Integrand size = 24, antiderivative size = 93 \[ \int e^{-2 \text {arctanh}(a x)} \sqrt {c-\frac {c}{a x}} \, dx=-\sqrt {c-\frac {c}{a x}} x+\frac {5 \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {c}}\right )}{a}-\frac {4 \sqrt {2} \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {2} \sqrt {c}}\right )}{a} \]
5*arctanh((c-c/a/x)^(1/2)/c^(1/2))*c^(1/2)/a-4*arctanh(1/2*(c-c/a/x)^(1/2) *2^(1/2)/c^(1/2))*2^(1/2)*c^(1/2)/a-x*(c-c/a/x)^(1/2)
Time = 0.04 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.00 \[ \int e^{-2 \text {arctanh}(a x)} \sqrt {c-\frac {c}{a x}} \, dx=-\sqrt {c-\frac {c}{a x}} x+\frac {5 \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {c}}\right )}{a}-\frac {4 \sqrt {2} \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {2} \sqrt {c}}\right )}{a} \]
-(Sqrt[c - c/(a*x)]*x) + (5*Sqrt[c]*ArcTanh[Sqrt[c - c/(a*x)]/Sqrt[c]])/a - (4*Sqrt[2]*Sqrt[c]*ArcTanh[Sqrt[c - c/(a*x)]/(Sqrt[2]*Sqrt[c])])/a
Time = 0.38 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.15, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {6683, 1035, 281, 899, 109, 27, 174, 73, 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.
| \(\displaystyle \int e^{-2 \text {arctanh}(a x)} \sqrt {c-\frac {c}{a x}} \, dx\) |
| \(\Big \downarrow \) 6683 |
| \(\displaystyle \int \frac {(1-a x) \sqrt {c-\frac {c}{a x}}}{a x+1}dx\) |
| \(\Big \downarrow \) 1035 |
| \(\displaystyle \int \frac {\left (\frac {1}{x}-a\right ) \sqrt {c-\frac {c}{a x}}}{a+\frac {1}{x}}dx\) |
| \(\Big \downarrow \) 281 |
| \(\displaystyle -\frac {a \int \frac {\left (c-\frac {c}{a x}\right )^{3/2}}{a+\frac {1}{x}}dx}{c}\) |
| \(\Big \downarrow \) 899 |
| \(\displaystyle \frac {a \int \frac {\left (c-\frac {c}{a x}\right )^{3/2} x^2}{a+\frac {1}{x}}d\frac {1}{x}}{c}\) |
| \(\Big \downarrow \) 109 |
| \(\displaystyle \frac {a \left (-\frac {\int \frac {c^2 \left (5 a-\frac {3}{x}\right ) x}{2 a \left (a+\frac {1}{x}\right ) \sqrt {c-\frac {c}{a x}}}d\frac {1}{x}}{a}-\frac {c x \sqrt {c-\frac {c}{a x}}}{a}\right )}{c}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {a \left (-\frac {c^2 \int \frac {\left (5 a-\frac {3}{x}\right ) x}{\left (a+\frac {1}{x}\right ) \sqrt {c-\frac {c}{a x}}}d\frac {1}{x}}{2 a^2}-\frac {c x \sqrt {c-\frac {c}{a x}}}{a}\right )}{c}\) |
| \(\Big \downarrow \) 174 |
