3.38 \(\int \frac{\sec (e+f x)}{(a+a \sec (e+f x)) (c-c \sec (e+f x))} \, dx\)

Optimal. Leaf size=16 \[ \frac{\csc (e+f x)}{a c f} \]

[Out]

Csc[e + f*x]/(a*c*f)

________________________________________________________________________________________

Rubi [A]  time = 0.0890242, antiderivative size = 16, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.094, Rules used = {3958, 2606, 8} \[ \frac{\csc (e+f x)}{a c f} \]

Antiderivative was successfully verified.

[In]

Int[Sec[e + f*x]/((a + a*Sec[e + f*x])*(c - c*Sec[e + f*x])),x]

[Out]

Csc[e + f*x]/(a*c*f)

Rule 3958

Int[csc[(e_.) + (f_.)*(x_)]*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_.)*(csc[(e_.) + (f_.)*(x_)]*(d_.) + (c_)
)^(n_.), x_Symbol] :> Dist[(-(a*c))^m, Int[ExpandTrig[csc[e + f*x]*cot[e + f*x]^(2*m), (c + d*csc[e + f*x])^(n
- m), x], x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && IntegersQ[m,
n] && GeQ[n - m, 0] && GtQ[m*n, 0]

Rule 2606

Int[((a_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Dist[a/f, Subst[
Int[(a*x)^(m - 1)*(-1 + x^2)^((n - 1)/2), x], x, Sec[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n -
1)/2] &&  !(IntegerQ[m/2] && LtQ[0, m, n + 1])

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rubi steps

\begin{align*} \int \frac{\sec (e+f x)}{(a+a \sec (e+f x)) (c-c \sec (e+f x))} \, dx &=-\frac{\int \cot (e+f x) \csc (e+f x) \, dx}{a c}\\ &=\frac{\operatorname{Subst}(\int 1 \, dx,x,\csc (e+f x))}{a c f}\\ &=\frac{\csc (e+f x)}{a c f}\\ \end{align*}

Mathematica [A]  time = 0.0284897, size = 16, normalized size = 1. \[ \frac{\csc (e+f x)}{a c f} \]

Antiderivative was successfully verified.

[In]

Integrate[Sec[e + f*x]/((a + a*Sec[e + f*x])*(c - c*Sec[e + f*x])),x]

[Out]

Csc[e + f*x]/(a*c*f)

________________________________________________________________________________________

Maple [F]  time = 180., size = 0, normalized size = 0. \begin{align*} \int{\frac{\sec \left ( fx+e \right ) }{ \left ( a+a\sec \left ( fx+e \right ) \right ) \left ( c-c\sec \left ( fx+e \right ) \right ) }}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sec(f*x+e)/(a+a*sec(f*x+e))/(c-c*sec(f*x+e)),x)

[Out]

int(sec(f*x+e)/(a+a*sec(f*x+e))/(c-c*sec(f*x+e)),x)

________________________________________________________________________________________

Maxima [A]  time = 0.961591, size = 24, normalized size = 1.5 \begin{align*} \frac{1}{a c f \sin \left (f x + e\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(f*x+e)/(a+a*sec(f*x+e))/(c-c*sec(f*x+e)),x, algorithm="maxima")

[Out]

1/(a*c*f*sin(f*x + e))

________________________________________________________________________________________

Fricas [A]  time = 0.433743, size = 32, normalized size = 2. \begin{align*} \frac{1}{a c f \sin \left (f x + e\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(f*x+e)/(a+a*sec(f*x+e))/(c-c*sec(f*x+e)),x, algorithm="fricas")

[Out]

1/(a*c*f*sin(f*x + e))

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} - \frac{\int \frac{\sec{\left (e + f x \right )}}{\sec ^{2}{\left (e + f x \right )} - 1}\, dx}{a c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(f*x+e)/(a+a*sec(f*x+e))/(c-c*sec(f*x+e)),x)

[Out]

-Integral(sec(e + f*x)/(sec(e + f*x)**2 - 1), x)/(a*c)

________________________________________________________________________________________

Giac [A]  time = 1.23839, size = 26, normalized size = 1.62 \begin{align*} \frac{1}{a c f \sin \left (f x + e\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(f*x+e)/(a+a*sec(f*x+e))/(c-c*sec(f*x+e)),x, algorithm="giac")

[Out]

1/(a*c*f*sin(f*x + e))