Integrand size = 8, antiderivative size = 54 \[ \int x \coth ^{-1}(a x)^2 \, dx=\frac {x \coth ^{-1}(a x)}{a}-\frac {\coth ^{-1}(a x)^2}{2 a^2}+\frac {1}{2} x^2 \coth ^{-1}(a x)^2+\frac {\log \left (1-a^2 x^2\right )}{2 a^2} \] Output:
x*arccoth(a*x)/a-1/2*arccoth(a*x)^2/a^2+1/2*x^2*arccoth(a*x)^2+1/2*ln(-a^2 *x^2+1)/a^2
Time = 0.01 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.80 \[ \int x \coth ^{-1}(a x)^2 \, dx=\frac {2 a x \coth ^{-1}(a x)+\left (-1+a^2 x^2\right ) \coth ^{-1}(a x)^2+\log \left (1-a^2 x^2\right )}{2 a^2} \] Input:
Integrate[x*ArcCoth[a*x]^2,x]
Output:
(2*a*x*ArcCoth[a*x] + (-1 + a^2*x^2)*ArcCoth[a*x]^2 + Log[1 - a^2*x^2])/(2 *a^2)
Time = 0.46 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.13, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.625, Rules used = {6453, 6543, 6437, 240, 6511}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int x \coth ^{-1}(a x)^2 \, dx\) |
\(\Big \downarrow \) 6453 |
\(\displaystyle \frac {1}{2} x^2 \coth ^{-1}(a x)^2-a \int \frac {x^2 \coth ^{-1}(a x)}{1-a^2 x^2}dx\) |
\(\Big \downarrow \) 6543 |
\(\displaystyle \frac {1}{2} x^2 \coth ^{-1}(a x)^2-a \left (\frac {\int \frac {\coth ^{-1}(a x)}{1-a^2 x^2}dx}{a^2}-\frac {\int \coth ^{-1}(a x)dx}{a^2}\right )\) |
\(\Big \downarrow \) 6437 |
\(\displaystyle \frac {1}{2} x^2 \coth ^{-1}(a x)^2-a \left (\frac {\int \frac {\coth ^{-1}(a x)}{1-a^2 x^2}dx}{a^2}-\frac {x \coth ^{-1}(a x)-a \int \frac {x}{1-a^2 x^2}dx}{a^2}\right )\) |
\(\Big \downarrow \) 240 |
\(\displaystyle \frac {1}{2} x^2 \coth ^{-1}(a x)^2-a \left (\frac {\int \frac {\coth ^{-1}(a x)}{1-a^2 x^2}dx}{a^2}-\frac {\frac {\log \left (1-a^2 x^2\right )}{2 a}+x \coth ^{-1}(a x)}{a^2}\right )\) |
\(\Big \downarrow \) 6511 |
\(\displaystyle \frac {1}{2} x^2 \coth ^{-1}(a x)^2-a \left (\frac {\coth ^{-1}(a x)^2}{2 a^3}-\frac {\frac {\log \left (1-a^2 x^2\right )}{2 a}+x \coth ^{-1}(a x)}{a^2}\right )\) |
Input:
Int[x*ArcCoth[a*x]^2,x]
Output:
(x^2*ArcCoth[a*x]^2)/2 - a*(ArcCoth[a*x]^2/(2*a^3) - (x*ArcCoth[a*x] + Log [1 - a^2*x^2]/(2*a))/a^2)
Int[(x_)/((a_) + (b_.)*(x_)^2), x_Symbol] :> Simp[Log[RemoveContent[a + b*x ^2, x]]/(2*b), x] /; FreeQ[{a, b}, x]
Int[((a_.) + ArcCoth[(c_.)*(x_)^(n_.)]*(b_.))^(p_.), x_Symbol] :> Simp[x*(a + b*ArcCoth[c*x^n])^p, x] - Simp[b*c*n*p Int[x^n*((a + b*ArcCoth[c*x^n]) ^(p - 1)/(1 - c^2*x^(2*n))), x], x] /; FreeQ[{a, b, c, n}, x] && IGtQ[p, 0] && (EqQ[n, 1] || EqQ[p, 1])
