3.110 \(\int \frac {\sec (c+d x)}{a+i a \tan (c+d x)} \, dx\)

Optimal. Leaf size=28 \[ \frac {i \sec (c+d x)}{d (a+i a \tan (c+d x))} \]

[Out]

I*sec(d*x+c)/d/(a+I*a*tan(d*x+c))

________________________________________________________________________________________

Rubi [A]  time = 0.02, antiderivative size = 28, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {3488} \[ \frac {i \sec (c+d x)}{d (a+i a \tan (c+d x))} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(I*Sec[c + d*x])/(d*(a + I*a*Tan[c + d*x]))

Rule 3488

Int[((d_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(b*(d
*Sec[e + f*x])^m*(a + b*Tan[e + f*x])^n)/(a*f*m), x] /; FreeQ[{a, b, d, e, f, m, n}, x] && EqQ[a^2 + b^2, 0] &
& EqQ[Simplify[m + n], 0]

Rubi steps

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

________________________________________________________________________________________

Mathematica [A]  time = 0.03, size = 25, normalized size = 0.89 \[ \frac {\sec (c+d x)}{a d (\tan (c+d x)-i)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

Sec[c + d*x]/(a*d*(-I + Tan[c + d*x]))

________________________________________________________________________________________

fricas [A]  time = 0.57, size = 17, normalized size = 0.61 \[ \frac {i \, e^{\left (-i \, d x - i \, c\right )}}{a d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

I*e^(-I*d*x - I*c)/(a*d)

________________________________________________________________________________________

giac [A]  time = 0.70, size = 21, normalized size = 0.75 \[ \frac {2}{a d {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - i\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/(a*d*(tan(1/2*d*x + 1/2*c) - I))

________________________________________________________________________________________

maple [A]  time = 0.20, size = 23, normalized size = 0.82 \[ \frac {2}{d a \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-i\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maxima [A]  time = 0.71, size = 29, normalized size = 1.04 \[ \frac {2}{{\left (-i \, a + \frac {a \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )} d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/((-I*a + a*sin(d*x + c)/(cos(d*x + c) + 1))*d)

________________________________________________________________________________________

mupad [B]  time = 3.35, size = 25, normalized size = 0.89 \[ \frac {2{}\mathrm {i}}{a\,d\,\left (1+\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,1{}\mathrm {i}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(cos(c + d*x)*(a + a*tan(c + d*x)*1i)),x)

[Out]

2i/(a*d*(tan(c/2 + (d*x)/2)*1i + 1))

________________________________________________________________________________________

sympy [A]  time = 0.64, size = 34, normalized size = 1.21 \[ \begin {cases} \frac {\sec {\left (c + d x \right )}}{a d \tan {\left (c + d x \right )} - i a d} & \text {for}\: d \neq 0 \\\frac {x \sec {\relax (c )}}{i a \tan {\relax (c )} + a} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((sec(c + d*x)/(a*d*tan(c + d*x) - I*a*d), Ne(d, 0)), (x*sec(c)/(I*a*tan(c) + a), True))

________________________________________________________________________________________