Integrand size = 27, antiderivative size = 172 \[ \int e^{-2 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}} x^3 \, dx=-\frac {149 \sqrt {c-\frac {c}{a x}} x}{64 a^3}+\frac {107 \sqrt {c-\frac {c}{a x}} x^2}{96 a^2}-\frac {17 \sqrt {c-\frac {c}{a x}} x^3}{24 a}+\frac {1}{4} \sqrt {c-\frac {c}{a x}} x^4+\frac {363 \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {c}}\right )}{64 a^4}-\frac {4 \sqrt {2} \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {2} \sqrt {c}}\right )}{a^4} \] Output:
-149/64*(c-c/a/x)^(1/2)*x/a^3+107/96*(c-c/a/x)^(1/2)*x^2/a^2-17/24*(c-c/a/ x)^(1/2)*x^3/a+1/4*(c-c/a/x)^(1/2)*x^4+363/64*c^(1/2)*arctanh((c-c/a/x)^(1 /2)/c^(1/2))/a^4-4*2^(1/2)*c^(1/2)*arctanh(1/2*(c-c/a/x)^(1/2)*2^(1/2)/c^( 1/2))/a^4
Time = 0.17 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.67 \[ \int e^{-2 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}} x^3 \, dx=\frac {a \sqrt {c-\frac {c}{a x}} x \left (-447+214 a x-136 a^2 x^2+48 a^3 x^3\right )+1089 \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {c}}\right )-768 \sqrt {2} \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {2} \sqrt {c}}\right )}{192 a^4} \] Input:
Integrate[(Sqrt[c - c/(a*x)]*x^3)/E^(2*ArcCoth[a*x]),x]
Output:
(a*Sqrt[c - c/(a*x)]*x*(-447 + 214*a*x - 136*a^2*x^2 + 48*a^3*x^3) + 1089* Sqrt[c]*ArcTanh[Sqrt[c - c/(a*x)]/Sqrt[c]] - 768*Sqrt[2]*Sqrt[c]*ArcTanh[S qrt[c - c/(a*x)]/(Sqrt[2]*Sqrt[c])])/(192*a^4)
Time = 1.34 (sec) , antiderivative size = 207, normalized size of antiderivative = 1.20, number of steps used = 17, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.593, Rules used = {6717, 6683, 1070, 281, 948, 109, 27, 168, 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.
| \(\displaystyle \int x^3 \sqrt {c-\frac {c}{a x}} e^{-2 \coth ^{-1}(a x)} \, dx\) |
| \(\Big \downarrow \) 6717 |
| \(\displaystyle -\int e^{-2 \text {arctanh}(a x)} \sqrt {c-\frac {c}{a x}} x^3dx\) |
| \(\Big \downarrow \) 6683 |
| \(\displaystyle -\int \frac {\sqrt {c-\frac {c}{a x}} x^3 (1-a x)}{a x+1}dx\) |
| \(\Big \downarrow \) 1070 |
| \(\displaystyle -\int \frac {\left (\frac {1}{x}-a\right ) \sqrt {c-\frac {c}{a x}} x^3}{a+\frac {1}{x}}dx\) |
| \(\Big \downarrow \) 281 |
| \(\displaystyle \frac {a \int \frac {\left (c-\frac {c}{a x}\right )^{3/2} x^3}{a+\frac {1}{x}}dx}{c}\) |
| \(\Big \downarrow \) 948 |
| \(\displaystyle -\frac {a \int \frac {\left (c-\frac {c}{a x}\right )^{3/2} x^5}{a+\frac {1}{x}}d\frac {1}{x}}{c}\) |
| \(\Big \downarrow \) 109 |
