Integrand size = 27, antiderivative size = 147 \[ \int e^{-2 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}} x^2 \, dx=\frac {19 \sqrt {c-\frac {c}{a x}} x}{8 a^2}-\frac {13 \sqrt {c-\frac {c}{a x}} x^2}{12 a}+\frac {1}{3} \sqrt {c-\frac {c}{a x}} x^3-\frac {45 \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {c}}\right )}{8 a^3}+\frac {4 \sqrt {2} \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {2} \sqrt {c}}\right )}{a^3} \]
-45/8*arctanh((c-c/a/x)^(1/2)/c^(1/2))*c^(1/2)/a^3+4*arctanh(1/2*(c-c/a/x) ^(1/2)*2^(1/2)/c^(1/2))*2^(1/2)*c^(1/2)/a^3+19/8*x*(c-c/a/x)^(1/2)/a^2-13/ 12*x^2*(c-c/a/x)^(1/2)/a+1/3*x^3*(c-c/a/x)^(1/2)
Time = 0.12 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.73 \[ \int e^{-2 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}} x^2 \, dx=\frac {a \sqrt {c-\frac {c}{a x}} x \left (57-26 a x+8 a^2 x^2\right )-135 \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {c}}\right )+96 \sqrt {2} \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {2} \sqrt {c}}\right )}{24 a^3} \]
(a*Sqrt[c - c/(a*x)]*x*(57 - 26*a*x + 8*a^2*x^2) - 135*Sqrt[c]*ArcTanh[Sqr t[c - c/(a*x)]/Sqrt[c]] + 96*Sqrt[2]*Sqrt[c]*ArcTanh[Sqrt[c - c/(a*x)]/(Sq rt[2]*Sqrt[c])])/(24*a^3)
Time = 0.79 (sec) , antiderivative size = 174, normalized size of antiderivative = 1.18, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.519, Rules used = {6717, 6683, 1070, 281, 948, 109, 27, 168, 27, 168, 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.
| \(\displaystyle \int x^2 \sqrt {c-\frac {c}{a x}} e^{-2 \coth ^{-1}(a x)} \, dx\) |
| \(\Big \downarrow \) 6717 |
| \(\displaystyle -\int e^{-2 \text {arctanh}(a x)} \sqrt {c-\frac {c}{a x}} x^2dx\) |
| \(\Big \downarrow \) 6683 |
| \(\displaystyle -\int \frac {\sqrt {c-\frac {c}{a x}} x^2 (1-a x)}{a x+1}dx\) |
| \(\Big \downarrow \) 1070 |
| \(\displaystyle -\int \frac {\left (\frac {1}{x}-a\right ) \sqrt {c-\frac {c}{a x}} x^2}{a+\frac {1}{x}}dx\) |
| \(\Big \downarrow \) 281 |
| \(\displaystyle \frac {a \int \frac {\left (c-\frac {c}{a x}\right )^{3/2} x^2}{a+\frac {1}{x}}dx}{c}\) |
| \(\Big \downarrow \) 948 |
| \(\displaystyle -\frac {a \int \frac {\left (c-\frac {c}{a x}\right )^{3/2} x^4}{a+\frac {1}{x}}d\frac {1}{x}}{c}\) |
| \(\Big \downarrow \) 109 |
| \(\displaystyle -\frac {a \left (-\frac {\int \frac {c^2 \left (13 a-\frac {11}{x}\right ) x^3}{2 a \left (a+\frac {1}{x}\right ) \sqrt {c-\frac {c}{a x}}}d\frac {1}{x}}{3 a}-\frac {c x^3 \sqrt {c-\frac {c}{a x}}}{3 a}\right )}{c}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {a \left (-\frac {c^2 \int \frac {\left (13 a-\frac {11}{x}\right ) x^3}{\left (a+\frac {1}{x}\right ) \sqrt {c-\frac {c}{a x}}}d\frac {1}{x}}{6 a^2}-\frac {c x^3 \sqrt {c-\frac {c}{a x}}}{3 a}\right )}{c}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle -\frac {a \left (-\frac {c^2 \left (-\frac {\int \frac {3 c \left (19 a-\frac {13}{x}\right ) x^2}{2 \left (a+\frac {1}{x}\right ) \sqrt {c-\frac {c}{a x}}}d\frac {1}{x}}{2 a c}-\frac {13 x^2 \sqrt {c-\frac {c}{a x}}}{2 c}\right )}{6 a^2}-\frac {c x^3 \sqrt {c-\frac {c}{a x}}}{3 a}\right )}{c}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {a \left (-\frac {c^2 \left (-\frac {3 \int \frac {\left (19 a-\frac {13}{x}\right ) x^2}{\left (a+\frac {1}{x}\right ) \sqrt {c-\frac {c}{a x}}}d\frac {1}{x}}{4 a}-\frac {13 x^2 \sqrt {c-\frac {c}{a x}}}{2 c}\right )}{6 a^2}-\frac {c x^3 \sqrt {c-\frac {c}{a x}}}{3 a}\right )}{c}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle -\frac {a \left (-\frac {c^2 \left (-\frac {3 \left (-\frac {\int \frac {c \left (45 a-\frac {19}{x}\right ) x}{2 \left (a+\frac {1}{x}\right ) \sqrt {c-\frac {c}{a x}}}d\frac {1}{x}}{a c}-\frac {19 x \sqrt {c-\frac {c}{a x}}}{c}\right )}{4 a}-\frac {13 x^2 \sqrt {c-\frac {c}{a x}}}{2 c}\right )}{6 a^2}-\frac {c x^3 \sqrt {c-\frac {c}{a x}}}{3 a}\right )}{c}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {a \left (-\frac {c^2 \left (-\frac {3 \left (-\frac {\int \frac {\left (45 a-\frac {19}{x}\right ) x}{\left (a+\frac {1}{x}\right ) \sqrt {c-\frac {c}{a x}}}d\frac {1}{x}}{2 a}-\frac {19 x \sqrt {c-\frac {c}{a x}}}{c}\right )}{4 a}-\frac {13 x^2 \sqrt {c-\frac {c}{a x}}}{2 c}\right )}{6 a^2}-\frac {c x^3 \sqrt {c-\frac {c}{a x}}}{3 a}\right )}{c}\) |
| \(\Big \downarrow \) 174 |
| \(\displaystyle -\frac {a \left (-\frac {c^2 \left (-\frac {3 \left (-\frac {45 \int \frac {x}{\sqrt {c-\frac {c}{a x}}}d\frac {1}{x}-64 \int \frac {1}{\left (a+\frac {1}{x}\right ) \sqrt {c-\frac {c}{a x}}}d\frac {1}{x}}{2 a}-\frac {19 x \sqrt {c-\frac {c}{a x}}}{c}\right )}{4 a}-\frac {13 x^2 \sqrt {c-\frac {c}{a x}}}{2 c}\right )}{6 a^2}-\frac {c x^3 \sqrt {c-\frac {c}{a x}}}{3 a}\right )}{c}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle -\frac {a \left (-\frac {c^2 \left (-\frac {3 \left (-\frac {\frac {128 a \int \frac {1}{2 a-\frac {a}{c x^2}}d\sqrt {c-\frac {c}{a x}}}{c}-\frac {90 a \int \frac {1}{a-\frac {a}{c x^2}}d\sqrt {c-\frac {c}{a x}}}{c}}{2 a}-\frac {19 x \sqrt {c-\frac {c}{a x}}}{c}\right )}{4 a}-\frac {13 x^2 \sqrt {c-\frac {c}{a x}}}{2 c}\right )}{6 a^2}-\frac {c x^3 \sqrt {c-\frac {c}{a x}}}{3 a}\right )}{c}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle -\frac {a \left (-\frac {c^2 \left (-\frac {3 \left (-\frac {\frac {64 \sqrt {2} \text {arctanh}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {2} \sqrt {c}}\right )}{\sqrt {c}}-\frac {90 \text {arctanh}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {c}}\right )}{\sqrt {c}}}{2 a}-\frac {19 x \sqrt {c-\frac {c}{a x}}}{c}\right )}{4 a}-\frac {13 x^2 \sqrt {c-\frac {c}{a x}}}{2 c}\right )}{6 a^2}-\frac {c x^3 \sqrt {c-\frac {c}{a x}}}{3 a}\right )}{c}\) |
