Integrand size = 23, antiderivative size = 135 \[ \int e^{\text {arctanh}(a x)} \sqrt {c-\frac {c}{a x}} x \, dx=\frac {\sqrt {c-\frac {c}{a x}} x \sqrt {1+a x}}{4 a \sqrt {1-a x}}+\frac {\sqrt {c-\frac {c}{a x}} x^2 \sqrt {1+a x}}{2 \sqrt {1-a x}}-\frac {\sqrt {c-\frac {c}{a x}} \sqrt {x} \text {arcsinh}\left (\sqrt {a} \sqrt {x}\right )}{4 a^{3/2} \sqrt {1-a x}} \]
-1/4*arcsinh(a^(1/2)*x^(1/2))*(c-c/a/x)^(1/2)*x^(1/2)/a^(3/2)/(-a*x+1)^(1/ 2)+1/4*x*(c-c/a/x)^(1/2)*(a*x+1)^(1/2)/a/(-a*x+1)^(1/2)+1/2*x^2*(c-c/a/x)^ (1/2)*(a*x+1)^(1/2)/(-a*x+1)^(1/2)
Time = 0.04 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.59 \[ \int e^{\text {arctanh}(a x)} \sqrt {c-\frac {c}{a x}} x \, dx=\frac {\sqrt {c-\frac {c}{a x}} \sqrt {x} \left (\sqrt {a} \sqrt {x} \sqrt {1+a x} (1+2 a x)-\text {arcsinh}\left (\sqrt {a} \sqrt {x}\right )\right )}{4 a^{3/2} \sqrt {1-a x}} \]
(Sqrt[c - c/(a*x)]*Sqrt[x]*(Sqrt[a]*Sqrt[x]*Sqrt[1 + a*x]*(1 + 2*a*x) - Ar cSinh[Sqrt[a]*Sqrt[x]]))/(4*a^(3/2)*Sqrt[1 - a*x])
Time = 0.45 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.68, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.304, Rules used = {6684, 6678, 516, 60, 60, 63, 222}
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.
| \(\displaystyle \int x e^{\text {arctanh}(a x)} \sqrt {c-\frac {c}{a x}} \, dx\) |
| \(\Big \downarrow \) 6684 |
| \(\displaystyle \frac {\sqrt {x} \sqrt {c-\frac {c}{a x}} \int e^{\text {arctanh}(a x)} \sqrt {x} \sqrt {1-a x}dx}{\sqrt {1-a x}}\) |
| \(\Big \downarrow \) 6678 |
| \(\displaystyle \frac {\sqrt {x} \sqrt {c-\frac {c}{a x}} \int \frac {\sqrt {x} \sqrt {1-a^2 x^2}}{\sqrt {1-a x}}dx}{\sqrt {1-a x}}\) |
| \(\Big \downarrow \) 516 |
| \(\displaystyle \frac {\sqrt {x} \sqrt {c-\frac {c}{a x}} \int \sqrt {x} \sqrt {a x+1}dx}{\sqrt {1-a x}}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {\sqrt {x} \sqrt {c-\frac {c}{a x}} \left (\frac {1}{4} \int \frac {\sqrt {x}}{\sqrt {a x+1}}dx+\frac {1}{2} x^{3/2} \sqrt {a x+1}\right )}{\sqrt {1-a x}}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {\sqrt {x} \sqrt {c-\frac {c}{a x}} \left (\frac {1}{4} \left (\frac {\sqrt {x} \sqrt {a x+1}}{a}-\frac {\int \frac {1}{\sqrt {x} \sqrt {a x+1}}dx}{2 a}\right )+\frac {1}{2} x^{3/2} \sqrt {a x+1}\right )}{\sqrt {1-a x}}\) |
| \(\Big \downarrow \) 63 |
| \(\displaystyle \frac {\sqrt {x} \sqrt {c-\frac {c}{a x}} \left (\frac {1}{4} \left (\frac {\sqrt {x} \sqrt {a x+1}}{a}-\frac {\int \frac {1}{\sqrt {a x+1}}d\sqrt {x}}{a}\right )+\frac {1}{2} x^{3/2} \sqrt {a x+1}\right )}{\sqrt {1-a x}}\) |
