Integrand size = 23, antiderivative size = 79 \[ \int \frac {e^{-2 \text {arctanh}(a x)} \sqrt {c-a c x}}{x^2} \, dx=-\frac {\sqrt {c-a c x}}{x}+5 a \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c-a c x}}{\sqrt {c}}\right )-4 \sqrt {2} a \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c-a c x}}{\sqrt {2} \sqrt {c}}\right ) \]
5*a*arctanh((-a*c*x+c)^(1/2)/c^(1/2))*c^(1/2)-4*a*arctanh(1/2*(-a*c*x+c)^( 1/2)*2^(1/2)/c^(1/2))*2^(1/2)*c^(1/2)-(-a*c*x+c)^(1/2)/x
Time = 0.03 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.00 \[ \int \frac {e^{-2 \text {arctanh}(a x)} \sqrt {c-a c x}}{x^2} \, dx=-\frac {\sqrt {c-a c x}}{x}+5 a \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c-a c x}}{\sqrt {c}}\right )-4 \sqrt {2} a \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c-a c x}}{\sqrt {2} \sqrt {c}}\right ) \]
-(Sqrt[c - a*c*x]/x) + 5*a*Sqrt[c]*ArcTanh[Sqrt[c - a*c*x]/Sqrt[c]] - 4*Sq rt[2]*a*Sqrt[c]*ArcTanh[Sqrt[c - a*c*x]/(Sqrt[2]*Sqrt[c])]
Time = 0.34 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.15, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.348, Rules used = {6680, 35, 109, 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.
\(\displaystyle \int \frac {e^{-2 \text {arctanh}(a x)} \sqrt {c-a c x}}{x^2} \, dx\) |
\(\Big \downarrow \) 6680 |
\(\displaystyle \int \frac {(1-a x) \sqrt {c-a c x}}{x^2 (a x+1)}dx\) |
\(\Big \downarrow \) 35 |
\(\displaystyle \frac {\int \frac {(c-a c x)^{3/2}}{x^2 (a x+1)}dx}{c}\) |
\(\Big \downarrow \) 109 |
\(\displaystyle \frac {-\int \frac {a c^2 (5-3 a x)}{2 x (a x+1) \sqrt {c-a c x}}dx-\frac {c \sqrt {c-a c x}}{x}}{c}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {-\frac {1}{2} a c^2 \int \frac {5-3 a x}{x (a x+1) \sqrt {c-a c x}}dx-\frac {c \sqrt {c-a c x}}{x}}{c}\) |
\(\Big \downarrow \) 174 |
\(\displaystyle \frac {-\frac {1}{2} a c^2 \left (5 \int \frac {1}{x \sqrt {c-a c x}}dx-8 a \int \frac {1}{(a x+1) \sqrt {c-a c x}}dx\right )-\frac {c \sqrt {c-a c x}}{x}}{c}\) |
\(\Big \downarrow \) 73 |
\(\displaystyle \frac {-\frac {1}{2} a c^2 \left (\frac {16 \int \frac {1}{2-\frac {c-a c x}{c}}d\sqrt {c-a c x}}{c}-\frac {10 \int \frac {1}{\frac {1}{a}-\frac {c-a c x}{a c}}d\sqrt {c-a c x}}{a c}\right )-\frac {c \sqrt {c-a c x}}{x}}{c}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {-\frac {1}{2} a c^2 \left (\frac {8 \sqrt {2} \text {arctanh}\left (\frac {\sqrt {c-a c x}}{\sqrt {2} \sqrt {c}}\right )}{\sqrt {c}}-\frac {10 \int \frac {1}{\frac {1}{a}-\frac {c-a c x}{a c}}d\sqrt {c-a c x}}{a c}\right )-\frac {c \sqrt {c-a c x}}{x}}{c}\) |
\(\Big \downarrow \) 221 |
