Integrand size = 16, antiderivative size = 123 \[ \int x^3 \left (a+b \text {arctanh}\left (\frac {c}{x}\right )\right )^2 \, dx=\frac {1}{12} b^2 c^2 x^2+\frac {1}{2} b c^3 x \left (a+b \coth ^{-1}\left (\frac {x}{c}\right )\right )+\frac {1}{6} b c x^3 \left (a+b \coth ^{-1}\left (\frac {x}{c}\right )\right )-\frac {1}{4} c^4 \left (a+b \coth ^{-1}\left (\frac {x}{c}\right )\right )^2+\frac {1}{4} x^4 \left (a+b \coth ^{-1}\left (\frac {x}{c}\right )\right )^2+\frac {1}{3} b^2 c^4 \log \left (1-\frac {c^2}{x^2}\right )+\frac {2}{3} b^2 c^4 \log (x) \]
1/12*b^2*c^2*x^2+1/2*b*c^3*x*(a+b*arccoth(x/c))+1/6*b*c*x^3*(a+b*arccoth(x /c))-1/4*c^4*(a+b*arccoth(x/c))^2+1/4*x^4*(a+b*arccoth(x/c))^2+1/3*b^2*c^4 *ln(1-c^2/x^2)+2/3*b^2*c^4*ln(x)
Time = 0.05 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.07 \[ \int x^3 \left (a+b \text {arctanh}\left (\frac {c}{x}\right )\right )^2 \, dx=\frac {1}{12} \left (6 a b c^3 x+b^2 c^2 x^2+2 a b c x^3+3 a^2 x^4+2 b x \left (3 a x^3+b c \left (3 c^2+x^2\right )\right ) \text {arctanh}\left (\frac {c}{x}\right )+3 b^2 \left (-c^4+x^4\right ) \text {arctanh}\left (\frac {c}{x}\right )^2+b (3 a+4 b) c^4 \log (-c+x)-3 a b c^4 \log (c+x)+4 b^2 c^4 \log (c+x)\right ) \]
(6*a*b*c^3*x + b^2*c^2*x^2 + 2*a*b*c*x^3 + 3*a^2*x^4 + 2*b*x*(3*a*x^3 + b* c*(3*c^2 + x^2))*ArcTanh[c/x] + 3*b^2*(-c^4 + x^4)*ArcTanh[c/x]^2 + b*(3*a + 4*b)*c^4*Log[-c + x] - 3*a*b*c^4*Log[c + x] + 4*b^2*c^4*Log[c + x])/12
Time = 1.01 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.14, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.875, 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.
\(\displaystyle \int x^3 \left (a+b \text {arctanh}\left (\frac {c}{x}\right )\right )^2 \, dx\) |
\(\Big \downarrow \) 6454 |
\(\displaystyle -\int x^5 \left (a+b \text {arctanh}\left (\frac {c}{x}\right )\right )^2d\frac {1}{x}\) |
\(\Big \downarrow \) 6452 |
\(\displaystyle \frac {1}{4} x^4 \left (a+b \text {arctanh}\left (\frac {c}{x}\right )\right )^2-\frac {1}{2} b c \int \frac {x^4 \left (a+b \text {arctanh}\left (\frac {c}{x}\right )\right )}{1-\frac {c^2}{x^2}}d\frac {1}{x}\) |
\(\Big \downarrow \) 6544 |
\(\displaystyle \frac {1}{4} x^4 \left (a+b \text {arctanh}\left (\frac {c}{x}\right )\right )^2-\frac {1}{2} b c \left (c^2 \int \frac {x^2 \left (a+b \text {arctanh}\left (\frac {c}{x}\right )\right )}{1-\frac {c^2}{x^2}}d\frac {1}{x}+\int x^4 \left (a+b \text {arctanh}\left (\frac {c}{x}\right )\right )d\frac {1}{x}\right )\) |
\(\Big \downarrow \) 6452 |
