Integrand size = 21, antiderivative size = 81 \[ \int (a \sin (e+f x))^m (b \sin (e+f x))^n \, dx=\frac {\cos (e+f x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{2} (1+m+n),\frac {1}{2} (3+m+n),\sin ^2(e+f x)\right ) (a \sin (e+f x))^{1+m} (b \sin (e+f x))^n}{a f (1+m+n) \sqrt {\cos ^2(e+f x)}} \] Output:
cos(f*x+e)*hypergeom([1/2, 1/2+1/2*m+1/2*n],[3/2+1/2*m+1/2*n],sin(f*x+e)^2 )*(a*sin(f*x+e))^(1+m)*(b*sin(f*x+e))^n/a/f/(1+m+n)/(cos(f*x+e)^2)^(1/2)
Time = 0.10 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.94 \[ \int (a \sin (e+f x))^m (b \sin (e+f x))^n \, dx=\frac {\sqrt {\cos ^2(e+f x)} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{2} (1+m+n),\frac {1}{2} (3+m+n),\sin ^2(e+f x)\right ) (a \sin (e+f x))^m (b \sin (e+f x))^n \tan (e+f x)}{f (1+m+n)} \] Input:
Integrate[(a*Sin[e + f*x])^m*(b*Sin[e + f*x])^n,x]
Output:
(Sqrt[Cos[e + f*x]^2]*Hypergeometric2F1[1/2, (1 + m + n)/2, (3 + m + n)/2, Sin[e + f*x]^2]*(a*Sin[e + f*x])^m*(b*Sin[e + f*x])^n*Tan[e + f*x])/(f*(1 + m + n))
Time = 0.43 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {2034, 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.
\(\displaystyle \int (a \sin (e+f x))^m (b \sin (e+f x))^n \, dx\) |
\(\Big \downarrow \) 2034 |
\(\displaystyle (a \sin (e+f x))^{-n} (b \sin (e+f x))^n \int (a \sin (e+f x))^{m+n}dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle (a \sin (e+f x))^{-n} (b \sin (e+f x))^n \int (a \sin (e+f x))^{m+n}dx\) |
\(\Big \downarrow \) 3122 |
\(\displaystyle \frac {\cos (e+f x) (a \sin (e+f x))^{m+1} (b \sin (e+f x))^n \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{2} (m+n+1),\frac {1}{2} (m+n+3),\sin ^2(e+f x)\right )}{a f (m+n+1) \sqrt {\cos ^2(e+f x)}}\) |
Input:
Int[(a*Sin[e + f*x])^m*(b*Sin[e + f*x])^n,x]
Output:
(Cos[e + f*x]*Hypergeometric2F1[1/2, (1 + m + n)/2, (3 + m + n)/2, Sin[e + f*x]^2]*(a*Sin[e + f*x])^(1 + m)*(b*Sin[e + f*x])^n)/(a*f*(1 + m + n)*Sqr t[Cos[e + f*x]^2])
Int[(Fx_.)*((a_.)*(v_))^(m_)*((b_.)*(v_))^(n_), x_Symbol] :> Simp[b^IntPart [n]*((b*v)^FracPart[n]/(a^IntPart[n]*(a*v)^FracPart[n])) Int[(a*v)^(m + n )*Fx, x], x] /; FreeQ[{a, b, m, n}, x] && !IntegerQ[m] && !IntegerQ[n] && !IntegerQ[m + n]
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]
\[\int \left (a \sin \left (f x +e \right )\right )^{m} \left (b \sin \left (f x +e \right )\right )^{n}d x\]
Input:
int((a*sin(f*x+e))^m*(b*sin(f*x+e))^n,x)
Output:
int((a*sin(f*x+e))^m*(b*sin(f*x+e))^n,x)
\[ \int (a \sin (e+f x))^m (b \sin (e+f x))^n \, dx=\int { \left (a \sin \left (f x + e\right )\right )^{m} \left (b \sin \left (f x + e\right )\right )^{n} \,d x } \] Input:
integrate((a*sin(f*x+e))^m*(b*sin(f*x+e))^n,x, algorithm="fricas")
Output:
integral((a*sin(f*x + e))^m*(b*sin(f*x + e))^n, x)
\[ \int (a \sin (e+f x))^m (b \sin (e+f x))^n \, dx=\int \left (a \sin {\left (e + f x \right )}\right )^{m} \left (b \sin {\left (e + f x \right )}\right )^{n}\, dx \] Input:
integrate((a*sin(f*x+e))**m*(b*sin(f*x+e))**n,x)
Output:
Integral((a*sin(e + f*x))**m*(b*sin(e + f*x))**n, x)
\[ \int (a \sin (e+f x))^m (b \sin (e+f x))^n \, dx=\int { \left (a \sin \left (f x + e\right )\right )^{m} \left (b \sin \left (f x + e\right )\right )^{n} \,d x } \] Input:
integrate((a*sin(f*x+e))^m*(b*sin(f*x+e))^n,x, algorithm="maxima")
Output:
integrate((a*sin(f*x + e))^m*(b*sin(f*x + e))^n, x)
\[ \int (a \sin (e+f x))^m (b \sin (e+f x))^n \, dx=\int { \left (a \sin \left (f x + e\right )\right )^{m} \left (b \sin \left (f x + e\right )\right )^{n} \,d x } \] Input:
integrate((a*sin(f*x+e))^m*(b*sin(f*x+e))^n,x, algorithm="giac")
Output:
integrate((a*sin(f*x + e))^m*(b*sin(f*x + e))^n, x)
Timed out. \[ \int (a \sin (e+f x))^m (b \sin (e+f x))^n \, dx=\int {\left (a\,\sin \left (e+f\,x\right )\right )}^m\,{\left (b\,\sin \left (e+f\,x\right )\right )}^n \,d x \] Input:
int((a*sin(e + f*x))^m*(b*sin(e + f*x))^n,x)
Output:
int((a*sin(e + f*x))^m*(b*sin(e + f*x))^n, x)
\[ \int (a \sin (e+f x))^m (b \sin (e+f x))^n \, dx=b^{n} a^{m} \left (\int \sin \left (f x +e \right )^{m +n}d x \right ) \] Input:
int((a*sin(f*x+e))^m*(b*sin(f*x+e))^n,x)
Output:
b**n*a**m*int(sin(e + f*x)**(m + n),x)