Integrand size = 23, antiderivative size = 74 \[ \int \frac {e^{-2 \text {arctanh}(a x)} \sqrt {c-a c x}}{x} \, dx=-2 \sqrt {c-a c x}-2 \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c-a c x}}{\sqrt {c}}\right )+4 \sqrt {2} \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c-a c x}}{\sqrt {2} \sqrt {c}}\right ) \]
-2*arctanh((-a*c*x+c)^(1/2)/c^(1/2))*c^(1/2)+4*arctanh(1/2*(-a*c*x+c)^(1/2 )*2^(1/2)/c^(1/2))*2^(1/2)*c^(1/2)-2*(-a*c*x+c)^(1/2)
Time = 0.02 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.00 \[ \int \frac {e^{-2 \text {arctanh}(a x)} \sqrt {c-a c x}}{x} \, dx=-2 \sqrt {c-a c x}-2 \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c-a c x}}{\sqrt {c}}\right )+4 \sqrt {2} \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c-a c x}}{\sqrt {2} \sqrt {c}}\right ) \]
-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 = 0.33 (sec) , antiderivative size = 84, normalized size of antiderivative = 1.14, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.348, Rules used = {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.
\(\displaystyle \int \frac {e^{-2 \text {arctanh}(a x)} \sqrt {c-a c x}}{x} \, dx\) |
\(\Big \downarrow \) 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 |
\(\displaystyle \frac {c^2 \left (\frac {4 \sqrt {2} \text {arctanh}\left (\frac {\sqrt {c-a c x}}{\sqrt {2} \sqrt {c}}\right )}{\sqrt {c}}-\frac {2 \text {arctanh}\left (\frac {\sqrt {c-a c x}}{\sqrt {c}}\right )}{\sqrt {c}}\right )-2 c \sqrt {c-a c x}}{c}\) |
(-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
3.5.23.3.1 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[(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])
Time = 0.12 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.78
method | result | size |
derivativedivides | \(-2 \,\operatorname {arctanh}\left (\frac {\sqrt {-a c x +c}}{\sqrt {c}}\right ) \sqrt {c}+4 \,\operatorname {arctanh}\left (\frac {\sqrt {-a c x +c}\, \sqrt {2}}{2 \sqrt {c}}\right ) \sqrt {2}\, \sqrt {c}-2 \sqrt {-a c x +c}\) | \(58\) |
default | \(-2 \,\operatorname {arctanh}\left (\frac {\sqrt {-a c x +c}}{\sqrt {c}}\right ) \sqrt {c}+4 \,\operatorname {arctanh}\left (\frac {\sqrt {-a c x +c}\, \sqrt {2}}{2 \sqrt {c}}\right ) \sqrt {2}\, \sqrt {c}-2 \sqrt {-a c x +c}\) | \(58\) |
pseudoelliptic | \(-2 \,\operatorname {arctanh}\left (\frac {\sqrt {-c \left (a x -1\right )}}{\sqrt {c}}\right ) \sqrt {c}+4 \sqrt {c}\, \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {-c \left (a x -1\right )}\, \sqrt {2}}{2 \sqrt {c}}\right )-2 \sqrt {-c \left (a x -1\right )}\) | \(61\) |
-2*arctanh((-a*c*x+c)^(1/2)/c^(1/2))*c^(1/2)+4*arctanh(1/2*(-a*c*x+c)^(1/2 )*2^(1/2)/c^(1/2))*2^(1/2)*c^(1/2)-2*(-a*c*x+c)^(1/2)
