Integrand size = 16, antiderivative size = 75 \[ \int x^2 \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right ) \, dx=\frac {b \sqrt {x}}{3 c^5}+\frac {b x^{3/2}}{9 c^3}+\frac {b x^{5/2}}{15 c}-\frac {b \text {arctanh}\left (c \sqrt {x}\right )}{3 c^6}+\frac {1}{3} x^3 \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right ) \]
1/9*b*x^(3/2)/c^3+1/15*b*x^(5/2)/c-1/3*b*arctanh(c*x^(1/2))/c^6+1/3*x^3*(a +b*arctanh(c*x^(1/2)))+1/3*b*x^(1/2)/c^5
Time = 0.03 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.35 \[ \int x^2 \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right ) \, dx=\frac {b \sqrt {x}}{3 c^5}+\frac {b x^{3/2}}{9 c^3}+\frac {b x^{5/2}}{15 c}+\frac {a x^3}{3}+\frac {1}{3} b x^3 \text {arctanh}\left (c \sqrt {x}\right )+\frac {b \log \left (1-c \sqrt {x}\right )}{6 c^6}-\frac {b \log \left (1+c \sqrt {x}\right )}{6 c^6} \]
(b*Sqrt[x])/(3*c^5) + (b*x^(3/2))/(9*c^3) + (b*x^(5/2))/(15*c) + (a*x^3)/3 + (b*x^3*ArcTanh[c*Sqrt[x]])/3 + (b*Log[1 - c*Sqrt[x]])/(6*c^6) - (b*Log[ 1 + c*Sqrt[x]])/(6*c^6)
Time = 0.24 (sec) , antiderivative size = 84, normalized size of antiderivative = 1.12, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {6452, 60, 60, 60, 73, 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.
\(\displaystyle \int x^2 \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right ) \, dx\) |
\(\Big \downarrow \) 6452 |
\(\displaystyle \frac {1}{3} x^3 \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )-\frac {1}{6} b c \int \frac {x^{5/2}}{1-c^2 x}dx\) |
\(\Big \downarrow \) 60 |
\(\displaystyle \frac {1}{3} x^3 \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )-\frac {1}{6} b c \left (\frac {\int \frac {x^{3/2}}{1-c^2 x}dx}{c^2}-\frac {2 x^{5/2}}{5 c^2}\right )\) |
\(\Big \downarrow \) 60 |
\(\displaystyle \frac {1}{3} x^3 \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )-\frac {1}{6} b c \left (\frac {\frac {\int \frac {\sqrt {x}}{1-c^2 x}dx}{c^2}-\frac {2 x^{3/2}}{3 c^2}}{c^2}-\frac {2 x^{5/2}}{5 c^2}\right )\) |
\(\Big \downarrow \) 60 |
\(\displaystyle \frac {1}{3} x^3 \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )-\frac {1}{6} b c \left (\frac {\frac {\frac {\int \frac {1}{\sqrt {x} \left (1-c^2 x\right )}dx}{c^2}-\frac {2 \sqrt {x}}{c^2}}{c^2}-\frac {2 x^{3/2}}{3 c^2}}{c^2}-\frac {2 x^{5/2}}{5 c^2}\right )\) |
\(\Big \downarrow \) 73 |
\(\displaystyle \frac {1}{3} x^3 \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )-\frac {1}{6} b c \left (\frac {\frac {\frac {2 \int \frac {1}{1-c^2 x}d\sqrt {x}}{c^2}-\frac {2 \sqrt {x}}{c^2}}{c^2}-\frac {2 x^{3/2}}{3 c^2}}{c^2}-\frac {2 x^{5/2}}{5 c^2}\right )\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {1}{3} x^3 \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right )-\frac {1}{6} b c \left (\frac {\frac {\frac {2 \text {arctanh}\left (c \sqrt {x}\right )}{c^3}-\frac {2 \sqrt {x}}{c^2}}{c^2}-\frac {2 x^{3/2}}{3 c^2}}{c^2}-\frac {2 x^{5/2}}{5 c^2}\right )\) |
(x^3*(a + b*ArcTanh[c*Sqrt[x]]))/3 - (b*c*((-2*x^(5/2))/(5*c^2) + ((-2*x^( 3/2))/(3*c^2) + ((-2*Sqrt[x])/c^2 + (2*ArcTanh[c*Sqrt[x]])/c^3)/c^2)/c^2)) /6
3.2.88.3.1 Defintions of rubi rules used
