3.1.34 \(\int \cos ^3(e+f x) (-3+2 \sec ^2(e+f x)) \, dx\) [34]

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

[Out]

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

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Rubi [A]
time = 0.02, antiderivative size = 19, normalized size of antiderivative = 1.00, 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 = {4128} \begin {gather*} -\frac {\sin (e+f x) \cos ^2(e+f x)}{f} \end {gather*}

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 4128

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*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(46\) vs. \(2(19)=38\).
time = 0.02, size = 46, normalized size = 2.42 \begin {gather*} \frac {2 \cos (f x) \sin (e)}{f}+\frac {2 \cos (e) \sin (f x)}{f}-\frac {3 \sin (e+f x)}{f}+\frac {\sin ^3(e+f x)}{f} \end {gather*}

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 = 0.43, size = 32, normalized size = 1.68

method	result	size
risch	\(-\frac {\sin \left (f x +e \right )}{4 f}-\frac {\sin \left (3 f x +3 e \right )}{4 f}\)	\(27\)
derivativedivides	\(\frac {-\left (2+\cos ^{2}\left (f x +e \right )\right ) \sin \left (f x +e \right )+2 \sin \left (f x +e \right )}{f}\)	\(32\)
default	\(\frac {-\left (2+\cos ^{2}\left (f x +e \right )\right ) \sin \left (f x +e \right )+2 \sin \left (f x +e \right )}{f}\)	\(32\)
norman	\(\frac {\frac {2 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{f}-\frac {6 \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{f}+\frac {6 \left (\tan ^{5}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{f}-\frac {2 \left (\tan ^{7}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{f}}{\left (1+\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )^{3} \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )}\)	\(95\)

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,method=_RETURNVERBOSE)

[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 = 0.29, size = 23, normalized size = 1.21 \begin {gather*} \frac {\sin \left (f x + e\right )^{3} - \sin \left (f x + e\right )}{f} \end {gather*}

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 = 3.16, size = 21, normalized size = 1.11 \begin {gather*} -\frac {\cos \left (f x + e\right )^{2} \sin \left (f x + e\right )}{f} \end {gather*}

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]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (2 \sec ^{2}{\left (e + f x \right )} - 3\right ) \cos ^{3}{\left (e + f x \right )}\, dx \end {gather*}

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]

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

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Giac [A]
time = 0.46, size = 21, normalized size = 1.11 \begin {gather*} \frac {\sin \left (f x + e\right )^{3} - \sin \left (f x + e\right )}{f} \end {gather*}

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

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Mupad [B]
time = 0.04, size = 22, normalized size = 1.16 \begin {gather*} -\frac {\sin \left (e+f\,x\right )-{\sin \left (e+f\,x\right )}^3}{f} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(sin(e + f*x) - sin(e + f*x)^3)/f

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