3.35 \(\int \cos ^4(e+f x) (-4+3 \sec ^2(e+f x)) \, dx\)

Optimal. Leaf size=19 \[ -\frac{\sin (e+f x) \cos ^3(e+f x)}{f} \]

[Out]

-((Cos[e + f*x]^3*Sin[e + f*x])/f)

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Rubi [A]  time = 0.0238439, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.048, Rules used = {4043} \[ -\frac{\sin (e+f x) \cos ^3(e+f x)}{f} \]

Antiderivative was successfully verified.

[In]

Int[Cos[e + f*x]^4*(-4 + 3*Sec[e + f*x]^2),x]

[Out]

-((Cos[e + f*x]^3*Sin[e + f*x])/f)

Rule 4043

Int[(csc[(e_.) + (f_.)*(x_)]*(b_.))^(m_.)*(csc[(e_.) + (f_.)*(x_)]^2*(C_.) + (A_)), x_Symbol] :> Simp[(A*Cot[e
+ f*x]*(b*Csc[e + f*x])^m)/(f*m), x] /; FreeQ[{b, e, f, A, C, m}, x] && EqQ[C*m + A*(m + 1), 0]

Rubi steps

\begin{align*} \int \cos ^4(e+f x) \left (-4+3 \sec ^2(e+f x)\right ) \, dx &=-\frac{\cos ^3(e+f x) \sin (e+f x)}{f}\\ \end{align*}

Mathematica [A]  time = 0.0288244, size = 31, normalized size = 1.63 \[ -\frac{\sin (2 (e+f x))}{4 f}-\frac{\sin (4 (e+f x))}{8 f} \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[e + f*x]^4*(-4 + 3*Sec[e + f*x]^2),x]

[Out]

-Sin[2*(e + f*x)]/(4*f) - Sin[4*(e + f*x)]/(8*f)

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Maple [B]  time = 0.047, size = 45, normalized size = 2.4 \begin{align*}{\frac{1}{f} \left ( - \left ( \left ( \cos \left ( fx+e \right ) \right ) ^{3}+{\frac{3\,\cos \left ( fx+e \right ) }{2}} \right ) \sin \left ( fx+e \right ) +{\frac{3\,\cos \left ( fx+e \right ) \sin \left ( fx+e \right ) }{2}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(f*x+e)^4*(-4+3*sec(f*x+e)^2),x)

[Out]

1/f*(-(cos(f*x+e)^3+3/2*cos(f*x+e))*sin(f*x+e)+3/2*cos(f*x+e)*sin(f*x+e))

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Maxima [A]  time = 0.925892, size = 45, normalized size = 2.37 \begin{align*} -\frac{\tan \left (f x + e\right )}{{\left (\tan \left (f x + e\right )^{4} + 2 \, \tan \left (f x + e\right )^{2} + 1\right )} f} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(f*x+e)^4*(-4+3*sec(f*x+e)^2),x, algorithm="maxima")

[Out]

-tan(f*x + e)/((tan(f*x + e)^4 + 2*tan(f*x + e)^2 + 1)*f)

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Fricas [A]  time = 0.469055, size = 43, normalized size = 2.26 \begin{align*} -\frac{\cos \left (f x + e\right )^{3} \sin \left (f x + e\right )}{f} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(f*x+e)^4*(-4+3*sec(f*x+e)^2),x, algorithm="fricas")

[Out]

-cos(f*x + e)^3*sin(f*x + e)/f

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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*(-4+3*sec(f*x+e)**2),x)

[Out]

Timed out

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Giac [A]  time = 1.1582, size = 34, normalized size = 1.79 \begin{align*} -\frac{\tan \left (f x + e\right )}{{\left (\tan \left (f x + e\right )^{2} + 1\right )}^{2} f} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(f*x+e)^4*(-4+3*sec(f*x+e)^2),x, algorithm="giac")

[Out]

-tan(f*x + e)/((tan(f*x + e)^2 + 1)^2*f)