Integrand size = 18, antiderivative size = 85 \[ \int \frac {\left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2}{x^2} \, dx=-\frac {2 b c \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )}{\sqrt {x}}+c^2 \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2-\frac {\left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2}{x}+b^2 c^2 \log (x)-b^2 c^2 \log \left (1-c^2 x\right ) \]
c^2*(a+b*arctanh(c*x^(1/2)))^2-(a+b*arctanh(c*x^(1/2)))^2/x+b^2*c^2*ln(x)- b^2*c^2*ln(-c^2*x+1)-2*b*c*(a+b*arctanh(c*x^(1/2)))/x^(1/2)
Time = 0.08 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.52 \[ \int \frac {\left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2}{x^2} \, dx=-\frac {a^2+2 a b c \sqrt {x}+2 b \left (a+b c \sqrt {x}\right ) \text {arctanh}\left (c \sqrt {x}\right )-b^2 \left (-1+c^2 x\right ) \text {arctanh}\left (c \sqrt {x}\right )^2+b (a+b) c^2 x \log \left (1-c \sqrt {x}\right )-a b c^2 x \log \left (1+c \sqrt {x}\right )+b^2 c^2 x \log \left (1+c \sqrt {x}\right )-b^2 c^2 x \log (x)}{x} \]
-((a^2 + 2*a*b*c*Sqrt[x] + 2*b*(a + b*c*Sqrt[x])*ArcTanh[c*Sqrt[x]] - b^2* (-1 + c^2*x)*ArcTanh[c*Sqrt[x]]^2 + b*(a + b)*c^2*x*Log[1 - c*Sqrt[x]] - a *b*c^2*x*Log[1 + c*Sqrt[x]] + b^2*c^2*x*Log[1 + c*Sqrt[x]] - b^2*c^2*x*Log [x])/x)
Time = 0.61 (sec) , antiderivative size = 89, normalized size of antiderivative = 1.05, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6454, 6452, 6544, 6452, 243, 47, 14, 16, 6510}
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.
| \(\displaystyle \int \frac {\left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2}{x^2} \, dx\) |
| \(\Big \downarrow \) 6454 |
| \(\displaystyle 2 \int \frac {\left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2}{x^{3/2}}d\sqrt {x}\) |
| \(\Big \downarrow \) 6452 |
| \(\displaystyle 2 \left (b c \int \frac {a+b \text {arctanh}\left (c \sqrt {x}\right )}{x \left (1-c^2 x\right )}d\sqrt {x}-\frac {\left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2}{2 x}\right )\) |
| \(\Big \downarrow \) 6544 |
| \(\displaystyle 2 \left (b c \left (c^2 \int \frac {a+b \text {arctanh}\left (c \sqrt {x}\right )}{1-c^2 x}d\sqrt {x}+\int \frac {a+b \text {arctanh}\left (c \sqrt {x}\right )}{x}d\sqrt {x}\right )-\frac {\left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2}{2 x}\right )\) |
| \(\Big \downarrow \) 6452 |
| \(\displaystyle 2 \left (b c \left (c^2 \int \frac {a+b \text {arctanh}\left (c \sqrt {x}\right )}{1-c^2 x}d\sqrt {x}+b c \int \frac {1}{\sqrt {x} \left (1-c^2 x\right )}d\sqrt {x}-\frac {a+b \text {arctanh}\left (c \sqrt {x}\right )}{\sqrt {x}}\right )-\frac {\left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2}{2 x}\right )\) |
| \(\Big \downarrow \) 243 |
