Integrand size = 14, antiderivative size = 61 \[ \int x^2 \left (a+b \text {arctanh}\left (\frac {c}{x^2}\right )\right ) \, dx=\frac {2 b c x}{3}-\frac {1}{3} b c^{3/2} \arctan \left (\frac {x}{\sqrt {c}}\right )+\frac {1}{3} x^3 \left (a+b \text {arctanh}\left (\frac {c}{x^2}\right )\right )-\frac {1}{3} b c^{3/2} \text {arctanh}\left (\frac {x}{\sqrt {c}}\right ) \]
2/3*b*c*x-1/3*b*c^(3/2)*arctan(x/c^(1/2))+1/3*x^3*(a+b*arctanh(c/x^2))-1/3 *b*c^(3/2)*arctanh(x/c^(1/2))
Time = 0.03 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.41 \[ \int x^2 \left (a+b \text {arctanh}\left (\frac {c}{x^2}\right )\right ) \, dx=\frac {2 b c x}{3}+\frac {a x^3}{3}-\frac {1}{3} b c^{3/2} \arctan \left (\frac {x}{\sqrt {c}}\right )+\frac {1}{3} b x^3 \text {arctanh}\left (\frac {c}{x^2}\right )+\frac {1}{6} b c^{3/2} \log \left (\sqrt {c}-x\right )-\frac {1}{6} b c^{3/2} \log \left (\sqrt {c}+x\right ) \]
(2*b*c*x)/3 + (a*x^3)/3 - (b*c^(3/2)*ArcTan[x/Sqrt[c]])/3 + (b*x^3*ArcTanh [c/x^2])/3 + (b*c^(3/2)*Log[Sqrt[c] - x])/6 - (b*c^(3/2)*Log[Sqrt[c] + x]) /6
Time = 0.24 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.07, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {6452, 772, 843, 756, 216, 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.
| \(\displaystyle \int x^2 \left (a+b \text {arctanh}\left (\frac {c}{x^2}\right )\right ) \, dx\) |
| \(\Big \downarrow \) 6452 |
| \(\displaystyle \frac {2}{3} b c \int \frac {1}{1-\frac {c^2}{x^4}}dx+\frac {1}{3} x^3 \left (a+b \text {arctanh}\left (\frac {c}{x^2}\right )\right )\) |
| \(\Big \downarrow \) 772 |
| \(\displaystyle \frac {2}{3} b c \int \frac {x^4}{x^4-c^2}dx+\frac {1}{3} x^3 \left (a+b \text {arctanh}\left (\frac {c}{x^2}\right )\right )\) |
| \(\Big \downarrow \) 843 |
| \(\displaystyle \frac {2}{3} b c \left (c^2 \int \frac {1}{x^4-c^2}dx+x\right )+\frac {1}{3} x^3 \left (a+b \text {arctanh}\left (\frac {c}{x^2}\right )\right )\) |
| \(\Big \downarrow \) 756 |
| \(\displaystyle \frac {2}{3} b c \left (c^2 \left (-\frac {\int \frac {1}{c-x^2}dx}{2 c}-\frac {\int \frac {1}{x^2+c}dx}{2 c}\right )+x\right )+\frac {1}{3} x^3 \left (a+b \text {arctanh}\left (\frac {c}{x^2}\right )\right )\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle \frac {2}{3} b c \left (c^2 \left (-\frac {\int \frac {1}{c-x^2}dx}{2 c}-\frac {\arctan \left (\frac {x}{\sqrt {c}}\right )}{2 c^{3/2}}\right )+x\right )+\frac {1}{3} x^3 \left (a+b \text {arctanh}\left (\frac {c}{x^2}\right )\right )\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {1}{3} x^3 \left (a+b \text {arctanh}\left (\frac {c}{x^2}\right )\right )+\frac {2}{3} b c \left (c^2 \left (-\frac {\arctan \left (\frac {x}{\sqrt {c}}\right )}{2 c^{3/2}}-\frac {\text {arctanh}\left (\frac {x}{\sqrt {c}}\right )}{2 c^{3/2}}\right )+x\right )\) |
