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

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

[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)

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Rubi [A]
time = 0.02, 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, 2722} \begin {gather*} \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)}} \end {gather*}

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 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*} \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*}

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Mathematica [A]
time = 0.06, size = 76, normalized size = 0.94 \begin {gather*} \frac {\sqrt {\cos ^2(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))^m (b \sin (e+f x))^n \tan (e+f x)}{f (1+m+n)} \end {gather*}

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))

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Maple [F]
time = 0.03, 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)

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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)

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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)

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

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)

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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)

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

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)

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