3.338 \(\int \cos ^n(e+f x) (b \sin (e+f x))^m \, dx\)

Optimal. Leaf size=83 \[ -\frac{b \sin ^2(e+f x)^{\frac{1-m}{2}} \cos ^{n+1}(e+f x) (b \sin (e+f x))^{m-1} \, _2F_1\left (\frac{1-m}{2},\frac{n+1}{2};\frac{n+3}{2};\cos ^2(e+f x)\right )}{f (n+1)} \]

[Out]

-((b*Cos[e + f*x]^(1 + n)*Hypergeometric2F1[(1 - m)/2, (1 + n)/2, (3 + n)/2, Cos[e + f*x]^2]*(b*Sin[e + f*x])^
(-1 + m)*(Sin[e + f*x]^2)^((1 - m)/2))/(f*(1 + n)))

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Rubi [A]  time = 0.0432349, antiderivative size = 83, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.053, Rules used = {2576} \[ -\frac{b \sin ^2(e+f x)^{\frac{1-m}{2}} \cos ^{n+1}(e+f x) (b \sin (e+f x))^{m-1} \, _2F_1\left (\frac{1-m}{2},\frac{n+1}{2};\frac{n+3}{2};\cos ^2(e+f x)\right )}{f (n+1)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((b*Cos[e + f*x]^(1 + n)*Hypergeometric2F1[(1 - m)/2, (1 + n)/2, (3 + n)/2, Cos[e + f*x]^2]*(b*Sin[e + f*x])^
(-1 + m)*(Sin[e + f*x]^2)^((1 - m)/2))/(f*(1 + n)))

Rule 2576

Int[(cos[(e_.) + (f_.)*(x_)]*(a_.))^(m_)*((b_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> -Simp[(b^(2*IntPar
t[(n - 1)/2] + 1)*(b*Sin[e + f*x])^(2*FracPart[(n - 1)/2])*(a*Cos[e + f*x])^(m + 1)*Hypergeometric2F1[(1 + m)/
2, (1 - n)/2, (3 + m)/2, Cos[e + f*x]^2])/(a*f*(m + 1)*(Sin[e + f*x]^2)^FracPart[(n - 1)/2]), x] /; FreeQ[{a,
b, e, f, m, n}, x] && SimplerQ[n, m]

Rubi steps

\begin{align*} \int \cos ^n(e+f x) (b \sin (e+f x))^m \, dx &=-\frac{b \cos ^{1+n}(e+f x) \, _2F_1\left (\frac{1-m}{2},\frac{1+n}{2};\frac{3+n}{2};\cos ^2(e+f x)\right ) (b \sin (e+f x))^{-1+m} \sin ^2(e+f x)^{\frac{1-m}{2}}}{f (1+n)}\\ \end{align*}

Mathematica [A]  time = 0.0763074, size = 85, normalized size = 1.02 \[ \frac{\sin (e+f x) \cos ^{n-1}(e+f x) \cos ^2(e+f x)^{\frac{1-n}{2}} (b \sin (e+f x))^m \, _2F_1\left (\frac{m+1}{2},\frac{1-n}{2};\frac{m+3}{2};\sin ^2(e+f x)\right )}{f (m+1)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Cos[e + f*x]^(-1 + n)*(Cos[e + f*x]^2)^((1 - n)/2)*Hypergeometric2F1[(1 + m)/2, (1 - n)/2, (3 + m)/2, Sin[e +
f*x]^2]*Sin[e + f*x]*(b*Sin[e + f*x])^m)/(f*(1 + m))

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Maple [F]  time = 0.399, size = 0, normalized size = 0. \begin{align*} \int \left ( \cos \left ( fx+e \right ) \right ) ^{n} \left ( b\sin \left ( fx+e \right ) \right ) ^{m}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (b \sin \left (f x + e\right )\right )^{m} \cos \left (f x + e\right )^{n}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\left (b \sin \left (f x + e\right )\right )^{m} \cos \left (f x + e\right )^{n}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (b \sin \left (f x + e\right )\right )^{m} \cos \left (f x + e\right )^{n}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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