-((a*(-1/3*(c*Sqrt[c - c/(a*x)]*x^3)/a - (c^2*((-13*Sqrt[c - c/(a*x)]*x^2) /(2*c) - (3*((-19*Sqrt[c - c/(a*x)]*x)/c - ((-90*ArcTanh[Sqrt[c - c/(a*x)] /Sqrt[c]])/Sqrt[c] + (64*Sqrt[2]*ArcTanh[Sqrt[c - c/(a*x)]/(Sqrt[2]*Sqrt[c ])])/Sqrt[c])/(2*a)))/(4*a)))/(6*a^2)))/c)
3.6.25.3.1 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[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_)*((g_.) + (h_.)*(x_)), x_] :> Simp[(b*g - a*h)*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] + S imp[1/((m + 1)*(b*c - a*d)*(b*e - a*f)) Int[(a + b*x)^(m + 1)*(c + d*x)^n *(e + f*x)^p*Simp[(a*d*f*g - b*(d*e + c*f)*g + b*c*e*h)*(m + 1) - (b*g - a* h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(m + n + p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && ILtQ[m, -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[(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[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_. ), x_Symbol] :> Simp[1/n Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^ p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] && NeQ [b*c - a*d, 0] && IntegerQ[Simplify[(m + 1)/n]]
Int[(x_)^(m_.)*((c_) + (d_.)*(x_)^(mn_.))^(q_.)*((a_.) + (b_.)*(x_)^(n_.))^ (p_.)*((e_) + (f_.)*(x_)^(n_.))^(r_.), x_Symbol] :> Int[x^(m + 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, m, 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]
Int[E^(ArcCoth[(a_.)*(x_)]*(n_))*(u_.), x_Symbol] :> Simp[(-1)^(n/2) Int[ u*E^(n*ArcTanh[a*x]), x], x] /; FreeQ[a, x] && IntegerQ[n/2]
Time = 0.49 (sec) , antiderivative size = 189, normalized size of antiderivative = 1.29
| method | result | size |
| risch | \(\frac {\left (8 a^{2} x^{2}-26 a x +57\right ) x \sqrt {\frac {c \left (a x -1\right )}{a x}}}{24 a^{2}}+\frac {\left (-\frac {45 \ln \left (\frac {-\frac {1}{2} a c +a^{2} c x}{\sqrt {a^{2} c}}+\sqrt {a^{2} c \,x^{2}-a c x}\right )}{16 a^{2} \sqrt {a^{2} c}}-\frac {2 \sqrt {2}\, \ln \left (\frac {4 c -3 \left (x +\frac {1}{a}\right ) a c +2 \sqrt {2}\, \sqrt {c}\, \sqrt {a^{2} c \left (x +\frac {1}{a}\right )^{2}-3 \left (x +\frac {1}{a}\right ) a c +2 c}}{x +\frac {1}{a}}\right )}{a^{3} \sqrt {c}}\right ) \sqrt {\frac {c \left (a x -1\right )}{a x}}\, \sqrt {c \left (a x -1\right ) a x}}{a x -1}\) | \(189\) |
