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

Optimal. Leaf size=19 \[ -\frac{\sin (e+f x) \cos ^2(e+f x)}{f} \]

[Out]

-((Cos[e + f*x]^2*Sin[e + f*x])/f)

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Rubi [A]  time = 0.0242992, 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} \[ -\frac{\sin (e+f x) \cos ^2(e+f x)}{f} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [B]  time = 0.0202027, size = 46, normalized size = 2.42 \[ \frac{\sin ^3(e+f x)}{f}-\frac{3 \sin (e+f x)}{f}+\frac{2 \sin (e) \cos (f x)}{f}+\frac{2 \cos (e) \sin (f x)}{f} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.049, size = 32, normalized size = 1.7 \begin{align*}{\frac{- \left ( 2+ \left ( \cos \left ( fx+e \right ) \right ) ^{2} \right ) \sin \left ( fx+e \right ) +2\,\sin \left ( fx+e \right ) }{f}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.92614, size = 28, normalized size = 1.47 \begin{align*} \frac{\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)^3*(-3+2*sec(f*x+e)^2),x, algorithm="maxima")

[Out]

(sin(f*x + e)^3 - sin(f*x + e))/f

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Fricas [A]  time = 0.464534, size = 43, normalized size = 2.26 \begin{align*} -\frac{\cos \left (f x + e\right )^{2} \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)^3*(-3+2*sec(f*x+e)^2),x, algorithm="fricas")

[Out]

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

[Out]

Timed out

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Giac [A]  time = 1.15735, size = 31, normalized size = 1.63 \begin{align*} \frac{\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)^3*(-3+2*sec(f*x+e)^2),x, algorithm="giac")

[Out]

(sin(f*x + e)^3 - sin(f*x + e))/f