3.26 \(\int \sec ^5(e+f x) (5-6 \sec ^2(e+f x)) \, dx\)

Optimal. Leaf size=19 \[ -\frac{\tan (e+f x) \sec ^5(e+f x)}{f} \]

[Out]

-((Sec[e + f*x]^5*Tan[e + f*x])/f)

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Rubi [A]  time = 0.0239556, 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} \[ -\frac{\tan (e+f x) \sec ^5(e+f x)}{f} \]

Antiderivative was successfully verified.

[In]

Int[Sec[e + f*x]^5*(5 - 6*Sec[e + f*x]^2),x]

[Out]

-((Sec[e + f*x]^5*Tan[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 \sec ^5(e+f x) \left (5-6 \sec ^2(e+f x)\right ) \, dx &=-\frac{\sec ^5(e+f x) \tan (e+f x)}{f}\\ \end{align*}

Mathematica [A]  time = 0.0603043, size = 19, normalized size = 1. \[ -\frac{\tan (e+f x) \sec ^5(e+f x)}{f} \]

Antiderivative was successfully verified.

[In]

Integrate[Sec[e + f*x]^5*(5 - 6*Sec[e + f*x]^2),x]

[Out]

-((Sec[e + f*x]^5*Tan[e + f*x])/f)

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Maple [B]  time = 0.03, size = 70, normalized size = 3.7 \begin{align*}{\frac{1}{f} \left ( -5\, \left ( -1/4\, \left ( \sec \left ( fx+e \right ) \right ) ^{3}-3/8\,\sec \left ( fx+e \right ) \right ) \tan \left ( fx+e \right ) +6\, \left ( -1/6\, \left ( \sec \left ( fx+e \right ) \right ) ^{5}-{\frac{5\, \left ( \sec \left ( fx+e \right ) \right ) ^{3}}{24}}-{\frac{5\,\sec \left ( fx+e \right ) }{16}} \right ) \tan \left ( fx+e \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sec(f*x+e)^5*(5-6*sec(f*x+e)^2),x)

[Out]

1/f*(-5*(-1/4*sec(f*x+e)^3-3/8*sec(f*x+e))*tan(f*x+e)+6*(-1/6*sec(f*x+e)^5-5/24*sec(f*x+e)^3-5/16*sec(f*x+e))*
tan(f*x+e))

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Maxima [B]  time = 0.931995, size = 57, normalized size = 3. \begin{align*} \frac{\sin \left (f x + e\right )}{{\left (\sin \left (f x + e\right )^{6} - 3 \, \sin \left (f x + e\right )^{4} + 3 \, \sin \left (f x + e\right )^{2} - 1\right )} f} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(f*x+e)^5*(5-6*sec(f*x+e)^2),x, algorithm="maxima")

[Out]

sin(f*x + e)/((sin(f*x + e)^6 - 3*sin(f*x + e)^4 + 3*sin(f*x + e)^2 - 1)*f)

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Fricas [A]  time = 0.465308, size = 46, normalized size = 2.42 \begin{align*} -\frac{\sin \left (f x + e\right )}{f \cos \left (f x + e\right )^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(f*x+e)^5*(5-6*sec(f*x+e)^2),x, algorithm="fricas")

[Out]

-sin(f*x + e)/(f*cos(f*x + e)^6)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} - \int - 5 \sec ^{5}{\left (e + f x \right )}\, dx - \int 6 \sec ^{7}{\left (e + f x \right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(f*x+e)**5*(5-6*sec(f*x+e)**2),x)

[Out]

-Integral(-5*sec(e + f*x)**5, x) - Integral(6*sec(e + f*x)**7, x)

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Giac [A]  time = 1.17626, size = 32, normalized size = 1.68 \begin{align*} \frac{\sin \left (f x + e\right )}{{\left (\sin \left (f x + e\right )^{2} - 1\right )}^{3} f} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(f*x+e)^5*(5-6*sec(f*x+e)^2),x, algorithm="giac")

[Out]

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