Integrand size = 18, antiderivative size = 101 \[ \int x^2 \left (a+b \text {arctanh}\left (c x^{3/2}\right )\right )^2 \, dx=\frac {2 a b x^{3/2}}{3 c}+\frac {2 b^2 x^{3/2} \text {arctanh}\left (c x^{3/2}\right )}{3 c}-\frac {\left (a+b \text {arctanh}\left (c x^{3/2}\right )\right )^2}{3 c^2}+\frac {1}{3} x^3 \left (a+b \text {arctanh}\left (c x^{3/2}\right )\right )^2+\frac {b^2 \log \left (1-c^2 x^3\right )}{3 c^2} \]
2/3*a*b*x^(3/2)/c+2/3*b^2*x^(3/2)*arctanh(c*x^(3/2))/c-1/3*(a+b*arctanh(c* x^(3/2)))^2/c^2+1/3*x^3*(a+b*arctanh(c*x^(3/2)))^2+1/3*b^2*ln(-c^2*x^3+1)/ c^2
Time = 0.07 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.21 \[ \int x^2 \left (a+b \text {arctanh}\left (c x^{3/2}\right )\right )^2 \, dx=\frac {2 a b c x^{3/2}+a^2 c^2 x^3+2 b c x^{3/2} \left (b+a c x^{3/2}\right ) \text {arctanh}\left (c x^{3/2}\right )+b^2 \left (-1+c^2 x^3\right ) \text {arctanh}\left (c x^{3/2}\right )^2+b (a+b) \log \left (1-c x^{3/2}\right )-a b \log \left (1+c x^{3/2}\right )+b^2 \log \left (1+c x^{3/2}\right )}{3 c^2} \]
(2*a*b*c*x^(3/2) + a^2*c^2*x^3 + 2*b*c*x^(3/2)*(b + a*c*x^(3/2))*ArcTanh[c *x^(3/2)] + b^2*(-1 + c^2*x^3)*ArcTanh[c*x^(3/2)]^2 + b*(a + b)*Log[1 - c* x^(3/2)] - a*b*Log[1 + c*x^(3/2)] + b^2*Log[1 + c*x^(3/2)])/(3*c^2)
Time = 0.51 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.01, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.278, Rules used = {6454, 6452, 6542, 2009, 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.
| \(\displaystyle \int x^2 \left (a+b \text {arctanh}\left (c x^{3/2}\right )\right )^2 \, dx\) |
| \(\Big \downarrow \) 6454 |
| \(\displaystyle \frac {2}{3} \int x^{3/2} \left (a+b \text {arctanh}\left (c x^{3/2}\right )\right )^2dx^{3/2}\) |
| \(\Big \downarrow \) 6452 |
| \(\displaystyle \frac {2}{3} \left (\frac {1}{2} x^3 \left (a+b \text {arctanh}\left (c x^{3/2}\right )\right )^2-b c \int \frac {x^3 \left (a+b \text {arctanh}\left (c x^{3/2}\right )\right )}{1-c^2 x^3}dx^{3/2}\right )\) |
| \(\Big \downarrow \) 6542 |
| \(\displaystyle \frac {2}{3} \left (\frac {1}{2} x^3 \left (a+b \text {arctanh}\left (c x^{3/2}\right )\right )^2-b c \left (\frac {\int \frac {a+b \text {arctanh}\left (c x^{3/2}\right )}{1-c^2 x^3}dx^{3/2}}{c^2}-\frac {\int \left (a+b \text {arctanh}\left (c x^{3/2}\right )\right )dx^{3/2}}{c^2}\right )\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {2}{3} \left (\frac {1}{2} x^3 \left (a+b \text {arctanh}\left (c x^{3/2}\right )\right )^2-b c \left (\frac {\int \frac {a+b \text {arctanh}\left (c x^{3/2}\right )}{1-c^2 x^3}dx^{3/2}}{c^2}-\frac {a x^{3/2}+b x^{3/2} \text {arctanh}\left (c x^{3/2}\right )+\frac {b \log \left (1-c^2 x^3\right )}{2 c}}{c^2}\right )\right )\) |
| \(\Big \downarrow \) 6510 |
| \(\displaystyle \frac {2}{3} \left (\frac {1}{2} x^3 \left (a+b \text {arctanh}\left (c x^{3/2}\right )\right )^2-b c \left (\frac {\left (a+b \text {arctanh}\left (c x^{3/2}\right )\right )^2}{2 b c^3}-\frac {a x^{3/2}+b x^{3/2} \text {arctanh}\left (c x^{3/2}\right )+\frac {b \log \left (1-c^2 x^3\right )}{2 c}}{c^2}\right )\right )\) |
(2*((x^3*(a + b*ArcTanh[c*x^(3/2)])^2)/2 - b*c*((a + b*ArcTanh[c*x^(3/2)]) ^2/(2*b*c^3) - (a*x^(3/2) + b*x^(3/2)*ArcTanh[c*x^(3/2)] + (b*Log[1 - c^2* x^3])/(2*c))/c^2)))/3
3.3.21.3.1 Defintions of rubi rules used
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[f^2/e Int[(f*x)^(m - 2)*(a + b*ArcTanh[c* x])^p, x], x] - Simp[d*(f^2/e) 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] && GtQ[m, 1]
Leaf count of result is larger than twice the leaf count of optimal. \(219\) vs. \(2(81)=162\).