| \(\displaystyle 2 \left (b c \left (c^2 \int \frac {a+b \text {arctanh}\left (c \sqrt {x}\right )}{1-c^2 x}d\sqrt {x}+\frac {1}{2} b c \int \frac {1}{\sqrt {x} \left (1-c^2 x\right )}dx-\frac {a+b \text {arctanh}\left (c \sqrt {x}\right )}{\sqrt {x}}\right )-\frac {\left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2}{2 x}\right )\) |
| \(\Big \downarrow \) 47 |
| \(\displaystyle 2 \left (b c \left (c^2 \int \frac {a+b \text {arctanh}\left (c \sqrt {x}\right )}{1-c^2 x}d\sqrt {x}+\frac {1}{2} b c \left (c^2 \int \frac {1}{1-c^2 x}dx+\int \frac {1}{\sqrt {x}}dx\right )-\frac {a+b \text {arctanh}\left (c \sqrt {x}\right )}{\sqrt {x}}\right )-\frac {\left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2}{2 x}\right )\) |
| \(\Big \downarrow \) 14 |
| \(\displaystyle 2 \left (b c \left (c^2 \int \frac {a+b \text {arctanh}\left (c \sqrt {x}\right )}{1-c^2 x}d\sqrt {x}+\frac {1}{2} b c \left (c^2 \int \frac {1}{1-c^2 x}dx+\log (x)\right )-\frac {a+b \text {arctanh}\left (c \sqrt {x}\right )}{\sqrt {x}}\right )-\frac {\left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2}{2 x}\right )\) |
| \(\Big \downarrow \) 16 |
| \(\displaystyle 2 \left (b c \left (c^2 \int \frac {a+b \text {arctanh}\left (c \sqrt {x}\right )}{1-c^2 x}d\sqrt {x}-\frac {a+b \text {arctanh}\left (c \sqrt {x}\right )}{\sqrt {x}}+\frac {1}{2} b c \left (\log (x)-\log \left (1-c^2 x\right )\right )\right )-\frac {\left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2}{2 x}\right )\) |
| \(\Big \downarrow \) 6510 |
| \(\displaystyle 2 \left (b c \left (\frac {c \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2}{2 b}-\frac {a+b \text {arctanh}\left (c \sqrt {x}\right )}{\sqrt {x}}+\frac {1}{2} b c \left (\log (x)-\log \left (1-c^2 x\right )\right )\right )-\frac {\left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2}{2 x}\right )\) |
2*(-1/2*(a + b*ArcTanh[c*Sqrt[x]])^2/x + b*c*(-((a + b*ArcTanh[c*Sqrt[x]]) /Sqrt[x]) + (c*(a + b*ArcTanh[c*Sqrt[x]])^2)/(2*b) + (b*c*(Log[x] - Log[1 - c^2*x]))/2))
3.2.100.3.1 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[(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_)^(n_)]*(b_.))^(p_.)*(x_)^(m_.), x_Symbol] :> Simp[1/n Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*ArcTanh[c*x])^p, x ], x, x^n], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 1] && IntegerQ[Simpl ify[(m + 1)/n]]
Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)^2), x_Symb ol] :> Simp[(a + b*ArcTanh[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_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_))/((d_) + ( e_.)*(x_)^2), x_Symbol] :> Simp[1/d Int[(f*x)^m*(a + b*ArcTanh[c*x])^p, x ], x] - Simp[e/(d*f^2) Int[(f*x)^(m + 2)*((a + b*ArcTanh[c*x])^p/(d + e*x ^2)), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && GtQ[p, 0] && LtQ[m, -1]
Leaf count of result is larger than twice the leaf count of optimal. \(230\) vs. \(2(77)=154\).
