\(\int (c (d \sin (e+f x))^p)^n (a+b \sin (e+f x))^2 \, dx\) [834]

Optimal result

Integrand size = 27, antiderivative size = 231 \[ \int \left (c (d \sin (e+f x))^p\right )^n (a+b \sin (e+f x))^2 \, dx=-\frac {b^2 \cos (e+f x) \sin (e+f x) \left (c (d \sin (e+f x))^p\right )^n}{f (2+n p)}+\frac {\left (b^2 (1+n p)+a^2 (2+n p)\right ) \cos (e+f x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{2} (1+n p),\frac {1}{2} (3+n p),\sin ^2(e+f x)\right ) \sin (e+f x) \left (c (d \sin (e+f x))^p\right )^n}{f (1+n p) (2+n p) \sqrt {\cos ^2(e+f x)}}+\frac {2 a b \cos (e+f x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{2} (2+n p),\frac {1}{2} (4+n p),\sin ^2(e+f x)\right ) \sin ^2(e+f x) \left (c (d \sin (e+f x))^p\right )^n}{f (2+n p) \sqrt {\cos ^2(e+f x)}} \] Output:

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

Mathematica [A] (verified)

Time = 0.38 (sec) , antiderivative size = 198, normalized size of antiderivative = 0.86 \[ \int \left (c (d \sin (e+f x))^p\right )^n (a+b \sin (e+f x))^2 \, dx=\frac {\sqrt {\cos ^2(e+f x)} \left (c (d \sin (e+f x))^p\right )^n \left (a^2 \left (6+5 n p+n^2 p^2\right ) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{2} (1+n p),\frac {1}{2} (3+n p),\sin ^2(e+f x)\right )+b (1+n p) \sin (e+f x) \left (2 a (3+n p) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},1+\frac {n p}{2},2+\frac {n p}{2},\sin ^2(e+f x)\right )+b (2+n p) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{2} (3+n p),\frac {1}{2} (5+n p),\sin ^2(e+f x)\right ) \sin (e+f x)\right )\right ) \tan (e+f x)}{f (1+n p) (2+n p) (3+n p)} \] Input:

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

Output:

(Sqrt[Cos[e + f*x]^2]*(c*(d*Sin[e + f*x])^p)^n*(a^2*(6 + 5*n*p + n^2*p^2)* 
Hypergeometric2F1[1/2, (1 + n*p)/2, (3 + n*p)/2, Sin[e + f*x]^2] + b*(1 + 
n*p)*Sin[e + f*x]*(2*a*(3 + n*p)*Hypergeometric2F1[1/2, 1 + (n*p)/2, 2 + ( 
n*p)/2, Sin[e + f*x]^2] + b*(2 + n*p)*Hypergeometric2F1[1/2, (3 + n*p)/2, 
(5 + n*p)/2, Sin[e + f*x]^2]*Sin[e + f*x]))*Tan[e + f*x])/(f*(1 + n*p)*(2 
+ n*p)*(3 + n*p))
 

Rubi [A] (verified)

Time = 0.76 (sec) , antiderivative size = 242, normalized size of antiderivative = 1.05, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {3042, 3305, 3042, 3268, 3042, 3122, 3493, 3042, 3122}

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.

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

\(\Big \downarrow \) 3042

\(\displaystyle \int (a+b \sin (e+f x))^2 \left (c (d \sin (e+f x))^p\right )^ndx\)

\(\Big \downarrow \) 3305

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3268

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3122

\(\displaystyle (d \sin (e+f x))^{-n p} \left (c (d \sin (e+f x))^p\right )^n \left (\int (d \sin (e+f x))^{n p} \left (a^2+b^2 \sin (e+f x)^2\right )dx+\frac {2 a b \cos (e+f x) (d \sin (e+f x))^{n p+2} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{2} (n p+2),\frac {1}{2} (n p+4),\sin ^2(e+f x)\right )}{d^2 f (n p+2) \sqrt {\cos ^2(e+f x)}}\right )\)

\(\Big \downarrow \) 3493

\(\displaystyle (d \sin (e+f x))^{-n p} \left (c (d \sin (e+f x))^p\right )^n \left (\left (a^2+\frac {b^2 (n p+1)}{n p+2}\right ) \int (d \sin (e+f x))^{n p}dx+\frac {2 a b \cos (e+f x) (d \sin (e+f x))^{n p+2} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{2} (n p+2),\frac {1}{2} (n p+4),\sin ^2(e+f x)\right )}{d^2 f (n p+2) \sqrt {\cos ^2(e+f x)}}-\frac {b^2 \cos (e+f x) (d \sin (e+f x))^{n p+1}}{d f (n p+2)}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle (d \sin (e+f x))^{-n p} \left (c (d \sin (e+f x))^p\right )^n \left (\left (a^2+\frac {b^2 (n p+1)}{n p+2}\right ) \int (d \sin (e+f x))^{n p}dx+\frac {2 a b \cos (e+f x) (d \sin (e+f x))^{n p+2} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{2} (n p+2),\frac {1}{2} (n p+4),\sin ^2(e+f x)\right )}{d^2 f (n p+2) \sqrt {\cos ^2(e+f x)}}-\frac {b^2 \cos (e+f x) (d \sin (e+f x))^{n p+1}}{d f (n p+2)}\right )\)