Time = 0.29 (sec) , antiderivative size = 157, normalized size of antiderivative = 2.12 \[ \int \frac {e^{-2 \text {arctanh}(a x)} \sqrt {c-a c x}}{x} \, dx=\left [2 \, \sqrt {2} \sqrt {c} \log \left (\frac {a c x - 2 \, \sqrt {2} \sqrt {-a c x + c} \sqrt {c} - 3 \, c}{a x + 1}\right ) + \sqrt {c} \log \left (\frac {a c x + 2 \, \sqrt {-a c x + c} \sqrt {c} - 2 \, c}{x}\right ) - 2 \, \sqrt {-a c x + c}, -4 \, \sqrt {2} \sqrt {-c} \arctan \left (\frac {\sqrt {2} \sqrt {-a c x + c} \sqrt {-c}}{2 \, c}\right ) + 2 \, \sqrt {-c} \arctan \left (\frac {\sqrt {-a c x + c} \sqrt {-c}}{c}\right ) - 2 \, \sqrt {-a c x + c}\right ] \]
[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(1/2*sqrt(2)*sqrt(-a*c*x + c) *sqrt(-c)/c) + 2*sqrt(-c)*arctan(sqrt(-a*c*x + c)*sqrt(-c)/c) - 2*sqrt(-a* c*x + c)]
Time = 8.52 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.65 \[ \int \frac {e^{-2 \text {arctanh}(a x)} \sqrt {c-a c x}}{x} \, dx=\begin {cases} \frac {2 c \operatorname {atan}{\left (\frac {\sqrt {- a c x + c}}{\sqrt {- c}} \right )}}{\sqrt {- c}} - \frac {4 \sqrt {2} c \operatorname {atan}{\left (\frac {\sqrt {2} \sqrt {- a c x + c}}{2 \sqrt {- c}} \right )}}{\sqrt {- c}} - 2 \sqrt {- a c x + c} & \text {for}\: a c \neq 0 \\\sqrt {c} \left (\frac {3 a \left (\frac {\log {\left (- \frac {2}{x} \right )}}{a} - \frac {\log {\left (2 a + \frac {2}{x} \right )}}{a}\right )}{2} - \frac {\log {\left (\frac {a}{x} + \frac {1}{x^{2}} \right )}}{2}\right ) & \text {otherwise} \end {cases} \]
Piecewise((2*c*atan(sqrt(-a*c*x + c)/sqrt(-c))/sqrt(-c) - 4*sqrt(2)*c*atan (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 = 0.27 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.31 \[ \int \frac {e^{-2 \text {arctanh}(a x)} \sqrt {c-a c x}}{x} \, dx=-2 \, \sqrt {2} \sqrt {c} \log \left (-\frac {\sqrt {2} \sqrt {c} - \sqrt {-a c x + c}}{\sqrt {2} \sqrt {c} + \sqrt {-a c x + c}}\right ) + \sqrt {c} \log \left (\frac {\sqrt {-a c x + c} - \sqrt {c}}{\sqrt {-a c x + c} + \sqrt {c}}\right ) - 2 \, \sqrt {-a c x + c} \]
-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 = 0.27 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.91 \[ \int \frac {e^{-2 \text {arctanh}(a x)} \sqrt {c-a c x}}{x} \, dx=-\frac {4 \, \sqrt {2} c \arctan \left (\frac {\sqrt {2} \sqrt {-a c x + c}}{2 \, \sqrt {-c}}\right )}{\sqrt {-c}} + \frac {2 \, c \arctan \left (\frac {\sqrt {-a c x + c}}{\sqrt {-c}}\right )}{\sqrt {-c}} - 2 \, \sqrt {-a c x + c} \]
-4*sqrt(2)*c*arctan(1/2*sqrt(2)*sqrt(-a*c*x + c)/sqrt(-c))/sqrt(-c) + 2*c* arctan(sqrt(-a*c*x + c)/sqrt(-c))/sqrt(-c) - 2*sqrt(-a*c*x + c)
Time = 3.55 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.77 \[ \int \frac {e^{-2 \text {arctanh}(a x)} \sqrt {c-a c x}}{x} \, dx=4\,\sqrt {2}\,\sqrt {c}\,\mathrm {atanh}\left (\frac {\sqrt {2}\,\sqrt {c-a\,c\,x}}{2\,\sqrt {c}}\right )-2\,\sqrt {c-a\,c\,x}-2\,\sqrt {c}\,\mathrm {atanh}\left (\frac {\sqrt {c-a\,c\,x}}{\sqrt {c}}\right ) \]