3.425 \(\int \cos ^4(e+f x) (a+b \sin ^4(e+f x))^p \, dx\)

Optimal. Leaf size=25 \[ \text{Unintegrable}\left (\cos ^4(e+f x) \left (a+b \sin ^4(e+f x)\right )^p,x\right ) \]

[Out]

Unintegrable[Cos[e + f*x]^4*(a + b*Sin[e + f*x]^4)^p, x]

________________________________________________________________________________________

Rubi [A]  time = 0.0399659, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int \cos ^4(e+f x) \left (a+b \sin ^4(e+f x)\right )^p \, dx \]

Verification is Not applicable to the result.

[In]

Int[Cos[e + f*x]^4*(a + b*Sin[e + f*x]^4)^p,x]

[Out]

Defer[Int][Cos[e + f*x]^4*(a + b*Sin[e + f*x]^4)^p, x]

Rubi steps

\begin{align*} \int \cos ^4(e+f x) \left (a+b \sin ^4(e+f x)\right )^p \, dx &=\int \cos ^4(e+f x) \left (a+b \sin ^4(e+f x)\right )^p \, dx\\ \end{align*}

Mathematica [A]  time = 5.38007, size = 0, normalized size = 0. \[ \int \cos ^4(e+f x) \left (a+b \sin ^4(e+f x)\right )^p \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[Cos[e + f*x]^4*(a + b*Sin[e + f*x]^4)^p,x]

[Out]

Integrate[Cos[e + f*x]^4*(a + b*Sin[e + f*x]^4)^p, x]

________________________________________________________________________________________

Maple [A]  time = 1.881, size = 0, normalized size = 0. \begin{align*} \int \left ( \cos \left ( fx+e \right ) \right ) ^{4} \left ( a+b \left ( \sin \left ( fx+e \right ) \right ) ^{4} \right ) ^{p}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(f*x+e)^4*(a+b*sin(f*x+e)^4)^p,x)

[Out]

int(cos(f*x+e)^4*(a+b*sin(f*x+e)^4)^p,x)

________________________________________________________________________________________

Maxima [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b \sin \left (f x + e\right )^{4} + a\right )}^{p} \cos \left (f x + e\right )^{4}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(f*x+e)^4*(a+b*sin(f*x+e)^4)^p,x, algorithm="maxima")

[Out]

integrate((b*sin(f*x + e)^4 + a)^p*cos(f*x + e)^4, x)

________________________________________________________________________________________

Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (b \cos \left (f x + e\right )^{4} - 2 \, b \cos \left (f x + e\right )^{2} + a + b\right )}^{p} \cos \left (f x + e\right )^{4}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(f*x+e)^4*(a+b*sin(f*x+e)^4)^p,x, algorithm="fricas")

[Out]

integral((b*cos(f*x + e)^4 - 2*b*cos(f*x + e)^2 + a + b)^p*cos(f*x + e)^4, x)

________________________________________________________________________________________

Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(f*x+e)**4*(a+b*sin(f*x+e)**4)**p,x)

[Out]

Timed out

________________________________________________________________________________________

Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b \sin \left (f x + e\right )^{4} + a\right )}^{p} \cos \left (f x + e\right )^{4}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(f*x+e)^4*(a+b*sin(f*x+e)^4)^p,x, algorithm="giac")

[Out]

integrate((b*sin(f*x + e)^4 + a)^p*cos(f*x + e)^4, x)