3.147 \(\int \frac {\tan ^2(a+i \log (x))}{x} \, dx\)

Optimal. Leaf size=18 \[ -\log (x)-i \tan (a+i \log (x)) \]

[Out]

-ln(x)-I*tan(a+I*ln(x))

________________________________________________________________________________________

Rubi [A]  time = 0.02, antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {3473, 8} \[ -\log (x)-i \tan (a+i \log (x)) \]

Antiderivative was successfully verified.

[In]

Int[Tan[a + I*Log[x]]^2/x,x]

[Out]

-Log[x] - I*Tan[a + I*Log[x]]

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rule 3473

Int[((b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(b*(b*Tan[c + d*x])^(n - 1))/(d*(n - 1)), x] - Dis
t[b^2, Int[(b*Tan[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1]

Rubi steps

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

________________________________________________________________________________________

Mathematica [A]  time = 0.04, size = 28, normalized size = 1.56 \[ i \tan ^{-1}(\tan (a+i \log (x)))-i \tan (a+i \log (x)) \]

Antiderivative was successfully verified.

[In]

Integrate[Tan[a + I*Log[x]]^2/x,x]

[Out]

I*ArcTan[Tan[a + I*Log[x]]] - I*Tan[a + I*Log[x]]

________________________________________________________________________________________

fricas [B]  time = 0.50, size = 30, normalized size = 1.67 \[ -\frac {{\left (x^{2} + e^{\left (2 i \, a\right )}\right )} \log \relax (x) + 2 \, e^{\left (2 i \, a\right )}}{x^{2} + e^{\left (2 i \, a\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-((x^2 + e^(2*I*a))*log(x) + 2*e^(2*I*a))/(x^2 + e^(2*I*a))

________________________________________________________________________________________

giac [B]  time = 0.25, size = 38, normalized size = 2.11 \[ \frac {i \, \tan \relax (a)^{2} + i}{{\left (\frac {i \, {\left (x^{2} - 1\right )} \tan \relax (a)}{x^{2} + 1} - 1\right )} \tan \relax (a)} - \log \relax (x) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maple [A]  time = 0.01, size = 23, normalized size = 1.28 \[ -i \tan \left (a +i \ln \relax (x )\right )+i \left (a +i \ln \relax (x )\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-I*tan(a+I*ln(x))+I*(a+I*ln(x))

________________________________________________________________________________________

maxima [A]  time = 0.43, size = 17, normalized size = 0.94 \[ i \, a - \log \relax (x) - i \, \tan \left (a + i \, \log \relax (x)\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

I*a - log(x) - I*tan(a + I*log(x))

________________________________________________________________________________________

mupad [B]  time = 2.38, size = 16, normalized size = 0.89 \[ -\ln \relax (x)-\mathrm {tan}\left (a+\ln \relax (x)\,1{}\mathrm {i}\right )\,1{}\mathrm {i} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(tan(a + log(x)*1i)^2/x,x)

[Out]

- tan(a + log(x)*1i)*1i - log(x)

________________________________________________________________________________________

sympy [A]  time = 0.30, size = 22, normalized size = 1.22 \[ - \log {\relax (x )} - \frac {2 e^{2 i a}}{x^{2} + e^{2 i a}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________