Time = 1.00 (sec) , antiderivative size = 231, normalized size of antiderivative = 2.72
| method | result | size |
| parts | \(-\frac {a^{2}}{x}+2 b^{2} c^{2} \left (-\frac {\operatorname {arctanh}\left (c \sqrt {x}\right )^{2}}{2 c^{2} x}-\frac {\operatorname {arctanh}\left (c \sqrt {x}\right )}{c \sqrt {x}}-\frac {\operatorname {arctanh}\left (c \sqrt {x}\right ) \ln \left (c \sqrt {x}-1\right )}{2}+\frac {\operatorname {arctanh}\left (c \sqrt {x}\right ) \ln \left (1+c \sqrt {x}\right )}{2}-\frac {\ln \left (c \sqrt {x}-1\right )^{2}}{8}+\frac {\ln \left (c \sqrt {x}-1\right ) \ln \left (\frac {c \sqrt {x}}{2}+\frac {1}{2}\right )}{4}-\frac {\ln \left (c \sqrt {x}-1\right )}{2}-\frac {\ln \left (1+c \sqrt {x}\right )}{2}+\ln \left (c \sqrt {x}\right )+\frac {\left (\ln \left (1+c \sqrt {x}\right )-\ln \left (\frac {c \sqrt {x}}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {c \sqrt {x}}{2}+\frac {1}{2}\right )}{4}-\frac {\ln \left (1+c \sqrt {x}\right )^{2}}{8}\right )+4 a b \,c^{2} \left (-\frac {\operatorname {arctanh}\left (c \sqrt {x}\right )}{2 c^{2} x}-\frac {1}{2 c \sqrt {x}}-\frac {\ln \left (c \sqrt {x}-1\right )}{4}+\frac {\ln \left (1+c \sqrt {x}\right )}{4}\right )\) | \(231\) |
| derivativedivides | \(2 c^{2} \left (-\frac {a^{2}}{2 c^{2} x}+b^{2} \left (-\frac {\operatorname {arctanh}\left (c \sqrt {x}\right )^{2}}{2 c^{2} x}-\frac {\operatorname {arctanh}\left (c \sqrt {x}\right )}{c \sqrt {x}}-\frac {\operatorname {arctanh}\left (c \sqrt {x}\right ) \ln \left (c \sqrt {x}-1\right )}{2}+\frac {\operatorname {arctanh}\left (c \sqrt {x}\right ) \ln \left (1+c \sqrt {x}\right )}{2}-\frac {\ln \left (c \sqrt {x}-1\right )^{2}}{8}+\frac {\ln \left (c \sqrt {x}-1\right ) \ln \left (\frac {c \sqrt {x}}{2}+\frac {1}{2}\right )}{4}-\frac {\ln \left (c \sqrt {x}-1\right )}{2}-\frac {\ln \left (1+c \sqrt {x}\right )}{2}+\ln \left (c \sqrt {x}\right )+\frac {\left (\ln \left (1+c \sqrt {x}\right )-\ln \left (\frac {c \sqrt {x}}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {c \sqrt {x}}{2}+\frac {1}{2}\right )}{4}-\frac {\ln \left (1+c \sqrt {x}\right )^{2}}{8}\right )+2 a b \left (-\frac {\operatorname {arctanh}\left (c \sqrt {x}\right )}{2 c^{2} x}-\frac {1}{2 c \sqrt {x}}-\frac {\ln \left (c \sqrt {x}-1\right )}{4}+\frac {\ln \left (1+c \sqrt {x}\right )}{4}\right )\right )\) | \(232\) |