\(\Big \downarrow \) 3122

\(\displaystyle (d \sin (e+f x))^{-n p} \left (c (d \sin (e+f x))^p\right )^n \left (\frac {\left (a^2+\frac {b^2 (n p+1)}{n p+2}\right ) \cos (e+f x) (d \sin (e+f x))^{n p+1} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{2} (n p+1),\frac {1}{2} (n p+3),\sin ^2(e+f x)\right )}{d f (n p+1) \sqrt {\cos ^2(e+f x)}}+\frac {2 a b \cos (e+f x) (d \sin (e+f x))^{n p+2} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{2} (n p+2),\frac {1}{2} (n p+4),\sin ^2(e+f x)\right )}{d^2 f (n p+2) \sqrt {\cos ^2(e+f x)}}-\frac {b^2 \cos (e+f x) (d \sin (e+f x))^{n p+1}}{d f (n p+2)}\right )\)

Input:

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

Output:

((c*(d*Sin[e + f*x])^p)^n*(-((b^2*Cos[e + f*x]*(d*Sin[e + f*x])^(1 + n*p)) 
/(d*f*(2 + n*p))) + ((a^2 + (b^2*(1 + n*p))/(2 + n*p))*Cos[e + f*x]*Hyperg 
eometric2F1[1/2, (1 + n*p)/2, (3 + n*p)/2, Sin[e + f*x]^2]*(d*Sin[e + f*x] 
)^(1 + n*p))/(d*f*(1 + n*p)*Sqrt[Cos[e + f*x]^2]) + (2*a*b*Cos[e + f*x]*Hy 
pergeometric2F1[1/2, (2 + n*p)/2, (4 + n*p)/2, Sin[e + f*x]^2]*(d*Sin[e + 
f*x])^(2 + n*p))/(d^2*f*(2 + n*p)*Sqrt[Cos[e + f*x]^2])))/(d*Sin[e + f*x]) 
^(n*p)
 

Defintions of rubi rules used

rule 3042

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

rule 3122

Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[Cos[c + d*x]*(( 
b*Sin[c + d*x])^(n + 1)/(b*d*(n + 1)*Sqrt[Cos[c + d*x]^2]))*Hypergeometric2 
F1[1/2, (n + 1)/2, (n + 3)/2, Sin[c + d*x]^2], x] /; FreeQ[{b, c, d, n}, x] 
 &&  !IntegerQ[2*n]
 

rule 3268

Int[((b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_) + (d_.)*sin[(e_.) + (f_.)*(x 
_)])^2, x_Symbol] :> Simp[2*c*(d/b)   Int[(b*Sin[e + f*x])^(m + 1), x], x] 
+ Int[(b*Sin[e + f*x])^m*(c^2 + d^2*Sin[e + f*x]^2), x] /; FreeQ[{b, c, d, 
e, f, m}, x]
 

rule 3305

Int[((c_.)*((d_.)*sin[(e_.) + (f_.)*(x_)])^(p_))^(n_)*((a_.) + (b_.)*sin[(e 
_.) + (f_.)*(x_)])^(m_.), x_Symbol] :> Simp[c^IntPart[n]*((c*(d*Sin[e + f*x 
])^p)^FracPart[n]/(d*Sin[e + f*x])^(p*FracPart[n]))   Int[(a + b*Sin[e + f* 
x])^m*(d*Sin[e + f*x])^(n*p), x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, 
x] &&  !IntegerQ[n]
 

rule 3493

Int[((b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((A_) + (C_.)*sin[(e_.) + (f_.)*( 
x_)]^2), x_Symbol] :> Simp[(-C)*Cos[e + f*x]*((b*Sin[e + f*x])^(m + 1)/(b*f 
*(m + 2))), x] + Simp[(A*(m + 2) + C*(m + 1))/(m + 2)   Int[(b*Sin[e + f*x] 
)^m, x], x] /; FreeQ[{b, e, f, A, C, m}, x] &&  !LtQ[m, -1]
 

Maple [F]

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

Input:

int((c*(d*sin(f*x+e))^p)^n*(a+b*sin(f*x+e))^2,x)
 

Output:

int((c*(d*sin(f*x+e))^p)^n*(a+b*sin(f*x+e))^2,x)
 

Fricas [F]

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

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

Output:

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

Sympy [F]

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

integrate((c*(d*sin(f*x+e))**p)**n*(a+b*sin(f*x+e))**2,x)
 

Output:

Integral((c*(d*sin(e + f*x))**p)**n*(a + b*sin(e + f*x))**2, x)
 

Maxima [F]

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

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

Output:

integrate((b*sin(f*x + e) + a)^2*((d*sin(f*x + e))^p*c)^n, x)
 

Giac [F]

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

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

Output:

integrate((b*sin(f*x + e) + a)^2*((d*sin(f*x + e))^p*c)^n, x)
 

Mupad [F(-1)]

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

int((c*(d*sin(e + f*x))^p)^n*(a + b*sin(e + f*x))^2,x)
 

Output:

int((c*(d*sin(e + f*x))^p)^n*(a + b*sin(e + f*x))^2, x)
 

Reduce [F]

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

int((c*(d*sin(f*x+e))^p)^n*(a+b*sin(f*x+e))^2,x)
 

Output:

d**(n*p)*c**n*(int(sin(e + f*x)**(n*p),x)*a**2 + int(sin(e + f*x)**(n*p)*s 
in(e + f*x)**2,x)*b**2 + 2*int(sin(e + f*x)**(n*p)*sin(e + f*x),x)*a*b)