Int[((a_.) + ArcCoth[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*(x_)^(m_.), x_Symbol] : > Simp[x^(m + 1)*((a + b*ArcCoth[c*x^n])^p/(m + 1)), x] - Simp[b*c*n*(p/(m + 1)) Int[x^(m + n)*((a + b*ArcCoth[c*x^n])^(p - 1)/(1 - c^2*x^(2*n))), x ], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0] && (EqQ[p, 1] || (EqQ[n, 1 ] && IntegerQ[m])) && NeQ[m, -1]
Int[((a_.) + ArcCoth[(c_.)*(x_)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)^2), x_Symb ol] :> Simp[(a + b*ArcCoth[c*x])^(p + 1)/(b*c*d*(p + 1)), x] /; FreeQ[{a, b , c, d, e, p}, x] && EqQ[c^2*d + e, 0] && NeQ[p, -1]
Int[(((a_.) + ArcCoth[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_))/((d_) + ( e_.)*(x_)^2), x_Symbol] :> Simp[f^2/e Int[(f*x)^(m - 2)*(a + b*ArcCoth[c* x])^p, x], x] - Simp[d*(f^2/e) Int[(f*x)^(m - 2)*((a + b*ArcCoth[c*x])^p/ (d + e*x^2)), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && GtQ[p, 0] && GtQ[m, 1]
Time = 0.17 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.91
method | result | size |
parallelrisch | \(-\frac {-x^{2} a^{2} \operatorname {arccoth}\left (x a \right )^{2}-2 x a \,\operatorname {arccoth}\left (x a \right )+\operatorname {arccoth}\left (x a \right )^{2}-2 \ln \left (x a -1\right )-2 \,\operatorname {arccoth}\left (x a \right )}{2 a^{2}}\) | \(49\) |
risch | \(\frac {\left (a^{2} x^{2}-1\right ) \ln \left (x a +1\right )^{2}}{8 a^{2}}-\frac {\left (x^{2} \ln \left (x a -1\right ) a^{2}-2 x a -\ln \left (x a -1\right )\right ) \ln \left (x a +1\right )}{4 a^{2}}+\frac {x^{2} \ln \left (x a -1\right )^{2}}{8}-\frac {x \ln \left (x a -1\right )}{2 a}-\frac {\ln \left (x a -1\right )^{2}}{8 a^{2}}+\frac {\ln \left (a^{2} x^{2}-1\right )}{2 a^{2}}\) | \(114\) |
parts | \(\frac {x^{2} \operatorname {arccoth}\left (x a \right )^{2}}{2}+\frac {x a \,\operatorname {arccoth}\left (x a \right )+\frac {\operatorname {arccoth}\left (x a \right ) \ln \left (x a -1\right )}{2}-\frac {\operatorname {arccoth}\left (x a \right ) \ln \left (x a +1\right )}{2}+\frac {\ln \left (x a -1\right )^{2}}{8}-\frac {\ln \left (x a -1\right ) \ln \left (\frac {x a}{2}+\frac {1}{2}\right )}{4}+\frac {\ln \left (x a -1\right )}{2}+\frac {\ln \left (x a +1\right )}{2}-\frac {\left (\ln \left (x a +1\right )-\ln \left (\frac {x a}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {x a}{2}+\frac {1}{2}\right )}{4}+\frac {\ln \left (x a +1\right )^{2}}{8}}{a^{2}}\) | \(125\) |
