Integrand size = 23, antiderivative size = 163 \[ \int \frac {e^{-3 \text {arctanh}(a x)} \sqrt {c-a c x}}{x^3} \, dx=\frac {47 a^2 c^2 (1-a x)^{3/2}}{4 \sqrt {1+a x} (c-a c x)^{3/2}}-\frac {c^2 (1-a x)^{3/2}}{2 x^2 \sqrt {1+a x} (c-a c x)^{3/2}}+\frac {13 a c^2 (1-a x)^{3/2}}{4 x \sqrt {1+a x} (c-a c x)^{3/2}}-\frac {47 a^2 c^2 (1-a x)^{3/2} \text {arctanh}\left (\sqrt {1+a x}\right )}{4 (c-a c x)^{3/2}} \]
-47/4*a^2*c^2*(-a*x+1)^(3/2)*arctanh((a*x+1)^(1/2))/(-a*c*x+c)^(3/2)+47/4* a^2*c^2*(-a*x+1)^(3/2)/(-a*c*x+c)^(3/2)/(a*x+1)^(1/2)-1/2*c^2*(-a*x+1)^(3/ 2)/x^2/(-a*c*x+c)^(3/2)/(a*x+1)^(1/2)+13/4*a*c^2*(-a*x+1)^(3/2)/x/(-a*c*x+ c)^(3/2)/(a*x+1)^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.02 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.40 \[ \int \frac {e^{-3 \text {arctanh}(a x)} \sqrt {c-a c x}}{x^3} \, dx=\frac {c \sqrt {1-a x} \left (-2+13 a x+47 a^2 x^2 \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},1,\frac {1}{2},1+a x\right )\right )}{4 x^2 \sqrt {1+a x} \sqrt {c-a c x}} \]
(c*Sqrt[1 - a*x]*(-2 + 13*a*x + 47*a^2*x^2*Hypergeometric2F1[-1/2, 1, 1/2, 1 + a*x]))/(4*x^2*Sqrt[1 + a*x]*Sqrt[c - a*c*x])
Time = 0.31 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.54, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.348, Rules used = {6680, 37, 100, 27, 87, 61, 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.
\(\displaystyle \int \frac {e^{-3 \text {arctanh}(a x)} \sqrt {c-a c x}}{x^3} \, dx\) |
\(\Big \downarrow \) 6680 |
\(\displaystyle \int \frac {(1-a x)^{3/2} \sqrt {c-a c x}}{x^3 (a x+1)^{3/2}}dx\) |
\(\Big \downarrow \) 37 |
\(\displaystyle \frac {\sqrt {c-a c x} \int \frac {(1-a x)^2}{x^3 (a x+1)^{3/2}}dx}{\sqrt {1-a x}}\) |
\(\Big \downarrow \) 100 |
\(\displaystyle \frac {\sqrt {c-a c x} \left (\frac {1}{2} \int -\frac {a (13-4 a x)}{2 x^2 (a x+1)^{3/2}}dx-\frac {1}{2 x^2 \sqrt {a x+1}}\right )}{\sqrt {1-a x}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\sqrt {c-a c x} \left (-\frac {1}{4} a \int \frac {13-4 a x}{x^2 (a x+1)^{3/2}}dx-\frac {1}{2 x^2 \sqrt {a x+1}}\right )}{\sqrt {1-a x}}\) |
\(\Big \downarrow \) 87 |
\(\displaystyle \frac {\sqrt {c-a c x} \left (-\frac {1}{4} a \left (-\frac {47}{2} a \int \frac {1}{x (a x+1)^{3/2}}dx-\frac {13}{x \sqrt {a x+1}}\right )-\frac {1}{2 x^2 \sqrt {a x+1}}\right )}{\sqrt {1-a x}}\) |
\(\Big \downarrow \) 61 |
\(\displaystyle \frac {\sqrt {c-a c x} \left (-\frac {1}{4} a \left (-\frac {47}{2} a \left (\int \frac {1}{x \sqrt {a x+1}}dx+\frac {2}{\sqrt {a x+1}}\right )-\frac {13}{x \sqrt {a x+1}}\right )-\frac {1}{2 x^2 \sqrt {a x+1}}\right )}{\sqrt {1-a x}}\) |
\(\Big \downarrow \) 73 |
