Integrand size = 27, antiderivative size = 279 \[ \int \frac {\left (c (d \sin (e+f x))^p\right )^n}{(3+3 \sin (e+f x))^2} \, dx=-\frac {n p (1-2 n p) \cos (e+f x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{2} (1+n p),\frac {1}{2} (3+n p),\sin ^2(e+f x)\right ) \sin (e+f x) \left (c (d \sin (e+f x))^p\right )^n}{27 f (1+n p) \sqrt {\cos ^2(e+f x)}}+\frac {2 \left (1-n^2 p^2\right ) \cos (e+f x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{2} (2+n p),\frac {1}{2} (4+n p),\sin ^2(e+f x)\right ) \sin ^2(e+f x) \left (c (d \sin (e+f x))^p\right )^n}{27 f (2+n p) \sqrt {\cos ^2(e+f x)}}+\frac {2 (1-n p) \cos (e+f x) \sin (e+f x) \left (c (d \sin (e+f x))^p\right )^n}{27 f (1+\sin (e+f x))}+\frac {\cos (e+f x) \sin (e+f x) \left (c (d \sin (e+f x))^p\right )^n}{3 f (3+3 \sin (e+f x))^2} \]
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)
Time = 1.83 (sec) , antiderivative size = 197, normalized size of antiderivative = 0.71 \[ \int \frac {\left (c (d \sin (e+f x))^p\right )^n}{(3+3 \sin (e+f x))^2} \, dx=\frac {\left (c (d \sin (e+f x))^p\right )^n \left (-\frac {(-3+2 n p+2 (-1+n p) \sin (e+f x)) \sin (2 (e+f x))}{2 (1+\sin (e+f x))^2}+\frac {n p (-1+2 n p) \sqrt {\cos ^2(e+f x)} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{2} (1+n p),\frac {1}{2} (3+n p),\sin ^2(e+f x)\right ) \tan (e+f x)}{1+n p}-\frac {2 \left (-1+n^2 p^2\right ) \sqrt {\cos ^2(e+f x)} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},1+\frac {n p}{2},2+\frac {n p}{2},\sin ^2(e+f x)\right ) \sin (e+f x) \tan (e+f x)}{2+n p}\right )}{27 f} \]
((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]*Tan [e + f*x])/(1 + n*p) - (2*(-1 + n^2*p^2)*Sqrt[Cos[e + f*x]^2]*Hypergeometr ic2F1[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)
Time = 1.00 (sec) , antiderivative size = 305, normalized size of antiderivative = 1.09, 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 |
\(\displaystyle (d \sin (e+f x))^{-n p} \left (c (d \sin (e+f x))^p\right )^n \left (\frac {\frac {2 (1-n p) \cos (e+f x) (d \sin (e+f x))^{n p+1}}{f (\sin (e+f x)+1)}-\frac {\frac {a^2 d n p (1-2 n p) \cos (e+f x) (d \sin (e+f x))^{n p+1} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{2} (n p+1),\frac {1}{2} (n p+3),\sin ^2(e+f x)\right )}{f (n p+1) \sqrt {\cos ^2(e+f x)}}-\frac {2 a^2 (1-n p) (n p+1) \cos (e+f x) (d \sin (e+f x))^{n p+2} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{2} (n p+2),\frac {1}{2} (n p+4),\sin ^2(e+f x)\right )}{f (n p+2) \sqrt {\cos ^2(e+f x)}}}{a^2 d}}{3 a^2 d}+\frac {\cos (e+f x) (d \sin (e+f x))^{n p+1}}{3 d f (a \sin (e+f x)+a)^2}\right )\) |
((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)
3.9.24.3.1 Defintions of rubi rules used
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]
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]
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]))
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]
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])
\[\int \frac {\left (c \left (d \sin \left (f x +e \right )\right )^{p}\right )^{n}}{\left (a +a \sin \left (f x +e \right )\right )^{2}}d x\]
\[ \int \frac {\left (c (d \sin (e+f x))^p\right )^n}{(3+3 \sin (e+f x))^2} \, dx=\int { \frac {\left (\left (d \sin \left (f x + e\right )\right )^{p} c\right )^{n}}{{\left (a \sin \left (f x + e\right ) + a\right )}^{2}} \,d x } \]
\[ \int \frac {\left (c (d \sin (e+f x))^p\right )^n}{(3+3 \sin (e+f x))^2} \, dx=\frac {\int \frac {\left (c \left (d \sin {\left (e + f x \right )}\right )^{p}\right )^{n}}{\sin ^{2}{\left (e + f x \right )} + 2 \sin {\left (e + f x \right )} + 1}\, dx}{a^{2}} \]
\[ \int \frac {\left (c (d \sin (e+f x))^p\right )^n}{(3+3 \sin (e+f x))^2} \, dx=\int { \frac {\left (\left (d \sin \left (f x + e\right )\right )^{p} c\right )^{n}}{{\left (a \sin \left (f x + e\right ) + a\right )}^{2}} \,d x } \]
\[ \int \frac {\left (c (d \sin (e+f x))^p\right )^n}{(3+3 \sin (e+f x))^2} \, dx=\int { \frac {\left (\left (d \sin \left (f x + e\right )\right )^{p} c\right )^{n}}{{\left (a \sin \left (f x + e\right ) + a\right )}^{2}} \,d x } \]
Timed out. \[ \int \frac {\left (c (d \sin (e+f x))^p\right )^n}{(3+3 \sin (e+f x))^2} \, dx=\int \frac {{\left (c\,{\left (d\,\sin \left (e+f\,x\right )\right )}^p\right )}^n}{{\left (a+a\,\sin \left (e+f\,x\right )\right )}^2} \,d x \]