3.46 \(\int \frac{\sec (c+d x)}{a+a \sec (c+d x)} \, dx\)

Optimal. Leaf size=22 \[ \frac{\tan (c+d x)}{d (a \sec (c+d x)+a)} \]

[Out]

Tan[c + d*x]/(d*(a + a*Sec[c + d*x]))

________________________________________________________________________________________

Rubi [A]  time = 0.0238029, antiderivative size = 22, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.053, Rules used = {3794} \[ \frac{\tan (c+d x)}{d (a \sec (c+d x)+a)} \]

Antiderivative was successfully verified.

[In]

Int[Sec[c + d*x]/(a + a*Sec[c + d*x]),x]

[Out]

Tan[c + d*x]/(d*(a + a*Sec[c + d*x]))

Rule 3794

Int[csc[(e_.) + (f_.)*(x_)]/(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)), x_Symbol] :> -Simp[Cot[e + f*x]/(f*(b + a*
Csc[e + f*x])), x] /; FreeQ[{a, b, e, f}, x] && EqQ[a^2 - b^2, 0]

Rubi steps

\begin{align*} \int \frac{\sec (c+d x)}{a+a \sec (c+d x)} \, dx &=\frac{\tan (c+d x)}{d (a+a \sec (c+d x))}\\ \end{align*}

Mathematica [A]  time = 0.0257344, size = 17, normalized size = 0.77 \[ \frac{\tan \left (\frac{1}{2} (c+d x)\right )}{a d} \]

Antiderivative was successfully verified.

[In]

Integrate[Sec[c + d*x]/(a + a*Sec[c + d*x]),x]

[Out]

Tan[(c + d*x)/2]/(a*d)

________________________________________________________________________________________

Maple [A]  time = 0.027, size = 17, normalized size = 0.8 \begin{align*}{\frac{1}{da}\tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sec(d*x+c)/(a+a*sec(d*x+c)),x)

[Out]

1/a/d*tan(1/2*d*x+1/2*c)

________________________________________________________________________________________

Maxima [A]  time = 1.08566, size = 31, normalized size = 1.41 \begin{align*} \frac{\sin \left (d x + c\right )}{a d{\left (\cos \left (d x + c\right ) + 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)/(a+a*sec(d*x+c)),x, algorithm="maxima")

[Out]

sin(d*x + c)/(a*d*(cos(d*x + c) + 1))

________________________________________________________________________________________

Fricas [A]  time = 1.52776, size = 53, normalized size = 2.41 \begin{align*} \frac{\sin \left (d x + c\right )}{a d \cos \left (d x + c\right ) + a d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)/(a+a*sec(d*x+c)),x, algorithm="fricas")

[Out]

sin(d*x + c)/(a*d*cos(d*x + c) + a*d)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)/(a+a*sec(d*x+c)),x)

[Out]

Integral(sec(c + d*x)/(sec(c + d*x) + 1), x)/a

________________________________________________________________________________________

Giac [A]  time = 1.26239, size = 22, normalized size = 1. \begin{align*} \frac{\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )}{a d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)/(a+a*sec(d*x+c)),x, algorithm="giac")

[Out]

tan(1/2*d*x + 1/2*c)/(a*d)