| default | \(2 c^{2} \left (-\frac {a^{2}}{2 c^{2} x}+b^{2} \left (-\frac {\operatorname {arctanh}\left (c \sqrt {x}\right )^{2}}{2 c^{2} x}-\frac {\operatorname {arctanh}\left (c \sqrt {x}\right )}{c \sqrt {x}}-\frac {\operatorname {arctanh}\left (c \sqrt {x}\right ) \ln \left (c \sqrt {x}-1\right )}{2}+\frac {\operatorname {arctanh}\left (c \sqrt {x}\right ) \ln \left (1+c \sqrt {x}\right )}{2}-\frac {\ln \left (c \sqrt {x}-1\right )^{2}}{8}+\frac {\ln \left (c \sqrt {x}-1\right ) \ln \left (\frac {c \sqrt {x}}{2}+\frac {1}{2}\right )}{4}-\frac {\ln \left (c \sqrt {x}-1\right )}{2}-\frac {\ln \left (1+c \sqrt {x}\right )}{2}+\ln \left (c \sqrt {x}\right )+\frac {\left (\ln \left (1+c \sqrt {x}\right )-\ln \left (\frac {c \sqrt {x}}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {c \sqrt {x}}{2}+\frac {1}{2}\right )}{4}-\frac {\ln \left (1+c \sqrt {x}\right )^{2}}{8}\right )+2 a b \left (-\frac {\operatorname {arctanh}\left (c \sqrt {x}\right )}{2 c^{2} x}-\frac {1}{2 c \sqrt {x}}-\frac {\ln \left (c \sqrt {x}-1\right )}{4}+\frac {\ln \left (1+c \sqrt {x}\right )}{4}\right )\right )\) | \(232\) |
-a^2/x+2*b^2*c^2*(-1/2/c^2/x*arctanh(c*x^(1/2))^2-arctanh(c*x^(1/2))/c/x^( 1/2)-1/2*arctanh(c*x^(1/2))*ln(c*x^(1/2)-1)+1/2*arctanh(c*x^(1/2))*ln(1+c* x^(1/2))-1/8*ln(c*x^(1/2)-1)^2+1/4*ln(c*x^(1/2)-1)*ln(1/2*c*x^(1/2)+1/2)-1 /2*ln(c*x^(1/2)-1)-1/2*ln(1+c*x^(1/2))+ln(c*x^(1/2))+1/4*(ln(1+c*x^(1/2))- ln(1/2*c*x^(1/2)+1/2))*ln(-1/2*c*x^(1/2)+1/2)-1/8*ln(1+c*x^(1/2))^2)+4*a*b *c^2*(-1/2/c^2/x*arctanh(c*x^(1/2))-1/2/c/x^(1/2)-1/4*ln(c*x^(1/2)-1)+1/4* ln(1+c*x^(1/2)))
Leaf count of result is larger than twice the leaf count of optimal. 157 vs. \(2 (77) = 154\).
Time = 0.27 (sec) , antiderivative size = 157, normalized size of antiderivative = 1.85 \[ \int \frac {\left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2}{x^2} \, dx=\frac {8 \, b^{2} c^{2} x \log \left (\sqrt {x}\right ) + 4 \, {\left (a b - b^{2}\right )} c^{2} x \log \left (c \sqrt {x} + 1\right ) - 4 \, {\left (a b + b^{2}\right )} c^{2} x \log \left (c \sqrt {x} - 1\right ) - 8 \, a b c \sqrt {x} + {\left (b^{2} c^{2} x - b^{2}\right )} \log \left (-\frac {c^{2} x + 2 \, c \sqrt {x} + 1}{c^{2} x - 1}\right )^{2} - 4 \, a^{2} - 4 \, {\left (b^{2} c \sqrt {x} + a b\right )} \log \left (-\frac {c^{2} x + 2 \, c \sqrt {x} + 1}{c^{2} x - 1}\right )}{4 \, x} \]
1/4*(8*b^2*c^2*x*log(sqrt(x)) + 4*(a*b - b^2)*c^2*x*log(c*sqrt(x) + 1) - 4 *(a*b + b^2)*c^2*x*log(c*sqrt(x) - 1) - 8*a*b*c*sqrt(x) + (b^2*c^2*x - b^2 )*log(-(c^2*x + 2*c*sqrt(x) + 1)/(c^2*x - 1))^2 - 4*a^2 - 4*(b^2*c*sqrt(x) + a*b)*log(-(c^2*x + 2*c*sqrt(x) + 1)/(c^2*x - 1)))/x
Leaf count of result is larger than twice the leaf count of optimal. 680 vs. \(2 (78) = 156\).
