3.1.28 \(\int \sec ^3(e+f x) (3-4 \sec ^2(e+f x)) \, dx\) [28]

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

[Out]

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

Antiderivative was successfully verified.

[In]

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

[Out]

-((Sec[e + f*x]^3*Tan[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 \sec ^3(e+f x) \left (3-4 \sec ^2(e+f x)\right ) \, dx &=-\frac {\sec ^3(e+f x) \tan (e+f x)}{f}\\ \end {align*}

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Mathematica [A]
time = 0.02, size = 19, normalized size = 1.00 \begin {gather*} -\frac {\sec ^3(e+f x) \tan (e+f x)}{f} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-((Sec[e + f*x]^3*Tan[e + f*x])/f)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(46\) vs. \(2(19)=38\).
time = 0.40, size = 47, normalized size = 2.47

method	result	size
risch	\(\frac {8 i \left ({\mathrm e}^{5 i \left (f x +e \right )}-{\mathrm e}^{3 i \left (f x +e \right )}\right )}{f \left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right )^{4}}\)	\(41\)
derivativedivides	\(\frac {\frac {3 \sec \left (f x +e \right ) \tan \left (f x +e \right )}{2}+4 \left (-\frac {\left (\sec ^{3}\left (f x +e \right )\right )}{4}-\frac {3 \sec \left (f x +e \right )}{8}\right ) \tan \left (f x +e \right )}{f}\)	\(47\)
default	\(\frac {\frac {3 \sec \left (f x +e \right ) \tan \left (f x +e \right )}{2}+4 \left (-\frac {\left (\sec ^{3}\left (f x +e \right )\right )}{4}-\frac {3 \sec \left (f x +e \right )}{8}\right ) \tan \left (f x +e \right )}{f}\)	\(47\)
norman	\(\frac {-\frac {2 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{f}-\frac {6 \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{f}-\frac {6 \left (\tan ^{5}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{f}-\frac {2 \left (\tan ^{7}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{f}}{\left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{4}}\)	\(80\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sec(f*x+e)^3*(3-4*sec(f*x+e)^2),x,method=_RETURNVERBOSE)

[Out]

1/f*(3/2*sec(f*x+e)*tan(f*x+e)+4*(-1/4*sec(f*x+e)^3-3/8*sec(f*x+e))*tan(f*x+e))

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Maxima [A]
time = 0.29, size = 36, normalized size = 1.89 \begin {gather*} -\frac {\sin \left (f x + e\right )}{{\left (\sin \left (f x + e\right )^{4} - 2 \, \sin \left (f x + e\right )^{2} + 1\right )} f} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]
time = 4.05, size = 21, normalized size = 1.11 \begin {gather*} -\frac {\sin \left (f x + e\right )}{f \cos \left (f x + e\right )^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-sin(f*x + e)/(f*cos(f*x + e)^4)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - \int \left (- 3 \sec ^{3}{\left (e + f x \right )}\right )\, dx - \int 4 \sec ^{5}{\left (e + f x \right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-Integral(-3*sec(e + f*x)**3, x) - Integral(4*sec(e + f*x)**5, x)

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Giac [A]
time = 0.48, size = 23, normalized size = 1.21 \begin {gather*} -\frac {\sin \left (f x + e\right )}{{\left (\sin \left (f x + e\right )^{2} - 1\right )}^{2} f} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [B]
time = 2.37, size = 23, normalized size = 1.21 \begin {gather*} -\frac {\sin \left (e+f\,x\right )}{f\,{\left ({\sin \left (e+f\,x\right )}^2-1\right )}^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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