Integrand size = 23, antiderivative size = 119 \[ \int \frac {e^{3 \text {arctanh}(a x)} \sqrt {c-a c x}}{x} \, dx=-\frac {2 \sqrt {1+a x} (c-a c x)^{3/2}}{c (1-a x)^{3/2}}-\frac {2 (c-a c x)^{3/2} \text {arctanh}\left (\sqrt {1+a x}\right )}{c (1-a x)^{3/2}}+\frac {4 \sqrt {2} (c-a c x)^{3/2} \text {arctanh}\left (\frac {\sqrt {1+a x}}{\sqrt {2}}\right )}{c (1-a x)^{3/2}} \]
-2*(-a*c*x+c)^(3/2)*arctanh((a*x+1)^(1/2))/c/(-a*x+1)^(3/2)+4*(-a*c*x+c)^( 3/2)*arctanh(1/2*(a*x+1)^(1/2)*2^(1/2))*2^(1/2)/c/(-a*x+1)^(3/2)-2*(-a*c*x +c)^(3/2)*(a*x+1)^(1/2)/c/(-a*x+1)^(3/2)
Time = 0.03 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.55 \[ \int \frac {e^{3 \text {arctanh}(a x)} \sqrt {c-a c x}}{x} \, dx=-\frac {2 \sqrt {c-a c x} \left (\sqrt {1+a x}+\text {arctanh}\left (\sqrt {1+a x}\right )-2 \sqrt {2} \text {arctanh}\left (\frac {\sqrt {1+a x}}{\sqrt {2}}\right )\right )}{\sqrt {1-a x}} \]
(-2*Sqrt[c - a*c*x]*(Sqrt[1 + a*x] + ArcTanh[Sqrt[1 + a*x]] - 2*Sqrt[2]*Ar cTanh[Sqrt[1 + a*x]/Sqrt[2]]))/Sqrt[1 - a*x]
Time = 0.31 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.58, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.391, Rules used = {6680, 37, 95, 25, 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.
\(\displaystyle \int \frac {e^{3 \text {arctanh}(a x)} \sqrt {c-a c x}}{x} \, dx\) |
\(\Big \downarrow \) 6680 |
\(\displaystyle \int \frac {(a x+1)^{3/2} \sqrt {c-a c x}}{x (1-a x)^{3/2}}dx\) |
\(\Big \downarrow \) 37 |
\(\displaystyle \frac {\sqrt {c-a c x} \int \frac {(a x+1)^{3/2}}{x (1-a x)}dx}{\sqrt {1-a x}}\) |
\(\Big \downarrow \) 95 |
\(\displaystyle \frac {\sqrt {c-a c x} \left (-\frac {\int -\frac {a (3 a x+1)}{x (1-a x) \sqrt {a x+1}}dx}{a}-2 \sqrt {a x+1}\right )}{\sqrt {1-a x}}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {\sqrt {c-a c x} \left (\frac {\int \frac {a (3 a x+1)}{x (1-a x) \sqrt {a x+1}}dx}{a}-2 \sqrt {a x+1}\right )}{\sqrt {1-a x}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\sqrt {c-a c x} \left (\int \frac {3 a x+1}{x (1-a x) \sqrt {a x+1}}dx-2 \sqrt {a x+1}\right )}{\sqrt {1-a x}}\) |
\(\Big \downarrow \) 174 |
\(\displaystyle \frac {\sqrt {c-a c x} \left (\int \frac {1}{x \sqrt {a x+1}}dx+4 a \int \frac {1}{(1-a x) \sqrt {a x+1}}dx-2 \sqrt {a x+1}\right )}{\sqrt {1-a x}}\) |
\(\Big \downarrow \) 73 |
\(\displaystyle \frac {\sqrt {c-a c x} \left (8 \int \frac {1}{1-a x}d\sqrt {a x+1}+\frac {2 \int \frac {1}{\frac {a x+1}{a}-\frac {1}{a}}d\sqrt {a x+1}}{a}-2 \sqrt {a x+1}\right )}{\sqrt {1-a x}}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {\sqrt {c-a c x} \left (\frac {2 \int \frac {1}{\frac {a x+1}{a}-\frac {1}{a}}d\sqrt {a x+1}}{a}+4 \sqrt {2} \text {arctanh}\left (\frac {\sqrt {a x+1}}{\sqrt {2}}\right )-2 \sqrt {a x+1}\right )}{\sqrt {1-a x}}\) |
\(\Big \downarrow \) 221 |
\(\displaystyle \frac {\left (-2 \text {arctanh}\left (\sqrt {a x+1}\right )+4 \sqrt {2} \text {arctanh}\left (\frac {\sqrt {a x+1}}{\sqrt {2}}\right )-2 \sqrt {a x+1}\right ) \sqrt {c-a c x}}{\sqrt {1-a x}}\) |
(Sqrt[c - a*c*x]*(-2*Sqrt[1 + a*x] - 2*ArcTanh[Sqrt[1 + a*x]] + 4*Sqrt[2]* ArcTanh[Sqrt[1 + a*x]/Sqrt[2]]))/Sqrt[1 - a*x]
