Integrand size = 23, antiderivative size = 238 \[ \int \frac {e^{-3 \coth ^{-1}(a x)} \sqrt {c-a c x}}{x^4} \, dx=-\frac {8 \sqrt {c-a c x}}{\sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}} x^3}-\frac {\sqrt {1+\frac {1}{a x}} \sqrt {c-a c x}}{3 \sqrt {1-\frac {1}{a x}} x^3}+\frac {119 a \sqrt {1+\frac {1}{a x}} \sqrt {c-a c x}}{12 \sqrt {1-\frac {1}{a x}} x^2}-\frac {119 a^2 \sqrt {1+\frac {1}{a x}} \sqrt {c-a c x}}{8 \sqrt {1-\frac {1}{a x}} x}+\frac {119 a^{5/2} \sqrt {\frac {1}{x}} \sqrt {c-a c x} \text {arcsinh}\left (\frac {\sqrt {\frac {1}{x}}}{\sqrt {a}}\right )}{8 \sqrt {1-\frac {1}{a x}}} \]
-8*(-a*c*x+c)^(1/2)/x^3/(1-1/a/x)^(1/2)/(1+1/a/x)^(1/2)-1/3*(1+1/a/x)^(1/2 )*(-a*c*x+c)^(1/2)/x^3/(1-1/a/x)^(1/2)+119/12*a*(1+1/a/x)^(1/2)*(-a*c*x+c) ^(1/2)/x^2/(1-1/a/x)^(1/2)-119/8*a^2*(1+1/a/x)^(1/2)*(-a*c*x+c)^(1/2)/x/(1 -1/a/x)^(1/2)+119/8*a^(5/2)*arcsinh((1/x)^(1/2)/a^(1/2))*(1/x)^(1/2)*(-a*c *x+c)^(1/2)/(1-1/a/x)^(1/2)
Time = 0.10 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.41 \[ \int \frac {e^{-3 \coth ^{-1}(a x)} \sqrt {c-a c x}}{x^4} \, dx=\frac {\sqrt {c-a c x} \left (-8+38 a x-119 a^2 x^2-357 a^3 x^3+\frac {357 a^{7/2} \sqrt {1+\frac {1}{a x}} \text {arcsinh}\left (\frac {\sqrt {\frac {1}{x}}}{\sqrt {a}}\right )}{\left (\frac {1}{x}\right )^{7/2}}\right )}{24 a \sqrt {1-\frac {1}{a^2 x^2}} x^4} \]
(Sqrt[c - a*c*x]*(-8 + 38*a*x - 119*a^2*x^2 - 357*a^3*x^3 + (357*a^(7/2)*S qrt[1 + 1/(a*x)]*ArcSinh[Sqrt[x^(-1)]/Sqrt[a]])/(x^(-1))^(7/2)))/(24*a*Sqr t[1 - 1/(a^2*x^2)]*x^4)
Time = 0.34 (sec) , antiderivative size = 172, normalized size of antiderivative = 0.72, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.391, Rules used = {6730, 27, 100, 27, 90, 60, 60, 63, 222}
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.
