Integrand size = 27, antiderivative size = 147 \[ \int \frac {e^{3 \text {arctanh}(a x)} \sqrt {c-\frac {c}{a x}}}{x^2} \, dx=-\frac {4 a \sqrt {c-\frac {c}{a x}} \sqrt {1+a x}}{\sqrt {1-a x}}-\frac {2 \sqrt {c-\frac {c}{a x}} (1+a x)^{3/2}}{3 x \sqrt {1-a x}}+\frac {4 \sqrt {2} a^{3/2} \sqrt {c-\frac {c}{a x}} \sqrt {x} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {x}}{\sqrt {1+a x}}\right )}{\sqrt {1-a x}} \]
-2/3*(a*x+1)^(3/2)*(c-c/a/x)^(1/2)/x/(-a*x+1)^(1/2)+4*a^(3/2)*arctanh(2^(1 /2)*a^(1/2)*x^(1/2)/(a*x+1)^(1/2))*2^(1/2)*(c-c/a/x)^(1/2)*x^(1/2)/(-a*x+1 )^(1/2)-4*a*(c-c/a/x)^(1/2)*(a*x+1)^(1/2)/(-a*x+1)^(1/2)
Time = 0.04 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.63 \[ \int \frac {e^{3 \text {arctanh}(a x)} \sqrt {c-\frac {c}{a x}}}{x^2} \, dx=\frac {2 \sqrt {c-\frac {c}{a x}} \left (-\sqrt {1+a x} (1+7 a x)+6 \sqrt {2} a^{3/2} x^{3/2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {x}}{\sqrt {1+a x}}\right )\right )}{3 x \sqrt {1-a x}} \]
(2*Sqrt[c - c/(a*x)]*(-(Sqrt[1 + a*x]*(1 + 7*a*x)) + 6*Sqrt[2]*a^(3/2)*x^( 3/2)*ArcTanh[(Sqrt[2]*Sqrt[a]*Sqrt[x])/Sqrt[1 + a*x]]))/(3*x*Sqrt[1 - a*x] )
Time = 0.47 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.73, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {6684, 6679, 105, 105, 104, 219}
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.
\(\displaystyle \int \frac {e^{3 \text {arctanh}(a x)} \sqrt {c-\frac {c}{a x}}}{x^2} \, dx\) |
\(\Big \downarrow \) 6684 |
\(\displaystyle \frac {\sqrt {x} \sqrt {c-\frac {c}{a x}} \int \frac {e^{3 \text {arctanh}(a x)} \sqrt {1-a x}}{x^{5/2}}dx}{\sqrt {1-a x}}\) |
\(\Big \downarrow \) 6679 |
\(\displaystyle \frac {\sqrt {x} \sqrt {c-\frac {c}{a x}} \int \frac {(a x+1)^{3/2}}{x^{5/2} (1-a x)}dx}{\sqrt {1-a x}}\) |
\(\Big \downarrow \) 105 |
\(\displaystyle \frac {\sqrt {x} \sqrt {c-\frac {c}{a x}} \left (2 a \int \frac {\sqrt {a x+1}}{x^{3/2} (1-a x)}dx-\frac {2 (a x+1)^{3/2}}{3 x^{3/2}}\right )}{\sqrt {1-a x}}\) |
\(\Big \downarrow \) 105 |
\(\displaystyle \frac {\sqrt {x} \sqrt {c-\frac {c}{a x}} \left (2 a \left (2 a \int \frac {1}{\sqrt {x} (1-a x) \sqrt {a x+1}}dx-\frac {2 \sqrt {a x+1}}{\sqrt {x}}\right )-\frac {2 (a x+1)^{3/2}}{3 x^{3/2}}\right )}{\sqrt {1-a x}}\) |
\(\Big \downarrow \) 104 |
\(\displaystyle \frac {\sqrt {x} \sqrt {c-\frac {c}{a x}} \left (2 a \left (4 a \int \frac {1}{1-\frac {2 a x}{a x+1}}d\frac {\sqrt {x}}{\sqrt {a x+1}}-\frac {2 \sqrt {a x+1}}{\sqrt {x}}\right )-\frac {2 (a x+1)^{3/2}}{3 x^{3/2}}\right )}{\sqrt {1-a x}}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {\sqrt {x} \left (2 a \left (2 \sqrt {2} \sqrt {a} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {x}}{\sqrt {a x+1}}\right )-\frac {2 \sqrt {a x+1}}{\sqrt {x}}\right )-\frac {2 (a x+1)^{3/2}}{3 x^{3/2}}\right ) \sqrt {c-\frac {c}{a x}}}{\sqrt {1-a x}}\) |