Time = 1.83 (sec) , antiderivative size = 220, normalized size of antiderivative = 2.18
| method | result | size |
| parts | \(\frac {a^{2} x^{3}}{3}+\frac {2 b^{2} \left (\frac {c^{2} x^{3} \operatorname {arctanh}\left (c \,x^{\frac {3}{2}}\right )^{2}}{2}+c \,x^{\frac {3}{2}} \operatorname {arctanh}\left (c \,x^{\frac {3}{2}}\right )+\frac {\operatorname {arctanh}\left (c \,x^{\frac {3}{2}}\right ) \ln \left (c \,x^{\frac {3}{2}}-1\right )}{2}-\frac {\operatorname {arctanh}\left (c \,x^{\frac {3}{2}}\right ) \ln \left (c \,x^{\frac {3}{2}}+1\right )}{2}-\frac {\ln \left (c \,x^{\frac {3}{2}}-1\right ) \ln \left (\frac {c \,x^{\frac {3}{2}}}{2}+\frac {1}{2}\right )}{4}+\frac {\ln \left (c \,x^{\frac {3}{2}}-1\right )^{2}}{8}+\frac {\ln \left (c \,x^{\frac {3}{2}}-1\right )}{2}+\frac {\ln \left (c \,x^{\frac {3}{2}}+1\right )}{2}+\frac {\ln \left (c \,x^{\frac {3}{2}}+1\right )^{2}}{8}-\frac {\left (\ln \left (c \,x^{\frac {3}{2}}+1\right )-\ln \left (\frac {c \,x^{\frac {3}{2}}}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {c \,x^{\frac {3}{2}}}{2}+\frac {1}{2}\right )}{4}\right )}{3 c^{2}}+\frac {4 a b \left (\frac {c^{2} x^{3} \operatorname {arctanh}\left (c \,x^{\frac {3}{2}}\right )}{2}+\frac {c \,x^{\frac {3}{2}}}{2}+\frac {\ln \left (c \,x^{\frac {3}{2}}-1\right )}{4}-\frac {\ln \left (c \,x^{\frac {3}{2}}+1\right )}{4}\right )}{3 c^{2}}\) | \(220\) |
| derivativedivides | \(\frac {\frac {a^{2} c^{2} x^{3}}{3}+\frac {2 b^{2} \left (\frac {c^{2} x^{3} \operatorname {arctanh}\left (c \,x^{\frac {3}{2}}\right )^{2}}{2}+c \,x^{\frac {3}{2}} \operatorname {arctanh}\left (c \,x^{\frac {3}{2}}\right )+\frac {\operatorname {arctanh}\left (c \,x^{\frac {3}{2}}\right ) \ln \left (c \,x^{\frac {3}{2}}-1\right )}{2}-\frac {\operatorname {arctanh}\left (c \,x^{\frac {3}{2}}\right ) \ln \left (c \,x^{\frac {3}{2}}+1\right )}{2}-\frac {\ln \left (c \,x^{\frac {3}{2}}-1\right ) \ln \left (\frac {c \,x^{\frac {3}{2}}}{2}+\frac {1}{2}\right )}{4}+\frac {\ln \left (c \,x^{\frac {3}{2}}-1\right )^{2}}{8}+\frac {\ln \left (c \,x^{\frac {3}{2}}-1\right )}{2}+\frac {\ln \left (c \,x^{\frac {3}{2}}+1\right )}{2}+\frac {\ln \left (c \,x^{\frac {3}{2}}+1\right )^{2}}{8}-\frac {\left (\ln \left (c \,x^{\frac {3}{2}}+1\right )-\ln \left (\frac {c \,x^{\frac {3}{2}}}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {c \,x^{\frac {3}{2}}}{2}+\frac {1}{2}\right )}{4}\right )}{3}+\frac {4 a b \left (\frac {c^{2} x^{3} \operatorname {arctanh}\left (c \,x^{\frac {3}{2}}\right )}{2}+\frac {c \,x^{\frac {3}{2}}}{2}+\frac {\ln \left (c \,x^{\frac {3}{2}}-1\right )}{4}-\frac {\ln \left (c \,x^{\frac {3}{2}}+1\right )}{4}\right )}{3}}{c^{2}}\) | \(221\) |
