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

Optimal. Leaf size=68 \[ \frac {x^3}{1-e^{2 a} x^4}+\frac {3}{2} e^{-3 a/2} \tan ^{-1}\left (e^{a/2} x\right )-\frac {3}{2} e^{-3 a/2} \tanh ^{-1}\left (e^{a/2} x\right )+\frac {x^3}{3} \]

[Out]

1/3*x^3+x^3/(1-exp(2*a)*x^4)+3/2*arctan(exp(1/2*a)*x)/exp(3/2*a)-3/2*arctanh(exp(1/2*a)*x)/exp(3/2*a)

________________________________________________________________________________________

Rubi [F]  time = 0.05, 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 = {} \[ \int x^2 \coth ^2(a+2 \log (x)) \, dx \]

Verification is Not applicable to the result.

[In]

Int[x^2*Coth[a + 2*Log[x]]^2,x]

[Out]

Defer[Int][x^2*Coth[a + 2*Log[x]]^2, x]

Rubi steps

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

________________________________________________________________________________________

Mathematica [C]  time = 2.95, size = 154, normalized size = 2.26 \[ \frac {16 e^{2 a} x^7 \left (e^{2 a} x^4+1\right )^2 \, _4F_3\left (\frac {7}{4},2,2,2;1,1,\frac {19}{4};e^{2 a} x^4\right )}{1155}+\frac {e^{-4 a} \left (7 \left (27 e^{8 a} x^{16}-632 e^{6 a} x^{12}-398 e^{4 a} x^8+1976 e^{2 a} x^4+1331\right ) \, _2F_1\left (\frac {3}{4},1;\frac {7}{4};e^{2 a} x^4\right )+1481 e^{6 a} x^{12}-4787 e^{4 a} x^8-17825 e^{2 a} x^4-9317\right )}{2688 x^5} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[x^2*Coth[a + 2*Log[x]]^2,x]

[Out]

(-9317 - 17825*E^(2*a)*x^4 - 4787*E^(4*a)*x^8 + 1481*E^(6*a)*x^12 + 7*(1331 + 1976*E^(2*a)*x^4 - 398*E^(4*a)*x
^8 - 632*E^(6*a)*x^12 + 27*E^(8*a)*x^16)*Hypergeometric2F1[3/4, 1, 7/4, E^(2*a)*x^4])/(2688*E^(4*a)*x^5) + (16
*E^(2*a)*x^7*(1 + E^(2*a)*x^4)^2*HypergeometricPFQ[{7/4, 2, 2, 2}, {1, 1, 19/4}, E^(2*a)*x^4])/1155

________________________________________________________________________________________

fricas [B]  time = 0.41, size = 104, normalized size = 1.53 \[ \frac {4 \, x^{7} e^{\left (4 \, a\right )} - 16 \, x^{3} e^{\left (2 \, a\right )} + 18 \, {\left (x^{4} e^{\left (2 \, a\right )} - 1\right )} \arctan \left (x e^{\left (\frac {1}{2} \, a\right )}\right ) e^{\left (\frac {1}{2} \, a\right )} + 9 \, {\left (x^{4} e^{\left (2 \, a\right )} - 1\right )} e^{\left (\frac {1}{2} \, a\right )} \log \left (\frac {x^{2} e^{a} - 2 \, x e^{\left (\frac {1}{2} \, a\right )} + 1}{x^{2} e^{a} - 1}\right )}{12 \, {\left (x^{4} e^{\left (4 \, a\right )} - e^{\left (2 \, a\right )}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*(4*x^7*e^(4*a) - 16*x^3*e^(2*a) + 18*(x^4*e^(2*a) - 1)*arctan(x*e^(1/2*a))*e^(1/2*a) + 9*(x^4*e^(2*a) - 1
)*e^(1/2*a)*log((x^2*e^a - 2*x*e^(1/2*a) + 1)/(x^2*e^a - 1)))/(x^4*e^(4*a) - e^(2*a))

