Integrand size = 18, antiderivative size = 133 \[ \int \frac {\left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2}{x^3} \, dx=-\frac {b^2 c^2}{6 x}-\frac {b c \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )}{3 x^{3/2}}-\frac {b c^3 \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )}{\sqrt {x}}+\frac {1}{2} c^4 \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2-\frac {\left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2}{2 x^2}+\frac {2}{3} b^2 c^4 \log (x)-\frac {2}{3} b^2 c^4 \log \left (1-c^2 x\right ) \]
-1/6*b^2*c^2/x-1/3*b*c*(a+b*arctanh(c*x^(1/2)))/x^(3/2)+1/2*c^4*(a+b*arcta nh(c*x^(1/2)))^2-1/2*(a+b*arctanh(c*x^(1/2)))^2/x^2+2/3*b^2*c^4*ln(x)-2/3* b^2*c^4*ln(-c^2*x+1)-b*c^3*(a+b*arctanh(c*x^(1/2)))/x^(1/2)
Time = 0.09 (sec) , antiderivative size = 178, normalized size of antiderivative = 1.34 \[ \int \frac {\left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2}{x^3} \, dx=-\frac {3 a^2+2 a b c \sqrt {x}+b^2 c^2 x+6 a b c^3 x^{3/2}+2 b \left (3 a+b c \sqrt {x} \left (1+3 c^2 x\right )\right ) \text {arctanh}\left (c \sqrt {x}\right )-3 b^2 \left (-1+c^4 x^2\right ) \text {arctanh}\left (c \sqrt {x}\right )^2+b (3 a+4 b) c^4 x^2 \log \left (1-c \sqrt {x}\right )-3 a b c^4 x^2 \log \left (1+c \sqrt {x}\right )+4 b^2 c^4 x^2 \log \left (1+c \sqrt {x}\right )-4 b^2 c^4 x^2 \log (x)}{6 x^2} \]
-1/6*(3*a^2 + 2*a*b*c*Sqrt[x] + b^2*c^2*x + 6*a*b*c^3*x^(3/2) + 2*b*(3*a + b*c*Sqrt[x]*(1 + 3*c^2*x))*ArcTanh[c*Sqrt[x]] - 3*b^2*(-1 + c^4*x^2)*ArcT anh[c*Sqrt[x]]^2 + b*(3*a + 4*b)*c^4*x^2*Log[1 - c*Sqrt[x]] - 3*a*b*c^4*x^ 2*Log[1 + c*Sqrt[x]] + 4*b^2*c^4*x^2*Log[1 + c*Sqrt[x]] - 4*b^2*c^4*x^2*Lo g[x])/x^2
Time = 1.05 (sec) , antiderivative size = 152, normalized size of antiderivative = 1.14, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.778, Rules used = {6454, 6452, 6544, 6452, 243, 54, 2009, 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.
| \(\displaystyle \int \frac {\left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2}{x^3} \, dx\) |
| \(\Big \downarrow \) 6454 |
| \(\displaystyle 2 \int \frac {\left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2}{x^{5/2}}d\sqrt {x}\) |
| \(\Big \downarrow \) 6452 |
| \(\displaystyle 2 \left (\frac {1}{2} b c \int \frac {a+b \text {arctanh}\left (c \sqrt {x}\right )}{x^2 \left (1-c^2 x\right )}d\sqrt {x}-\frac {\left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2}{4 x^2}\right )\) |
| \(\Big \downarrow \) 6544 |
| \(\displaystyle 2 \left (\frac {1}{2} b c \left (c^2 \int \frac {a+b \text {arctanh}\left (c \sqrt {x}\right )}{x \left (1-c^2 x\right )}d\sqrt {x}+\int \frac {a+b \text {arctanh}\left (c \sqrt {x}\right )}{x^2}d\sqrt {x}\right )-\frac {\left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2}{4 x^2}\right )\) |
| \(\Big \downarrow \) 6452 |
