3.135 \(\int x^3 \tan (a+i \log (x)) \, dx\)

Optimal. Leaf size=47 \[ -i e^{2 i a} x^2+i e^{4 i a} \log \left (x^2+e^{2 i a}\right )+\frac{i x^4}{4} \]

[Out]

(-I)*E^((2*I)*a)*x^2 + (I/4)*x^4 + I*E^((4*I)*a)*Log[E^((2*I)*a) + x^2]

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Rubi [F]  time = 0.0300232, 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 = {} \[ \int x^3 \tan (a+i \log (x)) \, dx \]

Verification is Not applicable to the result.

[In]

Int[x^3*Tan[a + I*Log[x]],x]

[Out]

Defer[Int][x^3*Tan[a + I*Log[x]], x]

Rubi steps

\begin{align*} \int x^3 \tan (a+i \log (x)) \, dx &=\int x^3 \tan (a+i \log (x)) \, dx\\ \end{align*}

Mathematica [B]  time = 0.0323966, size = 132, normalized size = 2.81 \[ x^2 \sin (2 a)-i x^2 \cos (2 a)+\frac{1}{2} i \cos (4 a) \log \left (2 x^2 \cos (2 a)+x^4+1\right )-\frac{1}{2} \sin (4 a) \log \left (2 x^2 \cos (2 a)+x^4+1\right )+\cos (4 a) \tan ^{-1}\left (\frac{\left (x^2+1\right ) \cos (a)}{\sin (a)-x^2 \sin (a)}\right )+i \sin (4 a) \tan ^{-1}\left (\frac{\left (x^2+1\right ) \cos (a)}{\sin (a)-x^2 \sin (a)}\right )+\frac{i x^4}{4} \]

Antiderivative was successfully verified.

[In]

Integrate[x^3*Tan[a + I*Log[x]],x]

[Out]

(I/4)*x^4 - I*x^2*Cos[2*a] + ArcTan[((1 + x^2)*Cos[a])/(Sin[a] - x^2*Sin[a])]*Cos[4*a] + (I/2)*Cos[4*a]*Log[1
+ x^4 + 2*x^2*Cos[2*a]] + x^2*Sin[2*a] + I*ArcTan[((1 + x^2)*Cos[a])/(Sin[a] - x^2*Sin[a])]*Sin[4*a] - (Log[1
+ x^4 + 2*x^2*Cos[2*a]]*Sin[4*a])/2

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Maple [A]  time = 0.07, size = 49, normalized size = 1. \begin{align*} -{\frac{i}{4}}{x}^{4}-i \left ({x}^{2} \left ({{\rm e}^{ia}} \right ) ^{2}-{\frac{{x}^{4}}{2}}- \left ({{\rm e}^{ia}} \right ) ^{4}\ln \left ( \left ({{\rm e}^{ia}} \right ) ^{2}+{x}^{2} \right ) \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*tan(a+I*ln(x)),x)

[Out]

-1/4*I*x^4-I*(x^2*exp(I*a)^2-1/2*x^4-exp(I*a)^4*ln(exp(I*a)^2+x^2))

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Maxima [B]  time = 1.05784, size = 122, normalized size = 2.6 \begin{align*} \frac{1}{4} i \, x^{4} + x^{2}{\left (-i \, \cos \left (2 \, a\right ) + \sin \left (2 \, a\right )\right )} - \frac{1}{4} \,{\left (4 \, \cos \left (4 \, a\right ) + 4 i \, \sin \left (4 \, a\right )\right )} \arctan \left (\sin \left (2 \, a\right ), x^{2} + \cos \left (2 \, a\right )\right ) + \frac{1}{2} \,{\left (i \, \cos \left (4 \, a\right ) - \sin \left (4 \, a\right )\right )} \log \left (x^{4} + 2 \, x^{2} \cos \left (2 \, a\right ) + \cos \left (2 \, a\right )^{2} + \sin \left (2 \, a\right )^{2}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*tan(a+I*log(x)),x, algorithm="maxima")

[Out]

1/4*I*x^4 + x^2*(-I*cos(2*a) + sin(2*a)) - 1/4*(4*cos(4*a) + 4*I*sin(4*a))*arctan2(sin(2*a), x^2 + cos(2*a)) +
1/2*(I*cos(4*a) - sin(4*a))*log(x^4 + 2*x^2*cos(2*a) + cos(2*a)^2 + sin(2*a)^2)

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

[Out]

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

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Sympy [A]  time = 0.683258, size = 37, normalized size = 0.79 \begin{align*} \frac{i x^{4}}{4} - i x^{2} e^{2 i a} + i e^{4 i a} \log{\left (x^{2} + e^{2 i a} \right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**3*tan(a+I*ln(x)),x)

[Out]

I*x**4/4 - I*x**2*exp(2*I*a) + I*exp(4*I*a)*log(x**2 + exp(2*I*a))

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Giac [A]  time = 1.1721, size = 46, normalized size = 0.98 \begin{align*} \frac{1}{4} i \, x^{4} - i \, x^{2} e^{\left (2 i \, a\right )} + i \, e^{\left (4 i \, a\right )} \log \left (i \, x^{2} + i \, e^{\left (2 i \, a\right )}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*tan(a+I*log(x)),x, algorithm="giac")

[Out]

1/4*I*x^4 - I*x^2*e^(2*I*a) + I*e^(4*I*a)*log(I*x^2 + I*e^(2*I*a))