Integrand size = 14, antiderivative size = 46 \[ \int \frac {a+b \text {arctanh}\left (\frac {c}{x^2}\right )}{x^2} \, dx=\frac {b \arctan \left (\frac {x}{\sqrt {c}}\right )}{\sqrt {c}}-\frac {a+b \text {arctanh}\left (\frac {c}{x^2}\right )}{x}+\frac {b \text {arctanh}\left (\frac {x}{\sqrt {c}}\right )}{\sqrt {c}} \]
Time = 0.03 (sec) , antiderivative size = 72, normalized size of antiderivative = 1.57 \[ \int \frac {a+b \text {arctanh}\left (\frac {c}{x^2}\right )}{x^2} \, dx=-\frac {a}{x}+\frac {b \arctan \left (\frac {x}{\sqrt {c}}\right )}{\sqrt {c}}-\frac {b \text {arctanh}\left (\frac {c}{x^2}\right )}{x}-\frac {b \log \left (\sqrt {c}-x\right )}{2 \sqrt {c}}+\frac {b \log \left (\sqrt {c}+x\right )}{2 \sqrt {c}} \]
-(a/x) + (b*ArcTan[x/Sqrt[c]])/Sqrt[c] - (b*ArcTanh[c/x^2])/x - (b*Log[Sqr t[c] - x])/(2*Sqrt[c]) + (b*Log[Sqrt[c] + x])/(2*Sqrt[c])
Time = 0.23 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.20, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.357, Rules used = {6452, 795, 756, 216, 219}
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.
| \(\displaystyle \int \frac {a+b \text {arctanh}\left (\frac {c}{x^2}\right )}{x^2} \, dx\) |
| \(\Big \downarrow \) 6452 |
| \(\displaystyle -2 b c \int \frac {1}{\left (1-\frac {c^2}{x^4}\right ) x^4}dx-\frac {a+b \text {arctanh}\left (\frac {c}{x^2}\right )}{x}\) |
| \(\Big \downarrow \) 795 |
| \(\displaystyle -2 b c \int \frac {1}{x^4-c^2}dx-\frac {a+b \text {arctanh}\left (\frac {c}{x^2}\right )}{x}\) |
| \(\Big \downarrow \) 756 |
| \(\displaystyle -2 b c \left (-\frac {\int \frac {1}{c-x^2}dx}{2 c}-\frac {\int \frac {1}{x^2+c}dx}{2 c}\right )-\frac {a+b \text {arctanh}\left (\frac {c}{x^2}\right )}{x}\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle -2 b c \left (-\frac {\int \frac {1}{c-x^2}dx}{2 c}-\frac {\arctan \left (\frac {x}{\sqrt {c}}\right )}{2 c^{3/2}}\right )-\frac {a+b \text {arctanh}\left (\frac {c}{x^2}\right )}{x}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle -\frac {a+b \text {arctanh}\left (\frac {c}{x^2}\right )}{x}-2 b c \left (-\frac {\arctan \left (\frac {x}{\sqrt {c}}\right )}{2 c^{3/2}}-\frac {\text {arctanh}\left (\frac {x}{\sqrt {c}}\right )}{2 c^{3/2}}\right )\) |
-((a + b*ArcTanh[c/x^2])/x) - 2*b*c*(-1/2*ArcTan[x/Sqrt[c]]/c^(3/2) - ArcT anh[x/Sqrt[c]]/(2*c^(3/2)))
3.2.68.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*A rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a , 0] || GtQ[b, 0])
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] && (Gt Q[a, 0] || LtQ[b, 0])
Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 2 ]], s = Denominator[Rt[-a/b, 2]]}, Simp[r/(2*a) Int[1/(r - s*x^2), x], x] + Simp[r/(2*a) Int[1/(r + s*x^2), x], x]] /; FreeQ[{a, b}, x] && !GtQ[a /b, 0]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[x^(m + n*p)* (b + a/x^n)^p, x] /; FreeQ[{a, b, m, n}, x] && IntegerQ[p] && NegQ[n]