| \(\displaystyle 2 \left (\frac {1}{2} b c \left (c^2 \int \frac {a+b \text {arctanh}\left (c \sqrt {x}\right )}{x \left (1-c^2 x\right )}d\sqrt {x}+\frac {1}{3} b c \int \frac {1}{x^{3/2} \left (1-c^2 x\right )}d\sqrt {x}-\frac {a+b \text {arctanh}\left (c \sqrt {x}\right )}{3 x^{3/2}}\right )-\frac {\left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2}{4 x^2}\right )\) |
| \(\Big \downarrow \) 243 |
| \(\displaystyle 2 \left (\frac {1}{2} b c \left (c^2 \int \frac {a+b \text {arctanh}\left (c \sqrt {x}\right )}{x \left (1-c^2 x\right )}d\sqrt {x}+\frac {1}{6} b c \int \frac {1}{x \left (1-c^2 x\right )}dx-\frac {a+b \text {arctanh}\left (c \sqrt {x}\right )}{3 x^{3/2}}\right )-\frac {\left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2}{4 x^2}\right )\) |
| \(\Big \downarrow \) 54 |
| \(\displaystyle 2 \left (\frac {1}{2} b c \left (c^2 \int \frac {a+b \text {arctanh}\left (c \sqrt {x}\right )}{x \left (1-c^2 x\right )}d\sqrt {x}+\frac {1}{6} b c \int \left (-\frac {c^4}{c^2 x-1}+\frac {c^2}{\sqrt {x}}+\frac {1}{x}\right )dx-\frac {a+b \text {arctanh}\left (c \sqrt {x}\right )}{3 x^{3/2}}\right )-\frac {\left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2}{4 x^2}\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle 2 \left (\frac {1}{2} b c \left (c^2 \int \frac {a+b \text {arctanh}\left (c \sqrt {x}\right )}{x \left (1-c^2 x\right )}d\sqrt {x}-\frac {a+b \text {arctanh}\left (c \sqrt {x}\right )}{3 x^{3/2}}+\frac {1}{6} b c \left (c^2 \log (x)-c^2 \log \left (1-c^2 x\right )-\frac {1}{\sqrt {x}}\right )\right )-\frac {\left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2}{4 x^2}\right )\) |
| \(\Big \downarrow \) 6544 |
| \(\displaystyle 2 \left (\frac {1}{2} b c \left (c^2 \left (c^2 \int \frac {a+b \text {arctanh}\left (c \sqrt {x}\right )}{1-c^2 x}d\sqrt {x}+\int \frac {a+b \text {arctanh}\left (c \sqrt {x}\right )}{x}d\sqrt {x}\right )-\frac {a+b \text {arctanh}\left (c \sqrt {x}\right )}{3 x^{3/2}}+\frac {1}{6} b c \left (c^2 \log (x)-c^2 \log \left (1-c^2 x\right )-\frac {1}{\sqrt {x}}\right )\right )-\frac {\left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2}{4 x^2}\right )\) |
| \(\Big \downarrow \) 6452 |
| \(\displaystyle 2 \left (\frac {1}{2} b c \left (c^2 \left (c^2 \int \frac {a+b \text {arctanh}\left (c \sqrt {x}\right )}{1-c^2 x}d\sqrt {x}+b c \int \frac {1}{\sqrt {x} \left (1-c^2 x\right )}d\sqrt {x}-\frac {a+b \text {arctanh}\left (c \sqrt {x}\right )}{\sqrt {x}}\right )-\frac {a+b \text {arctanh}\left (c \sqrt {x}\right )}{3 x^{3/2}}+\frac {1}{6} b c \left (c^2 \log (x)-c^2 \log \left (1-c^2 x\right )-\frac {1}{\sqrt {x}}\right )\right )-\frac {\left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2}{4 x^2}\right )\) |
| \(\Big \downarrow \) 243 |
| \(\displaystyle 2 \left (\frac {1}{2} b c \left (c^2 \left (c^2 \int \frac {a+b \text {arctanh}\left (c \sqrt {x}\right )}{1-c^2 x}d\sqrt {x}+\frac {1}{2} b c \int \frac {1}{\sqrt {x} \left (1-c^2 x\right )}dx-\frac {a+b \text {arctanh}\left (c \sqrt {x}\right )}{\sqrt {x}}\right )-\frac {a+b \text {arctanh}\left (c \sqrt {x}\right )}{3 x^{3/2}}+\frac {1}{6} b c \left (c^2 \log (x)-c^2 \log \left (1-c^2 x\right )-\frac {1}{\sqrt {x}}\right )\right )-\frac {\left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2}{4 x^2}\right )\) |