\(\displaystyle \frac {\sqrt {c-a c x} \left (-\frac {1}{4} a \left (-\frac {47}{2} a \left (\frac {2 \int \frac {1}{\frac {a x+1}{a}-\frac {1}{a}}d\sqrt {a x+1}}{a}+\frac {2}{\sqrt {a x+1}}\right )-\frac {13}{x \sqrt {a x+1}}\right )-\frac {1}{2 x^2 \sqrt {a x+1}}\right )}{\sqrt {1-a x}}\) |
\(\Big \downarrow \) 221 |
\(\displaystyle \frac {\left (-\frac {1}{4} a \left (-\frac {47}{2} a \left (\frac {2}{\sqrt {a x+1}}-2 \text {arctanh}\left (\sqrt {a x+1}\right )\right )-\frac {13}{x \sqrt {a x+1}}\right )-\frac {1}{2 x^2 \sqrt {a x+1}}\right ) \sqrt {c-a c x}}{\sqrt {1-a x}}\) |
(Sqrt[c - a*c*x]*(-1/2*1/(x^2*Sqrt[1 + a*x]) - (a*(-13/(x*Sqrt[1 + a*x]) - (47*a*(2/Sqrt[1 + a*x] - 2*ArcTanh[Sqrt[1 + a*x]]))/2))/4))/Sqrt[1 - a*x]
3.5.34.3.1 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] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((b*c - a*d)*(m + 1))), x] - Simp[d*(( m + n + 2)/((b*c - a*d)*(m + 1))) Int[(a + b*x)^(m + 1)*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && LtQ[m, -1] && !(LtQ[n, -1] && (EqQ[a, 0 ] || (NeQ[c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && IntLinearQ[a, b, c, d , m, n, 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_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p _.), x_] :> Simp[(-(b*e - a*f))*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(f*(p + 1)*(c*f - d*e))), x] - Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e)) Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || Intege rQ[p] || !(IntegerQ[n] || !(EqQ[e, 0] || !(EqQ[c, 0] || LtQ[p, n]))))
Int[((a_.) + (b_.)*(x_))^2*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^( p_), x_] :> Simp[(b*c - a*d)^2*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d^2*(d *e - c*f)*(n + 1))), x] - Simp[1/(d^2*(d*e - c*f)*(n + 1)) Int[(c + d*x)^ (n + 1)*(e + f*x)^p*Simp[a^2*d^2*f*(n + p + 2) + b^2*c*(d*e*(n + 1) + c*f*( p + 1)) - 2*a*b*d*(d*e*(n + 1) + c*f*(p + 1)) - b^2*d*(d*e - c*f)*(n + 1)*x , x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && (LtQ[n, -1] || (EqQ[n + p + 3, 0] && NeQ[n, -1] && (SumSimplerQ[n, 1] || !SumSimplerQ[p, 1])))
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 = 100, normalized size of antiderivative = 0.61
method | result | size |
default | \(\frac {\sqrt {-c \left (a x -1\right )}\, \sqrt {-a^{2} x^{2}+1}\, \left (47 \,\operatorname {arctanh}\left (\frac {\sqrt {c \left (a x +1\right )}}{\sqrt {c}}\right ) a^{2} x^{2} \sqrt {c \left (a x +1\right )}-47 \sqrt {c}\, a^{2} x^{2}-13 \sqrt {c}\, a x +2 \sqrt {c}\right )}{4 \sqrt {c}\, \left (a x -1\right ) \left (a x +1\right ) x^{2}}\) | \(100\) |