| \(\displaystyle -\frac {a \left (-\frac {\int \frac {c^2 \left (17 a-\frac {15}{x}\right ) x^4}{2 a \left (a+\frac {1}{x}\right ) \sqrt {c-\frac {c}{a x}}}d\frac {1}{x}}{4 a}-\frac {c x^4 \sqrt {c-\frac {c}{a x}}}{4 a}\right )}{c}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {a \left (-\frac {c^2 \int \frac {\left (17 a-\frac {15}{x}\right ) x^4}{\left (a+\frac {1}{x}\right ) \sqrt {c-\frac {c}{a x}}}d\frac {1}{x}}{8 a^2}-\frac {c x^4 \sqrt {c-\frac {c}{a x}}}{4 a}\right )}{c}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle -\frac {a \left (-\frac {c^2 \left (-\frac {\int \frac {c \left (107 a-\frac {85}{x}\right ) x^3}{2 \left (a+\frac {1}{x}\right ) \sqrt {c-\frac {c}{a x}}}d\frac {1}{x}}{3 a c}-\frac {17 x^3 \sqrt {c-\frac {c}{a x}}}{3 c}\right )}{8 a^2}-\frac {c x^4 \sqrt {c-\frac {c}{a x}}}{4 a}\right )}{c}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {a \left (-\frac {c^2 \left (-\frac {\int \frac {\left (107 a-\frac {85}{x}\right ) x^3}{\left (a+\frac {1}{x}\right ) \sqrt {c-\frac {c}{a x}}}d\frac {1}{x}}{6 a}-\frac {17 x^3 \sqrt {c-\frac {c}{a x}}}{3 c}\right )}{8 a^2}-\frac {c x^4 \sqrt {c-\frac {c}{a x}}}{4 a}\right )}{c}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle -\frac {a \left (-\frac {c^2 \left (-\frac {-\frac {\int \frac {3 c \left (149 a-\frac {107}{x}\right ) x^2}{2 \left (a+\frac {1}{x}\right ) \sqrt {c-\frac {c}{a x}}}d\frac {1}{x}}{2 a c}-\frac {107 x^2 \sqrt {c-\frac {c}{a x}}}{2 c}}{6 a}-\frac {17 x^3 \sqrt {c-\frac {c}{a x}}}{3 c}\right )}{8 a^2}-\frac {c x^4 \sqrt {c-\frac {c}{a x}}}{4 a}\right )}{c}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {a \left (-\frac {c^2 \left (-\frac {-\frac {3 \int \frac {\left (149 a-\frac {107}{x}\right ) x^2}{\left (a+\frac {1}{x}\right ) \sqrt {c-\frac {c}{a x}}}d\frac {1}{x}}{4 a}-\frac {107 x^2 \sqrt {c-\frac {c}{a x}}}{2 c}}{6 a}-\frac {17 x^3 \sqrt {c-\frac {c}{a x}}}{3 c}\right )}{8 a^2}-\frac {c x^4 \sqrt {c-\frac {c}{a x}}}{4 a}\right )}{c}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle -\frac {a \left (-\frac {c^2 \left (-\frac {-\frac {3 \left (-\frac {\int \frac {c \left (363 a-\frac {149}{x}\right ) x}{2 \left (a+\frac {1}{x}\right ) \sqrt {c-\frac {c}{a x}}}d\frac {1}{x}}{a c}-\frac {149 x \sqrt {c-\frac {c}{a x}}}{c}\right )}{4 a}-\frac {107 x^2 \sqrt {c-\frac {c}{a x}}}{2 c}}{6 a}-\frac {17 x^3 \sqrt {c-\frac {c}{a x}}}{3 c}\right )}{8 a^2}-\frac {c x^4 \sqrt {c-\frac {c}{a x}}}{4 a}\right )}{c}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {a \left (-\frac {c^2 \left (-\frac {-\frac {3 \left (-\frac {\int \frac {\left (363 a-\frac {149}{x}\right ) x}{\left (a+\frac {1}{x}\right ) \sqrt {c-\frac {c}{a x}}}d\frac {1}{x}}{2 a}-\frac {149 x \sqrt {c-\frac {c}{a x}}}{c}\right )}{4 a}-\frac {107 x^2 \sqrt {c-\frac {c}{a x}}}{2 c}}{6 a}-\frac {17 x^3 \sqrt {c-\frac {c}{a x}}}{3 c}\right )}{8 a^2}-\frac {c x^4 \sqrt {c-\frac {c}{a x}}}{4 a}\right )}{c}\) |