| \(\Big \downarrow \) 47 |
| \(\displaystyle 2 \left (\frac {1}{2} b c \left (c^2 \left (c^2 \int \frac {a+b \text {arctanh}\left (c \sqrt {x}\right )}{1-c^2 x}d\sqrt {x}+\frac {1}{2} b c \left (c^2 \int \frac {1}{1-c^2 x}dx+\int \frac {1}{\sqrt {x}}dx\right )-\frac {a+b \text {arctanh}\left (c \sqrt {x}\right )}{\sqrt {x}}\right )-\frac {a+b \text {arctanh}\left (c \sqrt {x}\right )}{3 x^{3/2}}+\frac {1}{6} b c \left (c^2 \log (x)-c^2 \log \left (1-c^2 x\right )-\frac {1}{\sqrt {x}}\right )\right )-\frac {\left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2}{4 x^2}\right )\) |
| \(\Big \downarrow \) 14 |
| \(\displaystyle 2 \left (\frac {1}{2} b c \left (c^2 \left (c^2 \int \frac {a+b \text {arctanh}\left (c \sqrt {x}\right )}{1-c^2 x}d\sqrt {x}+\frac {1}{2} b c \left (c^2 \int \frac {1}{1-c^2 x}dx+\log (x)\right )-\frac {a+b \text {arctanh}\left (c \sqrt {x}\right )}{\sqrt {x}}\right )-\frac {a+b \text {arctanh}\left (c \sqrt {x}\right )}{3 x^{3/2}}+\frac {1}{6} b c \left (c^2 \log (x)-c^2 \log \left (1-c^2 x\right )-\frac {1}{\sqrt {x}}\right )\right )-\frac {\left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2}{4 x^2}\right )\) |
| \(\Big \downarrow \) 16 |
| \(\displaystyle 2 \left (\frac {1}{2} b c \left (c^2 \left (c^2 \int \frac {a+b \text {arctanh}\left (c \sqrt {x}\right )}{1-c^2 x}d\sqrt {x}-\frac {a+b \text {arctanh}\left (c \sqrt {x}\right )}{\sqrt {x}}+\frac {1}{2} b c \left (\log (x)-\log \left (1-c^2 x\right )\right )\right )-\frac {a+b \text {arctanh}\left (c \sqrt {x}\right )}{3 x^{3/2}}+\frac {1}{6} b c \left (c^2 \log (x)-c^2 \log \left (1-c^2 x\right )-\frac {1}{\sqrt {x}}\right )\right )-\frac {\left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2}{4 x^2}\right )\) |
| \(\Big \downarrow \) 6510 |
| \(\displaystyle 2 \left (\frac {1}{2} b c \left (c^2 \left (\frac {c \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2}{2 b}-\frac {a+b \text {arctanh}\left (c \sqrt {x}\right )}{\sqrt {x}}+\frac {1}{2} b c \left (\log (x)-\log \left (1-c^2 x\right )\right )\right )-\frac {a+b \text {arctanh}\left (c \sqrt {x}\right )}{3 x^{3/2}}+\frac {1}{6} b c \left (c^2 \log (x)-c^2 \log \left (1-c^2 x\right )-\frac {1}{\sqrt {x}}\right )\right )-\frac {\left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2}{4 x^2}\right )\) |
2*(-1/4*(a + b*ArcTanh[c*Sqrt[x]])^2/x^2 + (b*c*(-1/3*(a + b*ArcTanh[c*Sqr t[x]])/x^(3/2) + c^2*(-((a + b*ArcTanh[c*Sqrt[x]])/Sqrt[x]) + (c*(a + b*Ar cTanh[c*Sqrt[x]])^2)/(2*b) + (b*c*(Log[x] - Log[1 - c^2*x]))/2) + (b*c*(-( 1/Sqrt[x]) + c^2*Log[x] - c^2*Log[1 - c^2*x]))/6))/2)
3.3.1.3.1 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_)^(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. \(262\) vs. \(2(109)=218\).
