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(f*x+e)^(1+n)*hypergeom([1/2-1/2*m, 1/2+1/2*n],[3/2+1/2*n],cos(f*x+e)^2)*(b*sin(f*x+e))^(-1+m)*(sin(f*x+
e)^2)^(1/2-1/2*m)/f/(1+n)

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Rubi [A]  time = 0.04, antiderivative size = 83, normalized size of antiderivative = 1.00, 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*}

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Mathematica [A]  time = 0.07, 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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fricas [F]  time = 1.02, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\left (b \sin \left (f x + e\right )\right )^{m} \cos \left (f x + e\right )^{n}, x\right ) \]

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

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)

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maple [F]  time = 0.53, size = 0, normalized size = 0.00 \[ \int \left (\cos ^{n}\left (f x +e \right )\right ) \left (b \sin \left (f x +e \right )\right )^{m}\, dx \]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

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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