/(f*(1 + (b*Sin[e + f*x]^n)/a)^p)
 

Defintions of rubi rules used

rule 778

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^p*x*Hypergeometric2F 
1[-p, 1/n, 1/n + 1, (-b)*(x^n/a)], x] /; FreeQ[{a, b, n, p}, x] &&  !IGtQ[p 
, 0] &&  !IntegerQ[1/n] &&  !ILtQ[Simplify[1/n + p], 0] && (IntegerQ[p] || 
GtQ[a, 0])
 

rule 779

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

rule 3042

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

rule 3702

Int[cos[(e_.) + (f_.)*(x_)]^(m_.)*((a_) + (b_.)*((c_.)*sin[(e_.) + (f_.)*(x 
_)])^(n_))^(p_.), x_Symbol] :> With[{ff = FreeFactors[Sin[e + f*x], x]}, Si 
mp[ff/f   Subst[Int[(1 - ff^2*x^2)^((m - 1)/2)*(a + b*(c*ff*x)^n)^p, x], x, 
 Sin[e + f*x]/ff], x]] /; FreeQ[{a, b, c, e, f, n, p}, x] && IntegerQ[(m - 
1)/2] && (EqQ[n, 4] || GtQ[m, 0] || IGtQ[p, 0] || IntegersQ[m, p])
 

Maple [F]

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

Input:

int(cos(f*x+e)*(a+b*sin(f*x+e)^n)^p,x)
 

Output:

int(cos(f*x+e)*(a+b*sin(f*x+e)^n)^p,x)
 

Fricas [F]

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

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

Output:

integral((b*sin(f*x + e)^n + a)^p*cos(f*x + e), x)
 

Sympy [F]

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

integrate(cos(f*x+e)*(a+b*sin(f*x+e)**n)**p,x)
 

Output:

Integral((a + b*sin(e + f*x)**n)**p*cos(e + f*x), x)
 

Maxima [F]

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

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

Output:

integrate((b*sin(f*x + e)^n + a)^p*cos(f*x + e), x)
 

Giac [F]

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

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

Output:

integrate((b*sin(f*x + e)^n + a)^p*cos(f*x + e), x)
 

Mupad [B] (verification not implemented)

Time = 36.56 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.01 \[ \int \cos (e+f x) \left (a+b \sin ^n(e+f x)\right )^p \, dx=\frac {\sin \left (e+f\,x\right )\,{\left (a+b\,{\sin \left (e+f\,x\right )}^n\right )}^p\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{n},-p;\ \frac {1}{n}+1;\ -\frac {b\,{\sin \left (e+f\,x\right )}^n}{a}\right )}{f\,{\left (\frac {b\,{\sin \left (e+f\,x\right )}^n}{a}+1\right )}^p} \] Input:

int(cos(e + f*x)*(a + b*sin(e + f*x)^n)^p,x)
 

Output:

(sin(e + f*x)*(a + b*sin(e + f*x)^n)^p*hypergeom([1/n, -p], 1/n + 1, -(b*s 
in(e + f*x)^n)/a))/(f*((b*sin(e + f*x)^n)/a + 1)^p)
 

Reduce [F]

\[ \int \cos (e+f x) \left (a+b \sin ^n(e+f x)\right )^p \, dx=\frac {\left (\sin \left (f x +e \right )^{n} b +a \right )^{p} \sin \left (f x +e \right )+\left (\int \frac {\left (\sin \left (f x +e \right )^{n} b +a \right )^{p} \cos \left (f x +e \right )}{\sin \left (f x +e \right )^{n} b n p +\sin \left (f x +e \right )^{n} b +a n p +a}d x \right ) a f \,n^{2} p^{2}+\left (\int \frac {\left (\sin \left (f x +e \right )^{n} b +a \right )^{p} \cos \left (f x +e \right )}{\sin \left (f x +e \right )^{n} b n p +\sin \left (f x +e \right )^{n} b +a n p +a}d x \right ) a f n p}{f \left (n p +1\right )} \] Input:

int(cos(f*x+e)*(a+b*sin(f*x+e)^n)^p,x)
 

Output:

((sin(e + f*x)**n*b + a)**p*sin(e + f*x) + int(((sin(e + f*x)**n*b + a)**p 
*cos(e + f*x))/(sin(e + f*x)**n*b*n*p + sin(e + f*x)**n*b + a*n*p + a),x)* 
a*f*n**2*p**2 + int(((sin(e + f*x)**n*b + a)**p*cos(e + f*x))/(sin(e + f*x 
)**n*b*n*p + sin(e + f*x)**n*b + a*n*p + a),x)*a*f*n*p)/(f*(n*p + 1))