Integrand size = 18, antiderivative size = 58 \[ \int \frac {\left (1-a^2 x^2\right ) \text {arctanh}(a x)}{x^4} \, dx=-\frac {a}{6 x^2}-\frac {\text {arctanh}(a x)}{3 x^3}+\frac {a^2 \text {arctanh}(a x)}{x}-\frac {2}{3} a^3 \log (x)+\frac {1}{3} a^3 \log \left (1-a^2 x^2\right ) \]
Time = 0.02 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.00 \[ \int \frac {\left (1-a^2 x^2\right ) \text {arctanh}(a x)}{x^4} \, dx=-\frac {a}{6 x^2}-\frac {\text {arctanh}(a x)}{3 x^3}+\frac {a^2 \text {arctanh}(a x)}{x}-\frac {2}{3} a^3 \log (x)+\frac {1}{3} a^3 \log \left (1-a^2 x^2\right ) \]
-1/6*a/x^2 - ArcTanh[a*x]/(3*x^3) + (a^2*ArcTanh[a*x])/x - (2*a^3*Log[x])/ 3 + (a^3*Log[1 - a^2*x^2])/3
Time = 0.35 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.47, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.444, Rules used = {6576, 6452, 243, 47, 14, 16, 54, 2009}
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.
\(\displaystyle \int \frac {\left (1-a^2 x^2\right ) \text {arctanh}(a x)}{x^4} \, dx\) |
\(\Big \downarrow \) 6576 |
\(\displaystyle \int \frac {\text {arctanh}(a x)}{x^4}dx-a^2 \int \frac {\text {arctanh}(a x)}{x^2}dx\) |
\(\Big \downarrow \) 6452 |
\(\displaystyle -\left (a^2 \left (a \int \frac {1}{x \left (1-a^2 x^2\right )}dx-\frac {\text {arctanh}(a x)}{x}\right )\right )+\frac {1}{3} a \int \frac {1}{x^3 \left (1-a^2 x^2\right )}dx-\frac {\text {arctanh}(a x)}{3 x^3}\) |
\(\Big \downarrow \) 243 |
\(\displaystyle -\left (a^2 \left (\frac {1}{2} a \int \frac {1}{x^2 \left (1-a^2 x^2\right )}dx^2-\frac {\text {arctanh}(a x)}{x}\right )\right )+\frac {1}{6} a \int \frac {1}{x^4 \left (1-a^2 x^2\right )}dx^2-\frac {\text {arctanh}(a x)}{3 x^3}\) |
\(\Big \downarrow \) 47 |
\(\displaystyle -\left (a^2 \left (\frac {1}{2} a \left (a^2 \int \frac {1}{1-a^2 x^2}dx^2+\int \frac {1}{x^2}dx^2\right )-\frac {\text {arctanh}(a x)}{x}\right )\right )+\frac {1}{6} a \int \frac {1}{x^4 \left (1-a^2 x^2\right )}dx^2-\frac {\text {arctanh}(a x)}{3 x^3}\) |
\(\Big \downarrow \) 14 |
\(\displaystyle -\left (a^2 \left (\frac {1}{2} a \left (a^2 \int \frac {1}{1-a^2 x^2}dx^2+\log \left (x^2\right )\right )-\frac {\text {arctanh}(a x)}{x}\right )\right )+\frac {1}{6} a \int \frac {1}{x^4 \left (1-a^2 x^2\right )}dx^2-\frac {\text {arctanh}(a x)}{3 x^3}\) |
\(\Big \downarrow \) 16 |
\(\displaystyle \frac {1}{6} a \int \frac {1}{x^4 \left (1-a^2 x^2\right )}dx^2-\left (a^2 \left (\frac {1}{2} a \left (\log \left (x^2\right )-\log \left (1-a^2 x^2\right )\right )-\frac {\text {arctanh}(a x)}{x}\right )\right )-\frac {\text {arctanh}(a x)}{3 x^3}\) |