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]
Time = 0.67 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.96
| method | result | size |
| parts | \(-\frac {a}{x}-\frac {b \,\operatorname {arctanh}\left (\frac {c}{x^{2}}\right )}{x}+\frac {b \,\operatorname {arctanh}\left (\frac {\sqrt {c}}{x}\right )}{\sqrt {c}}+\frac {b \arctan \left (\frac {x}{\sqrt {c}}\right )}{\sqrt {c}}\) | \(44\) |
| derivativedivides | \(-\frac {a}{x}-\frac {b \,\operatorname {arctanh}\left (\frac {c}{x^{2}}\right )}{x}+\frac {b \,\operatorname {arctanh}\left (\frac {\sqrt {c}}{x}\right )}{\sqrt {c}}-\frac {b \arctan \left (\frac {\sqrt {c}}{x}\right )}{\sqrt {c}}\) | \(47\) |
| default | \(-\frac {a}{x}-\frac {b \,\operatorname {arctanh}\left (\frac {c}{x^{2}}\right )}{x}+\frac {b \,\operatorname {arctanh}\left (\frac {\sqrt {c}}{x}\right )}{\sqrt {c}}-\frac {b \arctan \left (\frac {\sqrt {c}}{x}\right )}{\sqrt {c}}\) | \(47\) |
| risch | \(-\frac {b \ln \left (x^{2}+c \right )}{2 x}+\frac {b \arctan \left (\frac {x}{\sqrt {c}}\right )}{\sqrt {c}}-\frac {-i b \pi {\operatorname {csgn}\left (\frac {i \left (x^{2}+c \right )}{x^{2}}\right )}^{3}+4 a +2 i b \pi {\operatorname {csgn}\left (\frac {i \left (-x^{2}+c \right )}{x^{2}}\right )}^{2}+i b \pi \,\operatorname {csgn}\left (\frac {i}{x^{2}}\right ) \operatorname {csgn}\left (i \left (-x^{2}+c \right )\right ) \operatorname {csgn}\left (\frac {i \left (-x^{2}+c \right )}{x^{2}}\right )-i b \pi \,\operatorname {csgn}\left (\frac {i}{x^{2}}\right ) \operatorname {csgn}\left (i \left (x^{2}+c \right )\right ) \operatorname {csgn}\left (\frac {i \left (x^{2}+c \right )}{x^{2}}\right )-i b \pi \,\operatorname {csgn}\left (\frac {i}{x^{2}}\right ) {\operatorname {csgn}\left (\frac {i \left (-x^{2}+c \right )}{x^{2}}\right )}^{2}+i b \pi \,\operatorname {csgn}\left (\frac {i}{x^{2}}\right ) {\operatorname {csgn}\left (\frac {i \left (x^{2}+c \right )}{x^{2}}\right )}^{2}-i b \pi \,\operatorname {csgn}\left (i \left (-x^{2}+c \right )\right ) {\operatorname {csgn}\left (\frac {i \left (-x^{2}+c \right )}{x^{2}}\right )}^{2}+i b \pi \,\operatorname {csgn}\left (i \left (x^{2}+c \right )\right ) {\operatorname {csgn}\left (\frac {i \left (x^{2}+c \right )}{x^{2}}\right )}^{2}-i b \pi {\operatorname {csgn}\left (\frac {i \left (-x^{2}+c \right )}{x^{2}}\right )}^{3}-2 i b \pi }{4 x}+\frac {b \ln \left (-x^{2}+c \right )}{2 x}+\frac {b \,\operatorname {arctanh}\left (\frac {x}{\sqrt {c}}\right )}{\sqrt {c}}\) | \(309\) |
Leaf count of result is larger than twice the leaf count of optimal. 84 vs. \(2 (38) = 76\).
