Integrand size = 20, antiderivative size = 81 \[ \int (-3)^{-1-m} (a+a \sin (e+f x))^m \, dx=-\frac {(-3)^{-1-m} 2^{\frac {1}{2}+m} \cos (e+f x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{2}-m,\frac {3}{2},\frac {1}{2} (1-\sin (e+f x))\right ) (1+\sin (e+f x))^{-\frac {1}{2}-m} (a+a \sin (e+f x))^m}{f} \] Output:
-(-3)^(-1-m)*2^(1/2+m)*cos(f*x+e)*hypergeom([1/2, 1/2-m],[3/2],1/2-1/2*sin (f*x+e))*(1+sin(f*x+e))^(-1/2-m)*(a+a*sin(f*x+e))^m/f
Result contains higher order function than in optimal. Order 9 vs. order 5 in optimal.
Time = 0.11 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.95 \[ \int (-3)^{-1-m} (a+a \sin (e+f x))^m \, dx=\frac {(-3)^{-1-m} 2^m B_{\frac {1}{2} (1+\sin (e+f x))}\left (\frac {1}{2}+m,\frac {1}{2}\right ) \sqrt {\cos ^2(e+f x)} \sec (e+f x) (1+\sin (e+f x))^{-m} (a (1+\sin (e+f x)))^m}{f} \] Input:
Integrate[(-3)^(-1 - m)*(a + a*Sin[e + f*x])^m,x]
Output:
((-3)^(-1 - m)*2^m*Beta[(1 + Sin[e + f*x])/2, 1/2 + m, 1/2]*Sqrt[Cos[e + f *x]^2]*Sec[e + f*x]*(a*(1 + Sin[e + f*x]))^m)/(f*(1 + Sin[e + f*x])^m)
Time = 0.28 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {27, 3042, 3131, 3042, 3130}
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.
\(\displaystyle \int (-3)^{-m-1} (a \sin (e+f x)+a)^m \, dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle (-3)^{-m-1} \int (\sin (e+f x) a+a)^mdx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle (-3)^{-m-1} \int (\sin (e+f x) a+a)^mdx\) |
\(\Big \downarrow \) 3131 |
\(\displaystyle (-3)^{-m-1} (\sin (e+f x)+1)^{-m} (a \sin (e+f x)+a)^m \int (\sin (e+f x)+1)^mdx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle (-3)^{-m-1} (\sin (e+f x)+1)^{-m} (a \sin (e+f x)+a)^m \int (\sin (e+f x)+1)^mdx\) |
\(\Big \downarrow \) 3130 |
\(\displaystyle -\frac {(-3)^{-m-1} 2^{m+\frac {1}{2}} \cos (e+f x) (\sin (e+f x)+1)^{-m-\frac {1}{2}} (a \sin (e+f x)+a)^m \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{2}-m,\frac {3}{2},\frac {1}{2} (1-\sin (e+f x))\right )}{f}\) |
Input:
Int[(-3)^(-1 - m)*(a + a*Sin[e + f*x])^m,x]
Output:
-(((-3)^(-1 - m)*2^(1/2 + m)*Cos[e + f*x]*Hypergeometric2F1[1/2, 1/2 - m, 3/2, (1 - Sin[e + f*x])/2]*(1 + Sin[e + f*x])^(-1/2 - m)*(a + a*Sin[e + f* x])^m)/f)
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(-2^(n + 1/2))*a^(n - 1/2)*b*(Cos[c + d*x]/(d*Sqrt[a + b*Sin[c + d*x]]))*Hypergeome tric2F1[1/2, 1/2 - n, 3/2, (1/2)*(1 - b*(Sin[c + d*x]/a))], x] /; FreeQ[{a, b, c, d, n}, x] && EqQ[a^2 - b^2, 0] && !IntegerQ[2*n] && GtQ[a, 0]
Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[a^IntPar t[n]*((a + b*Sin[c + d*x])^FracPart[n]/(1 + (b/a)*Sin[c + d*x])^FracPart[n] ) Int[(1 + (b/a)*Sin[c + d*x])^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && EqQ[a^2 - b^2, 0] && !IntegerQ[2*n] && !GtQ[a, 0]
\[\int \left (-3\right )^{-1-m} \left (a +a \sin \left (f x +e \right )\right )^{m}d x\]
Input:
int((-3)^(-1-m)*(a+a*sin(f*x+e))^m,x)
Output:
int((-3)^(-1-m)*(a+a*sin(f*x+e))^m,x)
\[ \int (-3)^{-1-m} (a+a \sin (e+f x))^m \, dx=\int { \left (-3\right )^{-m - 1} {\left (a \sin \left (f x + e\right ) + a\right )}^{m} \,d x } \] Input:
integrate((-3)^(-1-m)*(a+a*sin(f*x+e))^m,x, algorithm="fricas")
Output:
integral((-3)^(-m - 1)*(a*sin(f*x + e) + a)^m, x)
\[ \int (-3)^{-1-m} (a+a \sin (e+f x))^m \, dx=\left (-3\right )^{- m - 1} \int \left (a \sin {\left (e + f x \right )} + a\right )^{m}\, dx \] Input:
integrate((-3)**(-1-m)*(a+a*sin(f*x+e))**m,x)
Output:
(-3)**(-m - 1)*Integral((a*sin(e + f*x) + a)**m, x)
\[ \int (-3)^{-1-m} (a+a \sin (e+f x))^m \, dx=\int { \left (-3\right )^{-m - 1} {\left (a \sin \left (f x + e\right ) + a\right )}^{m} \,d x } \] Input:
integrate((-3)^(-1-m)*(a+a*sin(f*x+e))^m,x, algorithm="maxima")
Output:
(-3)^(-m - 1)*integrate((a*sin(f*x + e) + a)^m, x)
\[ \int (-3)^{-1-m} (a+a \sin (e+f x))^m \, dx=\int { \left (-3\right )^{-m - 1} {\left (a \sin \left (f x + e\right ) + a\right )}^{m} \,d x } \] Input:
integrate((-3)^(-1-m)*(a+a*sin(f*x+e))^m,x, algorithm="giac")
Output:
integrate((-3)^(-m - 1)*(a*sin(f*x + e) + a)^m, x)
Timed out. \[ \int (-3)^{-1-m} (a+a \sin (e+f x))^m \, dx=\int \frac {1}{{\left (-3\right )}^{m+1}}\,{\left (a+a\,\sin \left (e+f\,x\right )\right )}^m \,d x \] Input:
int(1/(-3)^(m + 1)*(a + a*sin(e + f*x))^m,x)
Output:
int(1/(-3)^(m + 1)*(a + a*sin(e + f*x))^m, x)
\[ \int (-3)^{-1-m} (a+a \sin (e+f x))^m \, dx=-\frac {\int \left (a +a \sin \left (f x +e \right )\right )^{m}d x}{3 \left (-1\right )^{m} 3^{m}} \] Input:
int((-3)^(-1-m)*(a+a*sin(f*x+e))^m,x)
Output:
( - int((sin(e + f*x)*a + a)**m,x))/(3*( - 1)**m*3**m)