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

Optimal. Leaf size=26 \[ \text {Int}\left (\cos ^m(e+f x) \left (a+b \sin ^n(e+f x)\right )^p,x\right ) \]

[Out]

Unintegrable(cos(f*x+e)^m*(a+b*sin(f*x+e)^n)^p,x)

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Rubi [A]  time = 0.05, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \cos ^m(e+f x) \left (a+b \sin ^n(e+f x)\right )^p \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

\begin {align*} \int \cos ^m(e+f x) \left (a+b \sin ^n(e+f x)\right )^p \, dx &=\int \cos ^m(e+f x) \left (a+b \sin ^n(e+f x)\right )^p \, dx\\ \end {align*}

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Mathematica [A]  time = 4.59, size = 0, normalized size = 0.00 \[ \int \cos ^m(e+f x) \left (a+b \sin ^n(e+f x)\right )^p \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.49, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (b \sin \left (f x + e\right )^{n} + a\right )}^{p} \cos \left (f x + e\right )^{m}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 2.61, size = 0, normalized size = 0.00 \[ \int \left (\cos ^{m}\left (f x +e \right )\right ) \left (a +b \left (\sin ^{n}\left (f x +e \right )\right )\right )^{p}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [A]  time = 0.00, size = -1, normalized size = -0.04 \[ \int {\cos \left (e+f\,x\right )}^m\,{\left (a+b\,{\sin \left (e+f\,x\right )}^n\right )}^p \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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