Time = 0.27 (sec) , antiderivative size = 159, normalized size of antiderivative = 3.46 \[ \int \frac {a+b \text {arctanh}\left (\frac {c}{x^2}\right )}{x^2} \, dx=\left [\frac {2 \, b \sqrt {c} x \arctan \left (\frac {x}{\sqrt {c}}\right ) + b \sqrt {c} x \log \left (\frac {x^{2} + 2 \, \sqrt {c} x + c}{x^{2} - c}\right ) - b c \log \left (\frac {x^{2} + c}{x^{2} - c}\right ) - 2 \, a c}{2 \, c x}, -\frac {2 \, b \sqrt {-c} x \arctan \left (\frac {\sqrt {-c} x}{c}\right ) + b \sqrt {-c} x \log \left (\frac {x^{2} - 2 \, \sqrt {-c} x - c}{x^{2} + c}\right ) + b c \log \left (\frac {x^{2} + c}{x^{2} - c}\right ) + 2 \, a c}{2 \, c x}\right ] \]
[1/2*(2*b*sqrt(c)*x*arctan(x/sqrt(c)) + b*sqrt(c)*x*log((x^2 + 2*sqrt(c)*x + c)/(x^2 - c)) - b*c*log((x^2 + c)/(x^2 - c)) - 2*a*c)/(c*x), -1/2*(2*b* sqrt(-c)*x*arctan(sqrt(-c)*x/c) + b*sqrt(-c)*x*log((x^2 - 2*sqrt(-c)*x - c )/(x^2 + c)) + b*c*log((x^2 + c)/(x^2 - c)) + 2*a*c)/(c*x)]
Leaf count of result is larger than twice the leaf count of optimal. 886 vs. \(2 (42) = 84\).
Time = 3.54 (sec) , antiderivative size = 886, normalized size of antiderivative = 19.26 \[ \int \frac {a+b \text {arctanh}\left (\frac {c}{x^2}\right )}{x^2} \, dx=\begin {cases} - \frac {a}{x} & \text {for}\: c = 0 \\- \frac {a - \infty b}{x} & \text {for}\: c = - x^{2} \\- \frac {a + \infty b}{x} & \text {for}\: c = x^{2} \\\frac {2 a c^{\frac {7}{2}} \sqrt {- c}}{- 2 c^{\frac {7}{2}} x \sqrt {- c} + 2 c^{\frac {3}{2}} x^{5} \sqrt {- c}} - \frac {2 a c^{\frac {3}{2}} x^{4} \sqrt {- c}}{- 2 c^{\frac {7}{2}} x \sqrt {- c} + 2 c^{\frac {3}{2}} x^{5} \sqrt {- c}} - \frac {b c^{\frac {7}{2}} x \log {\left (x - \sqrt {- c} \right )}}{- 2 c^{\frac {7}{2}} x \sqrt {- c} + 2 c^{\frac {3}{2}} x^{5} \sqrt {- c}} + \frac {b c^{\frac {7}{2}} x \log {\left (x + \sqrt {- c} \right )}}{- 2 c^{\frac {7}{2}} x \sqrt {- c} + 2 c^{\frac {3}{2}} x^{5} \sqrt {- c}} + \frac {2 b c^{\frac {7}{2}} \sqrt {- c} \operatorname {atanh}{\left (\frac {c}{x^{2}} \right )}}{- 2 c^{\frac {7}{2}} x \sqrt {- c} + 2 c^{\frac {3}{2}} x^{5} \sqrt {- c}} + \frac {b c^{\frac {3}{2}} x^{5} \log {\left (x - \sqrt {- c} \right )}}{- 2 c^{\frac {7}{2}} x \sqrt {- c} + 2 c^{\frac {3}{2}} x^{5} \sqrt {- c}} - \frac {b c^{\frac {3}{2}} x^{5} \log {\left (x + \sqrt {- c} \right )}}{- 2 c^{\frac {7}{2}} x \sqrt {- c} + 2 c^{\frac {3}{2}} x^{5} \sqrt {- c}} - \frac {2 b c^{\frac {3}{2}} x^{4} \sqrt {- c} \operatorname {atanh}{\left (\frac {c}{x^{2}} \right )}}{- 2 c^{\frac {7}{2}} x \sqrt {- c} + 2 c^{\frac {3}{2}} x^{5} \sqrt {- c}} + \frac {2 b c^{3} x \sqrt {- c} \log {\left (- \sqrt {c} + x \right )}}{- 2 c^{\frac {7}{2}} x \sqrt {- c} + 2 c^{\frac {3}{2}} x^{5} \sqrt {- c}} - \frac {b c^{3} x \sqrt {- c} \log {\left (x - \sqrt {- c} \right )}}{- 2 c^{\frac {7}{2}} x \sqrt {- c} + 2 c^{\frac {3}{2}} x^{5} \sqrt {- c}} - \frac {b c^{3} x \sqrt {- c} \log {\left (x + \sqrt {- c} \right )}}{- 2 c^{\frac {7}{2}} x \sqrt {- c} + 2 c^{\frac {3}{2}} x^{5} \sqrt {- c}} + \frac {2 b c^{3} x \sqrt {- c} \operatorname {atanh}{\left (\frac {c}{x^{2}} \right )}}{- 2 c^{\frac {7}{2}} x \sqrt {- c} + 2 c^{\frac {3}{2}} x^{5} \sqrt {- c}} - \frac {2 b c x^{5} \sqrt {- c} \log {\left (- \sqrt {c} + x \right )}}{- 2 c^{\frac {7}{2}} x \sqrt {- c} + 2 c^{\frac {3}{2}} x^{5} \sqrt {- c}} + \frac {b c x^{5} \sqrt {- c} \log {\left (x - \sqrt {- c} \right )}}{- 2 c^{\frac {7}{2}} x \sqrt {- c} + 2 c^{\frac {3}{2}} x^{5} \sqrt {- c}} + \frac {b c x^{5} \sqrt {- c} \log {\left (x + \sqrt {- c} \right )}}{- 2 c^{\frac {7}{2}} x \sqrt {- c} + 2 c^{\frac {3}{2}} x^{5} \sqrt {- c}} - \frac {2 b c x^{5} \sqrt {- c} \operatorname {atanh}{\left (\frac {c}{x^{2}} \right )}}{- 2 c^{\frac {7}{2}} x \sqrt {- c} + 2 c^{\frac {3}{2}} x^{5} \sqrt {- c}} & \text {otherwise} \end {cases} \]
Piecewise((-a/x, Eq(c, 0)), (-(a - oo*b)/x, Eq(c, -x**2)), (-(a + oo*b)/x, Eq(c, x**2)), (2*a*c**(7/2)*sqrt(-c)/(-2*c**(7/2)*x*sqrt(-c) + 2*c**(3/2) *x**5*sqrt(-c)) - 2*a*c**(3/2)*x**4*sqrt(-c)/(-2*c**(7/2)*x*sqrt(-c) + 2*c **(3/2)*x**5*sqrt(-c)) - b*c**(7/2)*x*log(x - sqrt(-c))/(-2*c**(7/2)*x*sqr t(-c) + 2*c**(3/2)*x**5*sqrt(-c)) + b*c**(7/2)*x*log(x + sqrt(-c))/(-2*c** (7/2)*x*sqrt(-c) + 2*c**(3/2)*x**5*sqrt(-c)) + 2*b*c**(7/2)*sqrt(-c)*atanh (c/x**2)/(-2*c**(7/2)*x*sqrt(-c) + 2*c**(3/2)*x**5*sqrt(-c)) + b*c**(3/2)* x**5*log(x - sqrt(-c))/(-2*c**(7/2)*x*sqrt(-c) + 2*c**(3/2)*x**5*sqrt(-c)) - b*c**(3/2)*x**5*log(x + sqrt(-c))/(-2*c**(7/2)*x*sqrt(-c) + 2*c**(3/2)* x**5*sqrt(-c)) - 2*b*c**(3/2)*x**4*sqrt(-c)*atanh(c/x**2)/(-2*c**(7/2)*x*s