\(\displaystyle \frac {1}{4} x^4 \left (a+b \text {arctanh}\left (\frac {c}{x}\right )\right )^2-\frac {1}{2} b c \left (c^2 \int \frac {x^2 \left (a+b \text {arctanh}\left (\frac {c}{x}\right )\right )}{1-\frac {c^2}{x^2}}d\frac {1}{x}+\frac {1}{3} b c \int \frac {x^3}{1-\frac {c^2}{x^2}}d\frac {1}{x}-\frac {1}{3} x^3 \left (a+b \text {arctanh}\left (\frac {c}{x}\right )\right )\right )\) |
\(\Big \downarrow \) 243 |
\(\displaystyle \frac {1}{4} x^4 \left (a+b \text {arctanh}\left (\frac {c}{x}\right )\right )^2-\frac {1}{2} b c \left (c^2 \int \frac {x^2 \left (a+b \text {arctanh}\left (\frac {c}{x}\right )\right )}{1-\frac {c^2}{x^2}}d\frac {1}{x}+\frac {1}{6} b c \int \frac {x^2}{1-\frac {c^2}{x^2}}d\frac {1}{x^2}-\frac {1}{3} x^3 \left (a+b \text {arctanh}\left (\frac {c}{x}\right )\right )\right )\) |
\(\Big \downarrow \) 54 |
\(\displaystyle \frac {1}{4} x^4 \left (a+b \text {arctanh}\left (\frac {c}{x}\right )\right )^2-\frac {1}{2} b c \left (c^2 \int \frac {x^2 \left (a+b \text {arctanh}\left (\frac {c}{x}\right )\right )}{1-\frac {c^2}{x^2}}d\frac {1}{x}+\frac {1}{6} b c \int \left (-\frac {c^4}{\frac {c^2}{x^2}-1}+x c^2+x^2\right )d\frac {1}{x^2}-\frac {1}{3} x^3 \left (a+b \text {arctanh}\left (\frac {c}{x}\right )\right )\right )\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{4} x^4 \left (a+b \text {arctanh}\left (\frac {c}{x}\right )\right )^2-\frac {1}{2} b c \left (c^2 \int \frac {x^2 \left (a+b \text {arctanh}\left (\frac {c}{x}\right )\right )}{1-\frac {c^2}{x^2}}d\frac {1}{x}-\frac {1}{3} x^3 \left (a+b \text {arctanh}\left (\frac {c}{x}\right )\right )+\frac {1}{6} b c \left (c^2 \left (-\log \left (1-\frac {c^2}{x^2}\right )\right )+c^2 \log \left (\frac {1}{x^2}\right )-x\right )\right )\) |
\(\Big \downarrow \) 6544 |
\(\displaystyle \frac {1}{4} x^4 \left (a+b \text {arctanh}\left (\frac {c}{x}\right )\right )^2-\frac {1}{2} b c \left (c^2 \left (c^2 \int \frac {a+b \text {arctanh}\left (\frac {c}{x}\right )}{1-\frac {c^2}{x^2}}d\frac {1}{x}+\int x^2 \left (a+b \text {arctanh}\left (\frac {c}{x}\right )\right )d\frac {1}{x}\right )-\frac {1}{3} x^3 \left (a+b \text {arctanh}\left (\frac {c}{x}\right )\right )+\frac {1}{6} b c \left (c^2 \left (-\log \left (1-\frac {c^2}{x^2}\right )\right )+c^2 \log \left (\frac {1}{x^2}\right )-x\right )\right )\) |
\(\Big \downarrow \) 6452 |
\(\displaystyle \frac {1}{4} x^4 \left (a+b \text {arctanh}\left (\frac {c}{x}\right )\right )^2-\frac {1}{2} b c \left (c^2 \left (c^2 \int \frac {a+b \text {arctanh}\left (\frac {c}{x}\right )}{1-\frac {c^2}{x^2}}d\frac {1}{x}+b c \int \frac {x}{1-\frac {c^2}{x^2}}d\frac {1}{x}-x \left (a+b \text {arctanh}\left (\frac {c}{x}\right )\right )\right )-\frac {1}{3} x^3 \left (a+b \text {arctanh}\left (\frac {c}{x}\right )\right )+\frac {1}{6} b c \left (c^2 \left (-\log \left (1-\frac {c^2}{x^2}\right )\right )+c^2 \log \left (\frac {1}{x^2}\right )-x\right )\right )\) |
