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

Optimal. Leaf size=43 \[ -2 i e^{2 i a} x+2 i e^{3 i a} \tan ^{-1}\left (e^{-i a} x\right )+\frac{i x^3}{3} \]

[Out]

(-2*I)*E^((2*I)*a)*x + (I/3)*x^3 + (2*I)*E^((3*I)*a)*ArcTan[x/E^(I*a)]

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Rubi [F]  time = 0.022892, 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 = {} \[ \int x^2 \tan (a+i \log (x)) \, dx \]

Verification is Not applicable to the result.

[In]

Int[x^2*Tan[a + I*Log[x]],x]

[Out]

Defer[Int][x^2*Tan[a + I*Log[x]], x]

Rubi steps

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

Mathematica [A]  time = 0.0177957, size = 66, normalized size = 1.53 \[ 2 x \sin (2 a)-2 i x \cos (2 a)+2 i \cos (3 a) \tan ^{-1}(x \cos (a)-i x \sin (a))-2 \sin (3 a) \tan ^{-1}(x \cos (a)-i x \sin (a))+\frac{i x^3}{3} \]

Antiderivative was successfully verified.

[In]

Integrate[x^2*Tan[a + I*Log[x]],x]

[Out]

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

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Maple [F]  time = 0.068, size = 0, normalized size = 0. \begin{align*} \int{x}^{2}\tan \left ( a+i\ln \left ( x \right ) \right ) \, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B]  time = 1.63307, size = 204, normalized size = 4.74 \begin{align*} \frac{1}{3} i \, x^{3} - 2 \, x{\left (i \, \cos \left (2 \, a\right ) - \sin \left (2 \, a\right )\right )} -{\left (i \, \cos \left (3 \, a\right ) - \sin \left (3 \, 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 ) + \frac{1}{6} \,{\left (3 \, \cos \left (3 \, a\right ) + 3 i \, \sin \left (3 \, 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 ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*I*x^3 - 2*x*(I*cos(2*a) - sin(2*a)) - (I*cos(3*a) - sin(3*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)) + 1/6*(3*cos(3*a) + 3*I
*sin(3*a))*log((x^2 + cos(a)^2 + 2*x*sin(a) + sin(a)^2)/(x^2 + cos(a)^2 - 2*x*sin(a) + sin(a)^2))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.69732, size = 44, normalized size = 1.02 \begin{align*} \frac{i x^{3}}{3} - 2 i x e^{2 i a} + \left (\log{\left (x - i e^{i a} \right )} - \log{\left (x + i e^{i a} \right )}\right ) e^{3 i a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

I*x**3/3 - 2*I*x*exp(2*I*a) + (log(x - I*exp(I*a)) - log(x + I*exp(I*a)))*exp(3*I*a)

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Giac [A]  time = 1.16951, size = 35, normalized size = 0.81 \begin{align*} \frac{1}{3} i \, x^{3} + 2 i \, \arctan \left (x e^{\left (-i \, a\right )}\right ) e^{\left (3 i \, a\right )} - 2 i \, x e^{\left (2 i \, a\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*I*x^3 + 2*I*arctan(x*e^(-I*a))*e^(3*I*a) - 2*I*x*e^(2*I*a)