________________________________________________________________________________________

giac [A]  time = 0.11, size = 72, normalized size = 1.06 \[ \frac {1}{3} \, x^{3} - \frac {x^{3}}{x^{4} e^{\left (2 \, a\right )} - 1} + \frac {3}{2} \, \arctan \left (x e^{\left (\frac {1}{2} \, a\right )}\right ) e^{\left (-\frac {3}{2} \, a\right )} + \frac {3}{4} \, e^{\left (-\frac {3}{2} \, a\right )} \log \left (\frac {{\left | 2 \, x e^{a} - 2 \, e^{\left (\frac {1}{2} \, a\right )} \right |}}{{\left | 2 \, x e^{a} + 2 \, e^{\left (\frac {1}{2} \, a\right )} \right |}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*x^3 - x^3/(x^4*e^(2*a) - 1) + 3/2*arctan(x*e^(1/2*a))*e^(-3/2*a) + 3/4*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)))

________________________________________________________________________________________

maple [A]  time = 0.12, size = 100, normalized size = 1.47 \[ \frac {x^{3}}{3}-\frac {x^{3}}{-1+{\mathrm e}^{2 a} x^{4}}+\frac {3 \ln \left (-{\mathrm e}^{2 a} x +\left (-{\mathrm e}^{a}\right )^{\frac {3}{2}}\right )}{4 \left (-{\mathrm e}^{a}\right )^{\frac {3}{2}}}-\frac {3 \ln \left ({\mathrm e}^{2 a} x +\left (-{\mathrm e}^{a}\right )^{\frac {3}{2}}\right )}{4 \left (-{\mathrm e}^{a}\right )^{\frac {3}{2}}}+\frac {3 \ln \left (-\sqrt {{\mathrm e}^{a}}\, x +1\right )}{4 \left ({\mathrm e}^{a}\right )^{\frac {3}{2}}}-\frac {3 \ln \left (\sqrt {{\mathrm e}^{a}}\, x +1\right )}{4 \left ({\mathrm e}^{a}\right )^{\frac {3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maxima [A]  time = 0.42, size = 66, normalized size = 0.97 \[ \frac {1}{3} \, x^{3} - \frac {x^{3}}{x^{4} e^{\left (2 \, a\right )} - 1} + \frac {3}{2} \, \arctan \left (x e^{\left (\frac {1}{2} \, a\right )}\right ) e^{\left (-\frac {3}{2} \, a\right )} + \frac {3}{4} \, e^{\left (-\frac {3}{2} \, a\right )} \log \left (\frac {x e^{a} - e^{\left (\frac {1}{2} \, a\right )}}{x e^{a} + e^{\left (\frac {1}{2} \, a\right )}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*x^3 - x^3/(x^4*e^(2*a) - 1) + 3/2*arctan(x*e^(1/2*a))*e^(-3/2*a) + 3/4*e^(-3/2*a)*log((x*e^a - e^(1/2*a))/
(x*e^a + e^(1/2*a)))

________________________________________________________________________________________

mupad [B]  time = 1.24, size = 60, normalized size = 0.88 \[ \frac {3\,\mathrm {atan}\left (x\,{\left ({\mathrm {e}}^{2\,a}\right )}^{1/4}\right )}{2\,{\left ({\mathrm {e}}^{2\,a}\right )}^{3/4}}-\frac {x^3}{x^4\,{\mathrm {e}}^{2\,a}-1}+\frac {x^3}{3}+\frac {\mathrm {atan}\left (x\,{\left ({\mathrm {e}}^{2\,a}\right )}^{1/4}\,1{}\mathrm {i}\right )\,3{}\mathrm {i}}{2\,{\left ({\mathrm {e}}^{2\,a}\right )}^{3/4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*coth(a + 2*log(x))^2,x)

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(x**2*coth(a + 2*log(x))**2, x)

________________________________________________________________________________________