| \(\Big \downarrow \) 174 |
| \(\displaystyle -\frac {a \left (-\frac {c^2 \left (-\frac {-\frac {3 \left (-\frac {363 \int \frac {x}{\sqrt {c-\frac {c}{a x}}}d\frac {1}{x}-512 \int \frac {1}{\left (a+\frac {1}{x}\right ) \sqrt {c-\frac {c}{a x}}}d\frac {1}{x}}{2 a}-\frac {149 x \sqrt {c-\frac {c}{a x}}}{c}\right )}{4 a}-\frac {107 x^2 \sqrt {c-\frac {c}{a x}}}{2 c}}{6 a}-\frac {17 x^3 \sqrt {c-\frac {c}{a x}}}{3 c}\right )}{8 a^2}-\frac {c x^4 \sqrt {c-\frac {c}{a x}}}{4 a}\right )}{c}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle -\frac {a \left (-\frac {c^2 \left (-\frac {-\frac {3 \left (-\frac {\frac {1024 a \int \frac {1}{2 a-\frac {a}{c x^2}}d\sqrt {c-\frac {c}{a x}}}{c}-\frac {726 a \int \frac {1}{a-\frac {a}{c x^2}}d\sqrt {c-\frac {c}{a x}}}{c}}{2 a}-\frac {149 x \sqrt {c-\frac {c}{a x}}}{c}\right )}{4 a}-\frac {107 x^2 \sqrt {c-\frac {c}{a x}}}{2 c}}{6 a}-\frac {17 x^3 \sqrt {c-\frac {c}{a x}}}{3 c}\right )}{8 a^2}-\frac {c x^4 \sqrt {c-\frac {c}{a x}}}{4 a}\right )}{c}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle -\frac {a \left (-\frac {c^2 \left (-\frac {-\frac {3 \left (-\frac {\frac {512 \sqrt {2} \text {arctanh}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {2} \sqrt {c}}\right )}{\sqrt {c}}-\frac {726 \text {arctanh}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {c}}\right )}{\sqrt {c}}}{2 a}-\frac {149 x \sqrt {c-\frac {c}{a x}}}{c}\right )}{4 a}-\frac {107 x^2 \sqrt {c-\frac {c}{a x}}}{2 c}}{6 a}-\frac {17 x^3 \sqrt {c-\frac {c}{a x}}}{3 c}\right )}{8 a^2}-\frac {c x^4 \sqrt {c-\frac {c}{a x}}}{4 a}\right )}{c}\) |
Input:
Int[(Sqrt[c - c/(a*x)]*x^3)/E^(2*ArcCoth[a*x]),x]
Output:
-((a*(-1/4*(c*Sqrt[c - c/(a*x)]*x^4)/a - (c^2*((-17*Sqrt[c - c/(a*x)]*x^3) /(3*c) - ((-107*Sqrt[c - c/(a*x)]*x^2)/(2*c) - (3*((-149*Sqrt[c - c/(a*x)] *x)/c - ((-726*ArcTanh[Sqrt[c - c/(a*x)]/Sqrt[c]])/Sqrt[c] + (512*Sqrt[2]* ArcTanh[Sqrt[c - c/(a*x)]/(Sqrt[2]*Sqrt[c])])/Sqrt[c])/(2*a)))/(4*a))/(6*a )))/(8*a^2)))/c)
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 = 0.17 (sec) , antiderivative size = 197, normalized size of antiderivative = 1.15
| method | result | size |