3.5.6.3.1 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] :> S imp[(a + b*x)^m/(c + d*x)^m Int[u*(c + d*x)^(m + n), x], x] /; FreeQ[{a, b, c, d, m, n}, x] && EqQ[b*c - a*d, 0] && !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 = 98, normalized size of antiderivative = 0.82
method | result | size |
default | \(-\frac {2 \sqrt {-a^{2} x^{2}+1}\, \sqrt {-c \left (a x -1\right )}\, \left (2 \sqrt {c}\, \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {c \left (a x +1\right )}\, \sqrt {2}}{2 \sqrt {c}}\right )-\sqrt {c}\, \operatorname {arctanh}\left (\frac {\sqrt {c \left (a x +1\right )}}{\sqrt {c}}\right )-\sqrt {c \left (a x +1\right )}\right )}{\left (a x -1\right ) \sqrt {c \left (a x +1\right )}}\) | \(98\) |
-2*(-a^2*x^2+1)^(1/2)*(-c*(a*x-1))^(1/2)*(2*c^(1/2)*2^(1/2)*arctanh(1/2*(c *(a*x+1))^(1/2)*2^(1/2)/c^(1/2))-c^(1/2)*arctanh((c*(a*x+1))^(1/2)/c^(1/2) )-(c*(a*x+1))^(1/2))/(a*x-1)/(c*(a*x+1))^(1/2)
Time = 0.29 (sec) , antiderivative size = 321, normalized size of antiderivative = 2.70 \[ \int \frac {e^{3 \text {arctanh}(a x)} \sqrt {c-a c x}}{x} \, dx=\left [\frac {2 \, \sqrt {2} {\left (a x - 1\right )} \sqrt {c} \log \left (-\frac {a^{2} c x^{2} + 2 \, a c x - 2 \, \sqrt {2} \sqrt {-a^{2} x^{2} + 1} \sqrt {-a c x + c} \sqrt {c} - 3 \, c}{a^{2} x^{2} - 2 \, a x + 1}\right ) + {\left (a x - 1\right )} \sqrt {c} \log \left (-\frac {a^{2} c x^{2} + a c x + 2 \, \sqrt {-a^{2} x^{2} + 1} \sqrt {-a c x + c} \sqrt {c} - 2 \, c}{a x^{2} - x}\right ) + 2 \, \sqrt {-a^{2} x^{2} + 1} \sqrt {-a c x + c}}{a x - 1}, \frac {2 \, {\left (2 \, \sqrt {2} {\left (a x - 1\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {2} \sqrt {-a^{2} x^{2} + 1} \sqrt {-a c x + c} \sqrt {-c}}{a^{2} c x^{2} - c}\right ) - {\left (a x - 1\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} \sqrt {-a c x + c} \sqrt {-c}}{a^{2} c x^{2} - c}\right ) + \sqrt {-a^{2} x^{2} + 1} \sqrt {-a c x + c}\right )}}{a x - 1}\right ] \]
[(2*sqrt(2)*(a*x - 1)*sqrt(c)*log(-(a^2*c*x^2 + 2*a*c*x - 2*sqrt(2)*sqrt(- a^2*x^2 + 1)*sqrt(-a*c*x + c)*sqrt(c) - 3*c)/(a^2*x^2 - 2*a*x + 1)) + (a*x - 1)*sqrt(c)*log(-(a^2*c*x^2 + a*c*x + 2*sqrt(-a^2*x^2 + 1)*sqrt(-a*c*x + c)*sqrt(c) - 2*c)/(a*x^2 - x)) + 2*sqrt(-a^2*x^2 + 1)*sqrt(-a*c*x + c))/( a*x - 1), 2*(2*sqrt(2)*(a*x - 1)*sqrt(-c)*arctan(sqrt(2)*sqrt(-a^2*x^2 + 1 )*sqrt(-a*c*x + c)*sqrt(-c)/(a^2*c*x^2 - c)) - (a*x - 1)*sqrt(-c)*arctan(s qrt(-a^2*x^2 + 1)*sqrt(-a*c*x + c)*sqrt(-c)/(a^2*c*x^2 - c)) + sqrt(-a^2*x ^2 + 1)*sqrt(-a*c*x + c))/(a*x - 1)]
\[ \int \frac {e^{3 \text {arctanh}(a x)} \sqrt {c-a c x}}{x} \, dx=\int \frac {\sqrt {- c \left (a x - 1\right )} \left (a x + 1\right )^{3}}{x \left (- \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {3}{2}}}\, dx \]
\[ \int \frac {e^{3 \text {arctanh}(a x)} \sqrt {c-a c x}}{x} \, dx=\int { \frac {\sqrt {-a c x + c} {\left (a x + 1\right )}^{3}}{{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} x} \,d x } \]
Exception generated. \[ \int \frac {e^{3 \text {arctanh}(a x)} \sqrt {c-a c x}}{x} \, dx=\text {Exception raised: TypeError} \]
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. \[ \int \frac {e^{3 \text {arctanh}(a x)} \sqrt {c-a c x}}{x} \, dx=\int \frac {\sqrt {c-a\,c\,x}\,{\left (a\,x+1\right )}^3}{x\,{\left (1-a^2\,x^2\right )}^{3/2}} \,d x \]