\(\Big \downarrow \) 243 |
\(\displaystyle \frac {1}{4} x^4 \left (a+b \text {arctanh}\left (\frac {c}{x}\right )\right )^2-\frac {1}{2} b c \left (c^2 \left (c^2 \int \frac {a+b \text {arctanh}\left (\frac {c}{x}\right )}{1-\frac {c^2}{x^2}}d\frac {1}{x}+\frac {1}{2} b c \int \frac {x}{1-\frac {c^2}{x^2}}d\frac {1}{x^2}-x \left (a+b \text {arctanh}\left (\frac {c}{x}\right )\right )\right )-\frac {1}{3} x^3 \left (a+b \text {arctanh}\left (\frac {c}{x}\right )\right )+\frac {1}{6} b c \left (c^2 \left (-\log \left (1-\frac {c^2}{x^2}\right )\right )+c^2 \log \left (\frac {1}{x^2}\right )-x\right )\right )\) |
\(\Big \downarrow \) 47 |
\(\displaystyle \frac {1}{4} x^4 \left (a+b \text {arctanh}\left (\frac {c}{x}\right )\right )^2-\frac {1}{2} b c \left (c^2 \left (c^2 \int \frac {a+b \text {arctanh}\left (\frac {c}{x}\right )}{1-\frac {c^2}{x^2}}d\frac {1}{x}+\frac {1}{2} b c \left (c^2 \int \frac {1}{1-\frac {c^2}{x^2}}d\frac {1}{x^2}+\int xd\frac {1}{x^2}\right )-x \left (a+b \text {arctanh}\left (\frac {c}{x}\right )\right )\right )-\frac {1}{3} x^3 \left (a+b \text {arctanh}\left (\frac {c}{x}\right )\right )+\frac {1}{6} b c \left (c^2 \left (-\log \left (1-\frac {c^2}{x^2}\right )\right )+c^2 \log \left (\frac {1}{x^2}\right )-x\right )\right )\) |
\(\Big \downarrow \) 14 |
\(\displaystyle \frac {1}{4} x^4 \left (a+b \text {arctanh}\left (\frac {c}{x}\right )\right )^2-\frac {1}{2} b c \left (c^2 \left (c^2 \int \frac {a+b \text {arctanh}\left (\frac {c}{x}\right )}{1-\frac {c^2}{x^2}}d\frac {1}{x}+\frac {1}{2} b c \left (c^2 \int \frac {1}{1-\frac {c^2}{x^2}}d\frac {1}{x^2}+\log \left (\frac {1}{x^2}\right )\right )-x \left (a+b \text {arctanh}\left (\frac {c}{x}\right )\right )\right )-\frac {1}{3} x^3 \left (a+b \text {arctanh}\left (\frac {c}{x}\right )\right )+\frac {1}{6} b c \left (c^2 \left (-\log \left (1-\frac {c^2}{x^2}\right )\right )+c^2 \log \left (\frac {1}{x^2}\right )-x\right )\right )\) |
\(\Big \downarrow \) 16 |
\(\displaystyle \frac {1}{4} x^4 \left (a+b \text {arctanh}\left (\frac {c}{x}\right )\right )^2-\frac {1}{2} b c \left (c^2 \left (c^2 \int \frac {a+b \text {arctanh}\left (\frac {c}{x}\right )}{1-\frac {c^2}{x^2}}d\frac {1}{x}-x \left (a+b \text {arctanh}\left (\frac {c}{x}\right )\right )+\frac {1}{2} b c \left (\log \left (\frac {1}{x^2}\right )-\log \left (1-\frac {c^2}{x^2}\right )\right )\right )-\frac {1}{3} x^3 \left (a+b \text {arctanh}\left (\frac {c}{x}\right )\right )+\frac {1}{6} b c \left (c^2 \left (-\log \left (1-\frac {c^2}{x^2}\right )\right )+c^2 \log \left (\frac {1}{x^2}\right )-x\right )\right )\) |