| \(\Big \downarrow \) 222 |
| \(\displaystyle \frac {\sqrt {x} \left (\frac {1}{4} \left (\frac {\sqrt {x} \sqrt {a x+1}}{a}-\frac {\text {arcsinh}\left (\sqrt {a} \sqrt {x}\right )}{a^{3/2}}\right )+\frac {1}{2} x^{3/2} \sqrt {a x+1}\right ) \sqrt {c-\frac {c}{a x}}}{\sqrt {1-a x}}\) |
(Sqrt[c - c/(a*x)]*Sqrt[x]*((x^(3/2)*Sqrt[1 + a*x])/2 + ((Sqrt[x]*Sqrt[1 + a*x])/a - ArcSinh[Sqrt[a]*Sqrt[x]]/a^(3/2))/4))/Sqrt[1 - a*x]
3.6.66.3.1 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[1/(Sqrt[(b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Simp[2/b S ubst[Int[1/Sqrt[c + d*(x^2/b)], x], x, Sqrt[b*x]], x] /; FreeQ[{b, c, d}, x ] && GtQ[c, 0]
Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSinh[Rt[b, 2]*(x/Sqrt [a])]/Rt[b, 2], x] /; FreeQ[{a, b}, x] && GtQ[a, 0] && PosQ[b]
Int[((e_.)*(x_))^(m_.)*((c_) + (d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_)^2)^(p_. ), x_Symbol] :> Int[(e*x)^m*(c + d*x)^(n + p)*(a/c + (b/d)*x)^p, x] /; Free Q[{a, b, c, d, e, m, n, p}, x] && EqQ[b*c^2 + a*d^2, 0] && (IntegerQ[p] || (GtQ[a, 0] && GtQ[c, 0] && !IntegerQ[n]))
Int[E^(ArcTanh[(a_.)*(x_)]*(n_.))*((c_) + (d_.)*(x_))^(p_.)*((e_.) + (f_.)* (x_))^(m_.), x_Symbol] :> Simp[c^n Int[(e + f*x)^m*(c + d*x)^(p - n)*(1 - a^2*x^2)^(n/2), x], x] /; FreeQ[{a, c, d, e, f, m, p}, x] && EqQ[a*c + d, 0] && IntegerQ[(n - 1)/2] && (IntegerQ[p] || EqQ[p, n/2] || EqQ[p - n/2 - 1 , 0]) && IntegerQ[2*p]
Int[E^(ArcTanh[(a_.)*(x_)]*(n_.))*(u_.)*((c_) + (d_.)/(x_))^(p_), x_Symbol] :> Simp[x^p*((c + d/x)^p/(1 + c*(x/d))^p) Int[u*(1 + c*(x/d))^p*(E^(n*Ar cTanh[a*x])/x^p), x], x] /; FreeQ[{a, c, d, n, p}, x] && EqQ[c^2 - a^2*d^2, 0] && !IntegerQ[p]
Time = 0.20 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.78
| method | result | size |
| default | \(-\frac {\sqrt {\frac {c \left (a x -1\right )}{a x}}\, x \sqrt {-a^{2} x^{2}+1}\, \left (4 a^{\frac {3}{2}} x \sqrt {-\left (a x +1\right ) x}+2 \sqrt {a}\, \sqrt {-\left (a x +1\right ) x}+\arctan \left (\frac {2 a x +1}{2 \sqrt {a}\, \sqrt {-\left (a x +1\right ) x}}\right )\right )}{8 a^{\frac {3}{2}} \left (a x -1\right ) \sqrt {-\left (a x +1\right ) x}}\) | \(105\) |
| risch | \(\frac {\left (2 a x +1\right ) \left (a x +1\right ) x \sqrt {\frac {c \left (a x -1\right )}{a x}}\, \sqrt {\frac {c a x \left (-a^{2} x^{2}+1\right )}{a x -1}}}{4 a \sqrt {-\left (a x +1\right ) a c x}\, \sqrt {-a^{2} x^{2}+1}}-\frac {\arctan \left (\frac {\sqrt {a^{2} c}\, \left (x +\frac {1}{2 a}\right )}{\sqrt {-a^{2} c \,x^{2}-a c x}}\right ) \sqrt {\frac {c \left (a x -1\right )}{a x}}\, \sqrt {\frac {c a x \left (-a^{2} x^{2}+1\right )}{a x -1}}}{8 a \sqrt {a^{2} c}\, \sqrt {-a^{2} x^{2}+1}}\) | \(176\) |
