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[c + d*x])/(d*(a + I*a*Tan[c + d*x]))

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Rubi [A]  time = 0.0228633, antiderivative size = 28, normalized size of antiderivative = 1., 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 = 0.0267033, 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]))

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Maple [A]  time = 0.034, size = 23, normalized size = 0.8 \begin{align*} 2\,{\frac{1}{ad \left ( \tan \left ( 1/2\,dx+c/2 \right ) -i \right ) }} \end{align*}

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)

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Maxima [A]  time = 0.985451, size = 39, normalized size = 1.39 \begin{align*} \frac{2}{{\left (-i \, a + \frac{a \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )} d} \end{align*}

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)

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Fricas [A]  time = 2.16242, size = 35, normalized size = 1.25 \begin{align*} \frac{i \, e^{\left (-i \, d x - i \, c\right )}}{a d} \end{align*}

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)

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Sympy [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: AttributeError} \end{align*}

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]

Exception raised: AttributeError

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Giac [A]  time = 1.1525, size = 28, normalized size = 1. \begin{align*} \frac{2}{a d{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - i\right )}} \end{align*}

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))