(Sqrt[c - c/(a*x)]*Sqrt[x]*((-2*(1 + a*x)^(3/2))/(3*x^(3/2)) + 2*a*((-2*Sq rt[1 + a*x])/Sqrt[x] + 2*Sqrt[2]*Sqrt[a]*ArcTanh[(Sqrt[2]*Sqrt[a]*Sqrt[x]) /Sqrt[1 + a*x]])))/Sqrt[1 - a*x]
3.6.87.3.1 Defintions of rubi rules used
Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x _)), x_] :> With[{q = Denominator[m]}, Simp[q Subst[Int[x^(q*(m + 1) - 1) /(b*e - a*f - (d*e - c*f)*x^q), x], x, (a + b*x)^(1/q)/(c + d*x)^(1/q)], x] ] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[m + n + 1, 0] && RationalQ[n] && L tQ[-1, m, 0] && SimplerQ[a + b*x, c + d*x]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Simp[(a + b*x)^(m + 1)*(c + d*x)^n*((e + f*x)^(p + 1)/((m + 1)*(b*e - a*f))), x] - Simp[n*((d*e - c*f)/((m + 1)*(b*e - a*f))) Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, p}, x] && EqQ[m + n + p + 2, 0] && GtQ[n, 0] && (SumSimplerQ[m, 1] || !SumSimplerQ[p, 1]) && NeQ[m, -1]
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[E^(ArcTanh[(a_.)*(x_)]*(n_.))*(u_.)*((c_) + (d_.)*(x_))^(p_.), x_Symbol ] :> Simp[c^p Int[u*(1 + d*(x/c))^p*((1 + a*x)^(n/2)/(1 - a*x)^(n/2)), x] , x] /; FreeQ[{a, c, d, n, p}, x] && EqQ[a^2*c^2 - d^2, 0] && (IntegerQ[p] || GtQ[c, 0])
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 = 0.12 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.02
method | result | size |
default | \(\frac {\sqrt {\frac {c \left (a x -1\right )}{a x}}\, \sqrt {-a^{2} x^{2}+1}\, \left (7 a \sqrt {2}\, \sqrt {-\frac {1}{a}}\, x \sqrt {-\left (a x +1\right ) x}+6 a \ln \left (\frac {2 \sqrt {2}\, \sqrt {-\frac {1}{a}}\, \sqrt {-\left (a x +1\right ) x}\, a -3 a x -1}{a x -1}\right ) x^{2}+\sqrt {2}\, \sqrt {-\frac {1}{a}}\, \sqrt {-\left (a x +1\right ) x}\right ) \sqrt {2}}{3 x \left (a x -1\right ) \sqrt {-\left (a x +1\right ) x}\, \sqrt {-\frac {1}{a}}}\) | \(150\) |
risch | \(-\frac {2 \left (7 a^{2} x^{2}+8 a x +1\right ) \sqrt {\frac {c \left (a x -1\right )}{a x}}\, \sqrt {\frac {c a x \left (-a^{2} x^{2}+1\right )}{a x -1}}}{3 x \sqrt {-\left (a x +1\right ) a c x}\, \sqrt {-a^{2} x^{2}+1}}+\frac {4 a \ln \left (\frac {-4 c -3 \left (x -\frac {1}{a}\right ) a c +2 \sqrt {-2 c}\, \sqrt {-a^{2} c \left (x -\frac {1}{a}\right )^{2}-3 \left (x -\frac {1}{a}\right ) a c -2 c}}{x -\frac {1}{a}}\right ) \sqrt {\frac {c \left (a x -1\right )}{a x}}\, \sqrt {\frac {c a x \left (-a^{2} x^{2}+1\right )}{a x -1}}}{\sqrt {-2 c}\, \sqrt {-a^{2} x^{2}+1}}\) | \(206\) |