| default | \(\frac {\sqrt {\frac {c \left (a x -1\right )}{a x}}\, x \left (16 \left (a \,x^{2}-x \right )^{\frac {3}{2}} a^{\frac {7}{2}} \sqrt {\frac {1}{a}}-36 \sqrt {a \,x^{2}-x}\, a^{\frac {7}{2}} \sqrt {\frac {1}{a}}\, x +96 \sqrt {\left (a x -1\right ) x}\, a^{\frac {5}{2}} \sqrt {\frac {1}{a}}+18 \sqrt {a \,x^{2}-x}\, a^{\frac {5}{2}} \sqrt {\frac {1}{a}}-96 a^{\frac {3}{2}} \sqrt {2}\, \ln \left (\frac {2 \sqrt {2}\, \sqrt {\frac {1}{a}}\, \sqrt {\left (a x -1\right ) x}\, a -3 a x +1}{a x +1}\right )-144 a^{2} \ln \left (\frac {2 \sqrt {\left (a x -1\right ) x}\, \sqrt {a}+2 a x -1}{2 \sqrt {a}}\right ) \sqrt {\frac {1}{a}}+9 \ln \left (\frac {2 \sqrt {a \,x^{2}-x}\, \sqrt {a}+2 a x -1}{2 \sqrt {a}}\right ) \sqrt {\frac {1}{a}}\, a^{2}\right )}{48 \sqrt {\left (a x -1\right ) x}\, a^{\frac {9}{2}} \sqrt {\frac {1}{a}}}\) | \(237\) |
1/24*(8*a^2*x^2-26*a*x+57)/a^2*x*(c*(a*x-1)/a/x)^(1/2)+(-45/16/a^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^3* 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)))/(a*x-1)*(c*(a*x-1)/a/x)^(1/2)*(c*(a*x-1 )*a*x)^(1/2)
Time = 0.26 (sec) , antiderivative size = 259, normalized size of antiderivative = 1.76 \[ \int e^{-2 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}} x^2 \, dx=\left [\frac {96 \, \sqrt {2} \sqrt {c} \log \left (-\frac {2 \, \sqrt {2} a \sqrt {c} x \sqrt {\frac {a c x - c}{a x}} + 3 \, a c x - c}{a x + 1}\right ) + 2 \, {\left (8 \, a^{3} x^{3} - 26 \, a^{2} x^{2} + 57 \, a x\right )} \sqrt {\frac {a c x - c}{a x}} + 135 \, \sqrt {c} \log \left (-2 \, a c x + 2 \, a \sqrt {c} x \sqrt {\frac {a c x - c}{a x}} + c\right )}{48 \, a^{3}}, -\frac {96 \, \sqrt {2} \sqrt {-c} \arctan \left (\frac {\sqrt {2} \sqrt {-c} \sqrt {\frac {a c x - c}{a x}}}{2 \, c}\right ) - {\left (8 \, a^{3} x^{3} - 26 \, a^{2} x^{2} + 57 \, a x\right )} \sqrt {\frac {a c x - c}{a x}} - 135 \, \sqrt {-c} \arctan \left (\frac {\sqrt {-c} \sqrt {\frac {a c x - c}{a x}}}{c}\right )}{24 \, a^{3}}\right ] \]
[1/48*(96*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)) + 2*(8*a^3*x^3 - 26*a^2*x^2 + 57*a*x)*sqrt(( a*c*x - c)/(a*x)) + 135*sqrt(c)*log(-2*a*c*x + 2*a*sqrt(c)*x*sqrt((a*c*x - c)/(a*x)) + c))/a^3, -1/24*(96*sqrt(2)*sqrt(-c)*arctan(1/2*sqrt(2)*sqrt(- c)*sqrt((a*c*x - c)/(a*x))/c) - (8*a^3*x^3 - 26*a^2*x^2 + 57*a*x)*sqrt((a* c*x - c)/(a*x)) - 135*sqrt(-c)*arctan(sqrt(-c)*sqrt((a*c*x - c)/(a*x))/c)) /a^3]
\[ \int e^{-2 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}} x^2 \, dx=\int \frac {x^{2} \sqrt {- c \left (-1 + \frac {1}{a x}\right )} \left (a x - 1\right )}{a x + 1}\, dx \]
\[ \int e^{-2 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}} x^2 \, dx=\int { \frac {{\left (a x - 1\right )} \sqrt {c - \frac {c}{a x}} x^{2}}{a x + 1} \,d x } \]
Exception generated. \[ \int e^{-2 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}} x^2 \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Error: Bad Argument Type
Timed out. \[ \int e^{-2 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}} x^2 \, dx=\int \frac {x^2\,\sqrt {c-\frac {c}{a\,x}}\,\left (a\,x-1\right )}{a\,x+1} \,d x \]