derivativedivides | \(\frac {\frac {x^{2} a^{2} \operatorname {arccoth}\left (x a \right )^{2}}{2}+x a \,\operatorname {arccoth}\left (x a \right )+\frac {\operatorname {arccoth}\left (x a \right ) \ln \left (x a -1\right )}{2}-\frac {\operatorname {arccoth}\left (x a \right ) \ln \left (x a +1\right )}{2}+\frac {\ln \left (x a -1\right )^{2}}{8}-\frac {\ln \left (x a -1\right ) \ln \left (\frac {x a}{2}+\frac {1}{2}\right )}{4}+\frac {\ln \left (x a -1\right )}{2}+\frac {\ln \left (x a +1\right )}{2}-\frac {\left (\ln \left (x a +1\right )-\ln \left (\frac {x a}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {x a}{2}+\frac {1}{2}\right )}{4}+\frac {\ln \left (x a +1\right )^{2}}{8}}{a^{2}}\) | \(127\) |
default | \(\frac {\frac {x^{2} a^{2} \operatorname {arccoth}\left (x a \right )^{2}}{2}+x a \,\operatorname {arccoth}\left (x a \right )+\frac {\operatorname {arccoth}\left (x a \right ) \ln \left (x a -1\right )}{2}-\frac {\operatorname {arccoth}\left (x a \right ) \ln \left (x a +1\right )}{2}+\frac {\ln \left (x a -1\right )^{2}}{8}-\frac {\ln \left (x a -1\right ) \ln \left (\frac {x a}{2}+\frac {1}{2}\right )}{4}+\frac {\ln \left (x a -1\right )}{2}+\frac {\ln \left (x a +1\right )}{2}-\frac {\left (\ln \left (x a +1\right )-\ln \left (\frac {x a}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {x a}{2}+\frac {1}{2}\right )}{4}+\frac {\ln \left (x a +1\right )^{2}}{8}}{a^{2}}\) | \(127\) |
Input:
int(x*arccoth(x*a)^2,x,method=_RETURNVERBOSE)
Output:
-1/2*(-x^2*a^2*arccoth(x*a)^2-2*x*a*arccoth(x*a)+arccoth(x*a)^2-2*ln(a*x-1 )-2*arccoth(x*a))/a^2
Time = 0.12 (sec) , antiderivative size = 62, normalized size of antiderivative = 1.15 \[ \int x \coth ^{-1}(a x)^2 \, dx=\frac {4 \, a x \log \left (\frac {a x + 1}{a x - 1}\right ) + {\left (a^{2} x^{2} - 1\right )} \log \left (\frac {a x + 1}{a x - 1}\right )^{2} + 4 \, \log \left (a^{2} x^{2} - 1\right )}{8 \, a^{2}} \] Input:
integrate(x*arccoth(a*x)^2,x, algorithm="fricas")
Output:
1/8*(4*a*x*log((a*x + 1)/(a*x - 1)) + (a^2*x^2 - 1)*log((a*x + 1)/(a*x - 1 ))^2 + 4*log(a^2*x^2 - 1))/a^2
Time = 0.24 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.11 \[ \int x \coth ^{-1}(a x)^2 \, dx=\begin {cases} \frac {x^{2} \operatorname {acoth}^{2}{\left (a x \right )}}{2} + \frac {x \operatorname {acoth}{\left (a x \right )}}{a} + \frac {\log {\left (a x + 1 \right )}}{a^{2}} - \frac {\operatorname {acoth}^{2}{\left (a x \right )}}{2 a^{2}} - \frac {\operatorname {acoth}{\left (a x \right )}}{a^{2}} & \text {for}\: a \neq 0 \\- \frac {\pi ^{2} x^{2}}{8} & \text {otherwise} \end {cases} \] Input:
integrate(x*acoth(a*x)**2,x)
Output:
Piecewise((x**2*acoth(a*x)**2/2 + x*acoth(a*x)/a + log(a*x + 1)/a**2 - aco th(a*x)**2/(2*a**2) - acoth(a*x)/a**2, Ne(a, 0)), (-pi**2*x**2/8, True))
Leaf count of result is larger than twice the leaf count of optimal. 97 vs. \(2 (48) = 96\).
