Integrand size = 23, antiderivative size = 74 \[ \int \frac {e^{-2 \coth ^{-1}(a x)} \sqrt {c-a c x}}{x} \, dx=2 \sqrt {c-a c x}+2 \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c-a c x}}{\sqrt {c}}\right )-4 \sqrt {2} \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c-a c x}}{\sqrt {2} \sqrt {c}}\right ) \] Output:
2*(-a*c*x+c)^(1/2)+2*c^(1/2)*arctanh((-a*c*x+c)^(1/2)/c^(1/2))-4*2^(1/2)*c ^(1/2)*arctanh(1/2*(-a*c*x+c)^(1/2)*2^(1/2)/c^(1/2))
Time = 0.06 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.00 \[ \int \frac {e^{-2 \coth ^{-1}(a x)} \sqrt {c-a c x}}{x} \, dx=2 \sqrt {c-a c x}+2 \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c-a c x}}{\sqrt {c}}\right )-4 \sqrt {2} \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c-a c x}}{\sqrt {2} \sqrt {c}}\right ) \] Input:
Integrate[Sqrt[c - a*c*x]/(E^(2*ArcCoth[a*x])*x),x]
Output:
2*Sqrt[c - a*c*x] + 2*Sqrt[c]*ArcTanh[Sqrt[c - a*c*x]/Sqrt[c]] - 4*Sqrt[2] *Sqrt[c]*ArcTanh[Sqrt[c - a*c*x]/(Sqrt[2]*Sqrt[c])]
Time = 0.74 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.15, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.391, Rules used = {6717, 6680, 35, 95, 27, 174, 73, 219, 221}
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.
\(\displaystyle \int \frac {\sqrt {c-a c x} e^{-2 \coth ^{-1}(a x)}}{x} \, dx\) |
\(\Big \downarrow \) 6717 |
\(\displaystyle -\int \frac {e^{-2 \text {arctanh}(a x)} \sqrt {c-a c x}}{x}dx\) |
\(\Big \downarrow \) 6680 |
\(\displaystyle -\int \frac {(1-a x) \sqrt {c-a c x}}{x (a x+1)}dx\) |
\(\Big \downarrow \) 35 |
\(\displaystyle -\frac {\int \frac {(c-a c x)^{3/2}}{x (a x+1)}dx}{c}\) |
\(\Big \downarrow \) 95 |
\(\displaystyle -\frac {\frac {\int \frac {a c^2 (1-3 a x)}{x (a x+1) \sqrt {c-a c x}}dx}{a}-2 c \sqrt {c-a c x}}{c}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {c^2 \int \frac {1-3 a x}{x (a x+1) \sqrt {c-a c x}}dx-2 c \sqrt {c-a c x}}{c}\) |
\(\Big \downarrow \) 174 |
\(\displaystyle -\frac {c^2 \left (\int \frac {1}{x \sqrt {c-a c x}}dx-4 a \int \frac {1}{(a x+1) \sqrt {c-a c x}}dx\right )-2 c \sqrt {c-a c x}}{c}\) |
\(\Big \downarrow \) 73 |
\(\displaystyle -\frac {c^2 \left (\frac {8 \int \frac {1}{2-\frac {c-a c x}{c}}d\sqrt {c-a c x}}{c}-\frac {2 \int \frac {1}{\frac {1}{a}-\frac {c-a c x}{a c}}d\sqrt {c-a c x}}{a c}\right )-2 c \sqrt {c-a c x}}{c}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle -\frac {c^2 \left (\frac {4 \sqrt {2} \text {arctanh}\left (\frac {\sqrt {c-a c x}}{\sqrt {2} \sqrt {c}}\right )}{\sqrt {c}}-\frac {2 \int \frac {1}{\frac {1}{a}-\frac {c-a c x}{a c}}d\sqrt {c-a c x}}{a c}\right )-2 c \sqrt {c-a c x}}{c}\) |
