3.146 \(\int \tan ^2(a+i \log (x)) \, dx\)

Optimal. Leaf size=46 \[ -\frac{2 e^{2 i a} x}{x^2+e^{2 i a}}+2 e^{i a} \tan ^{-1}\left (e^{-i a} x\right )-x \]

[Out]

-x - (2*E^((2*I)*a)*x)/(E^((2*I)*a) + x^2) + 2*E^(I*a)*ArcTan[x/E^(I*a)]

________________________________________________________________________________________

Rubi [F]  time = 0.0104115, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int \tan ^2(a+i \log (x)) \, dx \]

Verification is Not applicable to the result.

[In]

Int[Tan[a + I*Log[x]]^2,x]

[Out]

Defer[Int][Tan[a + I*Log[x]]^2, x]

Rubi steps

\begin{align*} \int \tan ^2(a+i \log (x)) \, dx &=\int \tan ^2(a+i \log (x)) \, dx\\ \end{align*}

Mathematica [A]  time = 0.0887918, size = 70, normalized size = 1.52 \[ \frac{-x \left (x^2+3\right ) \cos (a)+i x \left (x^2-3\right ) \sin (a)}{\left (x^2+1\right ) \cos (a)-i \left (x^2-1\right ) \sin (a)}+2 (\cos (a)+i \sin (a)) \tan ^{-1}(x (\cos (a)-i \sin (a))) \]

Antiderivative was successfully verified.

[In]

Integrate[Tan[a + I*Log[x]]^2,x]

[Out]

2*ArcTan[x*(Cos[a] - I*Sin[a])]*(Cos[a] + I*Sin[a]) + (-(x*(3 + x^2)*Cos[a]) + I*x*(-3 + x^2)*Sin[a])/((1 + x^
2)*Cos[a] - I*(-1 + x^2)*Sin[a])

________________________________________________________________________________________

Maple [F]  time = 0.061, size = 0, normalized size = 0. \begin{align*} \int \left ( \tan \left ( a+i\ln \left ( x \right ) \right ) \right ) ^{2}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(tan(a+I*ln(x))^2,x)

[Out]

int(tan(a+I*ln(x))^2,x)

________________________________________________________________________________________

Maxima [B]  time = 1.59874, size = 305, normalized size = 6.63 \begin{align*} -\frac{2 \, x^{3} + x{\left (6 \, \cos \left (2 \, a\right ) + 6 i \, \sin \left (2 \, a\right )\right )} +{\left (x^{2}{\left (2 \, \cos \left (a\right ) + 2 i \, \sin \left (a\right )\right )} +{\left (2 \, \cos \left (a\right ) + 2 i \, \sin \left (a\right )\right )} \cos \left (2 \, a\right ) - 2 \,{\left (-i \, \cos \left (a\right ) + \sin \left (a\right )\right )} \sin \left (2 \, a\right )\right )} \arctan \left (\frac{2 \, x \cos \left (a\right )}{x^{2} + \cos \left (a\right )^{2} - 2 \, x \sin \left (a\right ) + \sin \left (a\right )^{2}}, \frac{x^{2} - \cos \left (a\right )^{2} - \sin \left (a\right )^{2}}{x^{2} + \cos \left (a\right )^{2} - 2 \, x \sin \left (a\right ) + \sin \left (a\right )^{2}}\right ) +{\left (x^{2}{\left (i \, \cos \left (a\right ) - \sin \left (a\right )\right )} +{\left (i \, \cos \left (a\right ) - \sin \left (a\right )\right )} \cos \left (2 \, a\right ) -{\left (\cos \left (a\right ) + i \, \sin \left (a\right )\right )} \sin \left (2 \, a\right )\right )} \log \left (\frac{x^{2} + \cos \left (a\right )^{2} + 2 \, x \sin \left (a\right ) + \sin \left (a\right )^{2}}{x^{2} + \cos \left (a\right )^{2} - 2 \, x \sin \left (a\right ) + \sin \left (a\right )^{2}}\right )}{2 \, x^{2} + 2 \, \cos \left (2 \, a\right ) + 2 i \, \sin \left (2 \, a\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tan(a+I*log(x))^2,x, algorithm="maxima")