| \(\displaystyle \frac {a \left (-\frac {c^2 \left (5 \int \frac {x}{\sqrt {c-\frac {c}{a x}}}d\frac {1}{x}-8 \int \frac {1}{\left (a+\frac {1}{x}\right ) \sqrt {c-\frac {c}{a x}}}d\frac {1}{x}\right )}{2 a^2}-\frac {c x \sqrt {c-\frac {c}{a x}}}{a}\right )}{c}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {a \left (-\frac {c^2 \left (\frac {16 a \int \frac {1}{2 a-\frac {a}{c x^2}}d\sqrt {c-\frac {c}{a x}}}{c}-\frac {10 a \int \frac {1}{a-\frac {a}{c x^2}}d\sqrt {c-\frac {c}{a x}}}{c}\right )}{2 a^2}-\frac {c x \sqrt {c-\frac {c}{a x}}}{a}\right )}{c}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {a \left (-\frac {c^2 \left (\frac {8 \sqrt {2} \text {arctanh}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {2} \sqrt {c}}\right )}{\sqrt {c}}-\frac {10 \text {arctanh}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {c}}\right )}{\sqrt {c}}\right )}{2 a^2}-\frac {c x \sqrt {c-\frac {c}{a x}}}{a}\right )}{c}\) |
(a*(-((c*Sqrt[c - c/(a*x)]*x)/a) - (c^2*((-10*ArcTanh[Sqrt[c - c/(a*x)]/Sq rt[c]])/Sqrt[c] + (8*Sqrt[2]*ArcTanh[Sqrt[c - c/(a*x)]/(Sqrt[2]*Sqrt[c])]) /Sqrt[c]))/(2*a^2)))/c
3.6.48.3.1 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[((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[(Rt[-a/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
Int[(u_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_ Symbol] :> Simp[(b/d)^p Int[u*(c + d*x^n)^(p + q), x], x] /; FreeQ[{a, b, c, d, n, p, q}, x] && EqQ[b*c - a*d, 0] && IntegerQ[p] && !(IntegerQ[q] & & SimplerQ[a + b*x^n, c + d*x^n])
Int[((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol ] :> -Subst[Int[(a + b/x^n)^p*((c + d/x^n)^q/x^2), x], x, 1/x] /; FreeQ[{a, b, c, d, p, q}, x] && NeQ[b*c - a*d, 0] && ILtQ[n, 0]
Int[((c_) + (d_.)*(x_)^(mn_.))^(q_.)*((a_.) + (b_.)*(x_)^(n_.))^(p_.)*((e_) + (f_.)*(x_)^(n_.))^(r_.), x_Symbol] :> Int[x^(n*(p + r))*(b + a/x^n)^p*(c + d/x^n)^q*(f + e/x^n)^r, x] /; FreeQ[{a, b, c, d, e, f, n, q}, x] && EqQ[ mn, -n] && IntegerQ[p] && IntegerQ[r]
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, p}, x] && EqQ[c^2 - a^2*d^2, 0] && !IntegerQ[p] && IntegerQ[n/2] && !G tQ[c, 0]
Leaf count of result is larger than twice the leaf count of optimal. \(169\) vs. \(2(76)=152\).
Time = 0.11 (sec) , antiderivative size = 170, normalized size of antiderivative = 1.83
| method | result | size |
| risch | \(-x \sqrt {\frac {c \left (a x -1\right )}{a x}}-\frac {\left (-\frac {5 \ln \left (\frac {-\frac {1}{2} a c +a^{2} c x}{\sqrt {a^{2} c}}+\sqrt {a^{2} c \,x^{2}-a c x}\right )}{2 \sqrt {a^{2} c}}-\frac {2 \sqrt {2}\, \ln \left (\frac {4 c -3 \left (x +\frac {1}{a}\right ) a c +2 \sqrt {2}\, \sqrt {c}\, \sqrt {a^{2} c \left (x +\frac {1}{a}\right )^{2}-3 \left (x +\frac {1}{a}\right ) a c +2 c}}{x +\frac {1}{a}}\right )}{a \sqrt {c}}\right ) \sqrt {c \left (a x -1\right ) a x}\, \sqrt {\frac {c \left (a x -1\right )}{a x}}}{a x -1}\) | \(170\) |
