3.36 \(\int \cos ^5(e+f x) (-5+4 \sec ^2(e+f x)) \, dx\)

Optimal. Leaf size=19 \[ -\frac{\sin (e+f x) \cos ^4(e+f x)}{f} \]

[Out]

-((Cos[e + f*x]^4*Sin[e + f*x])/f)

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Rubi [A]  time = 0.0251379, 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} \[ -\frac{\sin (e+f x) \cos ^4(e+f x)}{f} \]

Antiderivative was successfully verified.

[In]

Int[Cos[e + f*x]^5*(-5 + 4*Sec[e + f*x]^2),x]

[Out]

-((Cos[e + f*x]^4*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 ^5(e+f x) \left (-5+4 \sec ^2(e+f x)\right ) \, dx &=-\frac{\cos ^4(e+f x) \sin (e+f x)}{f}\\ \end{align*}

Mathematica [A]  time = 0.0293399, size = 38, normalized size = 2. \[ -\frac{\sin ^5(e+f x)}{f}+\frac{2 \sin ^3(e+f x)}{f}-\frac{\sin (e+f x)}{f} \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[e + f*x]^5*(-5 + 4*Sec[e + f*x]^2),x]

[Out]

-(Sin[e + f*x]/f) + (2*Sin[e + f*x]^3)/f - Sin[e + f*x]^5/f

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Maple [B]  time = 0.068, size = 52, normalized size = 2.7 \begin{align*}{\frac{1}{f} \left ( - \left ({\frac{8}{3}}+ \left ( \cos \left ( fx+e \right ) \right ) ^{4}+{\frac{4\, \left ( \cos \left ( fx+e \right ) \right ) ^{2}}{3}} \right ) \sin \left ( fx+e \right ) +{\frac{ \left ( 8+4\, \left ( \cos \left ( fx+e \right ) \right ) ^{2} \right ) \sin \left ( fx+e \right ) }{3}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(f*x+e)^5*(-5+4*sec(f*x+e)^2),x)

[Out]

1/f*(-(8/3+cos(f*x+e)^4+4/3*cos(f*x+e)^2)*sin(f*x+e)+4/3*(2+cos(f*x+e)^2)*sin(f*x+e))

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Maxima [A]  time = 0.921924, size = 41, normalized size = 2.16 \begin{align*} -\frac{\sin \left (f x + e\right )^{5} - 2 \, \sin \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)^5*(-5+4*sec(f*x+e)^2),x, algorithm="maxima")

[Out]

-(sin(f*x + e)^5 - 2*sin(f*x + e)^3 + sin(f*x + e))/f

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Fricas [A]  time = 0.472974, size = 43, normalized size = 2.26 \begin{align*} -\frac{\cos \left (f x + e\right )^{4} \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)^5*(-5+4*sec(f*x+e)^2),x, algorithm="fricas")

[Out]

-cos(f*x + e)^4*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)**5*(-5+4*sec(f*x+e)**2),x)

[Out]

Timed out

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Giac [A]  time = 1.126, size = 45, normalized size = 2.37 \begin{align*} -\frac{\sin \left (f x + e\right )^{5} - 2 \, \sin \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)^5*(-5+4*sec(f*x+e)^2),x, algorithm="giac")

[Out]

-(sin(f*x + e)^5 - 2*sin(f*x + e)^3 + sin(f*x + e))/f