Integrand size = 22, antiderivative size = 157 \[ \int \frac {e^{\text {arctanh}(a x)}}{\sqrt {c-\frac {c}{a x}}} \, dx=-\frac {\sqrt {1-a x} \sqrt {1+a x}}{a \sqrt {c-\frac {c}{a x}}}-\frac {3 \sqrt {1-a x} \text {arcsinh}\left (\sqrt {a} \sqrt {x}\right )}{a^{3/2} \sqrt {c-\frac {c}{a x}} \sqrt {x}}+\frac {2 \sqrt {2} \sqrt {1-a x} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {x}}{\sqrt {1+a x}}\right )}{a^{3/2} \sqrt {c-\frac {c}{a x}} \sqrt {x}} \]
-3*arcsinh(a^(1/2)*x^(1/2))*(-a*x+1)^(1/2)/a^(3/2)/(c-c/a/x)^(1/2)/x^(1/2) +2*arctanh(2^(1/2)*a^(1/2)*x^(1/2)/(a*x+1)^(1/2))*2^(1/2)*(-a*x+1)^(1/2)/a ^(3/2)/(c-c/a/x)^(1/2)/x^(1/2)-(-a*x+1)^(1/2)*(a*x+1)^(1/2)/a/(c-c/a/x)^(1 /2)
Time = 0.04 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.67 \[ \int \frac {e^{\text {arctanh}(a x)}}{\sqrt {c-\frac {c}{a x}}} \, dx=-\frac {\sqrt {1-a x} \left (\sqrt {a} \sqrt {x} \sqrt {1+a x}+3 \text {arcsinh}\left (\sqrt {a} \sqrt {x}\right )-2 \sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {x}}{\sqrt {1+a x}}\right )\right )}{a^{3/2} \sqrt {c-\frac {c}{a x}} \sqrt {x}} \]
-((Sqrt[1 - a*x]*(Sqrt[a]*Sqrt[x]*Sqrt[1 + a*x] + 3*ArcSinh[Sqrt[a]*Sqrt[x ]] - 2*Sqrt[2]*ArcTanh[(Sqrt[2]*Sqrt[a]*Sqrt[x])/Sqrt[1 + a*x]]))/(a^(3/2) *Sqrt[c - c/(a*x)]*Sqrt[x]))
Time = 0.43 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.74, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.409, Rules used = {6684, 6679, 112, 27, 175, 63, 104, 219, 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.
| \(\displaystyle \int \frac {e^{\text {arctanh}(a x)}}{\sqrt {c-\frac {c}{a x}}} \, dx\) |
| \(\Big \downarrow \) 6684 |
| \(\displaystyle \frac {\sqrt {1-a x} \int \frac {e^{\text {arctanh}(a x)} \sqrt {x}}{\sqrt {1-a x}}dx}{\sqrt {x} \sqrt {c-\frac {c}{a x}}}\) |
| \(\Big \downarrow \) 6679 |
| \(\displaystyle \frac {\sqrt {1-a x} \int \frac {\sqrt {x} \sqrt {a x+1}}{1-a x}dx}{\sqrt {x} \sqrt {c-\frac {c}{a x}}}\) |
| \(\Big \downarrow \) 112 |
| \(\displaystyle \frac {\sqrt {1-a x} \left (\frac {\int \frac {3 a x+1}{2 \sqrt {x} (1-a x) \sqrt {a x+1}}dx}{a}-\frac {\sqrt {x} \sqrt {a x+1}}{a}\right )}{\sqrt {x} \sqrt {c-\frac {c}{a x}}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sqrt {1-a x} \left (\frac {\int \frac {3 a x+1}{\sqrt {x} (1-a x) \sqrt {a x+1}}dx}{2 a}-\frac {\sqrt {x} \sqrt {a x+1}}{a}\right )}{\sqrt {x} \sqrt {c-\frac {c}{a x}}}\) |
| \(\Big \downarrow \) 175 |
