Integrand size = 27, antiderivative size = 251 \[ \int e^{-3 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}} x^2 \, dx=\frac {119 \sqrt {c-\frac {c}{a x}}}{8 a^3 \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}+\frac {119 \sqrt {c-\frac {c}{a x}} x}{24 a^2 \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}-\frac {19 \sqrt {c-\frac {c}{a x}} x^2}{12 a \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}+\frac {\sqrt {c-\frac {c}{a x}} x^3}{3 \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}-\frac {119 \sqrt {c-\frac {c}{a x}} \text {arctanh}\left (\sqrt {1+\frac {1}{a x}}\right )}{8 a^3 \sqrt {1-\frac {1}{a x}}} \]
-119/8*arctanh((1+1/a/x)^(1/2))*(c-c/a/x)^(1/2)/a^3/(1-1/a/x)^(1/2)+119/8* (c-c/a/x)^(1/2)/a^3/(1-1/a/x)^(1/2)/(1+1/a/x)^(1/2)+119/24*x*(c-c/a/x)^(1/ 2)/a^2/(1-1/a/x)^(1/2)/(1+1/a/x)^(1/2)-19/12*x^2*(c-c/a/x)^(1/2)/a/(1-1/a/ x)^(1/2)/(1+1/a/x)^(1/2)+1/3*x^3*(c-c/a/x)^(1/2)/(1-1/a/x)^(1/2)/(1+1/a/x) ^(1/2)
Time = 0.95 (sec) , antiderivative size = 159, normalized size of antiderivative = 0.63 \[ \int e^{-3 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}} x^2 \, dx=\frac {\frac {2 a^2 \sqrt {1-\frac {1}{a^2 x^2}} \sqrt {c-\frac {c}{a x}} x^2 \left (357+119 a x-38 a^2 x^2+8 a^3 x^3\right )}{-1+a^2 x^2}+357 \sqrt {c} \log (1-a x)-357 \sqrt {c} \log \left (2 a^2 \sqrt {c} \sqrt {1-\frac {1}{a^2 x^2}} \sqrt {c-\frac {c}{a x}} x^2+c \left (-1-a x+2 a^2 x^2\right )\right )}{48 a^3} \]
((2*a^2*Sqrt[1 - 1/(a^2*x^2)]*Sqrt[c - c/(a*x)]*x^2*(357 + 119*a*x - 38*a^ 2*x^2 + 8*a^3*x^3))/(-1 + a^2*x^2) + 357*Sqrt[c]*Log[1 - a*x] - 357*Sqrt[c ]*Log[2*a^2*Sqrt[c]*Sqrt[1 - 1/(a^2*x^2)]*Sqrt[c - c/(a*x)]*x^2 + c*(-1 - a*x + 2*a^2*x^2)])/(48*a^3)
Time = 0.44 (sec) , antiderivative size = 144, normalized size of antiderivative = 0.57, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.370, Rules used = {6733, 585, 27, 100, 27, 87, 52, 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.
