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]

1/3*x^3+arctan(exp(1/2*a)*x)/exp(3/2*a)-arctanh(exp(1/2*a)*x)/exp(3/2*a)

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Rubi [F]  time = 0.02, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, 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*}

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Mathematica [C]  time = 0.24, 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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fricas [A]  time = 0.40, size = 62, normalized size = 1.38 \[ \frac {1}{6} \, {\left (2 \, x^{3} e^{\left (2 \, a\right )} + 6 \, \arctan \left (x e^{\left (\frac {1}{2} \, a\right )}\right ) e^{\left (\frac {1}{2} \, a\right )} + 3 \, e^{\left (\frac {1}{2} \, a\right )} \log \left (\frac {x^{2} e^{a} - 2 \, x e^{\left (\frac {1}{2} \, a\right )} + 1}{x^{2} e^{a} - 1}\right )\right )} e^{\left (-2 \, a\right )} \]

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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giac [A]  time = 0.14, size = 54, normalized size = 1.20 \[ \frac {1}{3} \, x^{3} + \arctan \left (x e^{\left (\frac {1}{2} \, a\right )}\right ) e^{\left (-\frac {3}{2} \, a\right )} + \frac {1}{2} \, e^{\left (-\frac {3}{2} \, a\right )} \log \left (\frac {{\left | 2 \, x e^{a} - 2 \, e^{\left (\frac {1}{2} \, a\right )} \right |}}{{\left | 2 \, x e^{a} + 2 \, e^{\left (\frac {1}{2} \, a\right )} \right |}}\right ) \]

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)))

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maple [B]  time = 0.15, size = 83, normalized size = 1.84 \[ \frac {x^{3}}{3}+\frac {\ln \left (-{\mathrm e}^{2 a} x +\left (-{\mathrm e}^{a}\right )^{\frac {3}{2}}\right )}{2 \left (-{\mathrm e}^{a}\right )^{\frac {3}{2}}}-\frac {\ln \left ({\mathrm e}^{2 a} x +\left (-{\mathrm e}^{a}\right )^{\frac {3}{2}}\right )}{2 \left (-{\mathrm e}^{a}\right )^{\frac {3}{2}}}+\frac {\ln \left (-\sqrt {{\mathrm e}^{a}}\, x +1\right )}{2 \left ({\mathrm e}^{a}\right )^{\frac {3}{2}}}-\frac {\ln \left (\sqrt {{\mathrm e}^{a}}\, x +1\right )}{2 \left ({\mathrm e}^{a}\right )^{\frac {3}{2}}} \]

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

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maxima [A]  time = 1.07, size = 48, normalized size = 1.07 \[ \frac {1}{3} \, x^{3} + \arctan \left (x e^{\left (\frac {1}{2} \, a\right )}\right ) e^{\left (-\frac {3}{2} \, a\right )} + \frac {1}{2} \, e^{\left (-\frac {3}{2} \, a\right )} \log \left (\frac {x e^{a} - e^{\left (\frac {1}{2} \, a\right )}}{x e^{a} + e^{\left (\frac {1}{2} \, a\right )}}\right ) \]

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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mupad [B]  time = 1.21, size = 39, normalized size = 0.87 \[ \frac {\mathrm {atan}\left (x\,{\left ({\mathrm {e}}^{2\,a}\right )}^{1/4}\right )}{{\left ({\mathrm {e}}^{2\,a}\right )}^{3/4}}-\frac {\mathrm {atanh}\left (x\,{\left ({\mathrm {e}}^{2\,a}\right )}^{1/4}\right )}{{\left ({\mathrm {e}}^{2\,a}\right )}^{3/4}}+\frac {x^3}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*coth(a + 2*log(x)),x)

[Out]

atan(x*exp(2*a)^(1/4))/exp(2*a)^(3/4) - atanh(x*exp(2*a)^(1/4))/exp(2*a)^(3/4) + x^3/3

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{2} \coth {\left (a + 2 \log {\relax (x )} \right )}\, dx \]

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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