Time = 0.99 (sec) , antiderivative size = 263, normalized size of antiderivative = 1.98
| method | result | size |
| parts | \(-\frac {a^{2}}{2 x^{2}}+2 b^{2} c^{4} \left (-\frac {\operatorname {arctanh}\left (c \sqrt {x}\right )^{2}}{4 c^{4} x^{2}}-\frac {\operatorname {arctanh}\left (c \sqrt {x}\right )}{6 c^{3} x^{\frac {3}{2}}}-\frac {\operatorname {arctanh}\left (c \sqrt {x}\right )}{2 c \sqrt {x}}-\frac {\operatorname {arctanh}\left (c \sqrt {x}\right ) \ln \left (c \sqrt {x}-1\right )}{4}+\frac {\operatorname {arctanh}\left (c \sqrt {x}\right ) \ln \left (1+c \sqrt {x}\right )}{4}+\frac {\ln \left (c \sqrt {x}-1\right ) \ln \left (\frac {c \sqrt {x}}{2}+\frac {1}{2}\right )}{8}-\frac {\ln \left (c \sqrt {x}-1\right )^{2}}{16}+\frac {\left (\ln \left (1+c \sqrt {x}\right )-\ln \left (\frac {c \sqrt {x}}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {c \sqrt {x}}{2}+\frac {1}{2}\right )}{8}-\frac {\ln \left (1+c \sqrt {x}\right )^{2}}{16}-\frac {1}{12 c^{2} x}+\frac {2 \ln \left (c \sqrt {x}\right )}{3}-\frac {\ln \left (c \sqrt {x}-1\right )}{3}-\frac {\ln \left (1+c \sqrt {x}\right )}{3}\right )+4 a b \,c^{4} \left (-\frac {\operatorname {arctanh}\left (c \sqrt {x}\right )}{4 c^{4} x^{2}}+\frac {\ln \left (1+c \sqrt {x}\right )}{8}-\frac {1}{12 c^{3} x^{\frac {3}{2}}}-\frac {1}{4 c \sqrt {x}}-\frac {\ln \left (c \sqrt {x}-1\right )}{8}\right )\) | \(263\) |
| derivativedivides | \(2 c^{4} \left (-\frac {a^{2}}{4 c^{4} x^{2}}+b^{2} \left (-\frac {\operatorname {arctanh}\left (c \sqrt {x}\right )^{2}}{4 c^{4} x^{2}}-\frac {\operatorname {arctanh}\left (c \sqrt {x}\right )}{6 c^{3} x^{\frac {3}{2}}}-\frac {\operatorname {arctanh}\left (c \sqrt {x}\right )}{2 c \sqrt {x}}-\frac {\operatorname {arctanh}\left (c \sqrt {x}\right ) \ln \left (c \sqrt {x}-1\right )}{4}+\frac {\operatorname {arctanh}\left (c \sqrt {x}\right ) \ln \left (1+c \sqrt {x}\right )}{4}+\frac {\ln \left (c \sqrt {x}-1\right ) \ln \left (\frac {c \sqrt {x}}{2}+\frac {1}{2}\right )}{8}-\frac {\ln \left (c \sqrt {x}-1\right )^{2}}{16}+\frac {\left (\ln \left (1+c \sqrt {x}\right )-\ln \left (\frac {c \sqrt {x}}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {c \sqrt {x}}{2}+\frac {1}{2}\right )}{8}-\frac {\ln \left (1+c \sqrt {x}\right )^{2}}{16}-\frac {1}{12 c^{2} x}+\frac {2 \ln \left (c \sqrt {x}\right )}{3}-\frac {\ln \left (c \sqrt {x}-1\right )}{3}-\frac {\ln \left (1+c \sqrt {x}\right )}{3}\right )+2 a b \left (-\frac {\operatorname {arctanh}\left (c \sqrt {x}\right )}{4 c^{4} x^{2}}+\frac {\ln \left (1+c \sqrt {x}\right )}{8}-\frac {1}{12 c^{3} x^{\frac {3}{2}}}-\frac {1}{4 c \sqrt {x}}-\frac {\ln \left (c \sqrt {x}-1\right )}{8}\right )\right )\) | \(264\) |