| default | \(-\frac {\sqrt {\frac {c \left (a x -1\right )}{a x}}\, x \left (4 \sqrt {\left (a x -1\right ) x}\, a^{\frac {3}{2}} \sqrt {\frac {1}{a}}-2 \sqrt {a \,x^{2}-x}\, a^{\frac {3}{2}} \sqrt {\frac {1}{a}}+\ln \left (\frac {2 \sqrt {a \,x^{2}-x}\, \sqrt {a}+2 a x -1}{2 \sqrt {a}}\right ) a \sqrt {\frac {1}{a}}-4 \sqrt {2}\, \ln \left (\frac {2 \sqrt {2}\, \sqrt {\frac {1}{a}}\, \sqrt {\left (a x -1\right ) x}\, a -3 a x +1}{a x +1}\right ) \sqrt {a}-6 \ln \left (\frac {2 \sqrt {\left (a x -1\right ) x}\, \sqrt {a}+2 a x -1}{2 \sqrt {a}}\right ) a \sqrt {\frac {1}{a}}\right )}{2 \sqrt {\left (a x -1\right ) x}\, a^{\frac {3}{2}} \sqrt {\frac {1}{a}}}\) | \(189\) |
-x*(c*(a*x-1)/a/x)^(1/2)-(-5/2*ln((-1/2*a*c+a^2*c*x)/(a^2*c)^(1/2)+(a^2*c* x^2-a*c*x)^(1/2))/(a^2*c)^(1/2)-2/a*2^(1/2)/c^(1/2)*ln((4*c-3*(x+1/a)*a*c+ 2*2^(1/2)*c^(1/2)*(a^2*c*(x+1/a)^2-3*(x+1/a)*a*c+2*c)^(1/2))/(x+1/a)))*(c* (a*x-1)*a*x)^(1/2)*(c*(a*x-1)/a/x)^(1/2)/(a*x-1)
Time = 0.26 (sec) , antiderivative size = 217, normalized size of antiderivative = 2.33 \[ \int e^{-2 \text {arctanh}(a x)} \sqrt {c-\frac {c}{a x}} \, dx=\left [-\frac {2 \, a x \sqrt {\frac {a c x - c}{a x}} - 4 \, \sqrt {2} \sqrt {c} \log \left (\frac {2 \, \sqrt {2} a \sqrt {c} x \sqrt {\frac {a c x - c}{a x}} - 3 \, a c x + c}{a x + 1}\right ) - 5 \, \sqrt {c} \log \left (-2 \, a c x - 2 \, a \sqrt {c} x \sqrt {\frac {a c x - c}{a x}} + c\right )}{2 \, a}, -\frac {a x \sqrt {\frac {a c x - c}{a x}} - 4 \, \sqrt {2} \sqrt {-c} \arctan \left (\frac {\sqrt {2} \sqrt {-c} \sqrt {\frac {a c x - c}{a x}}}{2 \, c}\right ) + 5 \, \sqrt {-c} \arctan \left (\frac {\sqrt {-c} \sqrt {\frac {a c x - c}{a x}}}{c}\right )}{a}\right ] \]
[-1/2*(2*a*x*sqrt((a*c*x - c)/(a*x)) - 4*sqrt(2)*sqrt(c)*log((2*sqrt(2)*a* sqrt(c)*x*sqrt((a*c*x - c)/(a*x)) - 3*a*c*x + c)/(a*x + 1)) - 5*sqrt(c)*lo g(-2*a*c*x - 2*a*sqrt(c)*x*sqrt((a*c*x - c)/(a*x)) + c))/a, -(a*x*sqrt((a* c*x - c)/(a*x)) - 4*sqrt(2)*sqrt(-c)*arctan(1/2*sqrt(2)*sqrt(-c)*sqrt((a*c *x - c)/(a*x))/c) + 5*sqrt(-c)*arctan(sqrt(-c)*sqrt((a*c*x - c)/(a*x))/c)) /a]
\[ \int e^{-2 \text {arctanh}(a x)} \sqrt {c-\frac {c}{a x}} \, dx=- \int \left (- \frac {\sqrt {c - \frac {c}{a x}}}{a x + 1}\right )\, dx - \int \frac {a x \sqrt {c - \frac {c}{a x}}}{a x + 1}\, dx \]
\[ \int e^{-2 \text {arctanh}(a x)} \sqrt {c-\frac {c}{a x}} \, dx=\int { -\frac {{\left (a^{2} x^{2} - 1\right )} \sqrt {c - \frac {c}{a x}}}{{\left (a x + 1\right )}^{2}} \,d x } \]
Exception generated. \[ \int e^{-2 \text {arctanh}(a x)} \sqrt {c-\frac {c}{a x}} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Error: Bad Argument Type
Timed out. \[ \int e^{-2 \text {arctanh}(a x)} \sqrt {c-\frac {c}{a x}} \, dx=-\int \frac {\sqrt {c-\frac {c}{a\,x}}\,\left (a^2\,x^2-1\right )}{{\left (a\,x+1\right )}^2} \,d x \]