\(\Big \downarrow \) 6510 |
\(\displaystyle \frac {1}{4} x^4 \left (a+b \text {arctanh}\left (\frac {c}{x}\right )\right )^2-\frac {1}{2} b c \left (c^2 \left (\frac {c \left (a+b \text {arctanh}\left (\frac {c}{x}\right )\right )^2}{2 b}-x \left (a+b \text {arctanh}\left (\frac {c}{x}\right )\right )+\frac {1}{2} b c \left (\log \left (\frac {1}{x^2}\right )-\log \left (1-\frac {c^2}{x^2}\right )\right )\right )-\frac {1}{3} x^3 \left (a+b \text {arctanh}\left (\frac {c}{x}\right )\right )+\frac {1}{6} b c \left (c^2 \left (-\log \left (1-\frac {c^2}{x^2}\right )\right )+c^2 \log \left (\frac {1}{x^2}\right )-x\right )\right )\) |
(x^4*(a + b*ArcTanh[c/x])^2)/4 - (b*c*(-1/3*(x^3*(a + b*ArcTanh[c/x])) + ( b*c*(-x - c^2*Log[1 - c^2/x^2] + c^2*Log[x^(-2)]))/6 + c^2*(-(x*(a + b*Arc Tanh[c/x])) + (c*(a + b*ArcTanh[c/x])^2)/(2*b) + (b*c*(-Log[1 - c^2/x^2] + Log[x^(-2)]))/2)))/2
3.2.43.3.1 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]
Time = 3.58 (sec) , antiderivative size = 161, normalized size of antiderivative = 1.31
method | result | size |
parallelrisch | \(\frac {x^{4} \operatorname {arctanh}\left (\frac {c}{x}\right )^{2} b^{2}}{4}-\frac {\operatorname {arctanh}\left (\frac {c}{x}\right )^{2} b^{2} c^{4}}{4}+\frac {2 b^{2} c^{4} \ln \left (x -c \right )}{3}+\frac {x^{4} \operatorname {arctanh}\left (\frac {c}{x}\right ) a b}{2}+\frac {x^{3} \operatorname {arctanh}\left (\frac {c}{x}\right ) b^{2} c}{6}+\frac {x \,\operatorname {arctanh}\left (\frac {c}{x}\right ) b^{2} c^{3}}{2}-\frac {\operatorname {arctanh}\left (\frac {c}{x}\right ) a b \,c^{4}}{2}+\frac {2 \,\operatorname {arctanh}\left (\frac {c}{x}\right ) b^{2} c^{4}}{3}+\frac {a^{2} x^{4}}{4}+\frac {a b c \,x^{3}}{6}+\frac {b^{2} c^{2} x^{2}}{12}+\frac {a b \,c^{3} x}{2}+\frac {b^{2} c^{4}}{12}\) | \(161\) |
parts | \(\frac {a^{2} x^{4}}{4}-b^{2} c^{4} \left (-\frac {x^{4} \operatorname {arctanh}\left (\frac {c}{x}\right )^{2}}{4 c^{4}}-\frac {\operatorname {arctanh}\left (\frac {c}{x}\right ) \ln \left (\frac {c}{x}-1\right )}{4}+\frac {\operatorname {arctanh}\left (\frac {c}{x}\right ) \ln \left (1+\frac {c}{x}\right )}{4}-\frac {x^{3} \operatorname {arctanh}\left (\frac {c}{x}\right )}{6 c^{3}}-\frac {x \,\operatorname {arctanh}\left (\frac {c}{x}\right )}{2 c}+\frac {\ln \left (\frac {c}{x}-1\right ) \ln \left (\frac {c}{2 x}+\frac {1}{2}\right )}{8}-\frac {\ln \left (\frac {c}{x}-1\right )^{2}}{16}+\frac {\left (\ln \left (1+\frac {c}{x}\right )-\ln \left (\frac {c}{2 x}+\frac {1}{2}\right )\right ) \ln \left (-\frac {c}{2 x}+\frac {1}{2}\right )}{8}-\frac {\ln \left (1+\frac {c}{x}\right )^{2}}{16}-\frac {\ln \left (\frac {c}{x}-1\right )}{3}-\frac {\ln \left (1+\frac {c}{x}\right )}{3}-\frac {x^{2}}{12 c^{2}}+\frac {2 \ln \left (\frac {c}{x}\right )}{3}\right )-2 a b \,c^{4} \left (-\frac {x^{4} \operatorname {arctanh}\left (\frac {c}{x}\right )}{4 c^{4}}+\frac {\ln \left (1+\frac {c}{x}\right )}{8}-\frac {x^{3}}{12 c^{3}}-\frac {x}{4 c}-\frac {\ln \left (\frac {c}{x}-1\right )}{8}\right )\) | \(259\) |
