Integrand size = 10, antiderivative size = 44 \[ \int \left (a+b \text {arctanh}\left (\frac {c}{x^2}\right )\right ) \, dx=a x+b \sqrt {c} \arctan \left (\frac {x}{\sqrt {c}}\right )+b x \text {arctanh}\left (\frac {c}{x^2}\right )-b \sqrt {c} \text {arctanh}\left (\frac {x}{\sqrt {c}}\right ) \]
Time = 0.01 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.23 \[ \int \left (a+b \text {arctanh}\left (\frac {c}{x^2}\right )\right ) \, dx=a x+b x \text {arctanh}\left (\frac {c}{x^2}\right )+\frac {1}{2} b \sqrt {c} \left (2 \arctan \left (\frac {x}{\sqrt {c}}\right )+\log \left (\sqrt {c}-x\right )-\log \left (\sqrt {c}+x\right )\right ) \]
a*x + b*x*ArcTanh[c/x^2] + (b*Sqrt[c]*(2*ArcTan[x/Sqrt[c]] + Log[Sqrt[c] - x] - Log[Sqrt[c] + x]))/2
Time = 0.18 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {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.
\(\displaystyle \int \left (a+b \text {arctanh}\left (\frac {c}{x^2}\right )\right ) \, dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle a x+b \sqrt {c} \arctan \left (\frac {x}{\sqrt {c}}\right )+b x \text {arctanh}\left (\frac {c}{x^2}\right )-b \sqrt {c} \text {arctanh}\left (\frac {x}{\sqrt {c}}\right )\) |
3.2.67.3.1 Defintions of rubi rules used
Time = 0.80 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.89
method | result | size |
default | \(a x -b \sqrt {c}\, \operatorname {arctanh}\left (\frac {\sqrt {c}}{x}\right )+b \arctan \left (\frac {x}{\sqrt {c}}\right ) \sqrt {c}+b x \,\operatorname {arctanh}\left (\frac {c}{x^{2}}\right )\) | \(39\) |
parts | \(a x -b \sqrt {c}\, \operatorname {arctanh}\left (\frac {\sqrt {c}}{x}\right )+b \arctan \left (\frac {x}{\sqrt {c}}\right ) \sqrt {c}+b x \,\operatorname {arctanh}\left (\frac {c}{x^{2}}\right )\) | \(39\) |
derivativedivides | \(a x +b x \,\operatorname {arctanh}\left (\frac {c}{x^{2}}\right )-b \sqrt {c}\, \operatorname {arctanh}\left (\frac {\sqrt {c}}{x}\right )-b \sqrt {c}\, \arctan \left (\frac {\sqrt {c}}{x}\right )\) | \(42\) |
risch | \(a x +\frac {b x \ln \left (x^{2}+c \right )}{2}-\frac {b x \ln \left (-x^{2}+c \right )}{2}+\frac {i b \pi x {\operatorname {csgn}\left (\frac {i \left (-x^{2}+c \right )}{x^{2}}\right )}^{2}}{2}-\frac {i b \pi x}{2}-\frac {i b \pi x {\operatorname {csgn}\left (\frac {i \left (x^{2}+c \right )}{x^{2}}\right )}^{3}}{4}-\frac {i b \pi x \,\operatorname {csgn}\left (i \left (-x^{2}+c \right )\right ) {\operatorname {csgn}\left (\frac {i \left (-x^{2}+c \right )}{x^{2}}\right )}^{2}}{4}+\frac {i b \pi x \,\operatorname {csgn}\left (\frac {i}{x^{2}}\right ) \operatorname {csgn}\left (i \left (-x^{2}+c \right )\right ) \operatorname {csgn}\left (\frac {i \left (-x^{2}+c \right )}{x^{2}}\right )}{4}+\frac {i b \pi x \,\operatorname {csgn}\left (i \left (x^{2}+c \right )\right ) {\operatorname {csgn}\left (\frac {i \left (x^{2}+c \right )}{x^{2}}\right )}^{2}}{4}-\frac {i b \pi x \,\operatorname {csgn}\left (\frac {i}{x^{2}}\right ) {\operatorname {csgn}\left (\frac {i \left (-x^{2}+c \right )}{x^{2}}\right )}^{2}}{4}-\frac {i b \pi x {\operatorname {csgn}\left (\frac {i \left (-x^{2}+c \right )}{x^{2}}\right )}^{3}}{4}+\frac {i b \pi x \,\operatorname {csgn}\left (\frac {i}{x^{2}}\right ) {\operatorname {csgn}\left (\frac {i \left (x^{2}+c \right )}{x^{2}}\right )}^{2}}{4}-\frac {i b \pi x \,\operatorname {csgn}\left (\frac {i}{x^{2}}\right ) \operatorname {csgn}\left (i \left (x^{2}+c \right )\right ) \operatorname {csgn}\left (\frac {i \left (x^{2}+c \right )}{x^{2}}\right )}{4}+\frac {b \sqrt {-c}\, \ln \left (x +\sqrt {-c}\right )}{2}-\frac {b \sqrt {-c}\, \ln \left (-\sqrt {-c}+x \right )}{2}+\frac {b \sqrt {c}\, \ln \left (-\sqrt {c}+x \right )}{2}-\frac {b \sqrt {c}\, \ln \left (x +\sqrt {c}\right )}{2}\) | \(347\) |
Leaf count of result is larger than twice the leaf count of optimal. 74 vs. \(2 (36) = 72\).
