Integrand size = 20, antiderivative size = 177 \[ \int \frac {e^{3 \coth ^{-1}(a x)}}{\sqrt {c-a c x}} \, dx=-\frac {6 \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}{\left (a-\frac {1}{x}\right ) \sqrt {c-a c x}}+\frac {2 a \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/2} x}{\left (a-\frac {1}{x}\right ) \sqrt {c-a c x}}-\frac {3 \sqrt {2} \sqrt {1-\frac {1}{a x}} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {\frac {1}{x}}}{\sqrt {a} \sqrt {1+\frac {1}{a x}}}\right )}{\sqrt {a} \sqrt {\frac {1}{x}} \sqrt {c-a c x}} \]
2*a*(1+1/a/x)^(3/2)*x*(1-1/a/x)^(1/2)/(a-1/x)/(-a*c*x+c)^(1/2)-6*(1-1/a/x) ^(1/2)*(1+1/a/x)^(1/2)/(a-1/x)/(-a*c*x+c)^(1/2)-3*arctanh(2^(1/2)*(1/x)^(1 /2)/a^(1/2)/(1+1/a/x)^(1/2))*2^(1/2)*(1-1/a/x)^(1/2)/a^(1/2)/(1/x)^(1/2)/( -a*c*x+c)^(1/2)
Time = 0.16 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.66 \[ \int \frac {e^{3 \coth ^{-1}(a x)}}{\sqrt {c-a c x}} \, dx=\frac {\sqrt {1-\frac {1}{a x}} x \left (2 \sqrt {a} \sqrt {1+\frac {1}{a x}} (-2+a x)-3 \sqrt {2} \sqrt {\frac {1}{x}} (-1+a x) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {\frac {1}{x}}}{\sqrt {a} \sqrt {1+\frac {1}{a x}}}\right )\right )}{\sqrt {a} (-1+a x) \sqrt {c-a c x}} \]
(Sqrt[1 - 1/(a*x)]*x*(2*Sqrt[a]*Sqrt[1 + 1/(a*x)]*(-2 + a*x) - 3*Sqrt[2]*S qrt[x^(-1)]*(-1 + a*x)*ArcTanh[(Sqrt[2]*Sqrt[x^(-1)])/(Sqrt[a]*Sqrt[1 + 1/ (a*x)])]))/(Sqrt[a]*(-1 + a*x)*Sqrt[c - a*c*x])
Time = 0.29 (sec) , antiderivative size = 148, normalized size of antiderivative = 0.84, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {6727, 27, 105, 105, 104, 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.
| \(\displaystyle \int \frac {e^{3 \coth ^{-1}(a x)}}{\sqrt {c-a c x}} \, dx\) |
| \(\Big \downarrow \) 6727 |
| \(\displaystyle -\frac {\sqrt {1-\frac {1}{a x}} \int \frac {a^2 \left (1+\frac {1}{a x}\right )^{3/2}}{\left (a-\frac {1}{x}\right )^2 \left (\frac {1}{x}\right )^{3/2}}d\frac {1}{x}}{\sqrt {\frac {1}{x}} \sqrt {c-a c x}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {a^2 \sqrt {1-\frac {1}{a x}} \int \frac {\left (1+\frac {1}{a x}\right )^{3/2}}{\left (a-\frac {1}{x}\right )^2 \left (\frac {1}{x}\right )^{3/2}}d\frac {1}{x}}{\sqrt {\frac {1}{x}} \sqrt {c-a c x}}\) |
| \(\Big \downarrow \) 105 |
| \(\displaystyle -\frac {a^2 \sqrt {1-\frac {1}{a x}} \left (\frac {3 \int \frac {\sqrt {1+\frac {1}{a x}}}{\left (a-\frac {1}{x}\right ) \left (\frac {1}{x}\right )^{3/2}}d\frac {1}{x}}{2 a}+\frac {\left (\frac {1}{a x}+1\right )^{3/2}}{a \sqrt {\frac {1}{x}} \left (a-\frac {1}{x}\right )}\right )}{\sqrt {\frac {1}{x}} \sqrt {c-a c x}}\) |
| \(\Big \downarrow \) 105 |
| \(\displaystyle -\frac {a^2 \sqrt {1-\frac {1}{a x}} \left (\frac {3 \left (\frac {2 \int \frac {1}{\left (a-\frac {1}{x}\right ) \sqrt {1+\frac {1}{a x}} \sqrt {\frac {1}{x}}}d\frac {1}{x}}{a}-\frac {2 \sqrt {\frac {1}{a x}+1}}{a \sqrt {\frac {1}{x}}}\right )}{2 a}+\frac {\left (\frac {1}{a x}+1\right )^{3/2}}{a \sqrt {\frac {1}{x}} \left (a-\frac {1}{x}\right )}\right )}{\sqrt {\frac {1}{x}} \sqrt {c-a c x}}\) |