| \(\displaystyle \frac {\sqrt {1-a x} \left (\frac {4 \int \frac {1}{\sqrt {x} (1-a x) \sqrt {a x+1}}dx-3 \int \frac {1}{\sqrt {x} \sqrt {a x+1}}dx}{2 a}-\frac {\sqrt {x} \sqrt {a x+1}}{a}\right )}{\sqrt {x} \sqrt {c-\frac {c}{a x}}}\) |
| \(\Big \downarrow \) 63 |
| \(\displaystyle \frac {\sqrt {1-a x} \left (\frac {4 \int \frac {1}{\sqrt {x} (1-a x) \sqrt {a x+1}}dx-6 \int \frac {1}{\sqrt {a x+1}}d\sqrt {x}}{2 a}-\frac {\sqrt {x} \sqrt {a x+1}}{a}\right )}{\sqrt {x} \sqrt {c-\frac {c}{a x}}}\) |
| \(\Big \downarrow \) 104 |
| \(\displaystyle \frac {\sqrt {1-a x} \left (\frac {8 \int \frac {1}{1-\frac {2 a x}{a x+1}}d\frac {\sqrt {x}}{\sqrt {a x+1}}-6 \int \frac {1}{\sqrt {a x+1}}d\sqrt {x}}{2 a}-\frac {\sqrt {x} \sqrt {a x+1}}{a}\right )}{\sqrt {x} \sqrt {c-\frac {c}{a x}}}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {\sqrt {1-a x} \left (\frac {\frac {4 \sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {x}}{\sqrt {a x+1}}\right )}{\sqrt {a}}-6 \int \frac {1}{\sqrt {a x+1}}d\sqrt {x}}{2 a}-\frac {\sqrt {x} \sqrt {a x+1}}{a}\right )}{\sqrt {x} \sqrt {c-\frac {c}{a x}}}\) |
| \(\Big \downarrow \) 222 |
| \(\displaystyle \frac {\sqrt {1-a x} \left (\frac {\frac {4 \sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {x}}{\sqrt {a x+1}}\right )}{\sqrt {a}}-\frac {6 \text {arcsinh}\left (\sqrt {a} \sqrt {x}\right )}{\sqrt {a}}}{2 a}-\frac {\sqrt {x} \sqrt {a x+1}}{a}\right )}{\sqrt {x} \sqrt {c-\frac {c}{a x}}}\) |
(Sqrt[1 - a*x]*(-((Sqrt[x]*Sqrt[1 + a*x])/a) + ((-6*ArcSinh[Sqrt[a]*Sqrt[x ]])/Sqrt[a] + (4*Sqrt[2]*ArcTanh[(Sqrt[2]*Sqrt[a]*Sqrt[x])/Sqrt[1 + a*x]]) /Sqrt[a])/(2*a)))/(Sqrt[c - c/(a*x)]*Sqrt[x])
3.6.15.3.1 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[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[(((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*(c + d*x)^n*((e + f*x)^(p + 1)/(f*(m + n + p + 1))), x] - Simp[1/(f*(m + n + p + 1)) Int[(a + b*x)^(m - 1)*(c + d*x) ^(n - 1)*(e + f*x)^p*Simp[c*m*(b*e - a*f) + a*n*(d*e - c*f) + (d*m*(b*e - a *f) + b*n*(d*e - c*f))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && GtQ[m, 0] && GtQ[n, 0] && NeQ[m + n + p + 1, 0] && (IntegersQ[2*m, 2*n, 2*p ] || (IntegersQ[m, n + p] || IntegersQ[p, m + n]))
