\(\int (a \sin (e+f x))^m (b \sin (e+f x))^n \, dx\) [41]

Optimal result

Integrand size = 21, antiderivative size = 81 \[ \int (a \sin (e+f x))^m (b \sin (e+f x))^n \, dx=\frac {\cos (e+f x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{2} (1+m+n),\frac {1}{2} (3+m+n),\sin ^2(e+f x)\right ) (a \sin (e+f x))^{1+m} (b \sin (e+f x))^n}{a f (1+m+n) \sqrt {\cos ^2(e+f x)}} \]

[Out]

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)

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {20, 2722} \[ \int (a \sin (e+f x))^m (b \sin (e+f x))^n \, dx=\frac {\cos (e+f x) (a \sin (e+f x))^{m+1} (b \sin (e+f x))^n \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{2} (m+n+1),\frac {1}{2} (m+n+3),\sin ^2(e+f x)\right )}{a f (m+n+1) \sqrt {\cos ^2(e+f x)}} \]

[In]

Int[(a*Sin[e + f*x])^m*(b*Sin[e + f*x])^n,x]

[Out]

(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)*Sqrt[Cos[e + f*x]^2])

Rule 20

Int[(u_.)*((a_.)*(v_))^(m_)*((b_.)*(v_))^(n_), x_Symbol] :> Dist[b^IntPart[n]*((b*v)^FracPart[n]/(a^IntPart[n]
*(a*v)^FracPart[n])), Int[u*(a*v)^(m + n), x], x] /; FreeQ[{a, b, m, n}, x] &&  !IntegerQ[m] &&  !IntegerQ[n]
&&  !IntegerQ[m + n]

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]

Rubi steps \begin{align*} \text {integral}& = \left ((a \sin (e+f x))^{-n} (b \sin (e+f x))^n\right ) \int (a \sin (e+f x))^{m+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)}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.94 \[ \int (a \sin (e+f x))^m (b \sin (e+f x))^n \, dx=\frac {\sqrt {\cos ^2(e+f x)} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{2} (1+m+n),\frac {1}{2} (3+m+n),\sin ^2(e+f x)\right ) (a \sin (e+f x))^m (b \sin (e+f x))^n \tan (e+f x)}{f (1+m+n)} \]

[In]

Integrate[(a*Sin[e + f*x])^m*(b*Sin[e + f*x])^n,x]

[Out]

(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))

Maple [F]

\[\int \left (a \sin \left (f x +e \right )\right )^{m} \left (b \sin \left (f x +e \right )\right )^{n}d x\]

[In]

int((a*sin(f*x+e))^m*(b*sin(f*x+e))^n,x)

[Out]

int((a*sin(f*x+e))^m*(b*sin(f*x+e))^n,x)

Fricas [F]

\[ \int (a \sin (e+f x))^m (b \sin (e+f x))^n \, dx=\int { \left (a \sin \left (f x + e\right )\right )^{m} \left (b \sin \left (f x + e\right )\right )^{n} \,d x } \]

[In]

integrate((a*sin(f*x+e))^m*(b*sin(f*x+e))^n,x, algorithm="fricas")

[Out]

integral((a*sin(f*x + e))^m*(b*sin(f*x + e))^n, x)

Sympy [F]

\[ \int (a \sin (e+f x))^m (b \sin (e+f x))^n \, dx=\int \left (a \sin {\left (e + f x \right )}\right )^{m} \left (b \sin {\left (e + f x \right )}\right )^{n}\, dx \]

[In]

integrate((a*sin(f*x+e))**m*(b*sin(f*x+e))**n,x)

[Out]

Integral((a*sin(e + f*x))**m*(b*sin(e + f*x))**n, x)

Maxima [F]

\[ \int (a \sin (e+f x))^m (b \sin (e+f x))^n \, dx=\int { \left (a \sin \left (f x + e\right )\right )^{m} \left (b \sin \left (f x + e\right )\right )^{n} \,d x } \]

[In]

integrate((a*sin(f*x+e))^m*(b*sin(f*x+e))^n,x, algorithm="maxima")

[Out]

integrate((a*sin(f*x + e))^m*(b*sin(f*x + e))^n, x)

Giac [F]

\[ \int (a \sin (e+f x))^m (b \sin (e+f x))^n \, dx=\int { \left (a \sin \left (f x + e\right )\right )^{m} \left (b \sin \left (f x + e\right )\right )^{n} \,d x } \]

[In]

integrate((a*sin(f*x+e))^m*(b*sin(f*x+e))^n,x, algorithm="giac")

[Out]

integrate((a*sin(f*x + e))^m*(b*sin(f*x + e))^n, x)

Mupad [F(-1)]

Timed out. \[ \int (a \sin (e+f x))^m (b \sin (e+f x))^n \, dx=\int {\left (a\,\sin \left (e+f\,x\right )\right )}^m\,{\left (b\,\sin \left (e+f\,x\right )\right )}^n \,d x \]

[In]

int((a*sin(e + f*x))^m*(b*sin(e + f*x))^n,x)

[Out]

int((a*sin(e + f*x))^m*(b*sin(e + f*x))^n, x)