Integrand size = 14, antiderivative size = 46 \[ \int \frac {a+b \text {arctanh}\left (\frac {c}{x^2}\right )}{x^2} \, dx=\frac {b \arctan \left (\frac {x}{\sqrt {c}}\right )}{\sqrt {c}}-\frac {a+b \text {arctanh}\left (\frac {c}{x^2}\right )}{x}+\frac {b \text {arctanh}\left (\frac {x}{\sqrt {c}}\right )}{\sqrt {c}} \] Output:
b*arctan(x/c^(1/2))/c^(1/2)-(a+b*arctanh(c/x^2))/x+b*arctanh(x/c^(1/2))/c^ (1/2)
Time = 0.03 (sec) , antiderivative size = 72, normalized size of antiderivative = 1.57 \[ \int \frac {a+b \text {arctanh}\left (\frac {c}{x^2}\right )}{x^2} \, dx=-\frac {a}{x}+\frac {b \arctan \left (\frac {x}{\sqrt {c}}\right )}{\sqrt {c}}-\frac {b \text {arctanh}\left (\frac {c}{x^2}\right )}{x}-\frac {b \log \left (\sqrt {c}-x\right )}{2 \sqrt {c}}+\frac {b \log \left (\sqrt {c}+x\right )}{2 \sqrt {c}} \] Input:
Integrate[(a + b*ArcTanh[c/x^2])/x^2,x]
Output:
-(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 = 0.22 (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 |
\(\displaystyle -\frac {a+b \text {arctanh}\left (\frac {c}{x^2}\right )}{x}-2 b c \left (-\frac {\arctan \left (\frac {x}{\sqrt {c}}\right )}{2 c^{3/2}}-\frac {\text {arctanh}\left (\frac {x}{\sqrt {c}}\right )}{2 c^{3/2}}\right )\) |
Input:
Int[(a + b*ArcTanh[c/x^2])/x^2,x]
Output:
-((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)))
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 = 0.39 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.96
method | result | size |
parts | \(-\frac {a}{x}-\frac {b \,\operatorname {arctanh}\left (\frac {c}{x^{2}}\right )}{x}+\frac {b \arctan \left (\frac {x}{\sqrt {c}}\right )}{\sqrt {c}}+\frac {b \,\operatorname {arctanh}\left (\frac {\sqrt {c}}{x}\right )}{\sqrt {c}}\) | \(44\) |
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 | \(-\frac {b \ln \left (x^{2}+c \right )}{2 x}-\frac {-i \pi b c \,\operatorname {csgn}\left (\frac {i}{x^{2}}\right ) {\operatorname {csgn}\left (\frac {i \left (-x^{2}+c \right )}{x^{2}}\right )}^{2}-2 i \pi b c -i \pi b c {\operatorname {csgn}\left (\frac {i \left (-x^{2}+c \right )}{x^{2}}\right )}^{3}+2 i \pi b c {\operatorname {csgn}\left (\frac {i \left (-x^{2}+c \right )}{x^{2}}\right )}^{2}-i \pi b c \,\operatorname {csgn}\left (i \left (-x^{2}+c \right )\right ) {\operatorname {csgn}\left (\frac {i \left (-x^{2}+c \right )}{x^{2}}\right )}^{2}-i \pi b c \,\operatorname {csgn}\left (\frac {i}{x^{2}}\right ) \operatorname {csgn}\left (i \left (x^{2}+c \right )\right ) \operatorname {csgn}\left (\frac {i \left (x^{2}+c \right )}{x^{2}}\right )+i \pi b c \,\operatorname {csgn}\left (\frac {i}{x^{2}}\right ) {\operatorname {csgn}\left (\frac {i \left (x^{2}+c \right )}{x^{2}}\right )}^{2}+i \pi b c \,\operatorname {csgn}\left (i \left (x^{2}+c \right )\right ) {\operatorname {csgn}\left (\frac {i \left (x^{2}+c \right )}{x^{2}}\right )}^{2}-i \pi b c {\operatorname {csgn}\left (\frac {i \left (x^{2}+c \right )}{x^{2}}\right )}^{3}+i \pi b c \,\operatorname {csgn}\left (\frac {i}{x^{2}}\right ) \operatorname {csgn}\left (i \left (-x^{2}+c \right )\right ) \operatorname {csgn}\left (\frac {i \left (-x^{2}+c \right )}{x^{2}}\right )-2 \sqrt {-c}\, b \ln \left (-\sqrt {-c}\, x +c \right ) x +2 \sqrt {-c}\, b \ln \left (-\sqrt {-c}\, x -c \right ) x -2 \sqrt {c}\, b \ln \left (-\sqrt {c}-x \right ) x +2 \sqrt {c}\, b \ln \left (\sqrt {c}-x \right ) x -2 \ln \left (-x^{2}+c \right ) b c +4 a c}{4 c x}\) | \(373\) |
Input:
int((a+b*arctanh(c/x^2))/x^2,x,method=_RETURNVERBOSE)
Output:
-a/x-b/x*arctanh(c/x^2)+b*arctan(x/c^(1/2))/c^(1/2)+b/c^(1/2)*arctanh(1/x* c^(1/2))
Leaf count of result is larger than twice the leaf count of optimal. 84 vs. \(2 (38) = 76\).