\(\displaystyle \int \frac {\sqrt {c-a c x} e^{-3 \coth ^{-1}(a x)}}{x^4} \, dx\) |
\(\Big \downarrow \) 6730 |
\(\displaystyle -\frac {\sqrt {\frac {1}{x}} \sqrt {c-a c x} \int \frac {\left (a-\frac {1}{x}\right )^2 \left (\frac {1}{x}\right )^{3/2}}{a^2 \left (1+\frac {1}{a x}\right )^{3/2}}d\frac {1}{x}}{\sqrt {1-\frac {1}{a x}}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {\sqrt {\frac {1}{x}} \sqrt {c-a c x} \int \frac {\left (a-\frac {1}{x}\right )^2 \left (\frac {1}{x}\right )^{3/2}}{\left (1+\frac {1}{a x}\right )^{3/2}}d\frac {1}{x}}{a^2 \sqrt {1-\frac {1}{a x}}}\) |
\(\Big \downarrow \) 100 |
\(\displaystyle -\frac {\sqrt {\frac {1}{x}} \sqrt {c-a c x} \left (\frac {8 a^2 \left (\frac {1}{x}\right )^{5/2}}{\sqrt {\frac {1}{a x}+1}}-2 a^2 \int \frac {\left (19 a-\frac {1}{x}\right ) \left (\frac {1}{x}\right )^{3/2}}{2 a \sqrt {1+\frac {1}{a x}}}d\frac {1}{x}\right )}{a^2 \sqrt {1-\frac {1}{a x}}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {\sqrt {\frac {1}{x}} \sqrt {c-a c x} \left (\frac {8 a^2 \left (\frac {1}{x}\right )^{5/2}}{\sqrt {\frac {1}{a x}+1}}-a \int \frac {\left (19 a-\frac {1}{x}\right ) \left (\frac {1}{x}\right )^{3/2}}{\sqrt {1+\frac {1}{a x}}}d\frac {1}{x}\right )}{a^2 \sqrt {1-\frac {1}{a x}}}\) |
\(\Big \downarrow \) 90 |
\(\displaystyle -\frac {\sqrt {\frac {1}{x}} \sqrt {c-a c x} \left (\frac {8 a^2 \left (\frac {1}{x}\right )^{5/2}}{\sqrt {\frac {1}{a x}+1}}-a \left (\frac {119}{6} a \int \frac {\left (\frac {1}{x}\right )^{3/2}}{\sqrt {1+\frac {1}{a x}}}d\frac {1}{x}-\frac {1}{3} a \left (\frac {1}{x}\right )^{5/2} \sqrt {\frac {1}{a x}+1}\right )\right )}{a^2 \sqrt {1-\frac {1}{a x}}}\) |
\(\Big \downarrow \) 60 |
\(\displaystyle -\frac {\sqrt {\frac {1}{x}} \sqrt {c-a c x} \left (\frac {8 a^2 \left (\frac {1}{x}\right )^{5/2}}{\sqrt {\frac {1}{a x}+1}}-a \left (\frac {119}{6} a \left (\frac {1}{2} a \left (\frac {1}{x}\right )^{3/2} \sqrt {\frac {1}{a x}+1}-\frac {3}{4} a \int \frac {\sqrt {\frac {1}{x}}}{\sqrt {1+\frac {1}{a x}}}d\frac {1}{x}\right )-\frac {1}{3} a \left (\frac {1}{x}\right )^{5/2} \sqrt {\frac {1}{a x}+1}\right )\right )}{a^2 \sqrt {1-\frac {1}{a x}}}\) |
\(\Big \downarrow \) 60 |
\(\displaystyle -\frac {\sqrt {\frac {1}{x}} \sqrt {c-a c x} \left (\frac {8 a^2 \left (\frac {1}{x}\right )^{5/2}}{\sqrt {\frac {1}{a x}+1}}-a \left (\frac {119}{6} a \left (\frac {1}{2} a \left (\frac {1}{x}\right )^{3/2} \sqrt {\frac {1}{a x}+1}-\frac {3}{4} a \left (a \sqrt {\frac {1}{x}} \sqrt {\frac {1}{a x}+1}-\frac {1}{2} a \int \frac {1}{\sqrt {1+\frac {1}{a x}} \sqrt {\frac {1}{x}}}d\frac {1}{x}\right )\right )-\frac {1}{3} a \left (\frac {1}{x}\right )^{5/2} \sqrt {\frac {1}{a x}+1}\right )\right )}{a^2 \sqrt {1-\frac {1}{a x}}}\) |
\(\Big \downarrow \) 63 |
