3.150 \(\int (e x)^m \tan (a+i \log (x)) \, dx\)

Optimal. Leaf size=71 \[ \frac{2 i (e x)^{m+1} \text{Hypergeometric2F1}\left (1,\frac{1}{2} (-m-1),\frac{1-m}{2},-\frac{e^{2 i a}}{x^2}\right )}{e (m+1)}-\frac{i (e x)^{m+1}}{e (m+1)} \]

[Out]

((-I)*(e*x)^(1 + m))/(e*(1 + m)) + ((2*I)*(e*x)^(1 + m)*Hypergeometric2F1[1, (-1 - m)/2, (1 - m)/2, -(E^((2*I)
*a)/x^2)])/(e*(1 + m))

________________________________________________________________________________________

Rubi [F]  time = 0.0436427, 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 = {} \[ \int (e x)^m \tan (a+i \log (x)) \, dx \]

Verification is Not applicable to the result.

[In]

Int[(e*x)^m*Tan[a + I*Log[x]],x]

[Out]

Defer[Int][(e*x)^m*Tan[a + I*Log[x]], x]

Rubi steps

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

Mathematica [A]  time = 0.184527, size = 124, normalized size = 1.75 \[ \frac{x (\cos (a)-i \sin (a)) (e x)^m \left ((m+1) x^2 (\sin (a)+i \cos (a)) \text{Hypergeometric2F1}\left (1,\frac{m+3}{2},\frac{m+5}{2},-x^2 (\cos (2 a)-i \sin (2 a))\right )+(m+3) (\sin (a)-i \cos (a)) \text{Hypergeometric2F1}\left (1,\frac{m+1}{2},\frac{m+3}{2},-x^2 (\cos (2 a)-i \sin (2 a))\right )\right )}{(m+1) (m+3)} \]

Antiderivative was successfully verified.

[In]

Integrate[(e*x)^m*Tan[a + I*Log[x]],x]

[Out]

(x*(e*x)^m*(Cos[a] - I*Sin[a])*((3 + m)*Hypergeometric2F1[1, (1 + m)/2, (3 + m)/2, -(x^2*(Cos[2*a] - I*Sin[2*a
]))]*((-I)*Cos[a] + Sin[a]) + (1 + m)*x^2*Hypergeometric2F1[1, (3 + m)/2, (5 + m)/2, -(x^2*(Cos[2*a] - I*Sin[2
*a]))]*(I*Cos[a] + Sin[a])))/((1 + m)*(3 + m))

________________________________________________________________________________________

Maple [F]  time = 0.201, size = 0, normalized size = 0. \begin{align*} \int \left ( ex \right ) ^{m}\tan \left ( a+i\ln \left ( x \right ) \right ) \, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (e x\right )^{m} \tan \left (a + i \, \log \left (x\right )\right )\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((e*x)^m*tan(a + I*log(x)), x)

________________________________________________________________________________________

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

[Out]

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

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (e x\right )^{m} \tan{\left (a + i \log{\left (x \right )} \right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x)**m*tan(a+I*ln(x)),x)

[Out]

Integral((e*x)**m*tan(a + I*log(x)), x)

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (e x\right )^{m} \tan \left (a + i \, \log \left (x\right )\right )\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((e*x)^m*tan(a + I*log(x)), x)