| default | \(2 c^{4} \left (-\frac {a^{2}}{4 c^{4} x^{2}}+b^{2} \left (-\frac {\operatorname {arctanh}\left (c \sqrt {x}\right )^{2}}{4 c^{4} x^{2}}-\frac {\operatorname {arctanh}\left (c \sqrt {x}\right )}{6 c^{3} x^{\frac {3}{2}}}-\frac {\operatorname {arctanh}\left (c \sqrt {x}\right )}{2 c \sqrt {x}}-\frac {\operatorname {arctanh}\left (c \sqrt {x}\right ) \ln \left (c \sqrt {x}-1\right )}{4}+\frac {\operatorname {arctanh}\left (c \sqrt {x}\right ) \ln \left (1+c \sqrt {x}\right )}{4}+\frac {\ln \left (c \sqrt {x}-1\right ) \ln \left (\frac {c \sqrt {x}}{2}+\frac {1}{2}\right )}{8}-\frac {\ln \left (c \sqrt {x}-1\right )^{2}}{16}+\frac {\left (\ln \left (1+c \sqrt {x}\right )-\ln \left (\frac {c \sqrt {x}}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {c \sqrt {x}}{2}+\frac {1}{2}\right )}{8}-\frac {\ln \left (1+c \sqrt {x}\right )^{2}}{16}-\frac {1}{12 c^{2} x}+\frac {2 \ln \left (c \sqrt {x}\right )}{3}-\frac {\ln \left (c \sqrt {x}-1\right )}{3}-\frac {\ln \left (1+c \sqrt {x}\right )}{3}\right )+2 a b \left (-\frac {\operatorname {arctanh}\left (c \sqrt {x}\right )}{4 c^{4} x^{2}}+\frac {\ln \left (1+c \sqrt {x}\right )}{8}-\frac {1}{12 c^{3} x^{\frac {3}{2}}}-\frac {1}{4 c \sqrt {x}}-\frac {\ln \left (c \sqrt {x}-1\right )}{8}\right )\right )\) | \(264\) |
-1/2*a^2/x^2+2*b^2*c^4*(-1/4/c^4/x^2*arctanh(c*x^(1/2))^2-1/6*arctanh(c*x^ (1/2))/c^3/x^(3/2)-1/2*arctanh(c*x^(1/2))/c/x^(1/2)-1/4*arctanh(c*x^(1/2)) *ln(c*x^(1/2)-1)+1/4*arctanh(c*x^(1/2))*ln(1+c*x^(1/2))+1/8*ln(c*x^(1/2)-1 )*ln(1/2*c*x^(1/2)+1/2)-1/16*ln(c*x^(1/2)-1)^2+1/8*(ln(1+c*x^(1/2))-ln(1/2 *c*x^(1/2)+1/2))*ln(-1/2*c*x^(1/2)+1/2)-1/16*ln(1+c*x^(1/2))^2-1/12/c^2/x+ 2/3*ln(c*x^(1/2))-1/3*ln(c*x^(1/2)-1)-1/3*ln(1+c*x^(1/2)))+4*a*b*c^4*(-1/4 /c^4/x^2*arctanh(c*x^(1/2))+1/8*ln(1+c*x^(1/2))-1/12/c^3/x^(3/2)-1/4/c/x^( 1/2)-1/8*ln(c*x^(1/2)-1))
Time = 0.27 (sec) , antiderivative size = 201, normalized size of antiderivative = 1.51 \[ \int \frac {\left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2}{x^3} \, dx=\frac {32 \, b^{2} c^{4} x^{2} \log \left (\sqrt {x}\right ) + 4 \, {\left (3 \, a b - 4 \, b^{2}\right )} c^{4} x^{2} \log \left (c \sqrt {x} + 1\right ) - 4 \, {\left (3 \, a b + 4 \, b^{2}\right )} c^{4} x^{2} \log \left (c \sqrt {x} - 1\right ) - 4 \, b^{2} c^{2} x + 3 \, {\left (b^{2} c^{4} x^{2} - b^{2}\right )} \log \left (-\frac {c^{2} x + 2 \, c \sqrt {x} + 1}{c^{2} x - 1}\right )^{2} - 12 \, a^{2} - 4 \, {\left (3 \, a b + {\left (3 \, b^{2} c^{3} x + b^{2} c\right )} \sqrt {x}\right )} \log \left (-\frac {c^{2} x + 2 \, c \sqrt {x} + 1}{c^{2} x - 1}\right ) - 8 \, {\left (3 \, a b c^{3} x + a b c\right )} \sqrt {x}}{24 \, x^{2}} \]
