\(\int \frac {(c (d \sin (e+f x))^p)^n}{(a+a \sin (e+f x))^2} \, dx\) [831]

Optimal result

Integrand size = 27, antiderivative size = 288 \[ \int \frac {\left (c (d \sin (e+f x))^p\right )^n}{(a+a \sin (e+f x))^2} \, dx=-\frac {n p (1-2 n p) \cos (e+f x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{2} (1+n p),\frac {1}{2} (3+n p),\sin ^2(e+f x)\right ) \sin (e+f x) \left (c (d \sin (e+f x))^p\right )^n}{3 a^2 f (1+n p) \sqrt {\cos ^2(e+f x)}}+\frac {2 \left (1-n^2 p^2\right ) \cos (e+f x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{2} (2+n p),\frac {1}{2} (4+n p),\sin ^2(e+f x)\right ) \sin ^2(e+f x) \left (c (d \sin (e+f x))^p\right )^n}{3 a^2 f (2+n p) \sqrt {\cos ^2(e+f x)}}+\frac {2 (1-n p) \cos (e+f x) \sin (e+f x) \left (c (d \sin (e+f x))^p\right )^n}{3 a^2 f (1+\sin (e+f x))}+\frac {\cos (e+f x) \sin (e+f x) \left (c (d \sin (e+f x))^p\right )^n}{3 f (a+a \sin (e+f x))^2} \] Output:

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

Mathematica [A] (verified)

Time = 2.65 (sec) , antiderivative size = 216, normalized size of antiderivative = 0.75 \[ \int \frac {\left (c (d \sin (e+f x))^p\right )^n}{(a+a \sin (e+f x))^2} \, dx=\frac {\left (c (d \sin (e+f x))^p\right )^n \left (\frac {1}{2} \sin (2 (e+f x))-\frac {(1+\sin (e+f x)) \left ((-1+n p) (1+n p) (2+n p) \sin (2 (e+f x))+\sqrt {\cos ^2(e+f x)} (1+\sin (e+f x)) \left (n p (1-2 n p) (2+n p) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{2} (1+n p),\frac {1}{2} (3+n p),\sin ^2(e+f x)\right )+2 (-1+n p) (1+n p)^2 \operatorname {Hypergeometric2F1}\left (\frac {1}{2},1+\frac {n p}{2},2+\frac {n p}{2},\sin ^2(e+f x)\right ) \sin (e+f x)\right ) \tan (e+f x)\right )}{(1+n p) (2+n p)}\right )}{3 a^2 f (1+\sin (e+f x))^2} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 1.06 (sec) , antiderivative size = 305, normalized size of antiderivative = 1.06, number of steps used = 11, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.407, Rules used = {3042, 3305, 3042, 3245, 3042, 3457, 25, 3042, 3227, 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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3305

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3245

\(\displaystyle (d \sin (e+f x))^{-n p} \left (c (d \sin (e+f x))^p\right )^n \left (\frac {\int \frac {(d \sin (e+f x))^{n p} (a d (2-n p)+a d n p \sin (e+f x))}{\sin (e+f x) a+a}dx}{3 a^2 d}+\frac {\cos (e+f x) (d \sin (e+f x))^{n p+1}}{3 d f (a \sin (e+f x)+a)^2}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle (d \sin (e+f x))^{-n p} \left (c (d \sin (e+f x))^p\right )^n \left (\frac {\int \frac {(d \sin (e+f x))^{n p} (a d (2-n p)+a d n p \sin (e+f x))}{\sin (e+f x) a+a}dx}{3 a^2 d}+\frac {\cos (e+f x) (d \sin (e+f x))^{n p+1}}{3 d f (a \sin (e+f x)+a)^2}\right )\)

\(\Big \downarrow \) 3457

\(\displaystyle (d \sin (e+f x))^{-n p} \left (c (d \sin (e+f x))^p\right )^n \left (\frac {\frac {\int -(d \sin (e+f x))^{n p} \left (a^2 d^2 n p (1-2 n p)-2 a^2 d^2 (1-n p) (n p+1) \sin (e+f x)\right )dx}{a^2 d}+\frac {2 (1-n p) \cos (e+f x) (d \sin (e+f x))^{n p+1}}{f (\sin (e+f x)+1)}}{3 a^2 d}+\frac {\cos (e+f x) (d \sin (e+f x))^{n p+1}}{3 d f (a \sin (e+f x)+a)^2}\right )\)

\(\Big \downarrow \) 25

\(\displaystyle (d \sin (e+f x))^{-n p} \left (c (d \sin (e+f x))^p\right )^n \left (\frac {\frac {2 (1-n p) \cos (e+f x) (d \sin (e+f x))^{n p+1}}{f (\sin (e+f x)+1)}-\frac {\int (d \sin (e+f x))^{n p} \left (a^2 d^2 n p (1-2 n p)-2 a^2 d^2 (1-n p) (n p+1) \sin (e+f x)\right )dx}{a^2 d}}{3 a^2 d}+\frac {\cos (e+f x) (d \sin (e+f x))^{n p+1}}{3 d f (a \sin (e+f x)+a)^2}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle (d \sin (e+f x))^{-n p} \left (c (d \sin (e+f x))^p\right )^n \left (\frac {\frac {2 (1-n p) \cos (e+f x) (d \sin (e+f x))^{n p+1}}{f (\sin (e+f x)+1)}-\frac {\int (d \sin (e+f x))^{n p} \left (a^2 d^2 n p (1-2 n p)-2 a^2 d^2 (1-n p) (n p+1) \sin (e+f x)\right )dx}{a^2 d}}{3 a^2 d}+\frac {\cos (e+f x) (d \sin (e+f x))^{n p+1}}{3 d f (a \sin (e+f x)+a)^2}\right )\)

