3.2.57 \(\int \frac {\coth (a+2 \log (x))}{x^3} \, dx\) [157]

Optimal. Leaf size=21 \[ \frac {1}{2 x^2}-e^a \tanh ^{-1}\left (e^a x^2\right ) \]

[Out]

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

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Rubi [A]
time = 0.02, antiderivative size = 21, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {5657, 464, 281, 212} \begin {gather*} \frac {1}{2 x^2}-e^a \tanh ^{-1}\left (e^a x^2\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Coth[a + 2*Log[x]]/x^3,x]

[Out]

1/(2*x^2) - E^a*ArcTanh[E^a*x^2]

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 281

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = GCD[m + 1, n]}, Dist[1/k, Subst[Int[x^((m
+ 1)/k - 1)*(a + b*x^(n/k))^p, x], x, x^k], x] /; k != 1] /; FreeQ[{a, b, p}, x] && IGtQ[n, 0] && IntegerQ[m]

Rule 464

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Simp[c*(e*x)^(m +
1)*((a + b*x^n)^(p + 1)/(a*e*(m + 1))), x] + Dist[(a*d*(m + 1) - b*c*(m + n*(p + 1) + 1))/(a*e^n*(m + 1)), In
t[(e*x)^(m + n)*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[b*c - a*d, 0] && (IntegerQ[n] ||
GtQ[e, 0]) && ((GtQ[n, 0] && LtQ[m, -1]) || (LtQ[n, 0] && GtQ[m + n, -1])) &&  !ILtQ[p, -1]

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

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Mathematica [A]
time = 0.11, size = 27, normalized size = 1.29 \begin {gather*} \frac {1}{2 x^2}-\tanh ^{-1}\left (\frac {\cosh (a)-\sinh (a)}{x^2}\right ) (\cosh (a)+\sinh (a)) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Coth[a + 2*Log[x]]/x^3,x]

[Out]

1/(2*x^2) - ArcTanh[(Cosh[a] - Sinh[a])/x^2]*(Cosh[a] + Sinh[a])

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Maple [A]
time = 0.51, size = 35, normalized size = 1.67

method	result	size
risch	\(\frac {1}{2 x^{2}}+\frac {{\mathrm e}^{a} \ln \left (-{\mathrm e}^{a} x^{2}+1\right )}{2}-\frac {{\mathrm e}^{a} \ln \left (-{\mathrm e}^{a} x^{2}-1\right )}{2}\)	\(35\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.26, size = 30, normalized size = 1.43 \begin {gather*} -\frac {1}{2} \, e^{a} \log \left (\frac {1}{x^{2}} + e^{a}\right ) + \frac {1}{2} \, e^{a} \log \left (\frac {1}{x^{2}} - e^{a}\right ) + \frac {1}{2 \, x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 38 vs. \(2 (17) = 34\).
time = 0.34, size = 38, normalized size = 1.81 \begin {gather*} -\frac {x^{2} e^{a} \log \left (x^{2} e^{a} + 1\right ) - x^{2} e^{a} \log \left (x^{2} e^{a} - 1\right ) - 1}{2 \, x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(coth(a + 2*log(x))/x**3, x)

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Giac [A]
time = 0.41, size = 33, normalized size = 1.57 \begin {gather*} -\frac {1}{2} \, e^{a} \log \left (x^{2} e^{a} + 1\right ) + \frac {1}{2} \, e^{a} \log \left ({\left | x^{2} e^{a} - 1 \right |}\right ) + \frac {1}{2 \, x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [B]
time = 1.21, size = 25, normalized size = 1.19 \begin {gather*} \frac {1}{2\,x^2}-\mathrm {atanh}\left (x^2\,\sqrt {{\mathrm {e}}^{2\,a}}\right )\,\sqrt {{\mathrm {e}}^{2\,a}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(coth(a + 2*log(x))/x^3,x)

[Out]

1/(2*x^2) - atanh(x^2*exp(2*a)^(1/2))*exp(2*a)^(1/2)

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