Time = 2.62 (sec) , antiderivative size = 680, normalized size of antiderivative = 8.00 \[ \int \frac {\left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2}{x^2} \, dx=\begin {cases} - \frac {a^{2}}{x} & \text {for}\: c = 0 \\- \frac {a^{2}}{x} + \frac {2 a b \operatorname {atanh}{\left (\sqrt {x} \sqrt {\frac {1}{x}} \right )}}{x} - \frac {b^{2} \operatorname {atanh}^{2}{\left (\sqrt {x} \sqrt {\frac {1}{x}} \right )}}{x} & \text {for}\: c = - \sqrt {\frac {1}{x}} \\- \frac {a^{2}}{x} - \frac {2 a b \operatorname {atanh}{\left (\sqrt {x} \sqrt {\frac {1}{x}} \right )}}{x} - \frac {b^{2} \operatorname {atanh}^{2}{\left (\sqrt {x} \sqrt {\frac {1}{x}} \right )}}{x} & \text {for}\: c = \sqrt {\frac {1}{x}} \\- \frac {a^{2} c^{2} x^{\frac {3}{2}}}{c^{2} x^{\frac {5}{2}} - x^{\frac {3}{2}}} + \frac {a^{2} \sqrt {x}}{c^{2} x^{\frac {5}{2}} - x^{\frac {3}{2}}} + \frac {2 a b c^{4} x^{\frac {5}{2}} \operatorname {atanh}{\left (c \sqrt {x} \right )}}{c^{2} x^{\frac {5}{2}} - x^{\frac {3}{2}}} - \frac {2 a b c^{3} x^{2}}{c^{2} x^{\frac {5}{2}} - x^{\frac {3}{2}}} - \frac {4 a b c^{2} x^{\frac {3}{2}} \operatorname {atanh}{\left (c \sqrt {x} \right )}}{c^{2} x^{\frac {5}{2}} - x^{\frac {3}{2}}} + \frac {2 a b c x}{c^{2} x^{\frac {5}{2}} - x^{\frac {3}{2}}} + \frac {2 a b \sqrt {x} \operatorname {atanh}{\left (c \sqrt {x} \right )}}{c^{2} x^{\frac {5}{2}} - x^{\frac {3}{2}}} + \frac {b^{2} c^{4} x^{\frac {5}{2}} \log {\left (x \right )}}{c^{2} x^{\frac {5}{2}} - x^{\frac {3}{2}}} - \frac {2 b^{2} c^{4} x^{\frac {5}{2}} \log {\left (\sqrt {x} - \frac {1}{c} \right )}}{c^{2} x^{\frac {5}{2}} - x^{\frac {3}{2}}} + \frac {b^{2} c^{4} x^{\frac {5}{2}} \operatorname {atanh}^{2}{\left (c \sqrt {x} \right )}}{c^{2} x^{\frac {5}{2}} - x^{\frac {3}{2}}} - \frac {2 b^{2} c^{4} x^{\frac {5}{2}} \operatorname {atanh}{\left (c \sqrt {x} \right )}}{c^{2} x^{\frac {5}{2}} - x^{\frac {3}{2}}} - \frac {2 b^{2} c^{3} x^{2} \operatorname {atanh}{\left (c \sqrt {x} \right )}}{c^{2} x^{\frac {5}{2}} - x^{\frac {3}{2}}} - \frac {b^{2} c^{2} x^{\frac {3}{2}} \log {\left (x \right )}}{c^{2} x^{\frac {5}{2}} - x^{\frac {3}{2}}} + \frac {2 b^{2} c^{2} x^{\frac {3}{2}} \log {\left (\sqrt {x} - \frac {1}{c} \right )}}{c^{2} x^{\frac {5}{2}} - x^{\frac {3}{2}}} - \frac {2 b^{2} c^{2} x^{\frac {3}{2}} \operatorname {atanh}^{2}{\left (c \sqrt {x} \right )}}{c^{2} x^{\frac {5}{2}} - x^{\frac {3}{2}}} + \frac {2 b^{2} c^{2} x^{\frac {3}{2}} \operatorname {atanh}{\left (c \sqrt {x} \right )}}{c^{2} x^{\frac {5}{2}} - x^{\frac {3}{2}}} + \frac {2 b^{2} c x \operatorname {atanh}{\left (c \sqrt {x} \right )}}{c^{2} x^{\frac {5}{2}} - x^{\frac {3}{2}}} + \frac {b^{2} \sqrt {x} \operatorname {atanh}^{2}{\left (c \sqrt {x} \right )}}{c^{2} x^{\frac {5}{2}} - x^{\frac {3}{2}}} & \text {otherwise} \end {cases} \]
