3.2.47 \(\int \frac {\tan ^2(a+i \log (x))}{x} \, dx\) [147]

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

[Out]

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

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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 = {3554, 8} \begin {gather*} -\log (x)-i \tan (a+i \log (x)) \end {gather*}

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 3554

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

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Mathematica [A]
time = 0.05, size = 28, normalized size = 1.56 \begin {gather*} i \text {ArcTan}(\tan (a+i \log (x)))-i \tan (a+i \log (x)) \end {gather*}

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]]

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Maple [A]
time = 0.04, size = 24, normalized size = 1.33

method	result	size
norman	\(-\ln \left (x \right )-i \tan \left (a +i \ln \left (x \right )\right )\)	\(17\)
risch	\(-\ln \left (x \right )+\frac {2}{1+\frac {{\mathrm e}^{2 i a}}{x^{2}}}\)	\(21\)
derivativedivides	\(-i \left (\tan \left (a +i \ln \left (x \right )\right )-\arctan \left (\tan \left (a +i \ln \left (x \right )\right )\right )\right )\)	\(24\)
default	\(-i \left (\tan \left (a +i \ln \left (x \right )\right )-\arctan \left (\tan \left (a +i \ln \left (x \right )\right )\right )\right )\)	\(24\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.47, size = 17, normalized size = 0.94 \begin {gather*} i \, a - \log \left (x\right ) - i \, \tan \left (a + i \, \log \left (x\right )\right ) \end {gather*}

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))

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Fricas [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 30 vs. \(2 (14) = 28\).
time = 3.76, size = 30, normalized size = 1.67 \begin {gather*} -\frac {{\left (x^{2} + e^{\left (2 i \, a\right )}\right )} \log \left (x\right ) + 2 \, e^{\left (2 i \, a\right )}}{x^{2} + e^{\left (2 i \, a\right )}} \end {gather*}

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))

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Sympy [A]
time = 0.17, size = 22, normalized size = 1.22 \begin {gather*} - \log {\left (x \right )} - \frac {2 e^{2 i a}}{x^{2} + e^{2 i a}} \end {gather*}

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))

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Giac [A]
time = 0.43, size = 17, normalized size = 0.94 \begin {gather*} i \, a - \log \left (x\right ) - i \, \tan \left (a + i \, \log \left (x\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

I*a - log(x) - I*tan(a + I*log(x))

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Mupad [B]
time = 2.38, size = 16, normalized size = 0.89 \begin {gather*} -\ln \left (x\right )-\mathrm {tan}\left (a+\ln \left (x\right )\,1{}\mathrm {i}\right )\,1{}\mathrm {i} \end {gather*}

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)

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