Integrand size = 24, antiderivative size = 88 \[ \int \frac {e^{-\text {arctanh}(a x)}}{\sqrt {c-\frac {c}{a x}}} \, dx=\frac {\sqrt {1-a x} \sqrt {1+a x}}{a \sqrt {c-\frac {c}{a x}}}-\frac {\sqrt {1-a x} \text {arcsinh}\left (\sqrt {a} \sqrt {x}\right )}{a^{3/2} \sqrt {c-\frac {c}{a x}} \sqrt {x}} \] Output:
(-a*x+1)^(1/2)*(a*x+1)^(1/2)/a/(c-c/a/x)^(1/2)-(-a*x+1)^(1/2)*arcsinh(a^(1 /2)*x^(1/2))/a^(3/2)/(c-c/a/x)^(1/2)/x^(1/2)
Time = 0.04 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.78 \[ \int \frac {e^{-\text {arctanh}(a x)}}{\sqrt {c-\frac {c}{a x}}} \, dx=\frac {\sqrt {1-a x} \left (\frac {\sqrt {x} \sqrt {1+a x}}{a}-\frac {\text {arcsinh}\left (\sqrt {a} \sqrt {x}\right )}{a^{3/2}}\right )}{\sqrt {c-\frac {c}{a x}} \sqrt {x}} \] Input:
Integrate[1/(E^ArcTanh[a*x]*Sqrt[c - c/(a*x)]),x]
Output:
(Sqrt[1 - a*x]*((Sqrt[x]*Sqrt[1 + a*x])/a - ArcSinh[Sqrt[a]*Sqrt[x]]/a^(3/ 2)))/(Sqrt[c - c/(a*x)]*Sqrt[x])
Time = 0.44 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.78, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {6684, 6678, 516, 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.
\(\displaystyle \int \frac {e^{-\text {arctanh}(a x)}}{\sqrt {c-\frac {c}{a x}}} \, dx\) |
\(\Big \downarrow \) 6684 |
\(\displaystyle \frac {\sqrt {1-a x} \int \frac {e^{-\text {arctanh}(a x)} \sqrt {x}}{\sqrt {1-a x}}dx}{\sqrt {x} \sqrt {c-\frac {c}{a x}}}\) |
\(\Big \downarrow \) 6678 |
\(\displaystyle \frac {\sqrt {1-a x} \int \frac {\sqrt {x} \sqrt {1-a x}}{\sqrt {1-a^2 x^2}}dx}{\sqrt {x} \sqrt {c-\frac {c}{a x}}}\) |
\(\Big \downarrow \) 516 |
\(\displaystyle \frac {\sqrt {1-a x} \int \frac {\sqrt {x}}{\sqrt {a x+1}}dx}{\sqrt {x} \sqrt {c-\frac {c}{a x}}}\) |
\(\Big \downarrow \) 60 |
\(\displaystyle \frac {\sqrt {1-a x} \left (\frac {\sqrt {x} \sqrt {a x+1}}{a}-\frac {\int \frac {1}{\sqrt {x} \sqrt {a x+1}}dx}{2 a}\right )}{\sqrt {x} \sqrt {c-\frac {c}{a x}}}\) |
\(\Big \downarrow \) 63 |
\(\displaystyle \frac {\sqrt {1-a x} \left (\frac {\sqrt {x} \sqrt {a x+1}}{a}-\frac {\int \frac {1}{\sqrt {a x+1}}d\sqrt {x}}{a}\right )}{\sqrt {x} \sqrt {c-\frac {c}{a x}}}\) |
\(\Big \downarrow \) 222 |
\(\displaystyle \frac {\sqrt {1-a x} \left (\frac {\sqrt {x} \sqrt {a x+1}}{a}-\frac {\text {arcsinh}\left (\sqrt {a} \sqrt {x}\right )}{a^{3/2}}\right )}{\sqrt {x} \sqrt {c-\frac {c}{a x}}}\) |