| default | \(\frac {\frac {a^{2} c^{2} x^{3}}{3}+\frac {2 b^{2} \left (\frac {c^{2} x^{3} \operatorname {arctanh}\left (c \,x^{\frac {3}{2}}\right )^{2}}{2}+c \,x^{\frac {3}{2}} \operatorname {arctanh}\left (c \,x^{\frac {3}{2}}\right )+\frac {\operatorname {arctanh}\left (c \,x^{\frac {3}{2}}\right ) \ln \left (c \,x^{\frac {3}{2}}-1\right )}{2}-\frac {\operatorname {arctanh}\left (c \,x^{\frac {3}{2}}\right ) \ln \left (c \,x^{\frac {3}{2}}+1\right )}{2}-\frac {\ln \left (c \,x^{\frac {3}{2}}-1\right ) \ln \left (\frac {c \,x^{\frac {3}{2}}}{2}+\frac {1}{2}\right )}{4}+\frac {\ln \left (c \,x^{\frac {3}{2}}-1\right )^{2}}{8}+\frac {\ln \left (c \,x^{\frac {3}{2}}-1\right )}{2}+\frac {\ln \left (c \,x^{\frac {3}{2}}+1\right )}{2}+\frac {\ln \left (c \,x^{\frac {3}{2}}+1\right )^{2}}{8}-\frac {\left (\ln \left (c \,x^{\frac {3}{2}}+1\right )-\ln \left (\frac {c \,x^{\frac {3}{2}}}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {c \,x^{\frac {3}{2}}}{2}+\frac {1}{2}\right )}{4}\right )}{3}+\frac {4 a b \left (\frac {c^{2} x^{3} \operatorname {arctanh}\left (c \,x^{\frac {3}{2}}\right )}{2}+\frac {c \,x^{\frac {3}{2}}}{2}+\frac {\ln \left (c \,x^{\frac {3}{2}}-1\right )}{4}-\frac {\ln \left (c \,x^{\frac {3}{2}}+1\right )}{4}\right )}{3}}{c^{2}}\) | \(221\) |
1/3*a^2*x^3+2/3*b^2/c^2*(1/2*c^2*x^3*arctanh(c*x^(3/2))^2+c*x^(3/2)*arctan h(c*x^(3/2))+1/2*arctanh(c*x^(3/2))*ln(c*x^(3/2)-1)-1/2*arctanh(c*x^(3/2)) *ln(c*x^(3/2)+1)-1/4*ln(c*x^(3/2)-1)*ln(1/2*c*x^(3/2)+1/2)+1/8*ln(c*x^(3/2 )-1)^2+1/2*ln(c*x^(3/2)-1)+1/2*ln(c*x^(3/2)+1)+1/8*ln(c*x^(3/2)+1)^2-1/4*( ln(c*x^(3/2)+1)-ln(1/2*c*x^(3/2)+1/2))*ln(-1/2*c*x^(3/2)+1/2))+4/3*a*b/c^2 *(1/2*c^2*x^3*arctanh(c*x^(3/2))+1/2*c*x^(3/2)+1/4*ln(c*x^(3/2)-1)-1/4*ln( c*x^(3/2)+1))
Leaf count of result is larger than twice the leaf count of optimal. 179 vs. \(2 (81) = 162\).
Time = 0.28 (sec) , antiderivative size = 179, normalized size of antiderivative = 1.77 \[ \int x^2 \left (a+b \text {arctanh}\left (c x^{3/2}\right )\right )^2 \, dx=\frac {4 \, a^{2} c^{2} x^{3} + 8 \, a b c x^{\frac {3}{2}} + {\left (b^{2} c^{2} x^{3} - b^{2}\right )} \log \left (-\frac {c^{2} x^{3} + 2 \, c x^{\frac {3}{2}} + 1}{c^{2} x^{3} - 1}\right )^{2} + 4 \, {\left (a b c^{2} - a b + b^{2}\right )} \log \left (c x^{\frac {3}{2}} + 1\right ) - 4 \, {\left (a b c^{2} - a b - b^{2}\right )} \log \left (c x^{\frac {3}{2}} - 1\right ) + 4 \, {\left (a b c^{2} x^{3} + b^{2} c x^{\frac {3}{2}} - a b c^{2}\right )} \log \left (-\frac {c^{2} x^{3} + 2 \, c x^{\frac {3}{2}} + 1}{c^{2} x^{3} - 1}\right )}{12 \, c^{2}} \]
1/12*(4*a^2*c^2*x^3 + 8*a*b*c*x^(3/2) + (b^2*c^2*x^3 - b^2)*log(-(c^2*x^3 + 2*c*x^(3/2) + 1)/(c^2*x^3 - 1))^2 + 4*(a*b*c^2 - a*b + b^2)*log(c*x^(3/2 ) + 1) - 4*(a*b*c^2 - a*b - b^2)*log(c*x^(3/2) - 1) + 4*(a*b*c^2*x^3 + b^2 *c*x^(3/2) - a*b*c^2)*log(-(c^2*x^3 + 2*c*x^(3/2) + 1)/(c^2*x^3 - 1)))/c^2
Timed out. \[ \int x^2 \left (a+b \text {arctanh}\left (c x^{3/2}\right )\right )^2 \, dx=\text {Timed out} \]
Leaf count of result is larger than twice the leaf count of optimal. 186 vs. \(2 (81) = 162\).