\(\Big \downarrow \) 221 |
\(\displaystyle -\frac {c^2 \left (\frac {4 \sqrt {2} \text {arctanh}\left (\frac {\sqrt {c-a c x}}{\sqrt {2} \sqrt {c}}\right )}{\sqrt {c}}-\frac {2 \text {arctanh}\left (\frac {\sqrt {c-a c x}}{\sqrt {c}}\right )}{\sqrt {c}}\right )-2 c \sqrt {c-a c x}}{c}\) |
Input:
Int[Sqrt[c - a*c*x]/(E^(2*ArcCoth[a*x])*x),x]
Output:
-((-2*c*Sqrt[c - a*c*x] + c^2*((-2*ArcTanh[Sqrt[c - a*c*x]/Sqrt[c]])/Sqrt[ c] + (4*Sqrt[2]*ArcTanh[Sqrt[c - a*c*x]/(Sqrt[2]*Sqrt[c])])/Sqrt[c]))/c)
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[(u_.)*((a_) + (b_.)*(x_))^(m_.)*((c_) + (d_.)*(x_))^(n_.), x_Symbol] :> Simp[(b/d)^m Int[u*(c + d*x)^(m + n), x], x] /; FreeQ[{a, b, c, d, m, n} , x] && EqQ[b*c - a*d, 0] && IntegerQ[m] && !(IntegerQ[n] && SimplerQ[a + b*x, c + d*x])
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ {p = Denominator[m]}, Simp[p/b Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL inearQ[a, b, c, d, m, n, x]
Int[((e_.) + (f_.)*(x_))^(p_)/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_] :> Simp[f*((e + f*x)^(p - 1)/(b*d*(p - 1))), x] + Simp[1/(b*d) Int[(b *d*e^2 - a*c*f^2 + f*(2*b*d*e - b*c*f - a*d*f)*x)*((e + f*x)^(p - 2)/((a + b*x)*(c + d*x))), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && GtQ[p, 1]
Int[(((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)))/(((a_.) + (b_.)*(x_))* ((c_.) + (d_.)*(x_))), x_] :> Simp[(b*g - a*h)/(b*c - a*d) Int[(e + f*x)^ p/(a + b*x), x], x] - Simp[(d*g - c*h)/(b*c - a*d) Int[(e + f*x)^p/(c + d *x), x], x] /; FreeQ[{a, b, c, d, e, f, g, h}, 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[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
Int[E^(ArcTanh[(a_.)*(x_)]*(n_.))*(u_.)*((c_) + (d_.)*(x_))^(p_.), x_Symbol ] :> Int[u*(c + d*x)^p*((1 + a*x)^(n/2)/(1 - a*x)^(n/2)), x] /; FreeQ[{a, c , d, n, p}, x] && EqQ[a^2*c^2 - d^2, 0] && !(IntegerQ[p] || GtQ[c, 0])
Int[E^(ArcCoth[(a_.)*(x_)]*(n_))*(u_.), x_Symbol] :> Simp[(-1)^(n/2) Int[ u*E^(n*ArcTanh[a*x]), x], x] /; FreeQ[a, x] && IntegerQ[n/2]
Time = 0.19 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.78
method | result | size |
derivativedivides | \(2 \sqrt {-a c x +c}+2 \sqrt {c}\, \operatorname {arctanh}\left (\frac {\sqrt {-a c x +c}}{\sqrt {c}}\right )-4 \sqrt {2}\, \sqrt {c}\, \operatorname {arctanh}\left (\frac {\sqrt {-a c x +c}\, \sqrt {2}}{2 \sqrt {c}}\right )\) | \(58\) |
default | \(2 \sqrt {-a c x +c}+2 \sqrt {c}\, \operatorname {arctanh}\left (\frac {\sqrt {-a c x +c}}{\sqrt {c}}\right )-4 \sqrt {2}\, \sqrt {c}\, \operatorname {arctanh}\left (\frac {\sqrt {-a c x +c}\, \sqrt {2}}{2 \sqrt {c}}\right )\) | \(58\) |