Input:
Int[1/(E^ArcTanh[a*x]*Sqrt[c - c/(a*x)]),x]
Output:
(Sqrt[1 - a*x]*((Sqrt[x]*Sqrt[1 + a*x])/a - ArcSinh[Sqrt[a]*Sqrt[x]]/a^(3/ 2)))/(Sqrt[c - c/(a*x)]*Sqrt[x])
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 = 0.21 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.05
method | result | size |
default | \(\frac {\sqrt {\frac {c \left (a x -1\right )}{a x}}\, x \sqrt {-a^{2} x^{2}+1}\, \left (2 \sqrt {a}\, \sqrt {-x \left (a x +1\right )}+\arctan \left (\frac {2 a x +1}{2 \sqrt {a}\, \sqrt {-x \left (a x +1\right )}}\right )\right )}{2 \sqrt {a}\, c \left (a x -1\right ) \sqrt {-x \left (a x +1\right )}}\) | \(92\) |
risch | \(-\frac {\left (a x +1\right ) \sqrt {\frac {c a x \left (-a^{2} x^{2}+1\right )}{a x -1}}\, \left (a x -1\right )}{a \sqrt {-\left (a x +1\right ) a c x}\, \sqrt {\frac {c \left (a x -1\right )}{a x}}\, \sqrt {-a^{2} x^{2}+1}}+\frac {\arctan \left (\frac {\sqrt {a^{2} c}\, \left (x +\frac {1}{2 a}\right )}{\sqrt {-a^{2} c \,x^{2}-a c x}}\right ) \sqrt {\frac {c a x \left (-a^{2} x^{2}+1\right )}{a x -1}}\, \left (a x -1\right )}{2 a \sqrt {a^{2} c}\, \sqrt {\frac {c \left (a x -1\right )}{a x}}\, x \sqrt {-a^{2} x^{2}+1}}\) | \(182\) |
Input:
int(1/(a*x+1)*(-a^2*x^2+1)^(1/2)/(c-c/a/x)^(1/2),x,method=_RETURNVERBOSE)
Output:
1/2*(c*(a*x-1)/a/x)^(1/2)*x/a^(1/2)/c*(-a^2*x^2+1)^(1/2)*(2*a^(1/2)*(-x*(a *x+1))^(1/2)+arctan(1/2/a^(1/2)*(2*a*x+1)/(-x*(a*x+1))^(1/2)))/(a*x-1)/(-x *(a*x+1))^(1/2)
Time = 0.11 (sec) , antiderivative size = 254, normalized size of antiderivative = 2.89 \[ \int \frac {e^{-\text {arctanh}(a x)}}{\sqrt {c-\frac {c}{a x}}} \, dx=\left [\frac {4 \, \sqrt {-a^{2} x^{2} + 1} a x \sqrt {\frac {a c x - c}{a x}} - {\left (a x - 1\right )} \sqrt {-c} \log \left (-\frac {8 \, a^{3} c x^{3} - 7 \, a c x - 4 \, {\left (2 \, a^{2} x^{2} + a x\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {-c} \sqrt {\frac {a c x - c}{a x}} - c}{a x - 1}\right )}{4 \, {\left (a^{2} c x - a c\right )}}, \frac {2 \, \sqrt {-a^{2} x^{2} + 1} a x \sqrt {\frac {a c x - c}{a x}} - {\left (a x - 1\right )} \sqrt {c} \arctan \left (\frac {2 \, \sqrt {-a^{2} x^{2} + 1} a \sqrt {c} x \sqrt {\frac {a c x - c}{a x}}}{2 \, a^{2} c x^{2} - a c x - c}\right )}{2 \, {\left (a^{2} c x - a c\right )}}\right ] \] Input:
integrate(1/(a*x+1)*(-a^2*x^2+1)^(1/2)/(c-c/a/x)^(1/2),x, algorithm="frica s")
Output:
[1/4*(4*sqrt(-a^2*x^2 + 1)*a*x*sqrt((a*c*x - c)/(a*x)) - (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)*sqr t(-c)*sqrt((a*c*x - c)/(a*x)) - c)/(a*x - 1)))/(a^2*c*x - a*c), 1/2*(2*sqr t(-a^2*x^2 + 1)*a*x*sqrt((a*c*x - c)/(a*x)) - (a*x - 1)*sqrt(c)*arctan(2*s qrt(-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)))/(a^2*c*x - a*c)]
\[ \int \frac {e^{-\text {arctanh}(a x)}}{\sqrt {c-\frac {c}{a x}}} \, dx=\int \frac {\sqrt {- \left (a x - 1\right ) \left (a x + 1\right )}}{\sqrt {- c \left (-1 + \frac {1}{a x}\right )} \left (a x + 1\right )}\, dx \] Input:
integrate(1/(a*x+1)*(-a**2*x**2+1)**(1/2)/(c-c/a/x)**(1/2),x)
Output:
Integral(sqrt(-(a*x - 1)*(a*x + 1))/(sqrt(-c*(-1 + 1/(a*x)))*(a*x + 1)), x )
\[ \int \frac {e^{-\text {arctanh}(a x)}}{\sqrt {c-\frac {c}{a x}}} \, dx=\int { \frac {\sqrt {-a^{2} x^{2} + 1}}{{\left (a x + 1\right )} \sqrt {c - \frac {c}{a x}}} \,d x } \] Input:
integrate(1/(a*x+1)*(-a^2*x^2+1)^(1/2)/(c-c/a/x)^(1/2),x, algorithm="maxim a")
Output:
integrate(sqrt(-a^2*x^2 + 1)/((a*x + 1)*sqrt(c - c/(a*x))), x)
\[ \int \frac {e^{-\text {arctanh}(a x)}}{\sqrt {c-\frac {c}{a x}}} \, dx=\int { \frac {\sqrt {-a^{2} x^{2} + 1}}{{\left (a x + 1\right )} \sqrt {c - \frac {c}{a x}}} \,d x } \] Input:
integrate(1/(a*x+1)*(-a^2*x^2+1)^(1/2)/(c-c/a/x)^(1/2),x, algorithm="giac" )
Output:
integrate(sqrt(-a^2*x^2 + 1)/((a*x + 1)*sqrt(c - c/(a*x))), x)
Timed out. \[ \int \frac {e^{-\text {arctanh}(a x)}}{\sqrt {c-\frac {c}{a x}}} \, dx=\int \frac {\sqrt {1-a^2\,x^2}}{\sqrt {c-\frac {c}{a\,x}}\,\left (a\,x+1\right )} \,d x \] Input:
int((1 - a^2*x^2)^(1/2)/((c - c/(a*x))^(1/2)*(a*x + 1)),x)
Output:
int((1 - a^2*x^2)^(1/2)/((c - c/(a*x))^(1/2)*(a*x + 1)), x)
Time = 0.16 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.50 \[ \int \frac {e^{-\text {arctanh}(a x)}}{\sqrt {c-\frac {c}{a x}}} \, dx=\frac {\sqrt {c}\, \left (-\mathit {atan} \left (\frac {\sqrt {x}\, \sqrt {a}\, \sqrt {a x +1}\, i}{a x +1}\right )+\sqrt {x}\, \sqrt {a}\, \sqrt {a x +1}\, i \right )}{a c} \] Input:
int(1/(a*x+1)*(-a^2*x^2+1)^(1/2)/(c-c/a/x)^(1/2),x)
Output:
(sqrt(c)*( - atan((sqrt(x)*sqrt(a)*sqrt(a*x + 1)*i)/(a*x + 1)) + sqrt(x)*s qrt(a)*sqrt(a*x + 1)*i))/(a*c)