Time = 0.27 (sec) , antiderivative size = 138, normalized size of antiderivative = 3.14 \[ \int \left (a+b \text {arctanh}\left (\frac {c}{x^2}\right )\right ) \, dx=\left [\frac {1}{2} \, b x \log \left (\frac {x^{2} + c}{x^{2} - c}\right ) + b \sqrt {c} \arctan \left (\frac {x}{\sqrt {c}}\right ) + \frac {1}{2} \, b \sqrt {c} \log \left (\frac {x^{2} - 2 \, \sqrt {c} x + c}{x^{2} - c}\right ) + a x, \frac {1}{2} \, b x \log \left (\frac {x^{2} + c}{x^{2} - c}\right ) + b \sqrt {-c} \arctan \left (\frac {\sqrt {-c} x}{c}\right ) + \frac {1}{2} \, b \sqrt {-c} \log \left (\frac {x^{2} + 2 \, \sqrt {-c} x - c}{x^{2} + c}\right ) + a x\right ] \]
[1/2*b*x*log((x^2 + c)/(x^2 - c)) + b*sqrt(c)*arctan(x/sqrt(c)) + 1/2*b*sq rt(c)*log((x^2 - 2*sqrt(c)*x + c)/(x^2 - c)) + a*x, 1/2*b*x*log((x^2 + c)/ (x^2 - c)) + b*sqrt(-c)*arctan(sqrt(-c)*x/c) + 1/2*b*sqrt(-c)*log((x^2 + 2 *sqrt(-c)*x - c)/(x^2 + c)) + a*x]
Time = 2.04 (sec) , antiderivative size = 632, normalized size of antiderivative = 14.36 \[ \int \left (a+b \text {arctanh}\left (\frac {c}{x^2}\right )\right ) \, dx=a x + b \left (\begin {cases} 0 & \text {for}\: c = 0 \\- \infty x & \text {for}\: c = - x^{2} \\\infty x & \text {for}\: c = x^{2} \\- \frac {2 c^{\frac {5}{2}} \sqrt {- c} \log {\left (- \sqrt {c} + x \right )}}{- 2 c^{2} \sqrt {- c} + 2 x^{4} \sqrt {- c}} + \frac {c^{\frac {5}{2}} \sqrt {- c} \log {\left (x - \sqrt {- c} \right )}}{- 2 c^{2} \sqrt {- c} + 2 x^{4} \sqrt {- c}} + \frac {c^{\frac {5}{2}} \sqrt {- c} \log {\left (x + \sqrt {- c} \right )}}{- 2 c^{2} \sqrt {- c} + 2 x^{4} \sqrt {- c}} - \frac {2 c^{\frac {5}{2}} \sqrt {- c} \operatorname {atanh}{\left (\frac {c}{x^{2}} \right )}}{- 2 c^{2} \sqrt {- c} + 2 x^{4} \sqrt {- c}} + \frac {2 \sqrt {c} x^{4} \sqrt {- c} \log {\left (- \sqrt {c} + x \right )}}{- 2 c^{2} \sqrt {- c} + 2 x^{4} \sqrt {- c}} - \frac {\sqrt {c} x^{4} \sqrt {- c} \log {\left (x - \sqrt {- c} \right )}}{- 2 c^{2} \sqrt {- c} + 2 x^{4} \sqrt {- c}} - \frac {\sqrt {c} x^{4} \sqrt {- c} \log {\left (x + \sqrt {- c} \right )}}{- 2 c^{2} \sqrt {- c} + 2 x^{4} \sqrt {- c}} + \frac {2 \sqrt {c} x^{4} \sqrt {- c} \operatorname {atanh}{\left (\frac {c}{x^{2}} \right )}}{- 2 c^{2} \sqrt {- c} + 2 x^{4} \sqrt {- c}} - \frac {c^{3} \log {\left (x - \sqrt {- c} \right )}}{- 2 c^{2} \sqrt {- c} + 2 x^{4} \sqrt {- c}} + \frac {c^{3} \log {\left (x + \sqrt {- c} \right )}}{- 2 c^{2} \sqrt {- c} + 2 x^{4} \sqrt {- c}} - \frac {2 c^{2} x \sqrt {- c} \operatorname {atanh}{\left (\frac {c}{x^{2}} \right )}}{- 2 c^{2} \sqrt {- c} + 2 x^{4} \sqrt {- c}} + \frac {c x^{4} \log {\left (x - \sqrt {- c} \right )}}{- 2 c^{2} \sqrt {- c} + 2 x^{4} \sqrt {- c}} - \frac {c x^{4} \log {\left (x + \sqrt {- c} \right )}}{- 2 c^{2} \sqrt {- c} + 2 x^{4} \sqrt {- c}} + \frac {2 x^{5} \sqrt {- c} \operatorname {atanh}{\left (\frac {c}{x^{2}} \right )}}{- 2 c^{2} \sqrt {- c} + 2 x^{4} \sqrt {- c}} & \text {otherwise} \end {cases}\right ) \]