derivativedivides | \(-c^{4} \left (-\frac {a^{2} x^{4}}{4 c^{4}}+b^{2} \left (-\frac {x^{4} \operatorname {arctanh}\left (\frac {c}{x}\right )^{2}}{4 c^{4}}-\frac {\operatorname {arctanh}\left (\frac {c}{x}\right ) \ln \left (\frac {c}{x}-1\right )}{4}+\frac {\operatorname {arctanh}\left (\frac {c}{x}\right ) \ln \left (1+\frac {c}{x}\right )}{4}-\frac {x^{3} \operatorname {arctanh}\left (\frac {c}{x}\right )}{6 c^{3}}-\frac {x \,\operatorname {arctanh}\left (\frac {c}{x}\right )}{2 c}+\frac {\ln \left (\frac {c}{x}-1\right ) \ln \left (\frac {c}{2 x}+\frac {1}{2}\right )}{8}-\frac {\ln \left (\frac {c}{x}-1\right )^{2}}{16}+\frac {\left (\ln \left (1+\frac {c}{x}\right )-\ln \left (\frac {c}{2 x}+\frac {1}{2}\right )\right ) \ln \left (-\frac {c}{2 x}+\frac {1}{2}\right )}{8}-\frac {\ln \left (1+\frac {c}{x}\right )^{2}}{16}-\frac {\ln \left (\frac {c}{x}-1\right )}{3}-\frac {\ln \left (1+\frac {c}{x}\right )}{3}-\frac {x^{2}}{12 c^{2}}+\frac {2 \ln \left (\frac {c}{x}\right )}{3}\right )+2 a b \left (-\frac {x^{4} \operatorname {arctanh}\left (\frac {c}{x}\right )}{4 c^{4}}+\frac {\ln \left (1+\frac {c}{x}\right )}{8}-\frac {x^{3}}{12 c^{3}}-\frac {x}{4 c}-\frac {\ln \left (\frac {c}{x}-1\right )}{8}\right )\right )\) | \(260\) |
default | \(-c^{4} \left (-\frac {a^{2} x^{4}}{4 c^{4}}+b^{2} \left (-\frac {x^{4} \operatorname {arctanh}\left (\frac {c}{x}\right )^{2}}{4 c^{4}}-\frac {\operatorname {arctanh}\left (\frac {c}{x}\right ) \ln \left (\frac {c}{x}-1\right )}{4}+\frac {\operatorname {arctanh}\left (\frac {c}{x}\right ) \ln \left (1+\frac {c}{x}\right )}{4}-\frac {x^{3} \operatorname {arctanh}\left (\frac {c}{x}\right )}{6 c^{3}}-\frac {x \,\operatorname {arctanh}\left (\frac {c}{x}\right )}{2 c}+\frac {\ln \left (\frac {c}{x}-1\right ) \ln \left (\frac {c}{2 x}+\frac {1}{2}\right )}{8}-\frac {\ln \left (\frac {c}{x}-1\right )^{2}}{16}+\frac {\left (\ln \left (1+\frac {c}{x}\right )-\ln \left (\frac {c}{2 x}+\frac {1}{2}\right )\right ) \ln \left (-\frac {c}{2 x}+\frac {1}{2}\right )}{8}-\frac {\ln \left (1+\frac {c}{x}\right )^{2}}{16}-\frac {\ln \left (\frac {c}{x}-1\right )}{3}-\frac {\ln \left (1+\frac {c}{x}\right )}{3}-\frac {x^{2}}{12 c^{2}}+\frac {2 \ln \left (\frac {c}{x}\right )}{3}\right )+2 a b \left (-\frac {x^{4} \operatorname {arctanh}\left (\frac {c}{x}\right )}{4 c^{4}}+\frac {\ln \left (1+\frac {c}{x}\right )}{8}-\frac {x^{3}}{12 c^{3}}-\frac {x}{4 c}-\frac {\ln \left (\frac {c}{x}-1\right )}{8}\right )\right )\) | \(260\) |