1/24*(32*b^2*c^4*x^2*log(sqrt(x)) + 4*(3*a*b - 4*b^2)*c^4*x^2*log(c*sqrt(x ) + 1) - 4*(3*a*b + 4*b^2)*c^4*x^2*log(c*sqrt(x) - 1) - 4*b^2*c^2*x + 3*(b ^2*c^4*x^2 - b^2)*log(-(c^2*x + 2*c*sqrt(x) + 1)/(c^2*x - 1))^2 - 12*a^2 - 4*(3*a*b + (3*b^2*c^3*x + b^2*c)*sqrt(x))*log(-(c^2*x + 2*c*sqrt(x) + 1)/ (c^2*x - 1)) - 8*(3*a*b*c^3*x + a*b*c)*sqrt(x))/x^2
Leaf count of result is larger than twice the leaf count of optimal. 972 vs. \(2 (122) = 244\).
Time = 7.30 (sec) , antiderivative size = 972, normalized size of antiderivative = 7.31 \[ \int \frac {\left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2}{x^3} \, dx=\text {Too large to display} \]
Piecewise((-a**2/(2*x**2), Eq(c, 0)), (-a**2/(2*x**2) + a*b*atanh(sqrt(x)* sqrt(1/x))/x**2 - b**2*atanh(sqrt(x)*sqrt(1/x))**2/(2*x**2), Eq(c, -sqrt(1 /x))), (-a**2/(2*x**2) - a*b*atanh(sqrt(x)*sqrt(1/x))/x**2 - b**2*atanh(sq rt(x)*sqrt(1/x))**2/(2*x**2), Eq(c, sqrt(1/x))), (-3*a**2*c**2*x**(3/2)/(6 *c**2*x**(7/2) - 6*x**(5/2)) + 3*a**2*sqrt(x)/(6*c**2*x**(7/2) - 6*x**(5/2 )) + 6*a*b*c**6*x**(7/2)*atanh(c*sqrt(x))/(6*c**2*x**(7/2) - 6*x**(5/2)) - 6*a*b*c**5*x**3/(6*c**2*x**(7/2) - 6*x**(5/2)) - 6*a*b*c**4*x**(5/2)*atan h(c*sqrt(x))/(6*c**2*x**(7/2) - 6*x**(5/2)) + 4*a*b*c**3*x**2/(6*c**2*x**( 7/2) - 6*x**(5/2)) - 6*a*b*c**2*x**(3/2)*atanh(c*sqrt(x))/(6*c**2*x**(7/2) - 6*x**(5/2)) + 2*a*b*c*x/(6*c**2*x**(7/2) - 6*x**(5/2)) + 6*a*b*sqrt(x)* atanh(c*sqrt(x))/(6*c**2*x**(7/2) - 6*x**(5/2)) + 4*b**2*c**6*x**(7/2)*log (x)/(6*c**2*x**(7/2) - 6*x**(5/2)) - 8*b**2*c**6*x**(7/2)*log(sqrt(x) - 1/ c)/(6*c**2*x**(7/2) - 6*x**(5/2)) + 3*b**2*c**6*x**(7/2)*atanh(c*sqrt(x))* *2/(6*c**2*x**(7/2) - 6*x**(5/2)) - 8*b**2*c**6*x**(7/2)*atanh(c*sqrt(x))/ (6*c**2*x**(7/2) - 6*x**(5/2)) - 6*b**2*c**5*x**3*atanh(c*sqrt(x))/(6*c**2 *x**(7/2) - 6*x**(5/2)) - 4*b**2*c**4*x**(5/2)*log(x)/(6*c**2*x**(7/2) - 6 *x**(5/2)) + 8*b**2*c**4*x**(5/2)*log(sqrt(x) - 1/c)/(6*c**2*x**(7/2) - 6* x**(5/2)) - 3*b**2*c**4*x**(5/2)*atanh(c*sqrt(x))**2/(6*c**2*x**(7/2) - 6* x**(5/2)) + 8*b**2*c**4*x**(5/2)*atanh(c*sqrt(x))/(6*c**2*x**(7/2) - 6*x** (5/2)) - b**2*c**4*x**(5/2)/(6*c**2*x**(7/2) - 6*x**(5/2)) + 4*b**2*c**...
