\(\int \cos ^3(e+f x) (a+b \sin ^n(e+f x))^p \, dx\) [363]

Optimal result

Integrand size = 23, antiderivative size = 148 \[ \int \cos ^3(e+f x) \left (a+b \sin ^n(e+f x)\right )^p \, dx=\frac {\operatorname {Hypergeometric2F1}\left (\frac {1}{n},-p,1+\frac {1}{n},-\frac {b \sin ^n(e+f x)}{a}\right ) \sin (e+f x) \left (a+b \sin ^n(e+f x)\right )^p \left (1+\frac {b \sin ^n(e+f x)}{a}\right )^{-p}}{f}-\frac {\operatorname {Hypergeometric2F1}\left (\frac {3}{n},-p,\frac {3+n}{n},-\frac {b \sin ^n(e+f x)}{a}\right ) \sin ^3(e+f x) \left (a+b \sin ^n(e+f x)\right )^p \left (1+\frac {b \sin ^n(e+f x)}{a}\right )^{-p}}{3 f} \] 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)-1/3*hypergeom([-p, 3/n],[(3+n)/n],-b*sin(f 
*x+e)^n/a)*sin(f*x+e)^3*(a+b*sin(f*x+e)^n)^p/f/((1+b*sin(f*x+e)^n/a)^p)
 

Mathematica [A] (verified)

Time = 0.11 (sec) , antiderivative size = 114, normalized size of antiderivative = 0.77 \[ \int \cos ^3(e+f x) \left (a+b \sin ^n(e+f x)\right )^p \, dx=-\frac {\sin (e+f x) \left (-3 \operatorname {Hypergeometric2F1}\left (\frac {1}{n},-p,1+\frac {1}{n},-\frac {b \sin ^n(e+f x)}{a}\right )+\operatorname {Hypergeometric2F1}\left (\frac {3}{n},-p,\frac {3+n}{n},-\frac {b \sin ^n(e+f x)}{a}\right ) \sin ^2(e+f x)\right ) \left (a+b \sin ^n(e+f x)\right )^p \left (1+\frac {b \sin ^n(e+f x)}{a}\right )^{-p}}{3 f} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.34 (sec) , antiderivative size = 146, normalized size of antiderivative = 0.99, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {3042, 3702, 2432, 2009}

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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3702

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

\(\Big \downarrow \) 2432

\(\displaystyle \frac {\int \left (\left (b \sin ^n(e+f x)+a\right )^p-\sin ^2(e+f x) \left (b \sin ^n(e+f x)+a\right )^p\right )d\sin (e+f x)}{f}\)

\(\Big \downarrow \) 2009

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

Input:

Int[Cos[e + f*x]^3*(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)/(1 + (b*Sin[e + f*x]^n)/a)^p - (Hypergeo 
metric2F1[3/n, -p, (3 + n)/n, -((b*Sin[e + f*x]^n)/a)]*Sin[e + f*x]^3*(a + 
 b*Sin[e + f*x]^n)^p)/(3*(1 + (b*Sin[e + f*x]^n)/a)^p))/f
 

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 2432

Int[(Pq_)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[ 
Pq*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, n, p}, x] && (PolyQ[Pq, x] || Poly 
Q[Pq, x^n])
 

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]

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

Input:

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

Output:

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

Fricas [F]

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

integrate(cos(f*x+e)^3*(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)^3, x)
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

integrate(cos(f*x+e)^3*(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)^3, x)
 

Giac [F]

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

integrate(cos(f*x+e)^3*(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)^3, x)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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