Time = 0.09 (sec) , antiderivative size = 159, normalized size of antiderivative = 3.46 \[ \int \frac {a+b \text {arctanh}\left (\frac {c}{x^2}\right )}{x^2} \, dx=\left [\frac {2 \, b \sqrt {c} x \arctan \left (\frac {x}{\sqrt {c}}\right ) + b \sqrt {c} x \log \left (\frac {x^{2} + 2 \, \sqrt {c} x + c}{x^{2} - c}\right ) - b c \log \left (\frac {x^{2} + c}{x^{2} - c}\right ) - 2 \, a c}{2 \, c x}, -\frac {2 \, b \sqrt {-c} x \arctan \left (\frac {\sqrt {-c} x}{c}\right ) + b \sqrt {-c} x \log \left (\frac {x^{2} - 2 \, \sqrt {-c} x - c}{x^{2} + c}\right ) + b c \log \left (\frac {x^{2} + c}{x^{2} - c}\right ) + 2 \, a c}{2 \, c x}\right ] \] Input:
integrate((a+b*arctanh(c/x^2))/x^2,x, algorithm="fricas")
Output:
[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. \(2 (41) = 82\).
Time = 3.53 (sec) , antiderivative size = 886, normalized size of antiderivative = 19.26 \[ \int \frac {a+b \text {arctanh}\left (\frac {c}{x^2}\right )}{x^2} \, dx =\text {Too large to display} \] Input:
integrate((a+b*atanh(c/x**2))/x**2,x)
Output:
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 = 0.11 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.24 \[ \int \frac {a+b \text {arctanh}\left (\frac {c}{x^2}\right )}{x^2} \, dx=\frac {1}{2} \, {\left (c {\left (\frac {2 \, \arctan \left (\frac {x}{\sqrt {c}}\right )}{c^{\frac {3}{2}}} - \frac {\log \left (\frac {x - \sqrt {c}}{x + \sqrt {c}}\right )}{c^{\frac {3}{2}}}\right )} - \frac {2 \, \operatorname {artanh}\left (\frac {c}{x^{2}}\right )}{x}\right )} b - \frac {a}{x} \] Input:
integrate((a+b*arctanh(c/x^2))/x^2,x, algorithm="maxima")
Output:
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 = 0.14 (sec) , antiderivative size = 62, normalized size of antiderivative = 1.35 \[ \int \frac {a+b \text {arctanh}\left (\frac {c}{x^2}\right )}{x^2} \, dx=-b c {\left (\frac {\arctan \left (\frac {x}{\sqrt {-c}}\right )}{\sqrt {-c} c} - \frac {\arctan \left (\frac {x}{\sqrt {c}}\right )}{c^{\frac {3}{2}}}\right )} - \frac {b \log \left (\frac {x^{2} + c}{x^{2} - c}\right )}{2 \, x} - \frac {a}{x} \] Input:
integrate((a+b*arctanh(c/x^2))/x^2,x, algorithm="giac")
Output:
-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 = 3.82 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.28 \[ \int \frac {a+b \text {arctanh}\left (\frac {c}{x^2}\right )}{x^2} \, dx=\frac {b\,\mathrm {atan}\left (\frac {x}{\sqrt {c}}\right )}{\sqrt {c}}-\frac {a}{x}-\frac {b\,\ln \left (x^2+c\right )}{2\,x}+\frac {b\,\ln \left (x^2-c\right )}{2\,x}-\frac {b\,\mathrm {atan}\left (\frac {x\,1{}\mathrm {i}}{\sqrt {c}}\right )\,1{}\mathrm {i}}{\sqrt {c}} \] Input:
int((a + b*atanh(c/x^2))/x^2,x)
Output:
(b*atan(x/c^(1/2)))/c^(1/2) - a/x - (b*atan((x*1i)/c^(1/2))*1i)/c^(1/2) - (b*log(c + x^2))/(2*x) + (b*log(x^2 - c))/(2*x)
Time = 0.17 (sec) , antiderivative size = 72, normalized size of antiderivative = 1.57 \[ \int \frac {a+b \text {arctanh}\left (\frac {c}{x^2}\right )}{x^2} \, dx=\frac {2 \sqrt {c}\, \mathit {atan} \left (\frac {x}{\sqrt {c}}\right ) b x -2 \sqrt {c}\, \mathit {atanh} \left (\frac {c}{x^{2}}\right ) b x -2 \mathit {atanh} \left (\frac {c}{x^{2}}\right ) b c -2 \sqrt {c}\, \mathrm {log}\left (\sqrt {c}-x \right ) b x +\sqrt {c}\, \mathrm {log}\left (x^{2}+c \right ) b x -2 a c}{2 c x} \] Input:
int((a+b*atanh(c/x^2))/x^2,x)
Output:
(2*sqrt(c)*atan(x/sqrt(c))*b*x - 2*sqrt(c)*atanh(c/x**2)*b*x - 2*atanh(c/x **2)*b*c - 2*sqrt(c)*log(sqrt(c) - x)*b*x + sqrt(c)*log(c + x**2)*b*x - 2* a*c)/(2*c*x)