-1/8*(c*(a*x-1)/a/x)^(1/2)*x*(-a^2*x^2+1)^(1/2)/a^(3/2)*(4*a^(3/2)*x*(-(a* x+1)*x)^(1/2)+2*a^(1/2)*(-(a*x+1)*x)^(1/2)+arctan(1/2/a^(1/2)*(2*a*x+1)/(- (a*x+1)*x)^(1/2)))/(a*x-1)/(-(a*x+1)*x)^(1/2)
Time = 0.29 (sec) , antiderivative size = 272, normalized size of antiderivative = 2.01 \[ \int e^{\text {arctanh}(a x)} \sqrt {c-\frac {c}{a x}} x \, dx=\left [\frac {{\left (a x - 1\right )} \sqrt {-c} \log \left (-\frac {8 \, a^{3} c x^{3} - 7 \, a c x - 4 \, {\left (2 \, a^{2} x^{2} + a x\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {-c} \sqrt {\frac {a c x - c}{a x}} - c}{a x - 1}\right ) - 4 \, {\left (2 \, a^{2} x^{2} + a x\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {\frac {a c x - c}{a x}}}{16 \, {\left (a^{3} x - a^{2}\right )}}, \frac {{\left (a x - 1\right )} \sqrt {c} \arctan \left (\frac {2 \, \sqrt {-a^{2} x^{2} + 1} a \sqrt {c} x \sqrt {\frac {a c x - c}{a x}}}{2 \, a^{2} c x^{2} - a c x - c}\right ) - 2 \, {\left (2 \, a^{2} x^{2} + a x\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {\frac {a c x - c}{a x}}}{8 \, {\left (a^{3} x - a^{2}\right )}}\right ] \]
[1/16*((a*x - 1)*sqrt(-c)*log(-(8*a^3*c*x^3 - 7*a*c*x - 4*(2*a^2*x^2 + a*x )*sqrt(-a^2*x^2 + 1)*sqrt(-c)*sqrt((a*c*x - c)/(a*x)) - c)/(a*x - 1)) - 4* (2*a^2*x^2 + a*x)*sqrt(-a^2*x^2 + 1)*sqrt((a*c*x - c)/(a*x)))/(a^3*x - a^2 ), 1/8*((a*x - 1)*sqrt(c)*arctan(2*sqrt(-a^2*x^2 + 1)*a*sqrt(c)*x*sqrt((a* c*x - c)/(a*x))/(2*a^2*c*x^2 - a*c*x - c)) - 2*(2*a^2*x^2 + a*x)*sqrt(-a^2 *x^2 + 1)*sqrt((a*c*x - c)/(a*x)))/(a^3*x - a^2)]
\[ \int e^{\text {arctanh}(a x)} \sqrt {c-\frac {c}{a x}} x \, dx=\int \frac {x \sqrt {- c \left (-1 + \frac {1}{a x}\right )} \left (a x + 1\right )}{\sqrt {- \left (a x - 1\right ) \left (a x + 1\right )}}\, dx \]
\[ \int e^{\text {arctanh}(a x)} \sqrt {c-\frac {c}{a x}} x \, dx=\int { \frac {{\left (a x + 1\right )} \sqrt {c - \frac {c}{a x}} x}{\sqrt {-a^{2} x^{2} + 1}} \,d x } \]
\[ \int e^{\text {arctanh}(a x)} \sqrt {c-\frac {c}{a x}} x \, dx=\int { \frac {{\left (a x + 1\right )} \sqrt {c - \frac {c}{a x}} x}{\sqrt {-a^{2} x^{2} + 1}} \,d x } \]
Timed out. \[ \int e^{\text {arctanh}(a x)} \sqrt {c-\frac {c}{a x}} x \, dx=\int \frac {x\,\sqrt {c-\frac {c}{a\,x}}\,\left (a\,x+1\right )}{\sqrt {1-a^2\,x^2}} \,d x \]