| \(\Big \downarrow \) 104 |
| \(\displaystyle -\frac {a^2 \sqrt {1-\frac {1}{a x}} \left (\frac {3 \left (\frac {4 \int \frac {1}{a-\frac {2}{x^2}}d\frac {\sqrt {\frac {1}{x}}}{\sqrt {1+\frac {1}{a x}}}}{a}-\frac {2 \sqrt {\frac {1}{a x}+1}}{a \sqrt {\frac {1}{x}}}\right )}{2 a}+\frac {\left (\frac {1}{a x}+1\right )^{3/2}}{a \sqrt {\frac {1}{x}} \left (a-\frac {1}{x}\right )}\right )}{\sqrt {\frac {1}{x}} \sqrt {c-a c x}}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle -\frac {a^2 \sqrt {1-\frac {1}{a x}} \left (\frac {3 \left (\frac {2 \sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {\frac {1}{x}}}{\sqrt {a} \sqrt {\frac {1}{a x}+1}}\right )}{a^{3/2}}-\frac {2 \sqrt {\frac {1}{a x}+1}}{a \sqrt {\frac {1}{x}}}\right )}{2 a}+\frac {\left (\frac {1}{a x}+1\right )^{3/2}}{a \sqrt {\frac {1}{x}} \left (a-\frac {1}{x}\right )}\right )}{\sqrt {\frac {1}{x}} \sqrt {c-a c x}}\) |
-((a^2*Sqrt[1 - 1/(a*x)]*((1 + 1/(a*x))^(3/2)/(a*(a - x^(-1))*Sqrt[x^(-1)] ) + (3*((-2*Sqrt[1 + 1/(a*x)])/(a*Sqrt[x^(-1)]) + (2*Sqrt[2]*ArcTanh[(Sqrt [2]*Sqrt[x^(-1)])/(Sqrt[a]*Sqrt[1 + 1/(a*x)])])/a^(3/2)))/(2*a)))/(Sqrt[x^ (-1)]*Sqrt[c - a*c*x]))
3.3.48.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x _)), x_] :> With[{q = Denominator[m]}, Simp[q Subst[Int[x^(q*(m + 1) - 1) /(b*e - a*f - (d*e - c*f)*x^q), x], x, (a + b*x)^(1/q)/(c + d*x)^(1/q)], x] ] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[m + n + 1, 0] && RationalQ[n] && L tQ[-1, m, 0] && SimplerQ[a + b*x, c + d*x]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Simp[(a + b*x)^(m + 1)*(c + d*x)^n*((e + f*x)^(p + 1)/((m + 1)*(b*e - a*f))), x] - Simp[n*((d*e - c*f)/((m + 1)*(b*e - a*f))) Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, p}, x] && EqQ[m + n + p + 2, 0] && GtQ[n, 0] && (SumSimplerQ[m, 1] || !SumSimplerQ[p, 1]) && NeQ[m, -1]
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[E^(ArcCoth[(a_.)*(x_)]*(n_.))*((c_) + (d_.)*(x_))^(p_), x_Symbol] :> Si mp[(-(1/x)^p)*((c + d*x)^p/(1 + c/(d*x))^p) Subst[Int[((1 + c*(x/d))^p*(( 1 + x/a)^(n/2)/x^(p + 2)))/(1 - x/a)^(n/2), x], x, 1/x], x] /; FreeQ[{a, c, d, n, p}, x] && EqQ[a^2*c^2 - d^2, 0] && !IntegerQ[p]
Time = 0.45 (sec) , antiderivative size = 136, normalized size of antiderivative = 0.77
| method | result | size |
| default | \(-\frac {\sqrt {-c \left (a x -1\right )}\, \left (-3 \sqrt {2}\, \arctan \left (\frac {\sqrt {-c \left (a x +1\right )}\, \sqrt {2}}{2 \sqrt {c}}\right ) a c x +2 a x \sqrt {c}\, \sqrt {-c \left (a x +1\right )}+3 \sqrt {2}\, \arctan \left (\frac {\sqrt {-c \left (a x +1\right )}\, \sqrt {2}}{2 \sqrt {c}}\right ) c -4 \sqrt {-c \left (a x +1\right )}\, \sqrt {c}\right )}{\left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}} \left (a x +1\right ) c^{\frac {3}{2}} \sqrt {-c \left (a x +1\right )}\, a}\) | \(136\) |