1/3*(c*(a*x-1)/a/x)^(1/2)/x*(-a^2*x^2+1)^(1/2)*(7*a*2^(1/2)*(-1/a)^(1/2)*x *(-(a*x+1)*x)^(1/2)+6*a*ln((2*2^(1/2)*(-1/a)^(1/2)*(-(a*x+1)*x)^(1/2)*a-3* a*x-1)/(a*x-1))*x^2+2^(1/2)*(-1/a)^(1/2)*(-(a*x+1)*x)^(1/2))*2^(1/2)/(a*x- 1)/(-(a*x+1)*x)^(1/2)/(-1/a)^(1/2)
Time = 0.29 (sec) , antiderivative size = 309, normalized size of antiderivative = 2.10 \[ \int \frac {e^{3 \text {arctanh}(a x)} \sqrt {c-\frac {c}{a x}}}{x^2} \, dx=\left [\frac {3 \, \sqrt {2} {\left (a^{2} x^{2} - a x\right )} \sqrt {-c} \log \left (-\frac {17 \, a^{3} c x^{3} - 3 \, a^{2} c x^{2} - 13 \, a c x + 4 \, \sqrt {2} {\left (3 \, a^{2} x^{2} + a x\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {-c} \sqrt {\frac {a c x - c}{a x}} - c}{a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1}\right ) + 2 \, \sqrt {-a^{2} x^{2} + 1} {\left (7 \, a x + 1\right )} \sqrt {\frac {a c x - c}{a x}}}{3 \, {\left (a x^{2} - x\right )}}, -\frac {2 \, {\left (3 \, \sqrt {2} {\left (a^{2} x^{2} - a x\right )} \sqrt {c} \arctan \left (\frac {2 \, \sqrt {2} \sqrt {-a^{2} x^{2} + 1} a \sqrt {c} x \sqrt {\frac {a c x - c}{a x}}}{3 \, a^{2} c x^{2} - 2 \, a c x - c}\right ) - \sqrt {-a^{2} x^{2} + 1} {\left (7 \, a x + 1\right )} \sqrt {\frac {a c x - c}{a x}}\right )}}{3 \, {\left (a x^{2} - x\right )}}\right ] \]
[1/3*(3*sqrt(2)*(a^2*x^2 - a*x)*sqrt(-c)*log(-(17*a^3*c*x^3 - 3*a^2*c*x^2 - 13*a*c*x + 4*sqrt(2)*(3*a^2*x^2 + a*x)*sqrt(-a^2*x^2 + 1)*sqrt(-c)*sqrt( (a*c*x - c)/(a*x)) - c)/(a^3*x^3 - 3*a^2*x^2 + 3*a*x - 1)) + 2*sqrt(-a^2*x ^2 + 1)*(7*a*x + 1)*sqrt((a*c*x - c)/(a*x)))/(a*x^2 - x), -2/3*(3*sqrt(2)* (a^2*x^2 - a*x)*sqrt(c)*arctan(2*sqrt(2)*sqrt(-a^2*x^2 + 1)*a*sqrt(c)*x*sq rt((a*c*x - c)/(a*x))/(3*a^2*c*x^2 - 2*a*c*x - c)) - sqrt(-a^2*x^2 + 1)*(7 *a*x + 1)*sqrt((a*c*x - c)/(a*x)))/(a*x^2 - x)]
\[ \int \frac {e^{3 \text {arctanh}(a x)} \sqrt {c-\frac {c}{a x}}}{x^2} \, dx=\int \frac {\sqrt {- c \left (-1 + \frac {1}{a x}\right )} \left (a x + 1\right )^{3}}{x^{2} \left (- \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {3}{2}}}\, dx \]
\[ \int \frac {e^{3 \text {arctanh}(a x)} \sqrt {c-\frac {c}{a x}}}{x^2} \, dx=\int { \frac {{\left (a x + 1\right )}^{3} \sqrt {c - \frac {c}{a x}}}{{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} x^{2}} \,d x } \]
Exception generated. \[ \int \frac {e^{3 \text {arctanh}(a x)} \sqrt {c-\frac {c}{a x}}}{x^2} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value
Timed out. \[ \int \frac {e^{3 \text {arctanh}(a x)} \sqrt {c-\frac {c}{a x}}}{x^2} \, dx=\int \frac {\sqrt {c-\frac {c}{a\,x}}\,{\left (a\,x+1\right )}^3}{x^2\,{\left (1-a^2\,x^2\right )}^{3/2}} \,d x \]