pseudoelliptic | \(2 \sqrt {-c \left (a x -1\right )}+2 \sqrt {c}\, \operatorname {arctanh}\left (\frac {\sqrt {-c \left (a x -1\right )}}{\sqrt {c}}\right )-4 \sqrt {c}\, \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {-c \left (a x -1\right )}\, \sqrt {2}}{2 \sqrt {c}}\right )\) | \(61\) |
Input:
int((-a*c*x+c)^(1/2)*(a*x-1)/(a*x+1)/x,x,method=_RETURNVERBOSE)
Output:
2*(-a*c*x+c)^(1/2)+2*c^(1/2)*arctanh((-a*c*x+c)^(1/2)/c^(1/2))-4*2^(1/2)*c ^(1/2)*arctanh(1/2*(-a*c*x+c)^(1/2)*2^(1/2)/c^(1/2))
Time = 0.11 (sec) , antiderivative size = 170, normalized size of antiderivative = 2.30 \[ \int \frac {e^{-2 \coth ^{-1}(a x)} \sqrt {c-a c x}}{x} \, dx=\left [2 \, \sqrt {2} \sqrt {c} \log \left (\frac {a c x + 2 \, \sqrt {2} \sqrt {-a c x + c} \sqrt {c} - 3 \, c}{a x + 1}\right ) + \sqrt {c} \log \left (\frac {a c x - 2 \, \sqrt {-a c x + c} \sqrt {c} - 2 \, c}{x}\right ) + 2 \, \sqrt {-a c x + c}, -4 \, \sqrt {2} \sqrt {-c} \arctan \left (\frac {\sqrt {2} \sqrt {-a c x + c} \sqrt {-c}}{a c x - c}\right ) + 2 \, \sqrt {-c} \arctan \left (\frac {\sqrt {-a c x + c} \sqrt {-c}}{a c x - c}\right ) + 2 \, \sqrt {-a c x + c}\right ] \] Input:
integrate((-a*c*x+c)^(1/2)*(a*x-1)/(a*x+1)/x,x, algorithm="fricas")
Output:
[2*sqrt(2)*sqrt(c)*log((a*c*x + 2*sqrt(2)*sqrt(-a*c*x + c)*sqrt(c) - 3*c)/ (a*x + 1)) + sqrt(c)*log((a*c*x - 2*sqrt(-a*c*x + c)*sqrt(c) - 2*c)/x) + 2 *sqrt(-a*c*x + c), -4*sqrt(2)*sqrt(-c)*arctan(sqrt(2)*sqrt(-a*c*x + c)*sqr t(-c)/(a*c*x - c)) + 2*sqrt(-c)*arctan(sqrt(-a*c*x + c)*sqrt(-c)/(a*c*x - c)) + 2*sqrt(-a*c*x + c)]
Time = 4.59 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.65 \[ \int \frac {e^{-2 \coth ^{-1}(a x)} \sqrt {c-a c x}}{x} \, dx=\begin {cases} - \frac {2 c \operatorname {atan}{\left (\frac {\sqrt {- a c x + c}}{\sqrt {- c}} \right )}}{\sqrt {- c}} + \frac {4 \sqrt {2} c \operatorname {atan}{\left (\frac {\sqrt {2} \sqrt {- a c x + c}}{2 \sqrt {- c}} \right )}}{\sqrt {- c}} + 2 \sqrt {- a c x + c} & \text {for}\: a c \neq 0 \\\sqrt {c} \left (- \frac {3 a \left (\frac {\log {\left (- \frac {2}{x} \right )}}{a} - \frac {\log {\left (2 a + \frac {2}{x} \right )}}{a}\right )}{2} + \frac {\log {\left (\frac {a}{x} + \frac {1}{x^{2}} \right )}}{2}\right ) & \text {otherwise} \end {cases} \] Input:
integrate((-a*c*x+c)**(1/2)*(a*x-1)/(a*x+1)/x,x)
Output:
Piecewise((-2*c*atan(sqrt(-a*c*x + c)/sqrt(-c))/sqrt(-c) + 4*sqrt(2)*c*ata n(sqrt(2)*sqrt(-a*c*x + c)/(2*sqrt(-c)))/sqrt(-c) + 2*sqrt(-a*c*x + c), Ne (a*c, 0)), (sqrt(c)*(-3*a*(log(-2/x)/a - log(2*a + 2/x)/a)/2 + log(a/x + x **(-2))/2), True))