| risch | \(\frac {2 a x -2}{a \sqrt {\frac {a x -1}{a x +1}}\, \sqrt {-c \left (a x -1\right )}}+\frac {\left (-\frac {2 \sqrt {-a c x -c}}{a \left (-a c x +c \right )}+\frac {3 \sqrt {2}\, \arctan \left (\frac {\sqrt {-a c x -c}\, \sqrt {2}}{2 \sqrt {c}}\right )}{a \sqrt {c}}\right ) \sqrt {-c \left (a x +1\right )}\, \left (a x -1\right )}{\left (a x +1\right ) \sqrt {\frac {a x -1}{a x +1}}\, \sqrt {-c \left (a x -1\right )}}\) | \(142\) |
-(-c*(a*x-1))^(1/2)*(-3*2^(1/2)*arctan(1/2*(-c*(a*x+1))^(1/2)*2^(1/2)/c^(1 /2))*a*c*x+2*a*x*c^(1/2)*(-c*(a*x+1))^(1/2)+3*2^(1/2)*arctan(1/2*(-c*(a*x+ 1))^(1/2)*2^(1/2)/c^(1/2))*c-4*(-c*(a*x+1))^(1/2)*c^(1/2))/((a*x-1)/(a*x+1 ))^(3/2)/(a*x+1)/c^(3/2)/(-c*(a*x+1))^(1/2)/a
Time = 0.27 (sec) , antiderivative size = 288, normalized size of antiderivative = 1.63 \[ \int \frac {e^{3 \coth ^{-1}(a x)}}{\sqrt {c-a c x}} \, dx=\left [\frac {3 \, \sqrt {2} {\left (a^{2} c x^{2} - 2 \, a c x + c\right )} \sqrt {-\frac {1}{c}} \log \left (-\frac {a^{2} x^{2} - 2 \, \sqrt {2} \sqrt {-a c x + c} {\left (a x + 1\right )} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {-\frac {1}{c}} + 2 \, a x - 3}{a^{2} x^{2} - 2 \, a x + 1}\right ) - 4 \, {\left (a^{2} x^{2} - a x - 2\right )} \sqrt {-a c x + c} \sqrt {\frac {a x - 1}{a x + 1}}}{2 \, {\left (a^{3} c x^{2} - 2 \, a^{2} c x + a c\right )}}, -\frac {2 \, {\left (a^{2} x^{2} - a x - 2\right )} \sqrt {-a c x + c} \sqrt {\frac {a x - 1}{a x + 1}} - \frac {3 \, \sqrt {2} {\left (a^{2} c x^{2} - 2 \, a c x + c\right )} \arctan \left (\frac {\sqrt {2} \sqrt {-a c x + c} \sqrt {\frac {a x - 1}{a x + 1}}}{{\left (a x - 1\right )} \sqrt {c}}\right )}{\sqrt {c}}}{a^{3} c x^{2} - 2 \, a^{2} c x + a c}\right ] \]
[1/2*(3*sqrt(2)*(a^2*c*x^2 - 2*a*c*x + c)*sqrt(-1/c)*log(-(a^2*x^2 - 2*sqr t(2)*sqrt(-a*c*x + c)*(a*x + 1)*sqrt((a*x - 1)/(a*x + 1))*sqrt(-1/c) + 2*a *x - 3)/(a^2*x^2 - 2*a*x + 1)) - 4*(a^2*x^2 - a*x - 2)*sqrt(-a*c*x + c)*sq rt((a*x - 1)/(a*x + 1)))/(a^3*c*x^2 - 2*a^2*c*x + a*c), -(2*(a^2*x^2 - a*x - 2)*sqrt(-a*c*x + c)*sqrt((a*x - 1)/(a*x + 1)) - 3*sqrt(2)*(a^2*c*x^2 - 2*a*c*x + c)*arctan(sqrt(2)*sqrt(-a*c*x + c)*sqrt((a*x - 1)/(a*x + 1))/((a *x - 1)*sqrt(c)))/sqrt(c))/(a^3*c*x^2 - 2*a^2*c*x + a*c)]
\[ \int \frac {e^{3 \coth ^{-1}(a x)}}{\sqrt {c-a c x}} \, dx=\int \frac {1}{\left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}} \sqrt {- c \left (a x - 1\right )}}\, dx \]
\[ \int \frac {e^{3 \coth ^{-1}(a x)}}{\sqrt {c-a c x}} \, dx=\int { \frac {1}{\sqrt {-a c x + c} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}}} \,d x } \]
Time = 0.30 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.42 \[ \int \frac {e^{3 \coth ^{-1}(a x)}}{\sqrt {c-a c x}} \, dx=-\frac {3 \, \sqrt {2} \sqrt {c} \arctan \left (\frac {\sqrt {2} \sqrt {-a c x - c}}{2 \, \sqrt {c}}\right ) - 2 \, \sqrt {-a c x - c} + \frac {2 \, \sqrt {-a c x - c} c}{a c x - c}}{a {\left | c \right |}} \]
-(3*sqrt(2)*sqrt(c)*arctan(1/2*sqrt(2)*sqrt(-a*c*x - c)/sqrt(c)) - 2*sqrt( -a*c*x - c) + 2*sqrt(-a*c*x - c)*c/(a*c*x - c))/(a*abs(c))
Timed out. \[ \int \frac {e^{3 \coth ^{-1}(a x)}}{\sqrt {c-a c x}} \, dx=\int \frac {1}{\sqrt {c-a\,c\,x}\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/2}} \,d x \]