Piecewise((-a**2/x, Eq(c, 0)), (-a**2/x + 2*a*b*atanh(sqrt(x)*sqrt(1/x))/x - b**2*atanh(sqrt(x)*sqrt(1/x))**2/x, Eq(c, -sqrt(1/x))), (-a**2/x - 2*a* b*atanh(sqrt(x)*sqrt(1/x))/x - b**2*atanh(sqrt(x)*sqrt(1/x))**2/x, Eq(c, s qrt(1/x))), (-a**2*c**2*x**(3/2)/(c**2*x**(5/2) - x**(3/2)) + a**2*sqrt(x) /(c**2*x**(5/2) - x**(3/2)) + 2*a*b*c**4*x**(5/2)*atanh(c*sqrt(x))/(c**2*x **(5/2) - x**(3/2)) - 2*a*b*c**3*x**2/(c**2*x**(5/2) - x**(3/2)) - 4*a*b*c **2*x**(3/2)*atanh(c*sqrt(x))/(c**2*x**(5/2) - x**(3/2)) + 2*a*b*c*x/(c**2 *x**(5/2) - x**(3/2)) + 2*a*b*sqrt(x)*atanh(c*sqrt(x))/(c**2*x**(5/2) - x* *(3/2)) + b**2*c**4*x**(5/2)*log(x)/(c**2*x**(5/2) - x**(3/2)) - 2*b**2*c* *4*x**(5/2)*log(sqrt(x) - 1/c)/(c**2*x**(5/2) - x**(3/2)) + b**2*c**4*x**( 5/2)*atanh(c*sqrt(x))**2/(c**2*x**(5/2) - x**(3/2)) - 2*b**2*c**4*x**(5/2) *atanh(c*sqrt(x))/(c**2*x**(5/2) - x**(3/2)) - 2*b**2*c**3*x**2*atanh(c*sq rt(x))/(c**2*x**(5/2) - x**(3/2)) - b**2*c**2*x**(3/2)*log(x)/(c**2*x**(5/ 2) - x**(3/2)) + 2*b**2*c**2*x**(3/2)*log(sqrt(x) - 1/c)/(c**2*x**(5/2) - x**(3/2)) - 2*b**2*c**2*x**(3/2)*atanh(c*sqrt(x))**2/(c**2*x**(5/2) - x**( 3/2)) + 2*b**2*c**2*x**(3/2)*atanh(c*sqrt(x))/(c**2*x**(5/2) - x**(3/2)) + 2*b**2*c*x*atanh(c*sqrt(x))/(c**2*x**(5/2) - x**(3/2)) + b**2*sqrt(x)*ata nh(c*sqrt(x))**2/(c**2*x**(5/2) - x**(3/2)), True))
Leaf count of result is larger than twice the leaf count of optimal. 174 vs. \(2 (77) = 154\).
Time = 0.20 (sec) , antiderivative size = 174, normalized size of antiderivative = 2.05 \[ \int \frac {\left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2}{x^2} \, dx={\left ({\left (c \log \left (c \sqrt {x} + 1\right ) - c \log \left (c \sqrt {x} - 1\right ) - \frac {2}{\sqrt {x}}\right )} c - \frac {2 \, \operatorname {artanh}\left (c \sqrt {x}\right )}{x}\right )} a b + \frac {1}{4} \, {\left ({\left (2 \, {\left (\log \left (c \sqrt {x} - 1\right ) - 2\right )} \log \left (c \sqrt {x} + 1\right ) - \log \left (c \sqrt {x} + 1\right )^{2} - \log \left (c \sqrt {x} - 1\right )^{2} - 4 \, \log \left (c \sqrt {x} - 1\right ) + 4 \, \log \left (x\right )\right )} c^{2} + 4 \, {\left (c \log \left (c \sqrt {x} + 1\right ) - c \log \left (c \sqrt {x} - 1\right ) - \frac {2}{\sqrt {x}}\right )} c \operatorname {artanh}\left (c \sqrt {x}\right )\right )} b^{2} - \frac {b^{2} \operatorname {artanh}\left (c \sqrt {x}\right )^{2}}{x} - \frac {a^{2}}{x} \]
