3.336 \(\int \frac{\sec ^2(c+d x)}{\sqrt{a+i a \tan (c+d x)}} \, dx\)

Optimal. Leaf size=27 \[ -\frac{2 i \sqrt{a+i a \tan (c+d x)}}{a d} \]

[Out]

((-2*I)*Sqrt[a + I*a*Tan[c + d*x]])/(a*d)

________________________________________________________________________________________

Rubi [A]  time = 0.0566053, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {3487, 32} \[ -\frac{2 i \sqrt{a+i a \tan (c+d x)}}{a d} \]

Antiderivative was successfully verified.

[In]

Int[Sec[c + d*x]^2/Sqrt[a + I*a*Tan[c + d*x]],x]

[Out]

((-2*I)*Sqrt[a + I*a*Tan[c + d*x]])/(a*d)

Rule 3487

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

Rule 32

Int[((a_.) + (b_.)*(x_))^(m_), x_Symbol] :> Simp[(a + b*x)^(m + 1)/(b*(m + 1)), x] /; FreeQ[{a, b, m}, x] && N
eQ[m, -1]

Rubi steps

\begin{align*} \int \frac{\sec ^2(c+d x)}{\sqrt{a+i a \tan (c+d x)}} \, dx &=-\frac{i \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+x}} \, dx,x,i a \tan (c+d x)\right )}{a d}\\ &=-\frac{2 i \sqrt{a+i a \tan (c+d x)}}{a d}\\ \end{align*}

Mathematica [A]  time = 0.143902, size = 32, normalized size = 1.19 \[ \frac{2 (\tan (c+d x)-i)}{d \sqrt{a+i a \tan (c+d x)}} \]

Antiderivative was successfully verified.

[In]

Integrate[Sec[c + d*x]^2/Sqrt[a + I*a*Tan[c + d*x]],x]

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0.043, size = 24, normalized size = 0.9 \begin{align*}{\frac{-2\,i}{ad}\sqrt{a+ia\tan \left ( dx+c \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sec(d*x+c)^2/(a+I*a*tan(d*x+c))^(1/2),x)

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 1.05822, size = 28, normalized size = 1.04 \begin{align*} -\frac{2 i \, \sqrt{i \, a \tan \left (d x + c\right ) + a}}{a d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)^2/(a+I*a*tan(d*x+c))^(1/2),x, algorithm="maxima")

[Out]

-2*I*sqrt(I*a*tan(d*x + c) + a)/(a*d)

________________________________________________________________________________________

Fricas [A]  time = 1.92316, size = 95, normalized size = 3.52 \begin{align*} -\frac{2 i \, \sqrt{2} \sqrt{\frac{a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} 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)^2/(a+I*a*tan(d*x+c))^(1/2),x, algorithm="fricas")

[Out]

-2*I*sqrt(2)*sqrt(a/(e^(2*I*d*x + 2*I*c) + 1))*e^(I*d*x + I*c)/(a*d)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)**2/(a+I*a*tan(d*x+c))**(1/2),x)

[Out]

Integral(sec(c + d*x)**2/sqrt(a*(I*tan(c + d*x) + 1)), x)

________________________________________________________________________________________

Giac [A]  time = 1.67694, size = 53, normalized size = 1.96 \begin{align*} -\frac{2 i \, \sqrt{a - \frac{2 i \, a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )}{\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 1}}}{a d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)^2/(a+I*a*tan(d*x+c))^(1/2),x, algorithm="giac")

[Out]

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