3.2.58 \(\int x^3 \coth ^2(a+2 \log (x)) \, dx\) [158]

Optimal. Leaf size=47 \[ \frac {x^4}{4}+\frac {e^{-2 a}}{1-e^{2 a} x^4}+e^{-2 a} \log \left (1-e^{2 a} x^4\right ) \]

[Out]

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

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Rubi [A]
time = 0.05, antiderivative size = 47, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {5657, 455, 45} \begin {gather*} \frac {e^{-2 a}}{1-e^{2 a} x^4}+e^{-2 a} \log \left (1-e^{2 a} x^4\right )+\frac {x^4}{4} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^3*Coth[a + 2*Log[x]]^2,x]

[Out]

x^4/4 + 1/(E^(2*a)*(1 - E^(2*a)*x^4)) + Log[1 - E^(2*a)*x^4]/E^(2*a)

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 455

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[1/n, Subst[Int
[(a + b*x)^p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] && NeQ[b*c - a*d, 0] && EqQ[m
- n + 1, 0]

Rule 5657

Int[Coth[((a_.) + Log[x_]*(b_.))*(d_.)]^(p_.)*((e_.)*(x_))^(m_.), x_Symbol] :> Int[(e*x)^m*((-1 - E^(2*a*d)*x^
(2*b*d))^p/(1 - E^(2*a*d)*x^(2*b*d))^p), x] /; FreeQ[{a, b, d, e, m, p}, x]

Rubi steps

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

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Mathematica [A]
time = 0.08, size = 86, normalized size = 1.83 \begin {gather*} \frac {x^4}{4}+\cosh (2 a) \log \left (\left (-1+x^4\right ) \cosh (a)+\left (1+x^4\right ) \sinh (a)\right )-\log \left (\left (-1+x^4\right ) \cosh (a)+\left (1+x^4\right ) \sinh (a)\right ) \sinh (2 a)+\frac {-\cosh (3 a)+\sinh (3 a)}{\left (-1+x^4\right ) \cosh (a)+\left (1+x^4\right ) \sinh (a)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^3*Coth[a + 2*Log[x]]^2,x]

[Out]

x^4/4 + Cosh[2*a]*Log[(-1 + x^4)*Cosh[a] + (1 + x^4)*Sinh[a]] - Log[(-1 + x^4)*Cosh[a] + (1 + x^4)*Sinh[a]]*Si
nh[2*a] + (-Cosh[3*a] + Sinh[3*a])/((-1 + x^4)*Cosh[a] + (1 + x^4)*Sinh[a])

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Maple [A]
time = 0.42, size = 41, normalized size = 0.87

method	result	size
risch	\(\frac {x^{4}}{4}-\frac {{\mathrm e}^{-2 a}}{-1+{\mathrm e}^{2 a} x^{4}}+{\mathrm e}^{-2 a} \ln \left (-1+{\mathrm e}^{2 a} x^{4}\right )\)	\(41\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*coth(a+2*ln(x))^2,x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.27, size = 53, normalized size = 1.13 \begin {gather*} \frac {1}{4} \, x^{4} + e^{\left (-2 \, a\right )} \log \left (x^{2} e^{a} + 1\right ) + e^{\left (-2 \, a\right )} \log \left (x^{2} e^{a} - 1\right ) - \frac {1}{x^{4} e^{\left (4 \, a\right )} - e^{\left (2 \, a\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]
time = 0.36, size = 61, normalized size = 1.30 \begin {gather*} \frac {x^{8} e^{\left (4 \, a\right )} - x^{4} e^{\left (2 \, a\right )} + 4 \, {\left (x^{4} e^{\left (2 \, a\right )} - 1\right )} \log \left (x^{4} e^{\left (2 \, a\right )} - 1\right ) - 4}{4 \, {\left (x^{4} e^{\left (4 \, a\right )} - e^{\left (2 \, a\right )}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*(x^8*e^(4*a) - x^4*e^(2*a) + 4*(x^4*e^(2*a) - 1)*log(x^4*e^(2*a) - 1) - 4)/(x^4*e^(4*a) - e^(2*a))

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int x^{3} \coth ^{2}{\left (a + 2 \log {\left (x \right )} \right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(x**3*coth(a + 2*log(x))**2, x)

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Giac [A]
time = 0.41, size = 40, normalized size = 0.85 \begin {gather*} \frac {1}{4} \, x^{4} - \frac {x^{4}}{x^{4} e^{\left (2 \, a\right )} - 1} + e^{\left (-2 \, a\right )} \log \left ({\left | x^{4} e^{\left (2 \, a\right )} - 1 \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [B]
time = 1.25, size = 40, normalized size = 0.85 \begin {gather*} \ln \left (x^4-{\mathrm {e}}^{-2\,a}\right )\,{\mathrm {e}}^{-2\,a}-\frac {{\mathrm {e}}^{-2\,a}}{x^4\,{\mathrm {e}}^{2\,a}-1}+\frac {x^4}{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*coth(a + 2*log(x))^2,x)

[Out]

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

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