a*x + b*Piecewise((0, Eq(c, 0)), (-oo*x, Eq(c, -x**2)), (oo*x, Eq(c, x**2) ), (-2*c**(5/2)*sqrt(-c)*log(-sqrt(c) + x)/(-2*c**2*sqrt(-c) + 2*x**4*sqrt (-c)) + c**(5/2)*sqrt(-c)*log(x - sqrt(-c))/(-2*c**2*sqrt(-c) + 2*x**4*sqr t(-c)) + c**(5/2)*sqrt(-c)*log(x + sqrt(-c))/(-2*c**2*sqrt(-c) + 2*x**4*sq rt(-c)) - 2*c**(5/2)*sqrt(-c)*atanh(c/x**2)/(-2*c**2*sqrt(-c) + 2*x**4*sqr t(-c)) + 2*sqrt(c)*x**4*sqrt(-c)*log(-sqrt(c) + x)/(-2*c**2*sqrt(-c) + 2*x **4*sqrt(-c)) - sqrt(c)*x**4*sqrt(-c)*log(x - sqrt(-c))/(-2*c**2*sqrt(-c) + 2*x**4*sqrt(-c)) - sqrt(c)*x**4*sqrt(-c)*log(x + sqrt(-c))/(-2*c**2*sqrt (-c) + 2*x**4*sqrt(-c)) + 2*sqrt(c)*x**4*sqrt(-c)*atanh(c/x**2)/(-2*c**2*s qrt(-c) + 2*x**4*sqrt(-c)) - c**3*log(x - sqrt(-c))/(-2*c**2*sqrt(-c) + 2* x**4*sqrt(-c)) + c**3*log(x + sqrt(-c))/(-2*c**2*sqrt(-c) + 2*x**4*sqrt(-c )) - 2*c**2*x*sqrt(-c)*atanh(c/x**2)/(-2*c**2*sqrt(-c) + 2*x**4*sqrt(-c)) + c*x**4*log(x - sqrt(-c))/(-2*c**2*sqrt(-c) + 2*x**4*sqrt(-c)) - c*x**4*l og(x + sqrt(-c))/(-2*c**2*sqrt(-c) + 2*x**4*sqrt(-c)) + 2*x**5*sqrt(-c)*at anh(c/x**2)/(-2*c**2*sqrt(-c) + 2*x**4*sqrt(-c)), True))
Time = 0.26 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.16 \[ \int \left (a+b \text {arctanh}\left (\frac {c}{x^2}\right )\right ) \, dx=\frac {1}{2} \, {\left (c {\left (\frac {2 \, \arctan \left (\frac {x}{\sqrt {c}}\right )}{\sqrt {c}} + \frac {\log \left (\frac {x - \sqrt {c}}{x + \sqrt {c}}\right )}{\sqrt {c}}\right )} + 2 \, x \operatorname {artanh}\left (\frac {c}{x^{2}}\right )\right )} b + a x \]
1/2*(c*(2*arctan(x/sqrt(c))/sqrt(c) + log((x - sqrt(c))/(x + sqrt(c)))/sqr t(c)) + 2*x*arctanh(c/x^2))*b + a*x
Time = 0.27 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.30 \[ \int \left (a+b \text {arctanh}\left (\frac {c}{x^2}\right )\right ) \, dx=\frac {1}{2} \, {\left (2 \, c {\left (\frac {\arctan \left (\frac {x}{\sqrt {-c}}\right )}{\sqrt {-c}} + \frac {\arctan \left (\frac {x}{\sqrt {c}}\right )}{\sqrt {c}}\right )} + x \log \left (-\frac {\frac {c}{x^{2}} + 1}{\frac {c}{x^{2}} - 1}\right )\right )} b + a x \]
1/2*(2*c*(arctan(x/sqrt(-c))/sqrt(-c) + arctan(x/sqrt(c))/sqrt(c)) + x*log (-(c/x^2 + 1)/(c/x^2 - 1)))*b + a*x
Time = 3.23 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.18 \[ \int \left (a+b \text {arctanh}\left (\frac {c}{x^2}\right )\right ) \, dx=a\,x+\frac {b\,x\,\ln \left (x^2+c\right )}{2}+b\,\sqrt {c}\,\mathrm {atan}\left (\frac {x}{\sqrt {c}}\right )-\frac {b\,x\,\ln \left (x^2-c\right )}{2}+b\,\sqrt {c}\,\mathrm {atan}\left (\frac {x\,1{}\mathrm {i}}{\sqrt {c}}\right )\,1{}\mathrm {i} \]