\(\int \frac {(c (d \sin (e+f x))^p)^n}{(3+3 \sin (e+f x))^2} \, dx\) [824]

Optimal result

Integrand size = 27, antiderivative size = 279 \[ \int \frac {\left (c (d \sin (e+f x))^p\right )^n}{(3+3 \sin (e+f x))^2} \, dx=-\frac {n p (1-2 n p) \cos (e+f x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{2} (1+n p),\frac {1}{2} (3+n p),\sin ^2(e+f x)\right ) \sin (e+f x) \left (c (d \sin (e+f x))^p\right )^n}{27 f (1+n p) \sqrt {\cos ^2(e+f x)}}+\frac {2 \left (1-n^2 p^2\right ) \cos (e+f x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{2} (2+n p),\frac {1}{2} (4+n p),\sin ^2(e+f x)\right ) \sin ^2(e+f x) \left (c (d \sin (e+f x))^p\right )^n}{27 f (2+n p) \sqrt {\cos ^2(e+f x)}}+\frac {2 (1-n p) \cos (e+f x) \sin (e+f x) \left (c (d \sin (e+f x))^p\right )^n}{27 f (1+\sin (e+f x))}+\frac {\cos (e+f x) \sin (e+f x) \left (c (d \sin (e+f x))^p\right )^n}{3 f (3+3 \sin (e+f x))^2} \]

[Out]

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))^2-1/3*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)

Rubi [A] (verified)

Time = 0.34 (sec) , antiderivative size = 288, normalized size of antiderivative = 1.03, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {2905, 2845, 3057, 2827, 2722} \[ \int \frac {\left (c (d \sin (e+f x))^p\right )^n}{(3+3 \sin (e+f x))^2} \, dx=\frac {2 \left (1-n^2 p^2\right ) \sin ^2(e+f x) \cos (e+f x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{2} (n p+2),\frac {1}{2} (n p+4),\sin ^2(e+f x)\right ) \left (c (d \sin (e+f x))^p\right )^n}{3 a^2 f (n p+2) \sqrt {\cos ^2(e+f x)}}-\frac {n p (1-2 n p) \sin (e+f x) \cos (e+f x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{2} (n p+1),\frac {1}{2} (n p+3),\sin ^2(e+f x)\right ) \left (c (d \sin (e+f x))^p\right )^n}{3 a^2 f (n p+1) \sqrt {\cos ^2(e+f x)}}+\frac {2 (1-n p) \sin (e+f x) \cos (e+f x) \left (c (d \sin (e+f x))^p\right )^n}{3 a^2 f (\sin (e+f x)+1)}+\frac {\sin (e+f x) \cos (e+f x) \left (c (d \sin (e+f x))^p\right )^n}{3 f (a \sin (e+f x)+a)^2} \]

[In]

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

[Out]

-1/3*(n*p*(1 - 2*n*p)*Cos[e + f*x]*Hypergeometric2F1[1/2, (1 + n*p)/2, (3 + n*p)/2, Sin[e + f*x]^2]*Sin[e + f*
x]*(c*(d*Sin[e + f*x])^p)^n)/(a^2*f*(1 + n*p)*Sqrt[Cos[e + f*x]^2]) + (2*(1 - n^2*p^2)*Cos[e + f*x]*Hypergeome
tric2F1[1/2, (2 + n*p)/2, (4 + n*p)/2, Sin[e + f*x]^2]*Sin[e + f*x]^2*(c*(d*Sin[e + f*x])^p)^n)/(3*a^2*f*(2 +
n*p)*Sqrt[Cos[e + f*x]^2]) + (2*(1 - n*p)*Cos[e + f*x]*Sin[e + f*x]*(c*(d*Sin[e + f*x])^p)^n)/(3*a^2*f*(1 + Si
n[e + f*x])) + (Cos[e + f*x]*Sin[e + f*x]*(c*(d*Sin[e + f*x])^p)^n)/(3*f*(a + a*Sin[e + f*x])^2)

Rule 2722

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]))*Hypergeometric2F1[1/2, (n + 1)/2, (n + 3)/2, Sin[c + d*x]^2], x] /; FreeQ[{b, c, d, n
}, x] &&  !IntegerQ[2*n]