((c*log(c*sqrt(x) + 1) - c*log(c*sqrt(x) - 1) - 2/sqrt(x))*c - 2*arctanh(c *sqrt(x))/x)*a*b + 1/4*((2*(log(c*sqrt(x) - 1) - 2)*log(c*sqrt(x) + 1) - l og(c*sqrt(x) + 1)^2 - log(c*sqrt(x) - 1)^2 - 4*log(c*sqrt(x) - 1) + 4*log( x))*c^2 + 4*(c*log(c*sqrt(x) + 1) - c*log(c*sqrt(x) - 1) - 2/sqrt(x))*c*ar ctanh(c*sqrt(x)))*b^2 - b^2*arctanh(c*sqrt(x))^2/x - a^2/x
\[ \int \frac {\left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2}{x^2} \, dx=\int { \frac {{\left (b \operatorname {artanh}\left (c \sqrt {x}\right ) + a\right )}^{2}}{x^{2}} \,d x } \]
Time = 4.23 (sec) , antiderivative size = 278, normalized size of antiderivative = 3.27 \[ \int \frac {\left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2}{x^2} \, dx=2\,b^2\,c^2\,\ln \left (\sqrt {x}\right )-\frac {a^2}{x}-b^2\,c^2\,\ln \left (c\,\sqrt {x}-1\right )-b^2\,c^2\,\ln \left (c\,\sqrt {x}+1\right )+\frac {b^2\,c^2\,{\ln \left (c\,\sqrt {x}+1\right )}^2}{4}+\frac {b^2\,c^2\,{\ln \left (1-c\,\sqrt {x}\right )}^2}{4}-\frac {b^2\,{\ln \left (c\,\sqrt {x}+1\right )}^2}{4\,x}-\frac {b^2\,{\ln \left (1-c\,\sqrt {x}\right )}^2}{4\,x}-a\,b\,c^2\,\ln \left (c\,\sqrt {x}-1\right )+a\,b\,c^2\,\ln \left (c\,\sqrt {x}+1\right )-\frac {2\,a\,b\,c}{\sqrt {x}}-\frac {a\,b\,\ln \left (c\,\sqrt {x}+1\right )}{x}+\frac {a\,b\,\ln \left (1-c\,\sqrt {x}\right )}{x}-\frac {b^2\,c^2\,\ln \left (c\,\sqrt {x}+1\right )\,\ln \left (1-c\,\sqrt {x}\right )}{2}-\frac {b^2\,c\,\ln \left (c\,\sqrt {x}+1\right )}{\sqrt {x}}+\frac {b^2\,c\,\ln \left (1-c\,\sqrt {x}\right )}{\sqrt {x}}+\frac {b^2\,\ln \left (c\,\sqrt {x}+1\right )\,\ln \left (1-c\,\sqrt {x}\right )}{2\,x} \]
2*b^2*c^2*log(x^(1/2)) - a^2/x - b^2*c^2*log(c*x^(1/2) - 1) - b^2*c^2*log( c*x^(1/2) + 1) + (b^2*c^2*log(c*x^(1/2) + 1)^2)/4 + (b^2*c^2*log(1 - c*x^( 1/2))^2)/4 - (b^2*log(c*x^(1/2) + 1)^2)/(4*x) - (b^2*log(1 - c*x^(1/2))^2) /(4*x) - a*b*c^2*log(c*x^(1/2) - 1) + a*b*c^2*log(c*x^(1/2) + 1) - (2*a*b* c)/x^(1/2) - (a*b*log(c*x^(1/2) + 1))/x + (a*b*log(1 - c*x^(1/2)))/x - (b^ 2*c^2*log(c*x^(1/2) + 1)*log(1 - c*x^(1/2)))/2 - (b^2*c*log(c*x^(1/2) + 1) )/x^(1/2) + (b^2*c*log(1 - c*x^(1/2)))/x^(1/2) + (b^2*log(c*x^(1/2) + 1)*l og(1 - c*x^(1/2)))/(2*x)