| \(\displaystyle \int x^2 \sqrt {c-\frac {c}{a x}} e^{-3 \coth ^{-1}(a x)} \, dx\) |
| \(\Big \downarrow \) 6733 |
| \(\displaystyle -\frac {\int \frac {\left (c-\frac {c}{a x}\right )^{7/2} x^4}{\left (1-\frac {1}{a^2 x^2}\right )^{3/2}}d\frac {1}{x}}{c^3}\) |
| \(\Big \downarrow \) 585 |
| \(\displaystyle -\frac {\sqrt {c-\frac {c}{a x}} \int \frac {\left (a-\frac {1}{x}\right )^2 x^4}{a^2 \left (1+\frac {1}{a x}\right )^{3/2}}d\frac {1}{x}}{\sqrt {1-\frac {1}{a x}}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\sqrt {c-\frac {c}{a x}} \int \frac {\left (a-\frac {1}{x}\right )^2 x^4}{\left (1+\frac {1}{a x}\right )^{3/2}}d\frac {1}{x}}{a^2 \sqrt {1-\frac {1}{a x}}}\) |
| \(\Big \downarrow \) 100 |
| \(\displaystyle -\frac {\sqrt {c-\frac {c}{a x}} \left (\frac {1}{3} \int -\frac {\left (19 a-\frac {6}{x}\right ) x^3}{2 \left (1+\frac {1}{a x}\right )^{3/2}}d\frac {1}{x}-\frac {a^2 x^3}{3 \sqrt {\frac {1}{a x}+1}}\right )}{a^2 \sqrt {1-\frac {1}{a x}}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\sqrt {c-\frac {c}{a x}} \left (-\frac {1}{6} \int \frac {\left (19 a-\frac {6}{x}\right ) x^3}{\left (1+\frac {1}{a x}\right )^{3/2}}d\frac {1}{x}-\frac {a^2 x^3}{3 \sqrt {\frac {1}{a x}+1}}\right )}{a^2 \sqrt {1-\frac {1}{a x}}}\) |
| \(\Big \downarrow \) 87 |
| \(\displaystyle -\frac {\sqrt {c-\frac {c}{a x}} \left (\frac {1}{6} \left (\frac {119}{4} \int \frac {x^2}{\left (1+\frac {1}{a x}\right )^{3/2}}d\frac {1}{x}+\frac {19 a x^2}{2 \sqrt {\frac {1}{a x}+1}}\right )-\frac {a^2 x^3}{3 \sqrt {\frac {1}{a x}+1}}\right )}{a^2 \sqrt {1-\frac {1}{a x}}}\) |
| \(\Big \downarrow \) 52 |
| \(\displaystyle -\frac {\sqrt {c-\frac {c}{a x}} \left (\frac {1}{6} \left (\frac {119}{4} \left (-\frac {3 \int \frac {x}{\left (1+\frac {1}{a x}\right )^{3/2}}d\frac {1}{x}}{2 a}-\frac {x}{\sqrt {\frac {1}{a x}+1}}\right )+\frac {19 a x^2}{2 \sqrt {\frac {1}{a x}+1}}\right )-\frac {a^2 x^3}{3 \sqrt {\frac {1}{a x}+1}}\right )}{a^2 \sqrt {1-\frac {1}{a x}}}\) |
| \(\Big \downarrow \) 61 |
| \(\displaystyle -\frac {\sqrt {c-\frac {c}{a x}} \left (\frac {1}{6} \left (\frac {119}{4} \left (-\frac {3 \left (\int \frac {x}{\sqrt {1+\frac {1}{a x}}}d\frac {1}{x}+\frac {2}{\sqrt {\frac {1}{a x}+1}}\right )}{2 a}-\frac {x}{\sqrt {\frac {1}{a x}+1}}\right )+\frac {19 a x^2}{2 \sqrt {\frac {1}{a x}+1}}\right )-\frac {a^2 x^3}{3 \sqrt {\frac {1}{a x}+1}}\right )}{a^2 \sqrt {1-\frac {1}{a x}}}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle -\frac {\sqrt {c-\frac {c}{a x}} \left (\frac {1}{6} \left (\frac {119}{4} \left (-\frac {3 \left (2 a \int \frac {1}{\frac {a}{x^2}-a}d\sqrt {1+\frac {1}{a x}}+\frac {2}{\sqrt {\frac {1}{a x}+1}}\right )}{2 a}-\frac {x}{\sqrt {\frac {1}{a x}+1}}\right )+\frac {19 a x^2}{2 \sqrt {\frac {1}{a x}+1}}\right )-\frac {a^2 x^3}{3 \sqrt {\frac {1}{a x}+1}}\right )}{a^2 \sqrt {1-\frac {1}{a x}}}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle -\frac {\left (\frac {1}{6} \left (\frac {119}{4} \left (-\frac {3 \left (\frac {2}{\sqrt {\frac {1}{a x}+1}}-2 \text {arctanh}\left (\sqrt {\frac {1}{a x}+1}\right )\right )}{2 a}-\frac {x}{\sqrt {\frac {1}{a x}+1}}\right )+\frac {19 a x^2}{2 \sqrt {\frac {1}{a x}+1}}\right )-\frac {a^2 x^3}{3 \sqrt {\frac {1}{a x}+1}}\right ) \sqrt {c-\frac {c}{a x}}}{a^2 \sqrt {1-\frac {1}{a x}}}\) |
-((Sqrt[c - c/(a*x)]*(-1/3*(a^2*x^3)/Sqrt[1 + 1/(a*x)] + ((19*a*x^2)/(2*Sq rt[1 + 1/(a*x)]) + (119*(-(x/Sqrt[1 + 1/(a*x)]) - (3*(2/Sqrt[1 + 1/(a*x)] - 2*ArcTanh[Sqrt[1 + 1/(a*x)]]))/(2*a)))/4)/6))/(a^2*Sqrt[1 - 1/(a*x)]))
3.6.34.3.1 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[((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] && ILtQ[m, -1] && FractionQ[n] && LtQ[n, 0]
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_.)*(x_))^(m_.)*((c_) + (d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_)^2)^(p_) , x_Symbol] :> Simp[a^p*c^IntPart[n]*((c + d*x)^FracPart[n]/(1 + d*(x/c))^F racPart[n]) Int[(e*x)^m*(1 - d*(x/c))^p*(1 + d*(x/c))^(n + p), x], x] /; FreeQ[{a, b, c, d, e, m, n, p}, x] && EqQ[b*c^2 + a*d^2, 0] && GtQ[a, 0]
Int[E^(ArcCoth[(a_.)*(x_)]*(n_.))*((c_) + (d_.)/(x_))^(p_.)*(x_)^(m_.), x_S ymbol] :> Simp[-c^n Subst[Int[(c + d*x)^(p - n)*((1 - x^2/a^2)^(n/2)/x^(m + 2)), x], x, 1/x], x] /; FreeQ[{a, c, d, p}, x] && EqQ[c + a*d, 0] && Int egerQ[(n - 1)/2] && IntegerQ[m] && IntegerQ[2*p]
Time = 0.15 (sec) , antiderivative size = 180, normalized size of antiderivative = 0.72
| method | result | size |
| default | \(\frac {\left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}} \left (a x +1\right ) \sqrt {\frac {c \left (a x -1\right )}{a x}}\, x \left (16 a^{\frac {7}{2}} x^{3} \sqrt {\left (a x +1\right ) x}-76 a^{\frac {5}{2}} x^{2} \sqrt {\left (a x +1\right ) x}+238 a^{\frac {3}{2}} x \sqrt {\left (a x +1\right ) x}-357 \ln \left (\frac {2 \sqrt {\left (a x +1\right ) x}\, \sqrt {a}+2 a x +1}{2 \sqrt {a}}\right ) a x +714 \sqrt {\left (a x +1\right ) x}\, \sqrt {a}-357 \ln \left (\frac {2 \sqrt {\left (a x +1\right ) x}\, \sqrt {a}+2 a x +1}{2 \sqrt {a}}\right )\right )}{48 \left (a x -1\right )^{2} a^{\frac {5}{2}} \sqrt {\left (a x +1\right ) x}}\) | \(180\) |
| risch | \(\frac {\left (8 a^{2} x^{2}-46 a x +165\right ) x \left (a x +1\right ) \sqrt {\frac {a x -1}{a x +1}}\, \sqrt {\frac {c \left (a x -1\right )}{a x}}}{24 a^{2} \left (a x -1\right )}+\frac {\left (-\frac {119 \ln \left (\frac {\frac {1}{2} a c +a^{2} c x}{\sqrt {a^{2} c}}+\sqrt {a^{2} c \,x^{2}+a c x}\right )}{16 a^{2} \sqrt {a^{2} c}}+\frac {8 \sqrt {a^{2} c \left (x +\frac {1}{a}\right )^{2}-\left (x +\frac {1}{a}\right ) a c}}{a^{4} c \left (x +\frac {1}{a}\right )}\right ) \sqrt {\frac {a x -1}{a x +1}}\, \sqrt {\frac {c \left (a x -1\right )}{a x}}\, \sqrt {\left (a x +1\right ) a c x}}{a x -1}\) | \(201\) |