Rule 2827

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

Rule 2845

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Sim
p[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] + Dis
t[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] && (IntegersQ[2*m, 2*n] || (IntegerQ
[m] && EqQ[c, 0]))

Rule 2905

Int[((c_.)*((d_.)*sin[(e_.) + (f_.)*(x_)])^(p_))^(n_)*((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.), x_Symbol]
:> Dist[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 3057

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(e_
.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[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] + Dist[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])

Rubi steps \begin{align*} \text {integral}& = \left ((d \sin (e+f x))^{-n p} \left (c (d \sin (e+f x))^p\right )^n\right ) \int \frac {(d \sin (e+f x))^{n p}}{(a+a \sin (e+f x))^2} \, dx \\ & = \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}+\frac {\left ((d \sin (e+f x))^{-n p} \left (c (d \sin (e+f x))^p\right )^n\right ) \int \frac {(d \sin (e+f x))^{n p} (a d (2-n p)+a d n p \sin (e+f x))}{a+a \sin (e+f x)} \, dx}{3 a^2 d} \\ & = \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}+\frac {\left ((d \sin (e+f x))^{-n p} \left (c (d \sin (e+f x))^p\right )^n\right ) \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) (1+n p) \sin (e+f x)\right ) \, dx}{3 a^4 d^2} \\ & = \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}-\frac {\left (n p (1-2 n p) (d \sin (e+f x))^{-n p} \left (c (d \sin (e+f x))^p\right )^n\right ) \int (d \sin (e+f x))^{n p} \, dx}{3 a^2}+\frac {\left (2 (1-n p) (1+n p) (d \sin (e+f x))^{-n p} \left (c (d \sin (e+f x))^p\right )^n\right ) \int (d \sin (e+f x))^{1+n p} \, dx}{3 a^2 d} \\ & = -\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} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.83 (sec) , antiderivative size = 197, normalized size of antiderivative = 0.71 \[ \int \frac {\left (c (d \sin (e+f x))^p\right )^n}{(3+3 \sin (e+f x))^2} \, dx=\frac {\left (c (d \sin (e+f x))^p\right )^n \left (-\frac {(-3+2 n p+2 (-1+n p) \sin (e+f x)) \sin (2 (e+f x))}{2 (1+\sin (e+f x))^2}+\frac {n p (-1+2 n p) \sqrt {\cos ^2(e+f x)} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{2} (1+n p),\frac {1}{2} (3+n p),\sin ^2(e+f x)\right ) \tan (e+f x)}{1+n p}-\frac {2 \left (-1+n^2 p^2\right ) \sqrt {\cos ^2(e+f x)} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},1+\frac {n p}{2},2+\frac {n p}{2},\sin ^2(e+f x)\right ) \sin (e+f x) \tan (e+f x)}{2+n p}\right )}{27 f} \]

[In]

Integrate[(c*(d*Sin[e + f*x])^p)^n/(3 + 3*Sin[e + f*x])^2,x]

[Out]

((c*(d*Sin[e + f*x])^p)^n*(-1/2*((-3 + 2*n*p + 2*(-1 + n*p)*Sin[e + f*x])*Sin[2*(e + f*x)])/(1 + Sin[e + f*x])
^2 + (n*p*(-1 + 2*n*p)*Sqrt[Cos[e + f*x]^2]*Hypergeometric2F1[1/2, (1 + n*p)/2, (3 + n*p)/2, Sin[e + f*x]^2]*T
an[e + f*x])/(1 + n*p) - (2*(-1 + n^2*p^2)*Sqrt[Cos[e + f*x]^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])/(2 + n*p)))/(27*f)

Maple [F]

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

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

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]

\[ \int \frac {\left (c (d \sin (e+f x))^p\right )^n}{(3+3 \sin (e+f x))^2} \, dx=\frac {\int \frac {\left (c \left (d \sin {\left (e + f x \right )}\right )^{p}\right )^{n}}{\sin ^{2}{\left (e + f x \right )} + 2 \sin {\left (e + f x \right )} + 1}\, dx}{a^{2}} \]

[In]

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

[Out]

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]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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