(x^3*(a + b*ArcTanh[c/x^2]))/3 + (2*b*c*(x + c^2*(-1/2*ArcTan[x/Sqrt[c]]/c ^(3/2) - ArcTanh[x/Sqrt[c]]/(2*c^(3/2)))))/3
3.2.66.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*A rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a , 0] || GtQ[b, 0])
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_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 2 ]], s = Denominator[Rt[-a/b, 2]]}, Simp[r/(2*a) Int[1/(r - s*x^2), x], x] + Simp[r/(2*a) Int[1/(r + s*x^2), x], x]] /; FreeQ[{a, b}, x] && !GtQ[a /b, 0]
Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[x^(n*p)*(b + a/x^n)^p, x] /; FreeQ[{a, b}, x] && ILtQ[n, 0] && IntegerQ[p]
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[c^(n - 1)*(c*x)^(m - n + 1)*((a + b*x^n)^(p + 1)/(b*(m + n*p + 1))), x] - Simp[ a*c^n*((m - n + 1)/(b*(m + n*p + 1))) Int[(c*x)^(m - n)*(a + b*x^n)^p, x] , x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n* p + 1, 0] && IntBinomialQ[a, b, c, n, m, p, x]
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 = 1.06 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.84
| method | result | size |
| parts | \(\frac {a \,x^{3}}{3}+\frac {b \,x^{3} \operatorname {arctanh}\left (\frac {c}{x^{2}}\right )}{3}+\frac {2 b c x}{3}-\frac {b \,c^{\frac {3}{2}} \operatorname {arctanh}\left (\frac {\sqrt {c}}{x}\right )}{3}-\frac {b \,c^{\frac {3}{2}} \arctan \left (\frac {x}{\sqrt {c}}\right )}{3}\) | \(51\) |
| derivativedivides | \(\frac {a \,x^{3}}{3}+\frac {b \,x^{3} \operatorname {arctanh}\left (\frac {c}{x^{2}}\right )}{3}-\frac {b \,c^{\frac {3}{2}} \operatorname {arctanh}\left (\frac {\sqrt {c}}{x}\right )}{3}+\frac {2 b c x}{3}+\frac {b \,c^{\frac {3}{2}} \arctan \left (\frac {\sqrt {c}}{x}\right )}{3}\) | \(53\) |
| default | \(\frac {a \,x^{3}}{3}+\frac {b \,x^{3} \operatorname {arctanh}\left (\frac {c}{x^{2}}\right )}{3}-\frac {b \,c^{\frac {3}{2}} \operatorname {arctanh}\left (\frac {\sqrt {c}}{x}\right )}{3}+\frac {2 b c x}{3}+\frac {b \,c^{\frac {3}{2}} \arctan \left (\frac {\sqrt {c}}{x}\right )}{3}\) | \(53\) |
| risch | \(\frac {b \,x^{3} \ln \left (x^{2}+c \right )}{6}-\frac {b \,x^{3} \ln \left (-x^{2}+c \right )}{6}+\frac {i \pi b \,x^{3} \operatorname {csgn}\left (\frac {i}{x^{2}}\right ) {\operatorname {csgn}\left (\frac {i \left (x^{2}+c \right )}{x^{2}}\right )}^{2}}{12}-\frac {i \pi b \,x^{3} {\operatorname {csgn}\left (\frac {i \left (-x^{2}+c \right )}{x^{2}}\right )}^{3}}{12}+\frac {i \pi b \,x^{3} \operatorname {csgn}\left (i \left (x^{2}+c \right )\right ) {\operatorname {csgn}\left (\frac {i \left (x^{2}+c \right )}{x^{2}}\right )}^{2}}{12}-\frac {i \pi b \,x^{3} \operatorname {csgn}\left (\frac {i}{x^{2}}\right ) \operatorname {csgn}\left (i \left (x^{2}+c \right )\right ) \operatorname {csgn}\left (\frac {i \left (x^{2}+c \right )}{x^{2}}\right )}{12}-\frac {i \pi b \,x^{3} \operatorname {csgn}\left (i \left (-x^{2}+c \right )\right ) {\operatorname {csgn}\left (\frac {i \left (-x^{2}+c \right )}{x^{2}}\right )}^{2}}{12}-\frac {i \pi b \,x^{3}}{6}+\frac {i \pi b \,x^{3} \operatorname {csgn}\left (\frac {i}{x^{2}}\right ) \operatorname {csgn}\left (i \left (-x^{2}+c \right )\right ) \operatorname {csgn}\left (\frac {i \left (-x^{2}+c \right )}{x^{2}}\right )}{12}+\frac {i \pi b \,x^{3} {\operatorname {csgn}\left (\frac {i \left (-x^{2}+c \right )}{x^{2}}\right )}^{2}}{6}-\frac {i \pi b \,x^{3} \operatorname {csgn}\left (\frac {i}{x^{2}}\right ) {\operatorname {csgn}\left (\frac {i \left (-x^{2}+c \right )}{x^{2}}\right )}^{2}}{12}-\frac {i \pi b \,x^{3} {\operatorname {csgn}\left (\frac {i \left (x^{2}+c \right )}{x^{2}}\right )}^{3}}{12}+\frac {a \,x^{3}}{3}+\frac {2 b c x}{3}+\frac {c^{\frac {3}{2}} b \ln \left (c^{\frac {3}{2}}-c x \right )}{6}-\frac {c^{\frac {3}{2}} b \ln \left (-c^{\frac {3}{2}}-c x \right )}{6}+\frac {b c \sqrt {-c}\, \ln \left (\sqrt {-c}\, c -c x \right )}{6}-\frac {b c \sqrt {-c}\, \ln \left (-\sqrt {-c}\, c -c x \right )}{6}\) | \(396\) |
1/3*a*x^3+1/3*b*x^3*arctanh(c/x^2)+2/3*b*c*x-1/3*b*c^(3/2)*arctanh(1/x*c^( 1/2))-1/3*b*c^(3/2)*arctan(x/c^(1/2))
Time = 0.27 (sec) , antiderivative size = 162, normalized size of antiderivative = 2.66 \[ \int x^2 \left (a+b \text {arctanh}\left (\frac {c}{x^2}\right )\right ) \, dx=\left [\frac {1}{6} \, b x^{3} \log \left (\frac {x^{2} + c}{x^{2} - c}\right ) + \frac {1}{3} \, a x^{3} - \frac {1}{3} \, b c^{\frac {3}{2}} \arctan \left (\frac {x}{\sqrt {c}}\right ) + \frac {1}{6} \, b c^{\frac {3}{2}} \log \left (\frac {x^{2} - 2 \, \sqrt {c} x + c}{x^{2} - c}\right ) + \frac {2}{3} \, b c x, \frac {1}{6} \, b x^{3} \log \left (\frac {x^{2} + c}{x^{2} - c}\right ) + \frac {1}{3} \, a x^{3} + \frac {1}{3} \, b \sqrt {-c} c \arctan \left (\frac {\sqrt {-c} x}{c}\right ) + \frac {1}{6} \, b \sqrt {-c} c \log \left (\frac {x^{2} - 2 \, \sqrt {-c} x - c}{x^{2} + c}\right ) + \frac {2}{3} \, b c x\right ] \]
[1/6*b*x^3*log((x^2 + c)/(x^2 - c)) + 1/3*a*x^3 - 1/3*b*c^(3/2)*arctan(x/s qrt(c)) + 1/6*b*c^(3/2)*log((x^2 - 2*sqrt(c)*x + c)/(x^2 - c)) + 2/3*b*c*x , 1/6*b*x^3*log((x^2 + c)/(x^2 - c)) + 1/3*a*x^3 + 1/3*b*sqrt(-c)*c*arctan (sqrt(-c)*x/c) + 1/6*b*sqrt(-c)*c*log((x^2 - 2*sqrt(-c)*x - c)/(x^2 + c)) + 2/3*b*c*x]
Leaf count of result is larger than twice the leaf count of optimal. 830 vs. \(2 (56) = 112\).
