3.160 \(\int (e x)^m \tanh (a+2 \log (x)) \, dx\)

Optimal. Leaf size=60 \[ \frac{(e x)^{m+1}}{e (m+1)}-\frac{2 (e x)^{m+1} \, _2F_1\left (1,\frac{m+1}{4};\frac{m+5}{4};-e^{2 a} x^4\right )}{e (m+1)} \]

[Out]

(e*x)^(1 + m)/(e*(1 + m)) - (2*(e*x)^(1 + m)*Hypergeometric2F1[1, (1 + m)/4, (5 + m)/4, -(E^(2*a)*x^4)])/(e*(1
+ m))

________________________________________________________________________________________

Rubi [F]  time = 0.0391039, 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 = {} \[ \int (e x)^m \tanh (a+2 \log (x)) \, dx \]

Verification is Not applicable to the result.

[In]

Int[(e*x)^m*Tanh[a + 2*Log[x]],x]

[Out]

Defer[Int][(e*x)^m*Tanh[a + 2*Log[x]], x]

Rubi steps

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

Mathematica [A]  time = 0.0895244, size = 47, normalized size = 0.78 \[ -\frac{x (e x)^m \left (2 \, _2F_1\left (1,\frac{m+1}{4};\frac{m+5}{4};-x^4 (\cosh (2 a)+\sinh (2 a))\right )-1\right )}{m+1} \]

Antiderivative was successfully verified.

[In]

Integrate[(e*x)^m*Tanh[a + 2*Log[x]],x]

[Out]

-((x*(e*x)^m*(-1 + 2*Hypergeometric2F1[1, (1 + m)/4, (5 + m)/4, -(x^4*(Cosh[2*a] + Sinh[2*a]))]))/(1 + m))

________________________________________________________________________________________

Maple [F]  time = 0.036, size = 0, normalized size = 0. \begin{align*} \int \left ( ex \right ) ^{m}\tanh \left ( a+2\,\ln \left ( x \right ) \right ) \, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x)^m*tanh(a+2*ln(x)),x)

[Out]

int((e*x)^m*tanh(a+2*ln(x)),x)

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (e x\right )^{m} \tanh \left (a + 2 \, \log \left (x\right )\right )\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x)^m*tanh(a+2*log(x)),x, algorithm="maxima")

[Out]

integrate((e*x)^m*tanh(a + 2*log(x)), x)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x)^m*tanh(a+2*log(x)),x, algorithm="fricas")

[Out]

integral((e*x)^m*tanh(a + 2*log(x)), x)

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (e x\right )^{m} \tanh{\left (a + 2 \log{\left (x \right )} \right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x)**m*tanh(a+2*ln(x)),x)

[Out]

Integral((e*x)**m*tanh(a + 2*log(x)), x)

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (e x\right )^{m} \tanh \left (a + 2 \, \log \left (x\right )\right )\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x)^m*tanh(a+2*log(x)),x, algorithm="giac")

[Out]

integrate((e*x)^m*tanh(a + 2*log(x)), x)