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

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Rubi [F]  time = 0.02, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, 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*}

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Mathematica [A]  time = 0.02, 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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fricas [A]  time = 0.54, size = 42, normalized size = 0.98 \[ \frac {1}{3} i \, x^{3} - 2 i \, x e^{\left (2 i \, a\right )} - e^{\left (3 i \, a\right )} \log \left (x + i \, e^{\left (i \, a\right )}\right ) + e^{\left (3 i \, a\right )} \log \left (x - i \, e^{\left (i \, a\right )}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.35, size = 26, normalized size = 0.60 \[ \frac {1}{3} i \, x^{3} + 2 i \, \arctan \left (x e^{\left (-i \, a\right )}\right ) e^{\left (3 i \, a\right )} - 2 i \, x e^{\left (2 i \, a\right )} \]

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)

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maple [A]  time = 0.06, size = 33, normalized size = 0.77 \[ \frac {i x^{3}}{3}-2 i {\mathrm e}^{2 i a} x +2 i \arctan \left (x \,{\mathrm e}^{-i a}\right ) {\mathrm e}^{3 i a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [B]  time = 0.45, size = 151, normalized size = 3.51 \[ \frac {1}{3} i \, x^{3} - 2 \, x {\left (i \, \cos \left (2 \, a\right ) - \sin \left (2 \, a\right )\right )} - {\left (i \, \cos \left (3 \, a\right ) - \sin \left (3 \, a\right )\right )} \arctan \left (\frac {2 \, x \cos \relax (a)}{x^{2} + \cos \relax (a)^{2} - 2 \, x \sin \relax (a) + \sin \relax (a)^{2}}, \frac {x^{2} - \cos \relax (a)^{2} - \sin \relax (a)^{2}}{x^{2} + \cos \relax (a)^{2} - 2 \, x \sin \relax (a) + \sin \relax (a)^{2}}\right ) + \frac {1}{6} \, {\left (3 \, \cos \left (3 \, a\right ) + 3 i \, \sin \left (3 \, a\right )\right )} \log \left (\frac {x^{2} + \cos \relax (a)^{2} + 2 \, x \sin \relax (a) + \sin \relax (a)^{2}}{x^{2} + \cos \relax (a)^{2} - 2 \, x \sin \relax (a) + \sin \relax (a)^{2}}\right ) \]

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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mupad [B]  time = 2.21, size = 36, normalized size = 0.84 \[ {\left ({\mathrm {e}}^{a\,2{}\mathrm {i}}\right )}^{3/2}\,\mathrm {atan}\left (\frac {x}{\sqrt {{\mathrm {e}}^{a\,2{}\mathrm {i}}}}\right )\,2{}\mathrm {i}+\frac {x^3\,1{}\mathrm {i}}{3}-x\,{\mathrm {e}}^{a\,2{}\mathrm {i}}\,2{}\mathrm {i} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*tan(a + log(x)*1i),x)

[Out]

exp(a*2i)^(3/2)*atan(x/exp(a*2i)^(1/2))*2i + (x^3*1i)/3 - x*exp(a*2i)*2i

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sympy [A]  time = 0.20, size = 61, normalized size = 1.42 \[ \frac {i x^{3}}{3} - 2 i x e^{2 i a} + \left (\log {\left (x e^{2 i a} - i e^{3 i a} \right )} - \log {\left (x e^{2 i a} + i e^{3 i a} \right )}\right ) e^{3 i a} \]

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

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