3.1.35 \(\int \cos ^4(e+f x) (-4+3 \sec ^2(e+f x)) \, dx\) [35]

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

[Out]

-cos(f*x+e)^3*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 ^3(e+f x)}{f} \end {gather*}

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 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 ^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*}

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Mathematica [A]
time = 0.03, size = 31, normalized size = 1.63 \begin {gather*} -\frac {\sin (2 (e+f x))}{4 f}-\frac {\sin (4 (e+f x))}{8 f} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/4*Sin[2*(e + f*x)]/f - Sin[4*(e + f*x)]/(8*f)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(44\) vs. \(2(19)=38\).
time = 0.45, size = 45, normalized size = 2.37

method	result	size
risch	\(-\frac {\sin \left (4 f x +4 e \right )}{8 f}-\frac {\sin \left (2 f x +2 e \right )}{4 f}\)	\(30\)
derivativedivides	\(\frac {-\left (\cos ^{3}\left (f x +e \right )+\frac {3 \cos \left (f x +e \right )}{2}\right ) \sin \left (f x +e \right )+\frac {3 \cos \left (f x +e \right ) \sin \left (f x +e \right )}{2}}{f}\)	\(45\)
default	\(\frac {-\left (\cos ^{3}\left (f x +e \right )+\frac {3 \cos \left (f x +e \right )}{2}\right ) \sin \left (f x +e \right )+\frac {3 \cos \left (f x +e \right ) \sin \left (f x +e \right )}{2}}{f}\)	\(45\)
norman	\(\frac {\frac {2 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{f}-\frac {8 \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{f}+\frac {12 \left (\tan ^{5}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{f}-\frac {8 \left (\tan ^{7}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{f}+\frac {2 \left (\tan ^{9}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{f}}{\left (1+\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )^{4} \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )}\)	\(111\)

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,method=_RETURNVERBOSE)

[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 = 0.28, size = 36, normalized size = 1.89 \begin {gather*} -\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 {gather*}

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 = 2.71, size = 21, normalized size = 1.11 \begin {gather*} -\frac {\cos \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)^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]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (3 \sec ^{2}{\left (e + f x \right )} - 4\right ) \cos ^{4}{\left (e + f x \right )}\, dx \end {gather*}

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]

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

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Giac [A]
time = 0.42, size = 23, normalized size = 1.21 \begin {gather*} -\frac {\tan \left (f x + e\right )}{{\left (\tan \left (f x + e\right )^{2} + 1\right )}^{2} f} \end {gather*}

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)

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Mupad [B]
time = 2.35, size = 19, normalized size = 1.00 \begin {gather*} -\frac {{\cos \left (e+f\,x\right )}^3\,\sin \left (e+f\,x\right )}{f} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(cos(e + f*x)^3*sin(e + f*x))/f

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