\(\displaystyle \frac {-\frac {1}{2} a c^2 \left (\frac {8 \sqrt {2} \text {arctanh}\left (\frac {\sqrt {c-a c x}}{\sqrt {2} \sqrt {c}}\right )}{\sqrt {c}}-\frac {10 \text {arctanh}\left (\frac {\sqrt {c-a c x}}{\sqrt {c}}\right )}{\sqrt {c}}\right )-\frac {c \sqrt {c-a c x}}{x}}{c}\) |
(-((c*Sqrt[c - a*c*x])/x) - (a*c^2*((-10*ArcTanh[Sqrt[c - a*c*x]/Sqrt[c]]) /Sqrt[c] + (8*Sqrt[2]*ArcTanh[Sqrt[c - a*c*x]/(Sqrt[2]*Sqrt[c])])/Sqrt[c]) )/2)/c
3.5.24.3.1 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[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Simp[(b*c - a*d)*(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*((e + f *x)^(p + 1)/(b*(b*e - a*f)*(m + 1))), x] + Simp[1/(b*(b*e - a*f)*(m + 1)) Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 2)*(e + f*x)^p*Simp[a*d*(d*e*(n - 1) + c*f*(p + 1)) + b*c*(d*e*(m - n + 2) - c*f*(m + p + 2)) + d*(a*d*f*(n + p) + b*(d*e*(m + 1) - c*f*(m + n + p + 1)))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && LtQ[m, -1] && GtQ[n, 1] && (IntegersQ[2*m, 2*n, 2*p] || IntegersQ[m, n + p] || IntegersQ[p, m + n])
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 = 0.13 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.89
method | result | size |
pseudoelliptic | \(-\frac {\sqrt {-c \left (a x -1\right )}\, \sqrt {c}+a c x \left (4 \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {-c \left (a x -1\right )}\, \sqrt {2}}{2 \sqrt {c}}\right )-5 \,\operatorname {arctanh}\left (\frac {\sqrt {-c \left (a x -1\right )}}{\sqrt {c}}\right )\right )}{\sqrt {c}\, x}\) | \(70\) |
derivativedivides | \(2 a c \left (-\frac {2 \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {-a c x +c}\, \sqrt {2}}{2 \sqrt {c}}\right )}{\sqrt {c}}-\frac {\sqrt {-a c x +c}}{2 a c x}+\frac {5 \,\operatorname {arctanh}\left (\frac {\sqrt {-a c x +c}}{\sqrt {c}}\right )}{2 \sqrt {c}}\right )\) | \(71\) |
default | \(-2 a c \left (\frac {2 \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {-a c x +c}\, \sqrt {2}}{2 \sqrt {c}}\right )}{\sqrt {c}}+\frac {\sqrt {-a c x +c}}{2 a c x}-\frac {5 \,\operatorname {arctanh}\left (\frac {\sqrt {-a c x +c}}{\sqrt {c}}\right )}{2 \sqrt {c}}\right )\) | \(71\) |
risch | \(\frac {\left (a x -1\right ) c}{x \sqrt {-c \left (a x -1\right )}}-\frac {a \left (\frac {8 \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {-a c x +c}\, \sqrt {2}}{2 \sqrt {c}}\right )}{\sqrt {c}}-\frac {10 \,\operatorname {arctanh}\left (\frac {\sqrt {-a c x +c}}{\sqrt {c}}\right )}{\sqrt {c}}\right ) c}{2}\) | \(72\) |
-((-c*(a*x-1))^(1/2)*c^(1/2)+a*c*x*(4*2^(1/2)*arctanh(1/2*(-c*(a*x-1))^(1/ 2)*2^(1/2)/c^(1/2))-5*arctanh((-c*(a*x-1))^(1/2)/c^(1/2))))/c^(1/2)/x