Time = 0.03 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.80 \[ \int x \coth ^{-1}(a x)^2 \, dx=\frac {1}{2} \, x^{2} \operatorname {arcoth}\left (a x\right )^{2} + \frac {1}{2} \, a {\left (\frac {2 \, x}{a^{2}} - \frac {\log \left (a x + 1\right )}{a^{3}} + \frac {\log \left (a x - 1\right )}{a^{3}}\right )} \operatorname {arcoth}\left (a x\right ) - \frac {2 \, {\left (\log \left (a x - 1\right ) - 2\right )} \log \left (a x + 1\right ) - \log \left (a x + 1\right )^{2} - \log \left (a x - 1\right )^{2} - 4 \, \log \left (a x - 1\right )}{8 \, a^{2}} \] Input:
integrate(x*arccoth(a*x)^2,x, algorithm="maxima")
Output:
1/2*x^2*arccoth(a*x)^2 + 1/2*a*(2*x/a^2 - log(a*x + 1)/a^3 + log(a*x - 1)/ a^3)*arccoth(a*x) - 1/8*(2*(log(a*x - 1) - 2)*log(a*x + 1) - log(a*x + 1)^ 2 - log(a*x - 1)^2 - 4*log(a*x - 1))/a^2
Leaf count of result is larger than twice the leaf count of optimal. 154 vs. \(2 (48) = 96\).
Time = 0.12 (sec) , antiderivative size = 154, normalized size of antiderivative = 2.85 \[ \int x \coth ^{-1}(a x)^2 \, dx=\frac {1}{2} \, a {\left (\frac {{\left (a x + 1\right )} \log \left (\frac {a x + 1}{a x - 1}\right )^{2}}{{\left (\frac {{\left (a x + 1\right )}^{2} a^{3}}{{\left (a x - 1\right )}^{2}} - \frac {2 \, {\left (a x + 1\right )} a^{3}}{a x - 1} + a^{3}\right )} {\left (a x - 1\right )}} + \frac {2 \, \log \left (\frac {a x + 1}{a x - 1}\right )}{\frac {{\left (a x + 1\right )} a^{3}}{a x - 1} - a^{3}} - \frac {2 \, \log \left (\frac {a x + 1}{a x - 1} - 1\right )}{a^{3}} + \frac {2 \, \log \left (\frac {a x + 1}{a x - 1}\right )}{a^{3}}\right )} \] Input:
integrate(x*arccoth(a*x)^2,x, algorithm="giac")
Output:
1/2*a*((a*x + 1)*log((a*x + 1)/(a*x - 1))^2/(((a*x + 1)^2*a^3/(a*x - 1)^2 - 2*(a*x + 1)*a^3/(a*x - 1) + a^3)*(a*x - 1)) + 2*log((a*x + 1)/(a*x - 1)) /((a*x + 1)*a^3/(a*x - 1) - a^3) - 2*log((a*x + 1)/(a*x - 1) - 1)/a^3 + 2* log((a*x + 1)/(a*x - 1))/a^3)
Time = 3.77 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.81 \[ \int x \coth ^{-1}(a x)^2 \, dx=\frac {x^2\,{\mathrm {acoth}\left (a\,x\right )}^2}{2}+\frac {-\frac {{\mathrm {acoth}\left (a\,x\right )}^2}{2}+a\,x\,\mathrm {acoth}\left (a\,x\right )+\frac {\ln \left (a^2\,x^2-1\right )}{2}}{a^2} \] Input:
int(x*acoth(a*x)^2,x)
Output:
(x^2*acoth(a*x)^2)/2 + (log(a^2*x^2 - 1)/2 - acoth(a*x)^2/2 + a*x*acoth(a* x))/a^2
Time = 0.16 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.98 \[ \int x \coth ^{-1}(a x)^2 \, dx=\frac {\mathit {acoth} \left (a x \right )^{2} a^{2} x^{2}-\mathit {acoth} \left (a x \right )^{2}-2 \mathit {acoth} \left (a x \right ) a x -2 \mathit {acoth} \left (a x \right )+2 \,\mathrm {log}\left (a^{2} x -a \right )}{2 a^{2}} \] Input:
int(x*acoth(a*x)^2,x)
Output:
(acoth(a*x)**2*a**2*x**2 - acoth(a*x)**2 - 2*acoth(a*x)*a*x - 2*acoth(a*x) + 2*log(a**2*x - a))/(2*a**2)