Int[(((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_ )))/((a_.) + (b_.)*(x_)), x_] :> Simp[h/b Int[(c + d*x)^n*(e + f*x)^p, x] , x] + Simp[(b*g - a*h)/b Int[(c + d*x)^n*((e + f*x)^p/(a + b*x)), x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, 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[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^(ArcTanh[(a_.)*(x_)]*(n_.))*(u_.)*((c_) + (d_.)*(x_))^(p_.), x_Symbol ] :> Simp[c^p Int[u*(1 + d*(x/c))^p*((1 + a*x)^(n/2)/(1 - a*x)^(n/2)), x] , x] /; FreeQ[{a, c, d, n, p}, x] && EqQ[a^2*c^2 - d^2, 0] && (IntegerQ[p] || GtQ[c, 0])
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 = 0.12 (sec) , antiderivative size = 168, normalized size of antiderivative = 1.07
| method | result | size |
| default | \(-\frac {\sqrt {\frac {c \left (a x -1\right )}{a x}}\, x \sqrt {-a^{2} x^{2}+1}\, \left (2 \sqrt {-\left (a x +1\right ) x}\, a^{\frac {3}{2}} \sqrt {2}\, \sqrt {-\frac {1}{a}}-3 \arctan \left (\frac {2 a x +1}{2 \sqrt {a}\, \sqrt {-\left (a x +1\right ) x}}\right ) a \sqrt {2}\, \sqrt {-\frac {1}{a}}+4 \ln \left (\frac {2 \sqrt {2}\, \sqrt {-\frac {1}{a}}\, \sqrt {-\left (a x +1\right ) x}\, a -3 a x -1}{a x -1}\right ) \sqrt {a}\right ) \sqrt {2}}{4 a^{\frac {3}{2}} c \left (a x -1\right ) \sqrt {-\left (a x +1\right ) x}\, \sqrt {-\frac {1}{a}}}\) | \(168\) |
| risch | \(\frac {\left (a x +1\right ) \sqrt {\frac {c a x \left (-a^{2} x^{2}+1\right )}{a x -1}}\, \left (a x -1\right )}{a \sqrt {-\left (a x +1\right ) a c x}\, \sqrt {\frac {c \left (a x -1\right )}{a x}}\, \sqrt {-a^{2} x^{2}+1}}+\frac {\left (\frac {3 \arctan \left (\frac {\sqrt {a^{2} c}\, \left (x +\frac {1}{2 a}\right )}{\sqrt {-a^{2} c \,x^{2}-a c x}}\right )}{2 a \sqrt {a^{2} c}}-\frac {2 \ln \left (\frac {-4 c -3 \left (x -\frac {1}{a}\right ) a c +2 \sqrt {-2 c}\, \sqrt {-a^{2} c \left (x -\frac {1}{a}\right )^{2}-3 \left (x -\frac {1}{a}\right ) a c -2 c}}{x -\frac {1}{a}}\right )}{a^{2} \sqrt {-2 c}}\right ) \sqrt {\frac {c a x \left (-a^{2} x^{2}+1\right )}{a x -1}}\, \left (a x -1\right )}{\sqrt {\frac {c \left (a x -1\right )}{a x}}\, x \sqrt {-a^{2} x^{2}+1}}\) | \(258\) |
-1/4*(c*(a*x-1)/a/x)^(1/2)*x*(-a^2*x^2+1)^(1/2)*(2*(-(a*x+1)*x)^(1/2)*a^(3 /2)*2^(1/2)*(-1/a)^(1/2)-3*arctan(1/2/a^(1/2)*(2*a*x+1)/(-(a*x+1)*x)^(1/2) )*a*2^(1/2)*(-1/a)^(1/2)+4*ln((2*2^(1/2)*(-1/a)^(1/2)*(-(a*x+1)*x)^(1/2)*a -3*a*x-1)/(a*x-1))*a^(1/2))*2^(1/2)/a^(3/2)/c/(a*x-1)/(-(a*x+1)*x)^(1/2)/( -1/a)^(1/2)