Time = 2.66 (sec) , antiderivative size = 830, normalized size of antiderivative = 13.61 \[ \int x^2 \left (a+b \text {arctanh}\left (\frac {c}{x^2}\right )\right ) \, dx=\begin {cases} \frac {a x^{3}}{3} & \text {for}\: c = 0 \\\frac {x^{3} \left (a - \infty b\right )}{3} & \text {for}\: c = - x^{2} \\\frac {x^{3} \left (a + \infty b\right )}{3} & \text {for}\: c = x^{2} \\- \frac {2 a c^{2} x^{3} \sqrt {- c}}{- 6 c^{2} \sqrt {- c} + 6 x^{4} \sqrt {- c}} + \frac {2 a x^{7} \sqrt {- c}}{- 6 c^{2} \sqrt {- c} + 6 x^{4} \sqrt {- c}} - \frac {2 b c^{\frac {7}{2}} \sqrt {- c} \log {\left (- \sqrt {c} + x \right )}}{- 6 c^{2} \sqrt {- c} + 6 x^{4} \sqrt {- c}} + \frac {b c^{\frac {7}{2}} \sqrt {- c} \log {\left (x - \sqrt {- c} \right )}}{- 6 c^{2} \sqrt {- c} + 6 x^{4} \sqrt {- c}} + \frac {b c^{\frac {7}{2}} \sqrt {- c} \log {\left (x + \sqrt {- c} \right )}}{- 6 c^{2} \sqrt {- c} + 6 x^{4} \sqrt {- c}} - \frac {2 b c^{\frac {7}{2}} \sqrt {- c} \operatorname {atanh}{\left (\frac {c}{x^{2}} \right )}}{- 6 c^{2} \sqrt {- c} + 6 x^{4} \sqrt {- c}} + \frac {2 b c^{\frac {3}{2}} x^{4} \sqrt {- c} \log {\left (- \sqrt {c} + x \right )}}{- 6 c^{2} \sqrt {- c} + 6 x^{4} \sqrt {- c}} - \frac {b c^{\frac {3}{2}} x^{4} \sqrt {- c} \log {\left (x - \sqrt {- c} \right )}}{- 6 c^{2} \sqrt {- c} + 6 x^{4} \sqrt {- c}} - \frac {b c^{\frac {3}{2}} x^{4} \sqrt {- c} \log {\left (x + \sqrt {- c} \right )}}{- 6 c^{2} \sqrt {- c} + 6 x^{4} \sqrt {- c}} + \frac {2 b c^{\frac {3}{2}} x^{4} \sqrt {- c} \operatorname {atanh}{\left (\frac {c}{x^{2}} \right )}}{- 6 c^{2} \sqrt {- c} + 6 x^{4} \sqrt {- c}} + \frac {b c^{4} \log {\left (x - \sqrt {- c} \right )}}{- 6 c^{2} \sqrt {- c} + 6 x^{4} \sqrt {- c}} - \frac {b c^{4} \log {\left (x + \sqrt {- c} \right )}}{- 6 c^{2} \sqrt {- c} + 6 x^{4} \sqrt {- c}} - \frac {4 b c^{3} x \sqrt {- c}}{- 6 c^{2} \sqrt {- c} + 6 x^{4} \sqrt {- c}} - \frac {b c^{2} x^{4} \log {\left (x - \sqrt {- c} \right )}}{- 6 c^{2} \sqrt {- c} + 6 x^{4} \sqrt {- c}} + \frac {b c^{2} x^{4} \log {\left (x + \sqrt {- c} \right )}}{- 6 c^{2} \sqrt {- c} + 6 x^{4} \sqrt {- c}} - \frac {2 b c^{2} x^{3} \sqrt {- c} \operatorname {atanh}{\left (\frac {c}{x^{2}} \right )}}{- 6 c^{2} \sqrt {- c} + 6 x^{4} \sqrt {- c}} + \frac {4 b c x^{5} \sqrt {- c}}{- 6 c^{2} \sqrt {- c} + 6 x^{4} \sqrt {- c}} + \frac {2 b x^{7} \sqrt {- c} \operatorname {atanh}{\left (\frac {c}{x^{2}} \right )}}{- 6 c^{2} \sqrt {- c} + 6 x^{4} \sqrt {- c}} & \text {otherwise} \end {cases} \]
