3.167 \(\int (e x)^m \coth ^3(a+2 \log (x)) \, dx\)

Optimal. Leaf size=177 \[ -\frac{\left (m^2+2 m+9\right ) (e x)^{m+1} \, _2F_1\left (1,\frac{m+1}{4};\frac{m+5}{4};e^{2 a} x^4\right )}{4 e (m+1)}-\frac{\left (e^{2 a} x^4+1\right )^2 (e x)^{m+1}}{4 e \left (1-e^{2 a} x^4\right )^2}-\frac{e^{-2 a} \left (e^{2 a} (3-m)-e^{4 a} (m+5) x^4\right ) (e x)^{m+1}}{8 e \left (1-e^{2 a} x^4\right )}+\frac{(m+3) (m+5) (e x)^{m+1}}{8 e (m+1)} \]

[Out]

((3 + m)*(5 + m)*(e*x)^(1 + m))/(8*e*(1 + m)) - ((e*x)^(1 + m)*(1 + E^(2*a)*x^4)^2)/(4*e*(1 - E^(2*a)*x^4)^2)
- ((e*x)^(1 + m)*(E^(2*a)*(3 - m) - E^(4*a)*(5 + m)*x^4))/(8*e*E^(2*a)*(1 - E^(2*a)*x^4)) - ((9 + 2*m + m^2)*(
e*x)^(1 + m)*Hypergeometric2F1[1, (1 + m)/4, (5 + m)/4, E^(2*a)*x^4])/(4*e*(1 + m))

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Rubi [F]  time = 0.0728497, 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 = {} \[ \int (e x)^m \coth ^3(a+2 \log (x)) \, dx \]

Verification is Not applicable to the result.

[In]

Int[(e*x)^m*Coth[a + 2*Log[x]]^3,x]

[Out]

Defer[Int][(e*x)^m*Coth[a + 2*Log[x]]^3, x]

Rubi steps

\begin{align*} \int (e x)^m \coth ^3(a+2 \log (x)) \, dx &=\int (e x)^m \coth ^3(a+2 \log (x)) \, dx\\ \end{align*}

Mathematica [A]  time = 0.825247, size = 215, normalized size = 1.21 \[ \frac{x (e x)^m \left (-\frac{x^8 (\sinh (2 a)+\cosh (2 a)) \left ((m+9) x^4 (\sinh (a)+\cosh (a)) \, _2F_1\left (3,\frac{m+13}{4};\frac{m+17}{4};x^4 (\cosh (2 a)+\sinh (2 a))\right )+3 (m+13) (\cosh (a)-\sinh (a)) \, _2F_1\left (3,\frac{m+9}{4};\frac{m+13}{4};x^4 (\cosh (2 a)+\sinh (2 a))\right )\right )}{(m+9) (m+13)}-\frac{3 x^4 (\cosh (a)-\sinh (a)) \, _2F_1\left (3,\frac{m+5}{4};\frac{m+9}{4};x^4 (\cosh (2 a)+\sinh (2 a))\right )}{m+5}-\frac{(\cosh (a)-\sinh (a))^3 \, _2F_1\left (3,\frac{m+1}{4};\frac{m+5}{4};x^4 (\cosh (2 a)+\sinh (2 a))\right )}{m+1}\right )}{(\cosh (a)-\sinh (a))^3} \]

Antiderivative was successfully verified.

[In]

Integrate[(e*x)^m*Coth[a + 2*Log[x]]^3,x]

[Out]

(x*(e*x)^m*((-3*x^4*Hypergeometric2F1[3, (5 + m)/4, (9 + m)/4, x^4*(Cosh[2*a] + Sinh[2*a])]*(Cosh[a] - Sinh[a]
))/(5 + m) - (Hypergeometric2F1[3, (1 + m)/4, (5 + m)/4, x^4*(Cosh[2*a] + Sinh[2*a])]*(Cosh[a] - Sinh[a])^3)/(
1 + m) - (x^8*(3*(13 + m)*Hypergeometric2F1[3, (9 + m)/4, (13 + m)/4, x^4*(Cosh[2*a] + Sinh[2*a])]*(Cosh[a] -
Sinh[a]) + (9 + m)*x^4*Hypergeometric2F1[3, (13 + m)/4, (17 + m)/4, x^4*(Cosh[2*a] + Sinh[2*a])]*(Cosh[a] + Si
nh[a]))*(Cosh[2*a] + Sinh[2*a]))/((9 + m)*(13 + m))))/(Cosh[a] - Sinh[a])^3

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Maple [F]  time = 0.037, size = 0, normalized size = 0. \begin{align*} \int \left ( ex \right ) ^{m} \left ({\rm coth} \left (a+2\,\ln \left ( x \right ) \right ) \right ) ^{3}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x)^m*coth(a+2*ln(x))^3,x)

[Out]

int((e*x)^m*coth(a+2*ln(x))^3,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (e x\right )^{m} \coth \left (a + 2 \, \log \left (x\right )\right )^{3}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x)^m*coth(a+2*log(x))^3,x, algorithm="maxima")

[Out]

integrate((e*x)^m*coth(a + 2*log(x))^3, x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\left (e x\right )^{m} \coth \left (a + 2 \, \log \left (x\right )\right )^{3}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x)^m*coth(a+2*log(x))^3,x, algorithm="fricas")

[Out]

integral((e*x)^m*coth(a + 2*log(x))^3, x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x)**m*coth(a+2*ln(x))**3,x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (e x\right )^{m} \coth \left (a + 2 \, \log \left (x\right )\right )^{3}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x)^m*coth(a+2*log(x))^3,x, algorithm="giac")

[Out]

integrate((e*x)^m*coth(a + 2*log(x))^3, x)