Leaf count of result is larger than twice the leaf count of optimal. 234 vs. \(2 (109) = 218\).
Time = 0.22 (sec) , antiderivative size = 234, normalized size of antiderivative = 1.76 \[ \int \frac {\left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2}{x^3} \, dx=\frac {1}{6} \, {\left ({\left (3 \, c^{3} \log \left (c \sqrt {x} + 1\right ) - 3 \, c^{3} \log \left (c \sqrt {x} - 1\right ) - \frac {2 \, {\left (3 \, c^{2} x + 1\right )}}{x^{\frac {3}{2}}}\right )} c - \frac {6 \, \operatorname {artanh}\left (c \sqrt {x}\right )}{x^{2}}\right )} a b + \frac {1}{24} \, {\left ({\left (16 \, c^{2} \log \left (x\right ) - \frac {3 \, c^{2} x \log \left (c \sqrt {x} + 1\right )^{2} + 3 \, c^{2} x \log \left (c \sqrt {x} - 1\right )^{2} + 16 \, c^{2} x \log \left (c \sqrt {x} - 1\right ) - 2 \, {\left (3 \, c^{2} x \log \left (c \sqrt {x} - 1\right ) - 8 \, c^{2} x\right )} \log \left (c \sqrt {x} + 1\right ) + 4}{x}\right )} c^{2} + 4 \, {\left (3 \, c^{3} \log \left (c \sqrt {x} + 1\right ) - 3 \, c^{3} \log \left (c \sqrt {x} - 1\right ) - \frac {2 \, {\left (3 \, c^{2} x + 1\right )}}{x^{\frac {3}{2}}}\right )} c \operatorname {artanh}\left (c \sqrt {x}\right )\right )} b^{2} - \frac {b^{2} \operatorname {artanh}\left (c \sqrt {x}\right )^{2}}{2 \, x^{2}} - \frac {a^{2}}{2 \, x^{2}} \]
1/6*((3*c^3*log(c*sqrt(x) + 1) - 3*c^3*log(c*sqrt(x) - 1) - 2*(3*c^2*x + 1 )/x^(3/2))*c - 6*arctanh(c*sqrt(x))/x^2)*a*b + 1/24*((16*c^2*log(x) - (3*c ^2*x*log(c*sqrt(x) + 1)^2 + 3*c^2*x*log(c*sqrt(x) - 1)^2 + 16*c^2*x*log(c* sqrt(x) - 1) - 2*(3*c^2*x*log(c*sqrt(x) - 1) - 8*c^2*x)*log(c*sqrt(x) + 1) + 4)/x)*c^2 + 4*(3*c^3*log(c*sqrt(x) + 1) - 3*c^3*log(c*sqrt(x) - 1) - 2* (3*c^2*x + 1)/x^(3/2))*c*arctanh(c*sqrt(x)))*b^2 - 1/2*b^2*arctanh(c*sqrt( x))^2/x^2 - 1/2*a^2/x^2
\[ \int \frac {\left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2}{x^3} \, dx=\int { \frac {{\left (b \operatorname {artanh}\left (c \sqrt {x}\right ) + a\right )}^{2}}{x^{3}} \,d x } \]