Time = 0.32 (sec) , antiderivative size = 447, normalized size of antiderivative = 2.85 \[ \int \frac {e^{\text {arctanh}(a x)}}{\sqrt {c-\frac {c}{a x}}} \, dx=\left [-\frac {4 \, \sqrt {-a^{2} x^{2} + 1} a x \sqrt {\frac {a c x - c}{a x}} - 2 \, \sqrt {2} {\left (a c x - c\right )} \sqrt {-\frac {1}{c}} \log \left (-\frac {17 \, a^{3} x^{3} - 3 \, a^{2} x^{2} - 4 \, \sqrt {2} {\left (3 \, a^{2} x^{2} + a x\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {-\frac {1}{c}} \sqrt {\frac {a c x - c}{a x}} - 13 \, a x - 1}{a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1}\right ) + 3 \, {\left (a x - 1\right )} \sqrt {-c} \log \left (-\frac {8 \, a^{3} c x^{3} - 7 \, a c x - 4 \, {\left (2 \, a^{2} x^{2} + a x\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {-c} \sqrt {\frac {a c x - c}{a x}} - c}{a x - 1}\right )}{4 \, {\left (a^{2} c x - a c\right )}}, -\frac {2 \, \sqrt {-a^{2} x^{2} + 1} a x \sqrt {\frac {a c x - c}{a x}} + 3 \, {\left (a x - 1\right )} \sqrt {c} \arctan \left (\frac {2 \, \sqrt {-a^{2} x^{2} + 1} a \sqrt {c} x \sqrt {\frac {a c x - c}{a x}}}{2 \, a^{2} c x^{2} - a c x - c}\right ) - \frac {2 \, \sqrt {2} {\left (a c x - c\right )} \arctan \left (\frac {2 \, \sqrt {2} \sqrt {-a^{2} x^{2} + 1} a x \sqrt {\frac {a c x - c}{a x}}}{{\left (3 \, a^{2} x^{2} - 2 \, a x - 1\right )} \sqrt {c}}\right )}{\sqrt {c}}}{2 \, {\left (a^{2} c x - a c\right )}}\right ] \]
[-1/4*(4*sqrt(-a^2*x^2 + 1)*a*x*sqrt((a*c*x - c)/(a*x)) - 2*sqrt(2)*(a*c*x - c)*sqrt(-1/c)*log(-(17*a^3*x^3 - 3*a^2*x^2 - 4*sqrt(2)*(3*a^2*x^2 + a*x )*sqrt(-a^2*x^2 + 1)*sqrt(-1/c)*sqrt((a*c*x - c)/(a*x)) - 13*a*x - 1)/(a^3 *x^3 - 3*a^2*x^2 + 3*a*x - 1)) + 3*(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)))/(a^2*c*x - a*c), -1/2*(2*sqrt(-a^2*x^2 + 1)*a*x*s qrt((a*c*x - c)/(a*x)) + 3*(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*sqrt(2)* (a*c*x - c)*arctan(2*sqrt(2)*sqrt(-a^2*x^2 + 1)*a*x*sqrt((a*c*x - c)/(a*x) )/((3*a^2*x^2 - 2*a*x - 1)*sqrt(c)))/sqrt(c))/(a^2*c*x - a*c)]
\[ \int \frac {e^{\text {arctanh}(a x)}}{\sqrt {c-\frac {c}{a x}}} \, dx=\int \frac {a x + 1}{\sqrt {- c \left (-1 + \frac {1}{a x}\right )} \sqrt {- \left (a x - 1\right ) \left (a x + 1\right )}}\, dx \]
\[ \int \frac {e^{\text {arctanh}(a x)}}{\sqrt {c-\frac {c}{a x}}} \, dx=\int { \frac {a x + 1}{\sqrt {-a^{2} x^{2} + 1} \sqrt {c - \frac {c}{a x}}} \,d x } \]
\[ \int \frac {e^{\text {arctanh}(a x)}}{\sqrt {c-\frac {c}{a x}}} \, dx=\int { \frac {a x + 1}{\sqrt {-a^{2} x^{2} + 1} \sqrt {c - \frac {c}{a x}}} \,d x } \]
Timed out. \[ \int \frac {e^{\text {arctanh}(a x)}}{\sqrt {c-\frac {c}{a x}}} \, dx=\int \frac {a\,x+1}{\sqrt {c-\frac {c}{a\,x}}\,\sqrt {1-a^2\,x^2}} \,d x \]