Time = 0.19 (sec) , antiderivative size = 186, normalized size of antiderivative = 1.84 \[ \int x^2 \left (a+b \text {arctanh}\left (c x^{3/2}\right )\right )^2 \, dx=\frac {1}{3} \, b^{2} x^{3} \operatorname {artanh}\left (c x^{\frac {3}{2}}\right )^{2} + \frac {1}{3} \, a^{2} x^{3} + \frac {1}{3} \, {\left (2 \, x^{3} \operatorname {artanh}\left (c x^{\frac {3}{2}}\right ) + c {\left (\frac {2 \, x^{\frac {3}{2}}}{c^{2}} - \frac {\log \left (c x^{\frac {3}{2}} + 1\right )}{c^{3}} + \frac {\log \left (c x^{\frac {3}{2}} - 1\right )}{c^{3}}\right )}\right )} a b + \frac {1}{12} \, {\left (4 \, c {\left (\frac {2 \, x^{\frac {3}{2}}}{c^{2}} - \frac {\log \left (c x^{\frac {3}{2}} + 1\right )}{c^{3}} + \frac {\log \left (c x^{\frac {3}{2}} - 1\right )}{c^{3}}\right )} \operatorname {artanh}\left (c x^{\frac {3}{2}}\right ) - \frac {2 \, {\left (\log \left (c x^{\frac {3}{2}} - 1\right ) - 2\right )} \log \left (c x^{\frac {3}{2}} + 1\right ) - \log \left (c x^{\frac {3}{2}} + 1\right )^{2} - \log \left (c x^{\frac {3}{2}} - 1\right )^{2} - 4 \, \log \left (c x^{\frac {3}{2}} - 1\right )}{c^{2}}\right )} b^{2} \]
1/3*b^2*x^3*arctanh(c*x^(3/2))^2 + 1/3*a^2*x^3 + 1/3*(2*x^3*arctanh(c*x^(3 /2)) + c*(2*x^(3/2)/c^2 - log(c*x^(3/2) + 1)/c^3 + log(c*x^(3/2) - 1)/c^3) )*a*b + 1/12*(4*c*(2*x^(3/2)/c^2 - log(c*x^(3/2) + 1)/c^3 + log(c*x^(3/2) - 1)/c^3)*arctanh(c*x^(3/2)) - (2*(log(c*x^(3/2) - 1) - 2)*log(c*x^(3/2) + 1) - log(c*x^(3/2) + 1)^2 - log(c*x^(3/2) - 1)^2 - 4*log(c*x^(3/2) - 1))/ c^2)*b^2
\[ \int x^2 \left (a+b \text {arctanh}\left (c x^{3/2}\right )\right )^2 \, dx=\int { {\left (b \operatorname {artanh}\left (c x^{\frac {3}{2}}\right ) + a\right )}^{2} x^{2} \,d x } \]
Time = 4.06 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.04 \[ \int x^2 \left (a+b \text {arctanh}\left (c x^{3/2}\right )\right )^2 \, dx=\frac {c\,\left (\frac {2\,b^2\,x^{3/2}\,\mathrm {atanh}\left (c\,x^{3/2}\right )}{3}+\frac {2\,a\,b\,x^{3/2}}{3}\right )-\frac {b^2\,{\mathrm {atanh}\left (c\,x^{3/2}\right )}^2}{3}+\frac {b^2\,\ln \left (c^2\,x^3-1\right )}{3}-\frac {2\,a\,b\,\mathrm {atanh}\left (c\,x^{3/2}\right )}{3}}{c^2}+\frac {a^2\,x^3}{3}+\frac {b^2\,x^3\,{\mathrm {atanh}\left (c\,x^{3/2}\right )}^2}{3}+\frac {2\,a\,b\,x^3\,\mathrm {atanh}\left (c\,x^{3/2}\right )}{3} \]