qrt(-c) + 2*c**(3/2)*x**5*sqrt(-c)) + 2*b*c**3*x*sqrt(-c)*log(-sqrt(c) + x )/(-2*c**(7/2)*x*sqrt(-c) + 2*c**(3/2)*x**5*sqrt(-c)) - b*c**3*x*sqrt(-c)* log(x - sqrt(-c))/(-2*c**(7/2)*x*sqrt(-c) + 2*c**(3/2)*x**5*sqrt(-c)) - b* c**3*x*sqrt(-c)*log(x + sqrt(-c))/(-2*c**(7/2)*x*sqrt(-c) + 2*c**(3/2)*x** 5*sqrt(-c)) + 2*b*c**3*x*sqrt(-c)*atanh(c/x**2)/(-2*c**(7/2)*x*sqrt(-c) + 2*c**(3/2)*x**5*sqrt(-c)) - 2*b*c*x**5*sqrt(-c)*log(-sqrt(c) + x)/(-2*c**( 7/2)*x*sqrt(-c) + 2*c**(3/2)*x**5*sqrt(-c)) + b*c*x**5*sqrt(-c)*log(x - sq rt(-c))/(-2*c**(7/2)*x*sqrt(-c) + 2*c**(3/2)*x**5*sqrt(-c)) + b*c*x**5*sqr t(-c)*log(x + sqrt(-c))/(-2*c**(7/2)*x*sqrt(-c) + 2*c**(3/2)*x**5*sqrt(-c) ) - 2*b*c*x**5*sqrt(-c)*atanh(c/x**2)/(-2*c**(7/2)*x*sqrt(-c) + 2*c**(3...
Time = 0.26 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.24 \[ \int \frac {a+b \text {arctanh}\left (\frac {c}{x^2}\right )}{x^2} \, dx=\frac {1}{2} \, {\left (c {\left (\frac {2 \, \arctan \left (\frac {x}{\sqrt {c}}\right )}{c^{\frac {3}{2}}} - \frac {\log \left (\frac {x - \sqrt {c}}{x + \sqrt {c}}\right )}{c^{\frac {3}{2}}}\right )} - \frac {2 \, \operatorname {artanh}\left (\frac {c}{x^{2}}\right )}{x}\right )} b - \frac {a}{x} \]
1/2*(c*(2*arctan(x/sqrt(c))/c^(3/2) - log((x - sqrt(c))/(x + sqrt(c)))/c^( 3/2)) - 2*arctanh(c/x^2)/x)*b - a/x
Time = 0.31 (sec) , antiderivative size = 62, normalized size of antiderivative = 1.35 \[ \int \frac {a+b \text {arctanh}\left (\frac {c}{x^2}\right )}{x^2} \, dx=-b c {\left (\frac {\arctan \left (\frac {x}{\sqrt {-c}}\right )}{\sqrt {-c} c} - \frac {\arctan \left (\frac {x}{\sqrt {c}}\right )}{c^{\frac {3}{2}}}\right )} - \frac {b \log \left (\frac {x^{2} + c}{x^{2} - c}\right )}{2 \, x} - \frac {a}{x} \]
-b*c*(arctan(x/sqrt(-c))/(sqrt(-c)*c) - arctan(x/sqrt(c))/c^(3/2)) - 1/2*b *log((x^2 + c)/(x^2 - c))/x - a/x
Time = 3.42 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.28 \[ \int \frac {a+b \text {arctanh}\left (\frac {c}{x^2}\right )}{x^2} \, dx=\frac {b\,\mathrm {atan}\left (\frac {x}{\sqrt {c}}\right )}{\sqrt {c}}-\frac {a}{x}-\frac {b\,\ln \left (x^2+c\right )}{2\,x}+\frac {b\,\ln \left (x^2-c\right )}{2\,x}-\frac {b\,\mathrm {atan}\left (\frac {x\,1{}\mathrm {i}}{\sqrt {c}}\right )\,1{}\mathrm {i}}{\sqrt {c}} \]