1/48*((a*x-1)/(a*x+1))^(3/2)*(a*x+1)/(a*x-1)^2*(c*(a*x-1)/a/x)^(1/2)*x*(16 *a^(7/2)*x^3*((a*x+1)*x)^(1/2)-76*a^(5/2)*x^2*((a*x+1)*x)^(1/2)+238*a^(3/2 )*x*((a*x+1)*x)^(1/2)-357*ln(1/2*(2*((a*x+1)*x)^(1/2)*a^(1/2)+2*a*x+1)/a^( 1/2))*a*x+714*((a*x+1)*x)^(1/2)*a^(1/2)-357*ln(1/2*(2*((a*x+1)*x)^(1/2)*a^ (1/2)+2*a*x+1)/a^(1/2)))/a^(5/2)/((a*x+1)*x)^(1/2)
Time = 0.29 (sec) , antiderivative size = 337, normalized size of antiderivative = 1.34 \[ \int e^{-3 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}} x^2 \, dx=\left [\frac {357 \, {\left (a x - 1\right )} \sqrt {c} \log \left (-\frac {8 \, a^{3} c x^{3} - 7 \, a c x - 4 \, {\left (2 \, a^{3} x^{3} + 3 \, a^{2} x^{2} + a x\right )} \sqrt {c} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}} - c}{a x - 1}\right ) + 4 \, {\left (8 \, a^{4} x^{4} - 38 \, a^{3} x^{3} + 119 \, a^{2} x^{2} + 357 \, a x\right )} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}}}{96 \, {\left (a^{4} x - a^{3}\right )}}, \frac {357 \, {\left (a x - 1\right )} \sqrt {-c} \arctan \left (\frac {2 \, {\left (a^{2} x^{2} + a x\right )} \sqrt {-c} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}}}{2 \, a^{2} c x^{2} - a c x - c}\right ) + 2 \, {\left (8 \, a^{4} x^{4} - 38 \, a^{3} x^{3} + 119 \, a^{2} x^{2} + 357 \, a x\right )} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}}}{48 \, {\left (a^{4} x - a^{3}\right )}}\right ] \]
[1/96*(357*(a*x - 1)*sqrt(c)*log(-(8*a^3*c*x^3 - 7*a*c*x - 4*(2*a^3*x^3 + 3*a^2*x^2 + a*x)*sqrt(c)*sqrt((a*x - 1)/(a*x + 1))*sqrt((a*c*x - c)/(a*x)) - c)/(a*x - 1)) + 4*(8*a^4*x^4 - 38*a^3*x^3 + 119*a^2*x^2 + 357*a*x)*sqrt ((a*x - 1)/(a*x + 1))*sqrt((a*c*x - c)/(a*x)))/(a^4*x - a^3), 1/48*(357*(a *x - 1)*sqrt(-c)*arctan(2*(a^2*x^2 + a*x)*sqrt(-c)*sqrt((a*x - 1)/(a*x + 1 ))*sqrt((a*c*x - c)/(a*x))/(2*a^2*c*x^2 - a*c*x - c)) + 2*(8*a^4*x^4 - 38* a^3*x^3 + 119*a^2*x^2 + 357*a*x)*sqrt((a*x - 1)/(a*x + 1))*sqrt((a*c*x - c )/(a*x)))/(a^4*x - a^3)]
Timed out. \[ \int e^{-3 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}} x^2 \, dx=\text {Timed out} \]
\[ \int e^{-3 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}} x^2 \, dx=\int { \sqrt {c - \frac {c}{a x}} x^{2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}} \,d x } \]
Exception generated. \[ \int e^{-3 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}} x^2 \, dx=\text {Exception raised: TypeError} \]
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. \[ \int e^{-3 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}} x^2 \, dx=\int x^2\,\sqrt {c-\frac {c}{a\,x}}\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/2} \,d x \]