Time = 5.36 (sec) , antiderivative size = 341, normalized size of antiderivative = 2.56 \[ \int \frac {\left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )^2}{x^3} \, dx=\frac {4\,b^2\,c^4\,\ln \left (\sqrt {x}\right )}{3}-\frac {a^2}{2\,x^2}-\frac {2\,b^2\,c^4\,\ln \left (c\,\sqrt {x}-1\right )}{3}-\frac {2\,b^2\,c^4\,\ln \left (c\,\sqrt {x}+1\right )}{3}+\frac {b^2\,c^4\,{\ln \left (c\,\sqrt {x}+1\right )}^2}{8}+\frac {b^2\,c^4\,{\ln \left (1-c\,\sqrt {x}\right )}^2}{8}-\frac {b^2\,c^2}{6\,x}-\frac {b^2\,{\ln \left (c\,\sqrt {x}+1\right )}^2}{8\,x^2}-\frac {b^2\,{\ln \left (1-c\,\sqrt {x}\right )}^2}{8\,x^2}-\frac {b^2\,c^3\,\ln \left (c\,\sqrt {x}+1\right )}{2\,\sqrt {x}}+\frac {b^2\,c^3\,\ln \left (1-c\,\sqrt {x}\right )}{2\,\sqrt {x}}-\frac {a\,b\,c^4\,\ln \left (c\,\sqrt {x}-1\right )}{2}+\frac {a\,b\,c^4\,\ln \left (c\,\sqrt {x}+1\right )}{2}-\frac {a\,b\,c}{3\,x^{3/2}}-\frac {a\,b\,\ln \left (c\,\sqrt {x}+1\right )}{2\,x^2}+\frac {a\,b\,\ln \left (1-c\,\sqrt {x}\right )}{2\,x^2}-\frac {b^2\,c^4\,\ln \left (c\,\sqrt {x}+1\right )\,\ln \left (1-c\,\sqrt {x}\right )}{4}-\frac {a\,b\,c^3}{\sqrt {x}}-\frac {b^2\,c\,\ln \left (c\,\sqrt {x}+1\right )}{6\,x^{3/2}}+\frac {b^2\,c\,\ln \left (1-c\,\sqrt {x}\right )}{6\,x^{3/2}}+\frac {b^2\,\ln \left (c\,\sqrt {x}+1\right )\,\ln \left (1-c\,\sqrt {x}\right )}{4\,x^2} \]
(4*b^2*c^4*log(x^(1/2)))/3 - a^2/(2*x^2) - (2*b^2*c^4*log(c*x^(1/2) - 1))/ 3 - (2*b^2*c^4*log(c*x^(1/2) + 1))/3 + (b^2*c^4*log(c*x^(1/2) + 1)^2)/8 + (b^2*c^4*log(1 - c*x^(1/2))^2)/8 - (b^2*c^2)/(6*x) - (b^2*log(c*x^(1/2) + 1)^2)/(8*x^2) - (b^2*log(1 - c*x^(1/2))^2)/(8*x^2) - (b^2*c^3*log(c*x^(1/2 ) + 1))/(2*x^(1/2)) + (b^2*c^3*log(1 - c*x^(1/2)))/(2*x^(1/2)) - (a*b*c^4* log(c*x^(1/2) - 1))/2 + (a*b*c^4*log(c*x^(1/2) + 1))/2 - (a*b*c)/(3*x^(3/2 )) - (a*b*log(c*x^(1/2) + 1))/(2*x^2) + (a*b*log(1 - c*x^(1/2)))/(2*x^2) - (b^2*c^4*log(c*x^(1/2) + 1)*log(1 - c*x^(1/2)))/4 - (a*b*c^3)/x^(1/2) - ( b^2*c*log(c*x^(1/2) + 1))/(6*x^(3/2)) + (b^2*c*log(1 - c*x^(1/2)))/(6*x^(3 /2)) + (b^2*log(c*x^(1/2) + 1)*log(1 - c*x^(1/2)))/(4*x^2)