[Out]

-(2*x^3 + x*(6*cos(2*a) + 6*I*sin(2*a)) + (x^2*(2*cos(a) + 2*I*sin(a)) + (2*cos(a) + 2*I*sin(a))*cos(2*a) - 2*
(-I*cos(a) + sin(a))*sin(2*a))*arctan2(2*x*cos(a)/(x^2 + cos(a)^2 - 2*x*sin(a) + sin(a)^2), (x^2 - cos(a)^2 -
sin(a)^2)/(x^2 + cos(a)^2 - 2*x*sin(a) + sin(a)^2)) + (x^2*(I*cos(a) - sin(a)) + (I*cos(a) - sin(a))*cos(2*a)
- (cos(a) + I*sin(a))*sin(2*a))*log((x^2 + cos(a)^2 + 2*x*sin(a) + sin(a)^2)/(x^2 + cos(a)^2 - 2*x*sin(a) + si
n(a)^2)))/(2*x^2 + 2*cos(2*a) + 2*I*sin(2*a))

________________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{{\left (e^{\left (2 i \, a - 2 \, \log \left (x\right )\right )} + 1\right )}{\rm integral}\left (-\frac{e^{\left (2 i \, a - 2 \, \log \left (x\right )\right )} + 3}{e^{\left (2 i \, a - 2 \, \log \left (x\right )\right )} + 1}, x\right ) + 2 \, x}{e^{\left (2 i \, a - 2 \, \log \left (x\right )\right )} + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tan(a+I*log(x))^2,x, algorithm="fricas")

[Out]

((e^(2*I*a - 2*log(x)) + 1)*integral(-(e^(2*I*a - 2*log(x)) + 3)/(e^(2*I*a - 2*log(x)) + 1), x) + 2*x)/(e^(2*I
*a - 2*log(x)) + 1)

________________________________________________________________________________________

Sympy [A]  time = 0.60995, size = 51, normalized size = 1.11 \begin{align*} - x - \frac{2 x e^{2 i a}}{x^{2} + e^{2 i a}} - \left (i \log{\left (x - i e^{i a} \right )} - i \log{\left (x + i e^{i a} \right )}\right ) e^{i a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tan(a+I*ln(x))**2,x)

[Out]

-x - 2*x*exp(2*I*a)/(x**2 + exp(2*I*a)) - (I*log(x - I*exp(I*a)) - I*log(x + I*exp(I*a)))*exp(I*a)

________________________________________________________________________________________

Giac [B]  time = 1.18675, size = 154, normalized size = 3.35 \begin{align*} -\frac{x^{3}}{x^{2} + \frac{e^{\left (4 i \, a\right )}}{x^{2}} + 2 \, e^{\left (2 i \, a\right )}} + 2 \,{\left (\arctan \left (x e^{\left (-i \, a\right )}\right ) e^{\left (-i \, a\right )} - \frac{x}{x^{2} + e^{\left (2 i \, a\right )}}\right )} e^{\left (2 i \, a\right )} - \frac{6 \, x e^{\left (2 i \, a\right )}}{x^{2} + \frac{e^{\left (4 i \, a\right )}}{x^{2}} + 2 \, e^{\left (2 i \, a\right )}} - \frac{5 \, e^{\left (4 i \, a\right )}}{{\left (x^{2} + \frac{e^{\left (4 i \, a\right )}}{x^{2}} + 2 \, e^{\left (2 i \, a\right )}\right )} x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tan(a+I*log(x))^2,x, algorithm="giac")

[Out]

-x^3/(x^2 + e^(4*I*a)/x^2 + 2*e^(2*I*a)) + 2*(arctan(x*e^(-I*a))*e^(-I*a) - x/(x^2 + e^(2*I*a)))*e^(2*I*a) - 6
*x*e^(2*I*a)/(x^2 + e^(4*I*a)/x^2 + 2*e^(2*I*a)) - 5*e^(4*I*a)/((x^2 + e^(4*I*a)/x^2 + 2*e^(2*I*a))*x)