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + n + 1))), x] + Simp[n*((b*c - a*d)/( b*(m + n + 1))) Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d}, x] && GtQ[n, 0] && NeQ[m + n + 1, 0] && !(IGtQ[m, 0] && ( !Integer Q[n] || (GtQ[m, 0] && LtQ[m - n, 0]))) && !ILtQ[m + n + 2, 0] && IntLinear Q[a, b, c, d, m, n, x]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ {p = Denominator[m]}, Simp[p/b Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL inearQ[a, b, c, d, m, n, x]
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_.) + 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 = 0.77 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.95
method | result | size |
parts | \(\frac {a \,x^{3}}{3}+\frac {2 b \left (\frac {c^{6} x^{3} \operatorname {arctanh}\left (c \sqrt {x}\right )}{6}+\frac {c^{5} x^{\frac {5}{2}}}{30}+\frac {c^{3} x^{\frac {3}{2}}}{18}+\frac {c \sqrt {x}}{6}+\frac {\ln \left (c \sqrt {x}-1\right )}{12}-\frac {\ln \left (1+c \sqrt {x}\right )}{12}\right )}{c^{6}}\) | \(71\) |
derivativedivides | \(\frac {\frac {a \,c^{6} x^{3}}{3}+2 b \left (\frac {c^{6} x^{3} \operatorname {arctanh}\left (c \sqrt {x}\right )}{6}+\frac {c^{5} x^{\frac {5}{2}}}{30}+\frac {c^{3} x^{\frac {3}{2}}}{18}+\frac {c \sqrt {x}}{6}+\frac {\ln \left (c \sqrt {x}-1\right )}{12}-\frac {\ln \left (1+c \sqrt {x}\right )}{12}\right )}{c^{6}}\) | \(75\) |
default | \(\frac {\frac {a \,c^{6} x^{3}}{3}+2 b \left (\frac {c^{6} x^{3} \operatorname {arctanh}\left (c \sqrt {x}\right )}{6}+\frac {c^{5} x^{\frac {5}{2}}}{30}+\frac {c^{3} x^{\frac {3}{2}}}{18}+\frac {c \sqrt {x}}{6}+\frac {\ln \left (c \sqrt {x}-1\right )}{12}-\frac {\ln \left (1+c \sqrt {x}\right )}{12}\right )}{c^{6}}\) | \(75\) |
1/3*a*x^3+2*b/c^6*(1/6*c^6*x^3*arctanh(c*x^(1/2))+1/30*c^5*x^(5/2)+1/18*c^ 3*x^(3/2)+1/6*c*x^(1/2)+1/12*ln(c*x^(1/2)-1)-1/12*ln(1+c*x^(1/2)))
Time = 0.27 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.07 \[ \int x^2 \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right ) \, dx=\frac {30 \, a c^{6} x^{3} + 15 \, {\left (b c^{6} x^{3} - b\right )} \log \left (-\frac {c^{2} x + 2 \, c \sqrt {x} + 1}{c^{2} x - 1}\right ) + 2 \, {\left (3 \, b c^{5} x^{2} + 5 \, b c^{3} x + 15 \, b c\right )} \sqrt {x}}{90 \, c^{6}} \]
1/90*(30*a*c^6*x^3 + 15*(b*c^6*x^3 - b)*log(-(c^2*x + 2*c*sqrt(x) + 1)/(c^ 2*x - 1)) + 2*(3*b*c^5*x^2 + 5*b*c^3*x + 15*b*c)*sqrt(x))/c^6
\[ \int x^2 \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right ) \, dx=\int x^{2} \left (a + b \operatorname {atanh}{\left (c \sqrt {x} \right )}\right )\, dx \]
Time = 0.19 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.04 \[ \int x^2 \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right ) \, dx=\frac {1}{3} \, a x^{3} + \frac {1}{90} \, {\left (30 \, x^{3} \operatorname {artanh}\left (c \sqrt {x}\right ) + c {\left (\frac {2 \, {\left (3 \, c^{4} x^{\frac {5}{2}} + 5 \, c^{2} x^{\frac {3}{2}} + 15 \, \sqrt {x}\right )}}{c^{6}} - \frac {15 \, \log \left (c \sqrt {x} + 1\right )}{c^{7}} + \frac {15 \, \log \left (c \sqrt {x} - 1\right )}{c^{7}}\right )}\right )} b \]
1/3*a*x^3 + 1/90*(30*x^3*arctanh(c*sqrt(x)) + c*(2*(3*c^4*x^(5/2) + 5*c^2* x^(3/2) + 15*sqrt(x))/c^6 - 15*log(c*sqrt(x) + 1)/c^7 + 15*log(c*sqrt(x) - 1)/c^7))*b
Leaf count of result is larger than twice the leaf count of optimal. 301 vs. \(2 (55) = 110\).