\(\Big \downarrow \) 54 |
\(\displaystyle \frac {1}{6} a \int \left (-\frac {a^4}{a^2 x^2-1}+\frac {a^2}{x^2}+\frac {1}{x^4}\right )dx^2-\left (a^2 \left (\frac {1}{2} a \left (\log \left (x^2\right )-\log \left (1-a^2 x^2\right )\right )-\frac {\text {arctanh}(a x)}{x}\right )\right )-\frac {\text {arctanh}(a x)}{3 x^3}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\left (a^2 \left (\frac {1}{2} a \left (\log \left (x^2\right )-\log \left (1-a^2 x^2\right )\right )-\frac {\text {arctanh}(a x)}{x}\right )\right )+\frac {1}{6} a \left (a^2 \log \left (x^2\right )-a^2 \log \left (1-a^2 x^2\right )-\frac {1}{x^2}\right )-\frac {\text {arctanh}(a x)}{3 x^3}\) |
-1/3*ArcTanh[a*x]/x^3 - a^2*(-(ArcTanh[a*x]/x) + (a*(Log[x^2] - Log[1 - a^ 2*x^2]))/2) + (a*(-x^(-2) + a^2*Log[x^2] - a^2*Log[1 - a^2*x^2]))/6
3.2.69.3.1 Defintions of rubi rules used
Int[(c_.)/((a_.) + (b_.)*(x_)), x_Symbol] :> Simp[c*(Log[RemoveContent[a + b*x, x]]/b), x] /; FreeQ[{a, b, c}, x]
Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Simp[b/(b*c - a*d) Int[1/(a + b*x), x], x] - Simp[d/(b*c - a*d) Int[1/(c + d*x), x ], x] /; FreeQ[{a, b, c, d}, x]
Int[((a_) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[E xpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && ILtQ[m, 0] && IntegerQ[n] && !(IGtQ[n, 0] && LtQ[m + n + 2, 0])
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[1/2 Subst[In t[x^((m - 1)/2)*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, m, p}, x] && I ntegerQ[(m - 1)/2]
Int[((a_.) + ArcTanh[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*(x_)^(m_.), x_Symbol] : > Simp[x^(m + 1)*((a + b*ArcTanh[c*x^n])^p/(m + 1)), x] - Simp[b*c*n*(p/(m + 1)) Int[x^(m + n)*((a + b*ArcTanh[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_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_)*((d_) + (e_ .)*(x_)^2)^(q_.), x_Symbol] :> Simp[d Int[(f*x)^m*(d + e*x^2)^(q - 1)*(a + b*ArcTanh[c*x])^p, x], x] - Simp[c^2*(d/f^2) Int[(f*x)^(m + 2)*(d + e*x ^2)^(q - 1)*(a + b*ArcTanh[c*x])^p, x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[c^2*d + e, 0] && GtQ[q, 0] && IGtQ[p, 0] && (RationalQ[m] || (EqQ [p, 1] && IntegerQ[q]))
Time = 0.53 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.02
method | result | size |
derivativedivides | \(a^{3} \left (\frac {\operatorname {arctanh}\left (a x \right )}{a x}-\frac {\operatorname {arctanh}\left (a x \right )}{3 a^{3} x^{3}}-\frac {1}{6 a^{2} x^{2}}-\frac {2 \ln \left (a x \right )}{3}+\frac {\ln \left (a x +1\right )}{3}+\frac {\ln \left (a x -1\right )}{3}\right )\) | \(59\) |
