3.152 \(\int x^2 \coth (a+2 \log (x)) \, dx\)

Optimal. Leaf size=45 \[ e^{-3 a/2} \tan ^{-1}\left (e^{a/2} x\right )-e^{-3 a/2} \tanh ^{-1}\left (e^{a/2} x\right )+\frac{x^3}{3} \]

[Out]

x^3/3 + ArcTan[E^(a/2)*x]/E^((3*a)/2) - ArcTanh[E^(a/2)*x]/E^((3*a)/2)

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Rubi [F]  time = 0.0213207, 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 = {} \[ \int x^2 \coth (a+2 \log (x)) \, dx \]

Verification is Not applicable to the result.

[In]

Int[x^2*Coth[a + 2*Log[x]],x]

[Out]

Defer[Int][x^2*Coth[a + 2*Log[x]], x]

Rubi steps

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

Mathematica [C]  time = 0.23153, size = 64, normalized size = 1.42 \[ \frac{1}{6} \left (3 (\sinh (2 a)-\cosh (2 a)) \text{RootSum}\left [\text{$\#$1}^4 \sinh (a)+\text{$\#$1}^4 \cosh (a)+\sinh (a)-\cosh (a)\& ,\frac{\log (x)-\log (x-\text{$\#$1})}{\text{$\#$1}}\& \right ]+2 x^3\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[x^2*Coth[a + 2*Log[x]],x]

[Out]

(2*x^3 + 3*RootSum[-Cosh[a] + Sinh[a] + Cosh[a]*#1^4 + Sinh[a]*#1^4 & , (Log[x] - Log[x - #1])/#1 & ]*(-Cosh[2
*a] + Sinh[2*a]))/6

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Maple [B]  time = 0.048, size = 83, normalized size = 1.8 \begin{align*}{\frac{{x}^{3}}{3}}+{\frac{1}{2}\ln \left ( -x\sqrt{{{\rm e}^{a}}}+1 \right ) \left ({{\rm e}^{a}} \right ) ^{-{\frac{3}{2}}}}-{\frac{1}{2}\ln \left ( x\sqrt{{{\rm e}^{a}}}+1 \right ) \left ({{\rm e}^{a}} \right ) ^{-{\frac{3}{2}}}}+{\frac{1}{2}\ln \left ( -{{\rm e}^{2\,a}}x+ \left ( -{{\rm e}^{a}} \right ) ^{{\frac{3}{2}}} \right ) \left ( -{{\rm e}^{a}} \right ) ^{-{\frac{3}{2}}}}-{\frac{1}{2}\ln \left ({{\rm e}^{2\,a}}x+ \left ( -{{\rm e}^{a}} \right ) ^{{\frac{3}{2}}} \right ) \left ( -{{\rm e}^{a}} \right ) ^{-{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*x^3+1/2/exp(a)^(3/2)*ln(-x*exp(a)^(1/2)+1)-1/2/exp(a)^(3/2)*ln(x*exp(a)^(1/2)+1)+1/2/(-exp(a))^(3/2)*ln(-e
xp(2*a)*x+(-exp(a))^(3/2))-1/2/(-exp(a))^(3/2)*ln(exp(2*a)*x+(-exp(a))^(3/2))

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Maxima [A]  time = 1.75688, size = 65, normalized size = 1.44 \begin{align*} \frac{1}{3} \, x^{3} + \arctan \left (x e^{\left (\frac{1}{2} \, a\right )}\right ) e^{\left (-\frac{3}{2} \, a\right )} + \frac{1}{2} \, e^{\left (-\frac{3}{2} \, a\right )} \log \left (\frac{x e^{a} - e^{\left (\frac{1}{2} \, a\right )}}{x e^{a} + e^{\left (\frac{1}{2} \, a\right )}}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 2.68469, size = 171, normalized size = 3.8 \begin{align*} \frac{1}{6} \,{\left (2 \, x^{3} e^{\left (2 \, a\right )} + 6 \, \arctan \left (x e^{\left (\frac{1}{2} \, a\right )}\right ) e^{\left (\frac{1}{2} \, a\right )} + 3 \, e^{\left (\frac{1}{2} \, a\right )} \log \left (\frac{x^{2} e^{a} - 2 \, x e^{\left (\frac{1}{2} \, a\right )} + 1}{x^{2} e^{a} - 1}\right )\right )} e^{\left (-2 \, a\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*(2*x^3*e^(2*a) + 6*arctan(x*e^(1/2*a))*e^(1/2*a) + 3*e^(1/2*a)*log((x^2*e^a - 2*x*e^(1/2*a) + 1)/(x^2*e^a
- 1)))*e^(-2*a)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**2*coth(a+2*ln(x)),x)

[Out]

Integral(x**2*coth(a + 2*log(x)), x)

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Giac [A]  time = 1.14067, size = 73, normalized size = 1.62 \begin{align*} \frac{1}{3} \, x^{3} + \arctan \left (x e^{\left (\frac{1}{2} \, a\right )}\right ) e^{\left (-\frac{3}{2} \, a\right )} + \frac{1}{2} \, e^{\left (-\frac{3}{2} \, a\right )} \log \left (\frac{{\left | 2 \, x e^{a} - 2 \, e^{\left (\frac{1}{2} \, a\right )} \right |}}{{\left | 2 \, x e^{a} + 2 \, e^{\left (\frac{1}{2} \, a\right )} \right |}}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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