3.2.51 \(\int x^3 \coth (a+2 \log (x)) \, dx\) [151]

Optimal. Leaf size=30 \[ \frac {x^4}{4}+\frac {1}{2} e^{-2 a} \log \left (1-e^{2 a} x^4\right ) \]

[Out]

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

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Rubi [A]
time = 0.03, antiderivative size = 30, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {5657, 455, 45} \begin {gather*} \frac {1}{2} 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]],x]

[Out]

x^4/4 + Log[1 - E^(2*a)*x^4]/(2*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 (a+2 \log (x)) \, dx &=\int x^3 \coth (a+2 \log (x)) \, dx\\ \end {align*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(64\) vs. \(2(30)=60\).
time = 0.02, size = 64, normalized size = 2.13 \begin {gather*} \frac {x^4}{4}+\frac {1}{2} \cosh (2 a) \log \left (-\cosh (a)+x^4 \cosh (a)+\sinh (a)+x^4 \sinh (a)\right )-\frac {1}{2} \log \left (-\cosh (a)+x^4 \cosh (a)+\sinh (a)+x^4 \sinh (a)\right ) \sinh (2 a) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^3*Coth[a + 2*Log[x]],x]

[Out]

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

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Maple [A]
time = 0.44, size = 24, normalized size = 0.80

method	result	size
risch	\(\frac {x^{4}}{4}+\frac {{\mathrm e}^{-2 a} \ln \left (-1+{\mathrm e}^{2 a} x^{4}\right )}{2}\)	\(24\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.27, size = 36, normalized size = 1.20 \begin {gather*} \frac {1}{4} \, x^{4} + \frac {1}{2} \, e^{\left (-2 \, a\right )} \log \left (x^{2} e^{a} + 1\right ) + \frac {1}{2} \, e^{\left (-2 \, a\right )} \log \left (x^{2} e^{a} - 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]
time = 0.37, size = 28, normalized size = 0.93 \begin {gather*} \frac {1}{4} \, {\left (x^{4} e^{\left (2 \, a\right )} + 2 \, \log \left (x^{4} e^{\left (2 \, a\right )} - 1\right )\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)),x, algorithm="fricas")

[Out]

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

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int x^{3} \coth {\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)),x)

[Out]

Integral(x**3*coth(a + 2*log(x)), x)

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Giac [A]
time = 0.41, size = 24, normalized size = 0.80 \begin {gather*} \frac {1}{4} \, x^{4} + \frac {1}{2} \, 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)),x, algorithm="giac")

[Out]

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

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Mupad [B]
time = 1.25, size = 23, normalized size = 0.77 \begin {gather*} \frac {\ln \left (x^4-{\mathrm {e}}^{-2\,a}\right )\,{\mathrm {e}}^{-2\,a}}{2}+\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)),x)

[Out]

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

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