Piecewise((a*x**3/3, Eq(c, 0)), (x**3*(a - oo*b)/3, Eq(c, -x**2)), (x**3*( a + oo*b)/3, Eq(c, x**2)), (-2*a*c**2*x**3*sqrt(-c)/(-6*c**2*sqrt(-c) + 6* x**4*sqrt(-c)) + 2*a*x**7*sqrt(-c)/(-6*c**2*sqrt(-c) + 6*x**4*sqrt(-c)) - 2*b*c**(7/2)*sqrt(-c)*log(-sqrt(c) + x)/(-6*c**2*sqrt(-c) + 6*x**4*sqrt(-c )) + b*c**(7/2)*sqrt(-c)*log(x - sqrt(-c))/(-6*c**2*sqrt(-c) + 6*x**4*sqrt (-c)) + b*c**(7/2)*sqrt(-c)*log(x + sqrt(-c))/(-6*c**2*sqrt(-c) + 6*x**4*s qrt(-c)) - 2*b*c**(7/2)*sqrt(-c)*atanh(c/x**2)/(-6*c**2*sqrt(-c) + 6*x**4* sqrt(-c)) + 2*b*c**(3/2)*x**4*sqrt(-c)*log(-sqrt(c) + x)/(-6*c**2*sqrt(-c) + 6*x**4*sqrt(-c)) - b*c**(3/2)*x**4*sqrt(-c)*log(x - sqrt(-c))/(-6*c**2* sqrt(-c) + 6*x**4*sqrt(-c)) - b*c**(3/2)*x**4*sqrt(-c)*log(x + sqrt(-c))/( -6*c**2*sqrt(-c) + 6*x**4*sqrt(-c)) + 2*b*c**(3/2)*x**4*sqrt(-c)*atanh(c/x **2)/(-6*c**2*sqrt(-c) + 6*x**4*sqrt(-c)) + b*c**4*log(x - sqrt(-c))/(-6*c **2*sqrt(-c) + 6*x**4*sqrt(-c)) - b*c**4*log(x + sqrt(-c))/(-6*c**2*sqrt(- c) + 6*x**4*sqrt(-c)) - 4*b*c**3*x*sqrt(-c)/(-6*c**2*sqrt(-c) + 6*x**4*sqr t(-c)) - b*c**2*x**4*log(x - sqrt(-c))/(-6*c**2*sqrt(-c) + 6*x**4*sqrt(-c) ) + b*c**2*x**4*log(x + sqrt(-c))/(-6*c**2*sqrt(-c) + 6*x**4*sqrt(-c)) - 2 *b*c**2*x**3*sqrt(-c)*atanh(c/x**2)/(-6*c**2*sqrt(-c) + 6*x**4*sqrt(-c)) + 4*b*c*x**5*sqrt(-c)/(-6*c**2*sqrt(-c) + 6*x**4*sqrt(-c)) + 2*b*x**7*sqrt( -c)*atanh(c/x**2)/(-6*c**2*sqrt(-c) + 6*x**4*sqrt(-c)), True))
Time = 0.26 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.00 \[ \int x^2 \left (a+b \text {arctanh}\left (\frac {c}{x^2}\right )\right ) \, dx=\frac {1}{3} \, a x^{3} + \frac {1}{6} \, {\left (2 \, x^{3} \operatorname {artanh}\left (\frac {c}{x^{2}}\right ) - {\left (2 \, \sqrt {c} \arctan \left (\frac {x}{\sqrt {c}}\right ) - \sqrt {c} \log \left (\frac {x - \sqrt {c}}{x + \sqrt {c}}\right ) - 4 \, x\right )} c\right )} b \]
1/3*a*x^3 + 1/6*(2*x^3*arctanh(c/x^2) - (2*sqrt(c)*arctan(x/sqrt(c)) - sqr t(c)*log((x - sqrt(c))/(x + sqrt(c))) - 4*x)*c)*b
Time = 0.28 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.13 \[ \int x^2 \left (a+b \text {arctanh}\left (\frac {c}{x^2}\right )\right ) \, dx=\frac {1}{3} \, b c^{3} {\left (\frac {\arctan \left (\frac {x}{\sqrt {-c}}\right )}{\sqrt {-c} c} - \frac {\arctan \left (\frac {x}{\sqrt {c}}\right )}{c^{\frac {3}{2}}}\right )} + \frac {1}{6} \, b x^{3} \log \left (\frac {x^{2} + c}{x^{2} - c}\right ) + \frac {1}{3} \, a x^{3} + \frac {2}{3} \, b c x \]
1/3*b*c^3*(arctan(x/sqrt(-c))/(sqrt(-c)*c) - arctan(x/sqrt(c))/c^(3/2)) + 1/6*b*x^3*log((x^2 + c)/(x^2 - c)) + 1/3*a*x^3 + 2/3*b*c*x
Time = 3.61 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.07 \[ \int x^2 \left (a+b \text {arctanh}\left (\frac {c}{x^2}\right )\right ) \, dx=\frac {a\,x^3}{3}-\frac {b\,c^{3/2}\,\mathrm {atan}\left (\frac {x}{\sqrt {c}}\right )}{3}+\frac {2\,b\,c\,x}{3}+\frac {b\,x^3\,\ln \left (x^2+c\right )}{6}-\frac {b\,x^3\,\ln \left (x^2-c\right )}{6}+\frac {b\,c^{3/2}\,\mathrm {atan}\left (\frac {x\,1{}\mathrm {i}}{\sqrt {c}}\right )\,1{}\mathrm {i}}{3} \]