\(\displaystyle -\frac {\sqrt {\frac {1}{x}} \sqrt {c-a c x} \left (\frac {8 a^2 \left (\frac {1}{x}\right )^{5/2}}{\sqrt {\frac {1}{a x}+1}}-a \left (\frac {119}{6} a \left (\frac {1}{2} a \left (\frac {1}{x}\right )^{3/2} \sqrt {\frac {1}{a x}+1}-\frac {3}{4} a \left (a \sqrt {\frac {1}{x}} \sqrt {\frac {1}{a x}+1}-a \int \frac {1}{\sqrt {1+\frac {1}{x^2 a}}}d\sqrt {\frac {1}{x}}\right )\right )-\frac {1}{3} a \left (\frac {1}{x}\right )^{5/2} \sqrt {\frac {1}{a x}+1}\right )\right )}{a^2 \sqrt {1-\frac {1}{a x}}}\) |
\(\Big \downarrow \) 222 |
\(\displaystyle -\frac {\sqrt {\frac {1}{x}} \left (\frac {8 a^2 \left (\frac {1}{x}\right )^{5/2}}{\sqrt {\frac {1}{a x}+1}}-a \left (\frac {119}{6} a \left (\frac {1}{2} a \left (\frac {1}{x}\right )^{3/2} \sqrt {\frac {1}{a x}+1}-\frac {3}{4} a \left (a \sqrt {\frac {1}{x}} \sqrt {\frac {1}{a x}+1}-a^{3/2} \text {arcsinh}\left (\frac {\sqrt {\frac {1}{x}}}{\sqrt {a}}\right )\right )\right )-\frac {1}{3} a \left (\frac {1}{x}\right )^{5/2} \sqrt {\frac {1}{a x}+1}\right )\right ) \sqrt {c-a c x}}{a^2 \sqrt {1-\frac {1}{a x}}}\) |
-((Sqrt[x^(-1)]*Sqrt[c - a*c*x]*((8*a^2*(x^(-1))^(5/2))/Sqrt[1 + 1/(a*x)] - a*(-1/3*(a*Sqrt[1 + 1/(a*x)]*(x^(-1))^(5/2)) + (119*a*((a*Sqrt[1 + 1/(a* x)]*(x^(-1))^(3/2))/2 - (3*a*(a*Sqrt[1 + 1/(a*x)]*Sqrt[x^(-1)] - a^(3/2)*A rcSinh[Sqrt[x^(-1)]/Sqrt[a]]))/4))/6)))/(a^2*Sqrt[1 - 1/(a*x)]))
3.4.57.3.1 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] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + n + 1))), x] + Simp[n*((b*c - a*d)/( b*(m + n + 1))) Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d}, x] && GtQ[n, 0] && NeQ[m + n + 1, 0] && !(IGtQ[m, 0] && ( !Integer Q[n] || (GtQ[m, 0] && LtQ[m - n, 0]))) && !ILtQ[m + n + 2, 0] && IntLinear Q[a, b, c, d, m, n, x]
Int[1/(Sqrt[(b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Simp[2/b S ubst[Int[1/Sqrt[c + d*(x^2/b)], x], x, Sqrt[b*x]], x] /; FreeQ[{b, c, d}, x ] && GtQ[c, 0]
Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p _.), x_] :> Simp[b*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + p + 2))), x] + Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(d*f*(n + p + 2)) Int[(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 2, 0]
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[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSinh[Rt[b, 2]*(x/Sqrt [a])]/Rt[b, 2], x] /; FreeQ[{a, b}, x] && GtQ[a, 0] && PosQ[b]
Int[E^(ArcCoth[(a_.)*(x_)]*(n_.))*((e_.)*(x_))^(m_.)*((c_) + (d_.)*(x_))^(p _), x_Symbol] :> Simp[(-(e*x)^m)*(1/x)^(m + p)*((c + d*x)^p/(1 + c/(d*x))^p ) Subst[Int[((1 + c*(x/d))^p*((1 + x/a)^(n/2)/x^(m + p + 2)))/(1 - x/a)^( n/2), x], x, 1/x], x] /; FreeQ[{a, c, d, e, m, n, p}, x] && EqQ[a^2*c^2 - d ^2, 0] && !IntegerQ[p]
Time = 0.47 (sec) , antiderivative size = 114, normalized size of antiderivative = 0.48
method | result | size |