default | \(a^{3} \left (\frac {\operatorname {arctanh}\left (a x \right )}{a x}-\frac {\operatorname {arctanh}\left (a x \right )}{3 a^{3} x^{3}}-\frac {1}{6 a^{2} x^{2}}-\frac {2 \ln \left (a x \right )}{3}+\frac {\ln \left (a x +1\right )}{3}+\frac {\ln \left (a x -1\right )}{3}\right )\) | \(59\) |
parts | \(-\frac {\operatorname {arctanh}\left (a x \right )}{3 x^{3}}+\frac {a^{2} \operatorname {arctanh}\left (a x \right )}{x}-\frac {a \left (-a^{2} \ln \left (a x +1\right )+\frac {1}{2 x^{2}}+2 a^{2} \ln \left (x \right )-a^{2} \ln \left (a x -1\right )\right )}{3}\) | \(60\) |
parallelrisch | \(-\frac {4 \ln \left (x \right ) a^{3} x^{3}-4 \ln \left (a x -1\right ) x^{3} a^{3}-4 a^{3} x^{3} \operatorname {arctanh}\left (a x \right )+a^{3} x^{3}-6 a^{2} x^{2} \operatorname {arctanh}\left (a x \right )+a x +2 \,\operatorname {arctanh}\left (a x \right )}{6 x^{3}}\) | \(71\) |
risch | \(\frac {\left (3 a^{2} x^{2}-1\right ) \ln \left (a x +1\right )}{6 x^{3}}-\frac {4 \ln \left (x \right ) a^{3} x^{3}-2 \ln \left (-a^{2} x^{2}+1\right ) a^{3} x^{3}+3 \ln \left (-a x +1\right ) a^{2} x^{2}+a x -\ln \left (-a x +1\right )}{6 x^{3}}\) | \(85\) |
meijerg | \(-\frac {a^{3} \left (\frac {2}{a^{2} x^{2}}+\frac {4}{9}-\frac {4 \ln \left (x \right )}{3}-\frac {4 \ln \left (i a \right )}{3}-\frac {2 \left (10 a^{2} x^{2}+30\right )}{45 a^{2} x^{2}}-\frac {2 \left (\ln \left (1-\sqrt {a^{2} x^{2}}\right )-\ln \left (1+\sqrt {a^{2} x^{2}}\right )\right )}{3 a^{2} x^{2} \sqrt {a^{2} x^{2}}}+\frac {2 \ln \left (-a^{2} x^{2}+1\right )}{3}\right )}{4}-\frac {a^{3} \left (4 \ln \left (x \right )+4 \ln \left (i a \right )+\frac {2 \ln \left (1-\sqrt {a^{2} x^{2}}\right )-2 \ln \left (1+\sqrt {a^{2} x^{2}}\right )}{\sqrt {a^{2} x^{2}}}-2 \ln \left (-a^{2} x^{2}+1\right )\right )}{4}\) | \(175\) |
a^3*(arctanh(a*x)/a/x-1/3*arctanh(a*x)/a^3/x^3-1/6/a^2/x^2-2/3*ln(a*x)+1/3 *ln(a*x+1)+1/3*ln(a*x-1))
Time = 0.25 (sec) , antiderivative size = 64, normalized size of antiderivative = 1.10 \[ \int \frac {\left (1-a^2 x^2\right ) \text {arctanh}(a x)}{x^4} \, dx=\frac {2 \, a^{3} x^{3} \log \left (a^{2} x^{2} - 1\right ) - 4 \, a^{3} x^{3} \log \left (x\right ) - a x + {\left (3 \, a^{2} x^{2} - 1\right )} \log \left (-\frac {a x + 1}{a x - 1}\right )}{6 \, x^{3}} \]
1/6*(2*a^3*x^3*log(a^2*x^2 - 1) - 4*a^3*x^3*log(x) - a*x + (3*a^2*x^2 - 1) *log(-(a*x + 1)/(a*x - 1)))/x^3