Time = 0.11 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.32 \[ \int \frac {e^{-2 \coth ^{-1}(a x)} \sqrt {c-a c x}}{x} \, dx=2 \, \sqrt {2} \sqrt {c} \log \left (-\frac {\sqrt {2} \sqrt {c} - \sqrt {-a c x + c}}{\sqrt {2} \sqrt {c} + \sqrt {-a c x + c}}\right ) - \sqrt {c} \log \left (\frac {\sqrt {-a c x + c} - \sqrt {c}}{\sqrt {-a c x + c} + \sqrt {c}}\right ) + 2 \, \sqrt {-a c x + c} \] Input:
integrate((-a*c*x+c)^(1/2)*(a*x-1)/(a*x+1)/x,x, algorithm="maxima")
Output:
2*sqrt(2)*sqrt(c)*log(-(sqrt(2)*sqrt(c) - sqrt(-a*c*x + c))/(sqrt(2)*sqrt( c) + sqrt(-a*c*x + c))) - sqrt(c)*log((sqrt(-a*c*x + c) - sqrt(c))/(sqrt(- a*c*x + c) + sqrt(c))) + 2*sqrt(-a*c*x + c)
Time = 0.14 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.91 \[ \int \frac {e^{-2 \coth ^{-1}(a x)} \sqrt {c-a c x}}{x} \, dx=\frac {4 \, \sqrt {2} c \arctan \left (\frac {\sqrt {2} \sqrt {-a c x + c}}{2 \, \sqrt {-c}}\right )}{\sqrt {-c}} - \frac {2 \, c \arctan \left (\frac {\sqrt {-a c x + c}}{\sqrt {-c}}\right )}{\sqrt {-c}} + 2 \, \sqrt {-a c x + c} \] Input:
integrate((-a*c*x+c)^(1/2)*(a*x-1)/(a*x+1)/x,x, algorithm="giac")
Output:
4*sqrt(2)*c*arctan(1/2*sqrt(2)*sqrt(-a*c*x + c)/sqrt(-c))/sqrt(-c) - 2*c*a rctan(sqrt(-a*c*x + c)/sqrt(-c))/sqrt(-c) + 2*sqrt(-a*c*x + c)
Time = 0.05 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.77 \[ \int \frac {e^{-2 \coth ^{-1}(a x)} \sqrt {c-a c x}}{x} \, dx=2\,\sqrt {c}\,\mathrm {atanh}\left (\frac {\sqrt {c-a\,c\,x}}{\sqrt {c}}\right )+2\,\sqrt {c-a\,c\,x}-4\,\sqrt {2}\,\sqrt {c}\,\mathrm {atanh}\left (\frac {\sqrt {2}\,\sqrt {c-a\,c\,x}}{2\,\sqrt {c}}\right ) \] Input:
int(((c - a*c*x)^(1/2)*(a*x - 1))/(x*(a*x + 1)),x)
Output:
2*c^(1/2)*atanh((c - a*c*x)^(1/2)/c^(1/2)) + 2*(c - a*c*x)^(1/2) - 4*2^(1/ 2)*c^(1/2)*atanh((2^(1/2)*(c - a*c*x)^(1/2))/(2*c^(1/2)))
Time = 0.17 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.91 \[ \int \frac {e^{-2 \coth ^{-1}(a x)} \sqrt {c-a c x}}{x} \, dx=\sqrt {c}\, \left (2 \sqrt {-a x +1}+2 \sqrt {2}\, \mathrm {log}\left (\sqrt {-a x +1}-\sqrt {2}\right )-2 \sqrt {2}\, \mathrm {log}\left (\sqrt {-a x +1}+\sqrt {2}\right )-\mathrm {log}\left (\sqrt {-a x +1}-1\right )+\mathrm {log}\left (\sqrt {-a x +1}+1\right )\right ) \] Input:
int((-a*c*x+c)^(1/2)*(a*x-1)/(a*x+1)/x,x)
Output:
sqrt(c)*(2*sqrt( - a*x + 1) + 2*sqrt(2)*log(sqrt( - a*x + 1) - sqrt(2)) - 2*sqrt(2)*log(sqrt( - a*x + 1) + sqrt(2)) - log(sqrt( - a*x + 1) - 1) + lo g(sqrt( - a*x + 1) + 1))