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]

x^3/3 + x^3/(1 - E^(2*a)*x^4) + (3*ArcTan[E^(a/2)*x])/(2*E^((3*a)/2)) - (3*ArcTanh[E^(a/2)*x])/(2*E^((3*a)/2))

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Rubi [F]  time = 0.0477565, 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 = {} \[ \int x^2 \coth ^2(a+2 \log (x)) \, dx \]

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 = 2.8989, size = 154, normalized size = 2.26 \[ \frac{16 e^{2 a} x^7 \left (e^{2 a} x^4+1\right )^2 \text{HypergeometricPFQ}\left (\left \{\frac{7}{4},2,2,2\right \},\left \{1,1,\frac{19}{4}\right \},e^{2 a} x^4\right )}{1155}+\frac{e^{-4 a} \left (7 \left (27 e^{8 a} x^{16}-632 e^{6 a} x^{12}-398 e^{4 a} x^8+1976 e^{2 a} x^4+1331\right ) \, _2F_1\left (\frac{3}{4},1;\frac{7}{4};e^{2 a} x^4\right )+1481 e^{6 a} x^{12}-4787 e^{4 a} x^8-17825 e^{2 a} x^4-9317\right )}{2688 x^5} \]

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

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Maple [A]  time = 0.046, size = 100, normalized size = 1.5 \begin{align*}{\frac{{x}^{3}}{3}}-{\frac{{x}^{3}}{{{\rm e}^{2\,a}}{x}^{4}-1}}+{\frac{3}{4}\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{3}{4}\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{3}{4}\ln \left ( -x\sqrt{{{\rm e}^{a}}}+1 \right ) \left ({{\rm e}^{a}} \right ) ^{-{\frac{3}{2}}}}-{\frac{3}{4}\ln \left ( x\sqrt{{{\rm e}^{a}}}+1 \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))^2,x)

[Out]

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

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Maxima [A]  time = 1.62442, size = 89, normalized size = 1.31 \begin{align*} \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 ) \end{align*}

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)))

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Fricas [B]  time = 2.59568, size = 266, normalized size = 3.91 \begin{align*} \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 )}} \end{align*}

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))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{2} \coth ^{2}{\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))**2,x)

[Out]

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

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Giac [A]  time = 1.14405, size = 97, normalized size = 1.43 \begin{align*} \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 ) \end{align*}

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)))