Time = 0.39 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.09 \[ \int \frac {\left (1-a^2 x^2\right ) \text {arctanh}(a x)}{x^4} \, dx=\begin {cases} - \frac {2 a^{3} \log {\left (x \right )}}{3} + \frac {2 a^{3} \log {\left (x - \frac {1}{a} \right )}}{3} + \frac {2 a^{3} \operatorname {atanh}{\left (a x \right )}}{3} + \frac {a^{2} \operatorname {atanh}{\left (a x \right )}}{x} - \frac {a}{6 x^{2}} - \frac {\operatorname {atanh}{\left (a x \right )}}{3 x^{3}} & \text {for}\: a \neq 0 \\0 & \text {otherwise} \end {cases} \]
Piecewise((-2*a**3*log(x)/3 + 2*a**3*log(x - 1/a)/3 + 2*a**3*atanh(a*x)/3 + a**2*atanh(a*x)/x - a/(6*x**2) - atanh(a*x)/(3*x**3), Ne(a, 0)), (0, Tru e))
Time = 0.19 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.91 \[ \int \frac {\left (1-a^2 x^2\right ) \text {arctanh}(a x)}{x^4} \, dx=\frac {1}{6} \, {\left (2 \, a^{2} \log \left (a^{2} x^{2} - 1\right ) - 2 \, a^{2} \log \left (x^{2}\right ) - \frac {1}{x^{2}}\right )} a + \frac {{\left (3 \, a^{2} x^{2} - 1\right )} \operatorname {artanh}\left (a x\right )}{3 \, x^{3}} \]
1/6*(2*a^2*log(a^2*x^2 - 1) - 2*a^2*log(x^2) - 1/x^2)*a + 1/3*(3*a^2*x^2 - 1)*arctanh(a*x)/x^3
Leaf count of result is larger than twice the leaf count of optimal. 204 vs. \(2 (50) = 100\).
Time = 0.29 (sec) , antiderivative size = 204, normalized size of antiderivative = 3.52 \[ \int \frac {\left (1-a^2 x^2\right ) \text {arctanh}(a x)}{x^4} \, dx=\frac {2}{3} \, {\left (a^{2} \log \left (\frac {{\left | -a x - 1 \right |}}{{\left | a x - 1 \right |}}\right ) - a^{2} \log \left ({\left | -\frac {a x + 1}{a x - 1} - 1 \right |}\right ) + \frac {{\left (a x + 1\right )} a^{2}}{{\left (a x - 1\right )} {\left (\frac {a x + 1}{a x - 1} + 1\right )}^{2}} - \frac {{\left (\frac {3 \, {\left (a x + 1\right )} a^{2}}{a x - 1} + a^{2}\right )} \log \left (-\frac {\frac {a {\left (\frac {a x + 1}{a x - 1} + 1\right )}}{\frac {{\left (a x + 1\right )} a}{a x - 1} - a} + 1}{\frac {a {\left (\frac {a x + 1}{a x - 1} + 1\right )}}{\frac {{\left (a x + 1\right )} a}{a x - 1} - a} - 1}\right )}{{\left (\frac {a x + 1}{a x - 1} + 1\right )}^{3}}\right )} a \]
2/3*(a^2*log(abs(-a*x - 1)/abs(a*x - 1)) - a^2*log(abs(-(a*x + 1)/(a*x - 1 ) - 1)) + (a*x + 1)*a^2/((a*x - 1)*((a*x + 1)/(a*x - 1) + 1)^2) - (3*(a*x + 1)*a^2/(a*x - 1) + a^2)*log(-(a*((a*x + 1)/(a*x - 1) + 1)/((a*x + 1)*a/( a*x - 1) - a) + 1)/(a*((a*x + 1)/(a*x - 1) + 1)/((a*x + 1)*a/(a*x - 1) - a ) - 1))/((a*x + 1)/(a*x - 1) + 1)^3)*a
Time = 3.56 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.84 \[ \int \frac {\left (1-a^2 x^2\right ) \text {arctanh}(a x)}{x^4} \, dx=\frac {a^3\,\ln \left (a^2\,x^2-1\right )}{3}-\frac {a}{6\,x^2}-\frac {\mathrm {atanh}\left (a\,x\right )}{3\,x^3}-\frac {2\,a^3\,\ln \left (x\right )}{3}+\frac {a^2\,\mathrm {atanh}\left (a\,x\right )}{x} \]