risch | \(-\frac {\left (15 a^{2} x^{2}+13 a x -2\right ) \sqrt {-\frac {\left (-a^{2} x^{2}+1\right ) c}{a x -1}}\, \left (a x -1\right ) c}{4 x^{2} \sqrt {c \left (a x +1\right )}\, \sqrt {-a^{2} x^{2}+1}\, \sqrt {-c \left (a x -1\right )}}-\frac {a^{2} \left (\frac {64}{\sqrt {a c x +c}}-\frac {94 \,\operatorname {arctanh}\left (\frac {\sqrt {a c x +c}}{\sqrt {c}}\right )}{\sqrt {c}}\right ) \sqrt {-\frac {\left (-a^{2} x^{2}+1\right ) c}{a x -1}}\, \left (a x -1\right ) c}{8 \sqrt {-a^{2} x^{2}+1}\, \sqrt {-c \left (a x -1\right )}}\) | \(164\) |
1/4*(-c*(a*x-1))^(1/2)*(-a^2*x^2+1)^(1/2)*(47*arctanh((c*(a*x+1))^(1/2)/c^ (1/2))*a^2*x^2*(c*(a*x+1))^(1/2)-47*c^(1/2)*a^2*x^2-13*c^(1/2)*a*x+2*c^(1/ 2))/c^(1/2)/(a*x-1)/(a*x+1)/x^2
Time = 0.27 (sec) , antiderivative size = 252, normalized size of antiderivative = 1.55 \[ \int \frac {e^{-3 \text {arctanh}(a x)} \sqrt {c-a c x}}{x^3} \, dx=\left [\frac {47 \, {\left (a^{4} x^{4} - a^{2} x^{2}\right )} \sqrt {c} \log \left (-\frac {a^{2} c x^{2} + a c x + 2 \, \sqrt {-a^{2} x^{2} + 1} \sqrt {-a c x + c} \sqrt {c} - 2 \, c}{a x^{2} - x}\right ) - 2 \, {\left (47 \, a^{2} x^{2} + 13 \, a x - 2\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {-a c x + c}}{8 \, {\left (a^{2} x^{4} - x^{2}\right )}}, -\frac {47 \, {\left (a^{4} x^{4} - a^{2} x^{2}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} \sqrt {-a c x + c} \sqrt {-c}}{a^{2} c x^{2} - c}\right ) + {\left (47 \, a^{2} x^{2} + 13 \, a x - 2\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {-a c x + c}}{4 \, {\left (a^{2} x^{4} - x^{2}\right )}}\right ] \]
[1/8*(47*(a^4*x^4 - a^2*x^2)*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*(47*a^2*x^2 + 1 3*a*x - 2)*sqrt(-a^2*x^2 + 1)*sqrt(-a*c*x + c))/(a^2*x^4 - x^2), -1/4*(47* (a^4*x^4 - a^2*x^2)*sqrt(-c)*arctan(sqrt(-a^2*x^2 + 1)*sqrt(-a*c*x + c)*sq rt(-c)/(a^2*c*x^2 - c)) + (47*a^2*x^2 + 13*a*x - 2)*sqrt(-a^2*x^2 + 1)*sqr t(-a*c*x + c))/(a^2*x^4 - x^2)]
\[ \int \frac {e^{-3 \text {arctanh}(a x)} \sqrt {c-a c x}}{x^3} \, dx=\int \frac {\sqrt {- c \left (a x - 1\right )} \left (- \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {3}{2}}}{x^{3} \left (a x + 1\right )^{3}}\, dx \]
\[ \int \frac {e^{-3 \text {arctanh}(a x)} \sqrt {c-a c x}}{x^3} \, dx=\int { \frac {{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} \sqrt {-a c x + c}}{{\left (a x + 1\right )}^{3} x^{3}} \,d x } \]
Exception generated. \[ \int \frac {e^{-3 \text {arctanh}(a x)} \sqrt {c-a c x}}{x^3} \, dx=\text {Exception raised: TypeError} \]
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. \[ \int \frac {e^{-3 \text {arctanh}(a x)} \sqrt {c-a c x}}{x^3} \, dx=\int \frac {{\left (1-a^2\,x^2\right )}^{3/2}\,\sqrt {c-a\,c\,x}}{x^3\,{\left (a\,x+1\right )}^3} \,d x \]