risch | \(\text {Expression too large to display}\) | \(14899\) |
1/4*x^4*arctanh(c/x)^2*b^2-1/4*arctanh(c/x)^2*b^2*c^4+2/3*b^2*c^4*ln(x-c)+ 1/2*x^4*arctanh(c/x)*a*b+1/6*x^3*arctanh(c/x)*b^2*c+1/2*x*arctanh(c/x)*b^2 *c^3-1/2*arctanh(c/x)*a*b*c^4+2/3*arctanh(c/x)*b^2*c^4+1/4*a^2*x^4+1/6*a*b *c*x^3+1/12*b^2*c^2*x^2+1/2*a*b*c^3*x+1/12*b^2*c^4
Time = 0.26 (sec) , antiderivative size = 149, normalized size of antiderivative = 1.21 \[ \int x^3 \left (a+b \text {arctanh}\left (\frac {c}{x}\right )\right )^2 \, dx=\frac {1}{2} \, a b c^{3} x + \frac {1}{12} \, b^{2} c^{2} x^{2} + \frac {1}{6} \, a b c x^{3} + \frac {1}{4} \, a^{2} x^{4} - \frac {1}{12} \, {\left (3 \, a b - 4 \, b^{2}\right )} c^{4} \log \left (c + x\right ) + \frac {1}{12} \, {\left (3 \, a b + 4 \, b^{2}\right )} c^{4} \log \left (-c + x\right ) - \frac {1}{16} \, {\left (b^{2} c^{4} - b^{2} x^{4}\right )} \log \left (-\frac {c + x}{c - x}\right )^{2} + \frac {1}{12} \, {\left (3 \, b^{2} c^{3} x + b^{2} c x^{3} + 3 \, a b x^{4}\right )} \log \left (-\frac {c + x}{c - x}\right ) \]
1/2*a*b*c^3*x + 1/12*b^2*c^2*x^2 + 1/6*a*b*c*x^3 + 1/4*a^2*x^4 - 1/12*(3*a *b - 4*b^2)*c^4*log(c + x) + 1/12*(3*a*b + 4*b^2)*c^4*log(-c + x) - 1/16*( b^2*c^4 - b^2*x^4)*log(-(c + x)/(c - x))^2 + 1/12*(3*b^2*c^3*x + b^2*c*x^3 + 3*a*b*x^4)*log(-(c + x)/(c - x))
Time = 0.23 (sec) , antiderivative size = 158, normalized size of antiderivative = 1.28 \[ \int x^3 \left (a+b \text {arctanh}\left (\frac {c}{x}\right )\right )^2 \, dx=\frac {a^{2} x^{4}}{4} - \frac {a b c^{4} \operatorname {atanh}{\left (\frac {c}{x} \right )}}{2} + \frac {a b c^{3} x}{2} + \frac {a b c x^{3}}{6} + \frac {a b x^{4} \operatorname {atanh}{\left (\frac {c}{x} \right )}}{2} + \frac {2 b^{2} c^{4} \log {\left (- c + x \right )}}{3} - \frac {b^{2} c^{4} \operatorname {atanh}^{2}{\left (\frac {c}{x} \right )}}{4} + \frac {2 b^{2} c^{4} \operatorname {atanh}{\left (\frac {c}{x} \right )}}{3} + \frac {b^{2} c^{3} x \operatorname {atanh}{\left (\frac {c}{x} \right )}}{2} + \frac {b^{2} c^{2} x^{2}}{12} + \frac {b^{2} c x^{3} \operatorname {atanh}{\left (\frac {c}{x} \right )}}{6} + \frac {b^{2} x^{4} \operatorname {atanh}^{2}{\left (\frac {c}{x} \right )}}{4} \]
a**2*x**4/4 - a*b*c**4*atanh(c/x)/2 + a*b*c**3*x/2 + a*b*c*x**3/6 + a*b*x* *4*atanh(c/x)/2 + 2*b**2*c**4*log(-c + x)/3 - b**2*c**4*atanh(c/x)**2/4 + 2*b**2*c**4*atanh(c/x)/3 + b**2*c**3*x*atanh(c/x)/2 + b**2*c**2*x**2/12 + b**2*c*x**3*atanh(c/x)/6 + b**2*x**4*atanh(c/x)**2/4