Time = 0.29 (sec) , antiderivative size = 175, normalized size of antiderivative = 2.22 \[ \int \frac {e^{-2 \text {arctanh}(a x)} \sqrt {c-a c x}}{x^2} \, dx=\left [\frac {4 \, \sqrt {2} a \sqrt {c} x \log \left (\frac {a c x + 2 \, \sqrt {2} \sqrt {-a c x + c} \sqrt {c} - 3 \, c}{a x + 1}\right ) + 5 \, a \sqrt {c} x \log \left (\frac {a c x - 2 \, \sqrt {-a c x + c} \sqrt {c} - 2 \, c}{x}\right ) - 2 \, \sqrt {-a c x + c}}{2 \, x}, \frac {4 \, \sqrt {2} a \sqrt {-c} x \arctan \left (\frac {\sqrt {2} \sqrt {-a c x + c} \sqrt {-c}}{2 \, c}\right ) - 5 \, a \sqrt {-c} x \arctan \left (\frac {\sqrt {-a c x + c} \sqrt {-c}}{c}\right ) - \sqrt {-a c x + c}}{x}\right ] \]
[1/2*(4*sqrt(2)*a*sqrt(c)*x*log((a*c*x + 2*sqrt(2)*sqrt(-a*c*x + c)*sqrt(c ) - 3*c)/(a*x + 1)) + 5*a*sqrt(c)*x*log((a*c*x - 2*sqrt(-a*c*x + c)*sqrt(c ) - 2*c)/x) - 2*sqrt(-a*c*x + c))/x, (4*sqrt(2)*a*sqrt(-c)*x*arctan(1/2*sq rt(2)*sqrt(-a*c*x + c)*sqrt(-c)/c) - 5*a*sqrt(-c)*x*arctan(sqrt(-a*c*x + c )*sqrt(-c)/c) - sqrt(-a*c*x + c))/x]
\[ \int \frac {e^{-2 \text {arctanh}(a x)} \sqrt {c-a c x}}{x^2} \, dx=- \int \left (- \frac {\sqrt {- a c x + c}}{a x^{3} + x^{2}}\right )\, dx - \int \frac {a x \sqrt {- a c x + c}}{a x^{3} + x^{2}}\, dx \]
-Integral(-sqrt(-a*c*x + c)/(a*x**3 + x**2), x) - Integral(a*x*sqrt(-a*c*x + c)/(a*x**3 + x**2), x)
Time = 0.28 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.41 \[ \int \frac {e^{-2 \text {arctanh}(a x)} \sqrt {c-a c x}}{x^2} \, dx=\frac {1}{2} \, a c {\left (\frac {4 \, \sqrt {2} \log \left (-\frac {\sqrt {2} \sqrt {c} - \sqrt {-a c x + c}}{\sqrt {2} \sqrt {c} + \sqrt {-a c x + c}}\right )}{\sqrt {c}} - \frac {5 \, \log \left (\frac {\sqrt {-a c x + c} - \sqrt {c}}{\sqrt {-a c x + c} + \sqrt {c}}\right )}{\sqrt {c}} - \frac {2 \, \sqrt {-a c x + c}}{a c x}\right )} \]
1/2*a*c*(4*sqrt(2)*log(-(sqrt(2)*sqrt(c) - sqrt(-a*c*x + c))/(sqrt(2)*sqrt (c) + sqrt(-a*c*x + c)))/sqrt(c) - 5*log((sqrt(-a*c*x + c) - sqrt(c))/(sqr t(-a*c*x + c) + sqrt(c)))/sqrt(c) - 2*sqrt(-a*c*x + c)/(a*c*x))
Time = 0.28 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.91 \[ \int \frac {e^{-2 \text {arctanh}(a x)} \sqrt {c-a c x}}{x^2} \, dx=\frac {4 \, \sqrt {2} a c \arctan \left (\frac {\sqrt {2} \sqrt {-a c x + c}}{2 \, \sqrt {-c}}\right )}{\sqrt {-c}} - \frac {5 \, a c \arctan \left (\frac {\sqrt {-a c x + c}}{\sqrt {-c}}\right )}{\sqrt {-c}} - \frac {\sqrt {-a c x + c}}{x} \]
4*sqrt(2)*a*c*arctan(1/2*sqrt(2)*sqrt(-a*c*x + c)/sqrt(-c))/sqrt(-c) - 5*a *c*arctan(sqrt(-a*c*x + c)/sqrt(-c))/sqrt(-c) - sqrt(-a*c*x + c)/x
Time = 0.13 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.78 \[ \int \frac {e^{-2 \text {arctanh}(a x)} \sqrt {c-a c x}}{x^2} \, dx=5\,a\,\sqrt {c}\,\mathrm {atanh}\left (\frac {\sqrt {c-a\,c\,x}}{\sqrt {c}}\right )-\frac {\sqrt {c-a\,c\,x}}{x}-4\,\sqrt {2}\,a\,\sqrt {c}\,\mathrm {atanh}\left (\frac {\sqrt {2}\,\sqrt {c-a\,c\,x}}{2\,\sqrt {c}}\right ) \]