default | \(-\frac {\left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}} \left (a x +1\right ) \sqrt {-c \left (a x -1\right )}\, \left (357 \arctan \left (\frac {\sqrt {-c \left (a x +1\right )}}{\sqrt {c}}\right ) a^{3} x^{3} \sqrt {-c \left (a x +1\right )}+357 a^{3} x^{3} \sqrt {c}+119 \sqrt {c}\, a^{2} x^{2}-38 \sqrt {c}\, a x +8 \sqrt {c}\right )}{24 \left (a x -1\right )^{2} \sqrt {c}\, x^{3}}\) | \(114\) |
risch | \(\frac {\left (165 a^{3} x^{3}+119 a^{2} x^{2}-38 a x +8\right ) c \sqrt {\frac {a x -1}{a x +1}}}{24 x^{3} \sqrt {-c \left (a x -1\right )}}-\frac {\left (-\frac {8 a^{3}}{\sqrt {-a c x -c}}-\frac {119 a^{3} \arctan \left (\frac {\sqrt {-a c x -c}}{\sqrt {c}}\right )}{8 \sqrt {c}}\right ) c \sqrt {\frac {a x -1}{a x +1}}\, \sqrt {-c \left (a x +1\right )}}{\sqrt {-c \left (a x -1\right )}}\) | \(134\) |
-1/24*((a*x-1)/(a*x+1))^(3/2)*(a*x+1)/(a*x-1)^2*(-c*(a*x-1))^(1/2)*(357*ar ctan((-c*(a*x+1))^(1/2)/c^(1/2))*a^3*x^3*(-c*(a*x+1))^(1/2)+357*a^3*x^3*c^ (1/2)+119*c^(1/2)*a^2*x^2-38*c^(1/2)*a*x+8*c^(1/2))/c^(1/2)/x^3
Time = 0.26 (sec) , antiderivative size = 278, normalized size of antiderivative = 1.17 \[ \int \frac {e^{-3 \coth ^{-1}(a x)} \sqrt {c-a c x}}{x^4} \, dx=\left [\frac {357 \, {\left (a^{4} x^{4} - a^{3} x^{3}\right )} \sqrt {-c} \log \left (-\frac {a^{2} c x^{2} + a c x - 2 \, \sqrt {-a c x + c} {\left (a x + 1\right )} \sqrt {-c} \sqrt {\frac {a x - 1}{a x + 1}} - 2 \, c}{a x^{2} - x}\right ) - 2 \, {\left (357 \, a^{3} x^{3} + 119 \, a^{2} x^{2} - 38 \, a x + 8\right )} \sqrt {-a c x + c} \sqrt {\frac {a x - 1}{a x + 1}}}{48 \, {\left (a x^{4} - x^{3}\right )}}, \frac {357 \, {\left (a^{4} x^{4} - a^{3} x^{3}\right )} \sqrt {c} \arctan \left (\frac {\sqrt {-a c x + c} \sqrt {c} \sqrt {\frac {a x - 1}{a x + 1}}}{a c x - c}\right ) - {\left (357 \, a^{3} x^{3} + 119 \, a^{2} x^{2} - 38 \, a x + 8\right )} \sqrt {-a c x + c} \sqrt {\frac {a x - 1}{a x + 1}}}{24 \, {\left (a x^{4} - x^{3}\right )}}\right ] \]
[1/48*(357*(a^4*x^4 - a^3*x^3)*sqrt(-c)*log(-(a^2*c*x^2 + a*c*x - 2*sqrt(- a*c*x + c)*(a*x + 1)*sqrt(-c)*sqrt((a*x - 1)/(a*x + 1)) - 2*c)/(a*x^2 - x) ) - 2*(357*a^3*x^3 + 119*a^2*x^2 - 38*a*x + 8)*sqrt(-a*c*x + c)*sqrt((a*x - 1)/(a*x + 1)))/(a*x^4 - x^3), 1/24*(357*(a^4*x^4 - a^3*x^3)*sqrt(c)*arct an(sqrt(-a*c*x + c)*sqrt(c)*sqrt((a*x - 1)/(a*x + 1))/(a*c*x - c)) - (357* a^3*x^3 + 119*a^2*x^2 - 38*a*x + 8)*sqrt(-a*c*x + c)*sqrt((a*x - 1)/(a*x + 1)))/(a*x^4 - x^3)]
Timed out. \[ \int \frac {e^{-3 \coth ^{-1}(a x)} \sqrt {c-a c x}}{x^4} \, dx=\text {Timed out} \]
\[ \int \frac {e^{-3 \coth ^{-1}(a x)} \sqrt {c-a c x}}{x^4} \, dx=\int { \frac {\sqrt {-a c x + c} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}}}{x^{4}} \,d x } \]
Exception generated. \[ \int \frac {e^{-3 \coth ^{-1}(a x)} \sqrt {c-a c x}}{x^4} \, dx=\text {Exception raised: TypeError} \]
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. \[ \int \frac {e^{-3 \coth ^{-1}(a x)} \sqrt {c-a c x}}{x^4} \, dx=\int \frac {\sqrt {c-a\,c\,x}\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/2}}{x^4} \,d x \]