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

Optimal. Leaf size=81 \[ \frac {\cos (e+f x) (a \sin (e+f x))^{m+1} (b \sin (e+f x))^n \, _2F_1\left (\frac {1}{2},\frac {1}{2} (m+n+1);\frac {1}{2} (m+n+3);\sin ^2(e+f x)\right )}{a f (m+n+1) \sqrt {\cos ^2(e+f x)}} \]

[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]  time = 0.03, antiderivative size = 81, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {20, 2643} \[ \frac {\cos (e+f x) (a \sin (e+f x))^{m+1} (b \sin (e+f x))^n \, _2F_1\left (\frac {1}{2},\frac {1}{2} (m+n+1);\frac {1}{2} (m+n+3);\sin ^2(e+f x)\right )}{a f (m+n+1) \sqrt {\cos ^2(e+f x)}} \]

Antiderivative was successfully verified.

[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 2643

Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(Cos[c + d*x]*(b*Sin[c + d*x])^(n + 1)*Hypergeomet
ric2F1[1/2, (n + 1)/2, (n + 3)/2, Sin[c + d*x]^2])/(b*d*(n + 1)*Sqrt[Cos[c + d*x]^2]), x] /; FreeQ[{b, c, d, n
}, x] &&  !IntegerQ[2*n]

Rubi steps

\begin {align*} \int (a \sin (e+f x))^m (b \sin (e+f x))^n \, dx &=\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) \, _2F_1\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]  time = 0.07, size = 76, normalized size = 0.94 \[ \frac {\sqrt {\cos ^2(e+f x)} \tan (e+f x) (a \sin (e+f x))^m (b \sin (e+f x))^n \, _2F_1\left (\frac {1}{2},\frac {1}{2} (m+n+1);\frac {1}{2} (m+n+3);\sin ^2(e+f x)\right )}{f (m+n+1)} \]

Antiderivative was successfully verified.

[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))

________________________________________________________________________________________

fricas [F]  time = 0.48, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\left (a \sin \left (f x + e\right )\right )^{m} \left (b \sin \left (f x + e\right )\right )^{n}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________

giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (a \sin \left (f x + e\right )\right )^{m} \left (b \sin \left (f x + e\right )\right )^{n}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________

maple [F]  time = 1.08, size = 0, normalized size = 0.00 \[ \int \left (a \sin \left (f x +e \right )\right )^{m} \left (b \sin \left (f x +e \right )\right )^{n}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (a \sin \left (f x + e\right )\right )^{m} \left (b \sin \left (f x + e\right )\right )^{n}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int {\left (a\,\sin \left (e+f\,x\right )\right )}^m\,{\left (b\,\sin \left (e+f\,x\right )\right )}^n \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (a \sin {\left (e + f x \right )}\right )^{m} \left (b \sin {\left (e + f x \right )}\right )^{n}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________