Time = 0.21 (sec) , antiderivative size = 189, normalized size of antiderivative = 1.54 \[ \int x^3 \left (a+b \text {arctanh}\left (\frac {c}{x}\right )\right )^2 \, dx=\frac {1}{4} \, b^{2} x^{4} \operatorname {artanh}\left (\frac {c}{x}\right )^{2} + \frac {1}{4} \, a^{2} x^{4} + \frac {1}{12} \, {\left (6 \, x^{4} \operatorname {artanh}\left (\frac {c}{x}\right ) - {\left (3 \, c^{3} \log \left (c + x\right ) - 3 \, c^{3} \log \left (-c + x\right ) - 6 \, c^{2} x - 2 \, x^{3}\right )} c\right )} a b + \frac {1}{48} \, {\left ({\left (3 \, c^{2} \log \left (c + x\right )^{2} + 3 \, c^{2} \log \left (-c + x\right )^{2} + 16 \, c^{2} \log \left (c + x\right ) + 4 \, x^{2} - 2 \, {\left (3 \, c^{2} \log \left (c + x\right ) - 8 \, c^{2}\right )} \log \left (-c + x\right )\right )} c^{2} - 4 \, {\left (3 \, c^{3} \log \left (c + x\right ) - 3 \, c^{3} \log \left (-c + x\right ) - 6 \, c^{2} x - 2 \, x^{3}\right )} c \operatorname {artanh}\left (\frac {c}{x}\right )\right )} b^{2} \]
1/4*b^2*x^4*arctanh(c/x)^2 + 1/4*a^2*x^4 + 1/12*(6*x^4*arctanh(c/x) - (3*c ^3*log(c + x) - 3*c^3*log(-c + x) - 6*c^2*x - 2*x^3)*c)*a*b + 1/48*((3*c^2 *log(c + x)^2 + 3*c^2*log(-c + x)^2 + 16*c^2*log(c + x) + 4*x^2 - 2*(3*c^2 *log(c + x) - 8*c^2)*log(-c + x))*c^2 - 4*(3*c^3*log(c + x) - 3*c^3*log(-c + x) - 6*c^2*x - 2*x^3)*c*arctanh(c/x))*b^2
Leaf count of result is larger than twice the leaf count of optimal. 552 vs. \(2 (109) = 218\).
Time = 0.27 (sec) , antiderivative size = 552, normalized size of antiderivative = 4.49 \[ \int x^3 \left (a+b \text {arctanh}\left (\frac {c}{x}\right )\right )^2 \, dx=-\frac {4 \, b^{2} c^{5} \log \left (-\frac {c + x}{c - x} - 1\right ) - 4 \, b^{2} c^{5} \log \left (-\frac {c + x}{c - x}\right ) + \frac {3 \, {\left (\frac {b^{2} {\left (c + x\right )}^{3} c^{5}}{{\left (c - x\right )}^{3}} + \frac {b^{2} {\left (c + x\right )} c^{5}}{c - x}\right )} \log \left (-\frac {c + x}{c - x}\right )^{2}}{\frac {{\left (c + x\right )}^{4}}{{\left (c - x\right )}^{4}} + \frac {4 \, {\left (c + x\right )}^{3}}{{\left (c - x\right )}^{3}} + \frac {6 \, {\left (c + x\right )}^{2}}{{\left (c - x\right )}^{2}} + \frac {4 \, {\left (c + x\right )}}{c - x} + 1} + \frac {2 \, {\left (2 \, b^{2} c^{5} + \frac {6 \, a b {\left (c + x\right )}^{3} c^{5}}{{\left (c - x\right )}^{3}} + \frac {3 \, b^{2} {\left (c + x\right )}^{3} c^{5}}{{\left (c - x\right )}^{3}} + \frac {6 \, b^{2} {\left (c + x\right )}^{2} c^{5}}{{\left (c - x\right )}^{2}} + \frac {6 \, a b {\left (c + x\right )} c^{5}}{c - x} + \frac {5 \, b^{2} {\left (c + x\right )} c^{5}}{c - x}\right )} \log \left (-\frac {c + x}{c - x}\right )}{\frac {{\left (c + x\right )}^{4}}{{\left (c - x\right )}^{4}} + \frac {4 \, {\left (c + x\right )}^{3}}{{\left (c - x\right )}^{3}} + \frac {6 \, {\left (c + x\right )}^{2}}{{\left (c - x\right )}^{2}} + \frac {4 \, {\left (c + x\right )}}{c - x} + 1} + \frac {2 \, {\left (4 \, a b c^{5} + \frac {6 \, a^{2} {\left (c + x\right )}^{3} c^{5}}{{\left (c - x\right )}^{3}} + \frac {6 \, a b {\left (c + x\right )}^{3} c^{5}}{{\left (c - x\right )}^{3}} + \frac {b^{2} {\left (c + x\right )}^{3} c^{5}}{{\left (c - x\right )}^{3}} + \frac {12 \, a b {\left (c + x\right )}^{2} c^{5}}{{\left (c - x\right )}^{2}} + \frac {2 \, b^{2} {\left (c + x\right )}^{2} c^{5}}{{\left (c - x\right )}^{2}} + \frac {6 \, a^{2} {\left (c + x\right )} c^{5}}{c - x} + \frac {10 \, a b {\left (c + x\right )} c^{5}}{c - x} + \frac {b^{2} {\left (c + x\right )} c^{5}}{c - x}\right )}}{\frac {{\left (c + x\right )}^{4}}{{\left (c - x\right )}^{4}} + \frac {4 \, {\left (c + x\right )}^{3}}{{\left (c - x\right )}^{3}} + \frac {6 \, {\left (c + x\right )}^{2}}{{\left (c - x\right )}^{2}} + \frac {4 \, {\left (c + x\right )}}{c - x} + 1}}{6 \, c} \]
-1/6*(4*b^2*c^5*log(-(c + x)/(c - x) - 1) - 4*b^2*c^5*log(-(c + x)/(c - x) ) + 3*(b^2*(c + x)^3*c^5/(c - x)^3 + b^2*(c + x)*c^5/(c - x))*log(-(c + x) /(c - x))^2/((c + x)^4/(c - x)^4 + 4*(c + x)^3/(c - x)^3 + 6*(c + x)^2/(c - x)^2 + 4*(c + x)/(c - x) + 1) + 2*(2*b^2*c^5 + 6*a*b*(c + x)^3*c^5/(c - x)^3 + 3*b^2*(c + x)^3*c^5/(c - x)^3 + 6*b^2*(c + x)^2*c^5/(c - x)^2 + 6*a *b*(c + x)*c^5/(c - x) + 5*b^2*(c + x)*c^5/(c - x))*log(-(c + x)/(c - x))/ ((c + x)^4/(c - x)^4 + 4*(c + x)^3/(c - x)^3 + 6*(c + x)^2/(c - x)^2 + 4*( c + x)/(c - x) + 1) + 2*(4*a*b*c^5 + 6*a^2*(c + x)^3*c^5/(c - x)^3 + 6*a*b *(c + x)^3*c^5/(c - x)^3 + b^2*(c + x)^3*c^5/(c - x)^3 + 12*a*b*(c + x)^2* c^5/(c - x)^2 + 2*b^2*(c + x)^2*c^5/(c - x)^2 + 6*a^2*(c + x)*c^5/(c - x) + 10*a*b*(c + x)*c^5/(c - x) + b^2*(c + x)*c^5/(c - x))/((c + x)^4/(c - x) ^4 + 4*(c + x)^3/(c - x)^3 + 6*(c + x)^2/(c - x)^2 + 4*(c + x)/(c - x) + 1 ))/c
Time = 3.36 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.15 \[ \int x^3 \left (a+b \text {arctanh}\left (\frac {c}{x}\right )\right )^2 \, dx=\frac {a^2\,x^4}{4}-\frac {b^2\,c^4\,{\mathrm {atanh}\left (\frac {c}{x}\right )}^2}{4}+\frac {b^2\,x^4\,{\mathrm {atanh}\left (\frac {c}{x}\right )}^2}{4}+\frac {b^2\,c^4\,\ln \left (x^2-c^2\right )}{3}+\frac {b^2\,c^2\,x^2}{12}+\frac {b^2\,c\,x^3\,\mathrm {atanh}\left (\frac {c}{x}\right )}{6}+\frac {b^2\,c^3\,x\,\mathrm {atanh}\left (\frac {c}{x}\right )}{2}+\frac {a\,b\,c\,x^3}{6}+\frac {a\,b\,c^3\,x}{2}-\frac {a\,b\,c^4\,\mathrm {atanh}\left (\frac {c}{x}\right )}{2}+\frac {a\,b\,x^4\,\mathrm {atanh}\left (\frac {c}{x}\right )}{2} \]