\(\Big \downarrow \) 3227

\(\displaystyle (d \sin (e+f x))^{-n p} \left (c (d \sin (e+f x))^p\right )^n \left (\frac {\frac {2 (1-n p) \cos (e+f x) (d \sin (e+f x))^{n p+1}}{f (\sin (e+f x)+1)}-\frac {a^2 d^2 n p (1-2 n p) \int (d \sin (e+f x))^{n p}dx-2 a^2 d (1-n p) (n p+1) \int (d \sin (e+f x))^{n p+1}dx}{a^2 d}}{3 a^2 d}+\frac {\cos (e+f x) (d \sin (e+f x))^{n p+1}}{3 d f (a \sin (e+f x)+a)^2}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle (d \sin (e+f x))^{-n p} \left (c (d \sin (e+f x))^p\right )^n \left (\frac {\frac {2 (1-n p) \cos (e+f x) (d \sin (e+f x))^{n p+1}}{f (\sin (e+f x)+1)}-\frac {a^2 d^2 n p (1-2 n p) \int (d \sin (e+f x))^{n p}dx-2 a^2 d (1-n p) (n p+1) \int (d \sin (e+f x))^{n p+1}dx}{a^2 d}}{3 a^2 d}+\frac {\cos (e+f x) (d \sin (e+f x))^{n p+1}}{3 d f (a \sin (e+f x)+a)^2}\right )\)

\(\Big \downarrow \) 3122

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

Input:

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

Output:

((c*(d*Sin[e + f*x])^p)^n*((Cos[e + f*x]*(d*Sin[e + f*x])^(1 + n*p))/(3*d* 
f*(a + a*Sin[e + f*x])^2) + ((2*(1 - n*p)*Cos[e + f*x]*(d*Sin[e + f*x])^(1 
 + n*p))/(f*(1 + Sin[e + f*x])) - ((a^2*d*n*p*(1 - 2*n*p)*Cos[e + f*x]*Hyp 
ergeometric2F1[1/2, (1 + n*p)/2, (3 + n*p)/2, Sin[e + f*x]^2]*(d*Sin[e + f 
*x])^(1 + n*p))/(f*(1 + n*p)*Sqrt[Cos[e + f*x]^2]) - (2*a^2*(1 - n*p)*(1 + 
 n*p)*Cos[e + f*x]*Hypergeometric2F1[1/2, (2 + n*p)/2, (4 + n*p)/2, Sin[e 
+ f*x]^2]*(d*Sin[e + f*x])^(2 + n*p))/(f*(2 + n*p)*Sqrt[Cos[e + f*x]^2]))/ 
(a^2*d))/(3*a^2*d)))/(d*Sin[e + f*x])^(n*p)
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

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 3227

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

rule 3245

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + 
(f_.)*(x_)])^(n_), x_Symbol] :> Simp[b^2*Cos[e + f*x]*(a + b*Sin[e + f*x])^ 
m*((c + d*Sin[e + f*x])^(n + 1)/(a*f*(2*m + 1)*(b*c - a*d))), x] + Simp[1/( 
a*(2*m + 1)*(b*c - a*d))   Int[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + 
f*x])^n*Simp[b*c*(m + 1) - a*d*(2*m + n + 2) + b*d*(m + n + 2)*Sin[e + f*x] 
, x], x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && EqQ 
[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && LtQ[m, -1] &&  !GtQ[n, 0] && (Intege 
rsQ[2*m, 2*n] || (IntegerQ[m] && EqQ[c, 0]))
 

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 3457

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*sin[(e_.) + 
(f_.)*(x_)])*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Sim 
p[b*(A*b - a*B)*Cos[e + f*x]*(a + b*Sin[e + f*x])^m*((c + d*Sin[e + f*x])^( 
n + 1)/(a*f*(2*m + 1)*(b*c - a*d))), x] + Simp[1/(a*(2*m + 1)*(b*c - a*d)) 
  Int[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^n*Simp[B*(a*c*m + b 
*d*(n + 1)) + A*(b*c*(m + 1) - a*d*(2*m + n + 2)) + d*(A*b - a*B)*(m + n + 
2)*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, n}, x] && NeQ 
[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && LtQ[m, -2^(-1)] 
 &&  !GtQ[n, 0] && IntegerQ[2*m] && (IntegerQ[2*n] || EqQ[c, 0])
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

Integral((c*(d*sin(e + f*x))**p)**n/(sin(e + f*x)**2 + 2*sin(e + f*x) + 1) 
, x)/a**2
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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