Time = 0.29 (sec) , antiderivative size = 301, normalized size of antiderivative = 4.01 \[ \int x^2 \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right ) \, dx=\frac {1}{3} \, a x^{3} + \frac {2}{45} \, b c {\left (\frac {\frac {45 \, {\left (c \sqrt {x} + 1\right )}^{4}}{{\left (c \sqrt {x} - 1\right )}^{4}} - \frac {90 \, {\left (c \sqrt {x} + 1\right )}^{3}}{{\left (c \sqrt {x} - 1\right )}^{3}} + \frac {140 \, {\left (c \sqrt {x} + 1\right )}^{2}}{{\left (c \sqrt {x} - 1\right )}^{2}} - \frac {70 \, {\left (c \sqrt {x} + 1\right )}}{c \sqrt {x} - 1} + 23}{c^{7} {\left (\frac {c \sqrt {x} + 1}{c \sqrt {x} - 1} - 1\right )}^{5}} + \frac {15 \, {\left (\frac {3 \, {\left (c \sqrt {x} + 1\right )}^{5}}{{\left (c \sqrt {x} - 1\right )}^{5}} + \frac {10 \, {\left (c \sqrt {x} + 1\right )}^{3}}{{\left (c \sqrt {x} - 1\right )}^{3}} + \frac {3 \, {\left (c \sqrt {x} + 1\right )}}{c \sqrt {x} - 1}\right )} \log \left (-\frac {\frac {c {\left (\frac {c \sqrt {x} + 1}{c \sqrt {x} - 1} + 1\right )}}{\frac {{\left (c \sqrt {x} + 1\right )} c}{c \sqrt {x} - 1} - c} + 1}{\frac {c {\left (\frac {c \sqrt {x} + 1}{c \sqrt {x} - 1} + 1\right )}}{\frac {{\left (c \sqrt {x} + 1\right )} c}{c \sqrt {x} - 1} - c} - 1}\right )}{c^{7} {\left (\frac {c \sqrt {x} + 1}{c \sqrt {x} - 1} - 1\right )}^{6}}\right )} \]
1/3*a*x^3 + 2/45*b*c*((45*(c*sqrt(x) + 1)^4/(c*sqrt(x) - 1)^4 - 90*(c*sqrt (x) + 1)^3/(c*sqrt(x) - 1)^3 + 140*(c*sqrt(x) + 1)^2/(c*sqrt(x) - 1)^2 - 7 0*(c*sqrt(x) + 1)/(c*sqrt(x) - 1) + 23)/(c^7*((c*sqrt(x) + 1)/(c*sqrt(x) - 1) - 1)^5) + 15*(3*(c*sqrt(x) + 1)^5/(c*sqrt(x) - 1)^5 + 10*(c*sqrt(x) + 1)^3/(c*sqrt(x) - 1)^3 + 3*(c*sqrt(x) + 1)/(c*sqrt(x) - 1))*log(-(c*((c*sq rt(x) + 1)/(c*sqrt(x) - 1) + 1)/((c*sqrt(x) + 1)*c/(c*sqrt(x) - 1) - c) + 1)/(c*((c*sqrt(x) + 1)/(c*sqrt(x) - 1) + 1)/((c*sqrt(x) + 1)*c/(c*sqrt(x) - 1) - c) - 1))/(c^7*((c*sqrt(x) + 1)/(c*sqrt(x) - 1) - 1)^6))
Time = 3.64 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.77 \[ \int x^2 \left (a+b \text {arctanh}\left (c \sqrt {x}\right )\right ) \, dx=\frac {a\,x^3}{3}+\frac {\frac {b\,c^3\,x^{3/2}}{9}-\frac {b\,\mathrm {atanh}\left (c\,\sqrt {x}\right )}{3}+\frac {b\,c^5\,x^{5/2}}{15}+\frac {b\,c\,\sqrt {x}}{3}}{c^6}+\frac {b\,x^3\,\mathrm {atanh}\left (c\,\sqrt {x}\right )}{3} \]