Integrand size = 21, antiderivative size = 91 \[ \int \frac {e^{\coth ^{-1}(a x)} \sqrt {c-a c x}}{x} \, dx=\frac {2 \sqrt {1+\frac {1}{a x}} \sqrt {c-a c x}}{\sqrt {1-\frac {1}{a x}}}-\frac {2 \sqrt {\frac {1}{a x}} \sqrt {c-a c x} \text {arcsinh}\left (\sqrt {\frac {1}{a x}}\right )}{\sqrt {1-\frac {1}{a x}}} \] Output:
2*(1+1/a/x)^(1/2)*(-a*c*x+c)^(1/2)/(1-1/a/x)^(1/2)-2*(1/a/x)^(1/2)*(-a*c*x +c)^(1/2)*arcsinh((1/a/x)^(1/2))/(1-1/a/x)^(1/2)
Time = 0.06 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.82 \[ \int \frac {e^{\coth ^{-1}(a x)} \sqrt {c-a c x}}{x} \, dx=\frac {2 \sqrt {c-a c x} \left (\sqrt {a} \sqrt {1+\frac {1}{a x}}-\sqrt {\frac {1}{x}} \text {arcsinh}\left (\frac {\sqrt {\frac {1}{x}}}{\sqrt {a}}\right )\right )}{\sqrt {a} \sqrt {1-\frac {1}{a x}}} \] Input:
Integrate[(E^ArcCoth[a*x]*Sqrt[c - a*c*x])/x,x]
Output:
(2*Sqrt[c - a*c*x]*(Sqrt[a]*Sqrt[1 + 1/(a*x)] - Sqrt[x^(-1)]*ArcSinh[Sqrt[ x^(-1)]/Sqrt[a]]))/(Sqrt[a]*Sqrt[1 - 1/(a*x)])
Time = 0.49 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.86, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {6730, 57, 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.
\(\displaystyle \int \frac {\sqrt {c-a c x} e^{\coth ^{-1}(a x)}}{x} \, dx\) |
\(\Big \downarrow \) 6730 |
\(\displaystyle -\frac {\sqrt {\frac {1}{x}} \sqrt {c-a c x} \int \frac {\sqrt {1+\frac {1}{a x}}}{\left (\frac {1}{x}\right )^{3/2}}d\frac {1}{x}}{\sqrt {1-\frac {1}{a x}}}\) |
\(\Big \downarrow \) 57 |
\(\displaystyle -\frac {\sqrt {\frac {1}{x}} \sqrt {c-a c x} \left (\frac {\int \frac {1}{\sqrt {1+\frac {1}{a x}} \sqrt {\frac {1}{x}}}d\frac {1}{x}}{a}-\frac {2 \sqrt {\frac {1}{a x}+1}}{\sqrt {\frac {1}{x}}}\right )}{\sqrt {1-\frac {1}{a x}}}\) |
\(\Big \downarrow \) 63 |
\(\displaystyle -\frac {\sqrt {\frac {1}{x}} \sqrt {c-a c x} \left (\frac {2 \int \frac {1}{\sqrt {1+\frac {1}{x^2 a}}}d\sqrt {\frac {1}{x}}}{a}-\frac {2 \sqrt {\frac {1}{a x}+1}}{\sqrt {\frac {1}{x}}}\right )}{\sqrt {1-\frac {1}{a x}}}\) |
\(\Big \downarrow \) 222 |
\(\displaystyle -\frac {\sqrt {\frac {1}{x}} \left (\frac {2 \text {arcsinh}\left (\frac {\sqrt {\frac {1}{x}}}{\sqrt {a}}\right )}{\sqrt {a}}-\frac {2 \sqrt {\frac {1}{a x}+1}}{\sqrt {\frac {1}{x}}}\right ) \sqrt {c-a c x}}{\sqrt {1-\frac {1}{a x}}}\) |
Input:
Int[(E^ArcCoth[a*x]*Sqrt[c - a*c*x])/x,x]
Output:
-((Sqrt[x^(-1)]*Sqrt[c - a*c*x]*((-2*Sqrt[1 + 1/(a*x)])/Sqrt[x^(-1)] + (2* ArcSinh[Sqrt[x^(-1)]/Sqrt[a]])/Sqrt[a]))/Sqrt[1 - 1/(a*x)])
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + 1))), x] - Simp[d*(n/(b*(m + 1))) Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d}, x] & & GtQ[n, 0] && LtQ[m, -1] && !(IntegerQ[n] && !IntegerQ[m]) && !(ILeQ[m + n + 2, 0] && (FractionQ[m] || GeQ[2*n + m + 1, 0])) && IntLinearQ[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^(ArcCoth[(a_.)*(x_)]*(n_.))*((e_.)*(x_))^(m_.)*((c_) + (d_.)*(x_))^(p _), x_Symbol] :> Simp[(-(e*x)^m)*(1/x)^(m + p)*((c + d*x)^p/(1 + c/(d*x))^p ) Subst[Int[((1 + c*(x/d))^p*((1 + x/a)^(n/2)/x^(m + p + 2)))/(1 - x/a)^( n/2), x], x, 1/x], x] /; FreeQ[{a, c, d, e, m, n, p}, x] && EqQ[a^2*c^2 - d ^2, 0] && !IntegerQ[p]
Time = 0.14 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.76
method | result | size |
default | \(\frac {2 \sqrt {-c \left (a x -1\right )}\, \left (-\sqrt {c}\, \arctan \left (\frac {\sqrt {-c \left (a x +1\right )}}{\sqrt {c}}\right )+\sqrt {-c \left (a x +1\right )}\right )}{\sqrt {\frac {a x -1}{a x +1}}\, \sqrt {-c \left (a x +1\right )}}\) | \(69\) |
Input:
int(1/((a*x-1)/(a*x+1))^(1/2)*(-a*c*x+c)^(1/2)/x,x,method=_RETURNVERBOSE)
Output:
2*(-c*(a*x-1))^(1/2)*(-c^(1/2)*arctan((-c*(a*x+1))^(1/2)/c^(1/2))+(-c*(a*x +1))^(1/2))/((a*x-1)/(a*x+1))^(1/2)/(-c*(a*x+1))^(1/2)
Time = 0.10 (sec) , antiderivative size = 212, normalized size of antiderivative = 2.33 \[ \int \frac {e^{\coth ^{-1}(a x)} \sqrt {c-a c x}}{x} \, dx=\left [\frac {{\left (a x - 1\right )} \sqrt {-c} \log \left (-\frac {a^{2} c x^{2} + a c x + 2 \, \sqrt {-a c x + c} {\left (a x + 1\right )} \sqrt {-c} \sqrt {\frac {a x - 1}{a x + 1}} - 2 \, c}{a x^{2} - x}\right ) + 2 \, \sqrt {-a c x + c} {\left (a x + 1\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{a x - 1}, -\frac {2 \, {\left ({\left (a x - 1\right )} \sqrt {c} \arctan \left (\frac {\sqrt {-a c x + c} {\left (a x + 1\right )} \sqrt {c} \sqrt {\frac {a x - 1}{a x + 1}}}{a c x - c}\right ) - \sqrt {-a c x + c} {\left (a x + 1\right )} \sqrt {\frac {a x - 1}{a x + 1}}\right )}}{a x - 1}\right ] \] Input:
integrate(1/((a*x-1)/(a*x+1))^(1/2)*(-a*c*x+c)^(1/2)/x,x, algorithm="frica s")
Output:
[((a*x - 1)*sqrt(-c)*log(-(a^2*c*x^2 + a*c*x + 2*sqrt(-a*c*x + c)*(a*x + 1 )*sqrt(-c)*sqrt((a*x - 1)/(a*x + 1)) - 2*c)/(a*x^2 - x)) + 2*sqrt(-a*c*x + c)*(a*x + 1)*sqrt((a*x - 1)/(a*x + 1)))/(a*x - 1), -2*((a*x - 1)*sqrt(c)* arctan(sqrt(-a*c*x + c)*(a*x + 1)*sqrt(c)*sqrt((a*x - 1)/(a*x + 1))/(a*c*x - c)) - sqrt(-a*c*x + c)*(a*x + 1)*sqrt((a*x - 1)/(a*x + 1)))/(a*x - 1)]
\[ \int \frac {e^{\coth ^{-1}(a x)} \sqrt {c-a c x}}{x} \, dx=\int \frac {\sqrt {- c \left (a x - 1\right )}}{x \sqrt {\frac {a x - 1}{a x + 1}}}\, dx \] Input:
integrate(1/((a*x-1)/(a*x+1))**(1/2)*(-a*c*x+c)**(1/2)/x,x)
Output:
Integral(sqrt(-c*(a*x - 1))/(x*sqrt((a*x - 1)/(a*x + 1))), x)
\[ \int \frac {e^{\coth ^{-1}(a x)} \sqrt {c-a c x}}{x} \, dx=\int { \frac {\sqrt {-a c x + c}}{x \sqrt {\frac {a x - 1}{a x + 1}}} \,d x } \] Input:
integrate(1/((a*x-1)/(a*x+1))^(1/2)*(-a*c*x+c)^(1/2)/x,x, algorithm="maxim a")
Output:
integrate(sqrt(-a*c*x + c)/(x*sqrt((a*x - 1)/(a*x + 1))), x)
Time = 0.13 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.97 \[ \int \frac {e^{\coth ^{-1}(a x)} \sqrt {c-a c x}}{x} \, dx=\frac {2 \, c^{3} {\left (\frac {\arctan \left (\frac {\sqrt {-a c x - c}}{\sqrt {c}}\right )}{c^{\frac {3}{2}}} - \frac {c \arctan \left (\frac {\sqrt {2} \sqrt {-c}}{\sqrt {c}}\right ) - \sqrt {2} \sqrt {-c} \sqrt {c}}{c^{\frac {5}{2}}} - \frac {\sqrt {-a c x - c}}{c^{2}}\right )}}{{\left | c \right |} \mathrm {sgn}\left (a x + 1\right )} \] Input:
integrate(1/((a*x-1)/(a*x+1))^(1/2)*(-a*c*x+c)^(1/2)/x,x, algorithm="giac" )
Output:
2*c^3*(arctan(sqrt(-a*c*x - c)/sqrt(c))/c^(3/2) - (c*arctan(sqrt(2)*sqrt(- c)/sqrt(c)) - sqrt(2)*sqrt(-c)*sqrt(c))/c^(5/2) - sqrt(-a*c*x - c)/c^2)/(a bs(c)*sgn(a*x + 1))
Timed out. \[ \int \frac {e^{\coth ^{-1}(a x)} \sqrt {c-a c x}}{x} \, dx=\int \frac {\sqrt {c-a\,c\,x}}{x\,\sqrt {\frac {a\,x-1}{a\,x+1}}} \,d x \] Input:
int((c - a*c*x)^(1/2)/(x*((a*x - 1)/(a*x + 1))^(1/2)),x)
Output:
int((c - a*c*x)^(1/2)/(x*((a*x - 1)/(a*x + 1))^(1/2)), x)
Time = 0.15 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.52 \[ \int \frac {e^{\coth ^{-1}(a x)} \sqrt {c-a c x}}{x} \, dx=\sqrt {c}\, i \left (-2 \sqrt {a x +1}-\mathrm {log}\left (\frac {2 \sqrt {a x +1}-2}{\sqrt {2}}\right )+\mathrm {log}\left (\frac {2 \sqrt {a x +1}+2}{\sqrt {2}}\right )\right ) \] Input:
int(1/((a*x-1)/(a*x+1))^(1/2)*(-a*c*x+c)^(1/2)/x,x)
Output:
sqrt(c)*i*( - 2*sqrt(a*x + 1) - log((2*sqrt(a*x + 1) - 2)/sqrt(2)) + log(( 2*sqrt(a*x + 1) + 2)/sqrt(2)))