| risch | \(\frac {\left (48 a^{3} x^{3}-136 a^{2} x^{2}+214 a x -447\right ) x \sqrt {\frac {c \left (a x -1\right )}{a x}}}{192 a^{3}}+\frac {\left (\frac {363 \ln \left (\frac {-\frac {1}{2} a c +a^{2} c x}{\sqrt {a^{2} c}}+\sqrt {a^{2} c \,x^{2}-a c x}\right )}{128 a^{3} \sqrt {a^{2} c}}+\frac {2 \sqrt {2}\, \ln \left (\frac {4 c -3 \left (x +\frac {1}{a}\right ) a c +2 \sqrt {2}\, \sqrt {c}\, \sqrt {\left (x +\frac {1}{a}\right )^{2} a^{2} c -3 \left (x +\frac {1}{a}\right ) a c +2 c}}{x +\frac {1}{a}}\right )}{a^{4} \sqrt {c}}\right ) \sqrt {\frac {c \left (a x -1\right )}{a x}}\, \sqrt {c \left (a x -1\right ) a x}}{a x -1}\) | \(197\) |
| default | \(-\frac {\sqrt {\frac {c \left (a x -1\right )}{a x}}\, x \left (-96 x \left (a \,x^{2}-x \right )^{\frac {3}{2}} a^{\frac {9}{2}} \sqrt {\frac {1}{a}}+176 \sqrt {\frac {1}{a}}\, \left (a \,x^{2}-x \right )^{\frac {3}{2}} a^{\frac {7}{2}}-252 \sqrt {\frac {1}{a}}\, \sqrt {a \,x^{2}-x}\, a^{\frac {7}{2}} x +768 \sqrt {x \left (a x -1\right )}\, \sqrt {\frac {1}{a}}\, a^{\frac {5}{2}}+126 \sqrt {\frac {1}{a}}\, \sqrt {a \,x^{2}-x}\, a^{\frac {5}{2}}-1152 \sqrt {\frac {1}{a}}\, a^{2} \ln \left (\frac {2 \sqrt {x \left (a x -1\right )}\, \sqrt {a}+2 a x -1}{2 \sqrt {a}}\right )-768 \ln \left (\frac {2 \sqrt {2}\, \sqrt {\frac {1}{a}}\, \sqrt {x \left (a x -1\right )}\, a -3 a x +1}{a x +1}\right ) a^{\frac {3}{2}} \sqrt {2}+63 \sqrt {\frac {1}{a}}\, \ln \left (\frac {2 \sqrt {a \,x^{2}-x}\, \sqrt {a}+2 a x -1}{2 \sqrt {a}}\right ) a^{2}\right )}{384 \sqrt {x \left (a x -1\right )}\, a^{\frac {11}{2}} \sqrt {\frac {1}{a}}}\) | \(259\) |
Input:
int((c-c/a/x)^(1/2)*x^3*(a*x-1)/(a*x+1),x,method=_RETURNVERBOSE)
Output:
1/192*(48*a^3*x^3-136*a^2*x^2+214*a*x-447)/a^3*x*(c*(a*x-1)/a/x)^(1/2)+(36 3/128/a^3*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^4*2^(1/2)/c^(1/2)*ln((4*c-3*(x+1/a)*a*c+2*2^(1/2)*c^(1/2)*( (x+1/a)^2*a^2*c-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 = 0.13 (sec) , antiderivative size = 288, normalized size of antiderivative = 1.67 \[ \int e^{-2 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}} x^3 \, dx=\left [\frac {768 \, \sqrt {2} \sqrt {c} \log \left (\frac {2 \, \sqrt {2} a \sqrt {c} x \sqrt {\frac {a c x - c}{a x}} - 3 \, a c x + c}{a x + 1}\right ) + 2 \, {\left (48 \, a^{4} x^{4} - 136 \, a^{3} x^{3} + 214 \, a^{2} x^{2} - 447 \, a x\right )} \sqrt {\frac {a c x - c}{a x}} + 1089 \, \sqrt {c} \log \left (-2 \, a c x - 2 \, a \sqrt {c} x \sqrt {\frac {a c x - c}{a x}} + c\right )}{384 \, a^{4}}, \frac {768 \, \sqrt {2} \sqrt {-c} \arctan \left (\frac {\sqrt {2} a \sqrt {-c} x \sqrt {\frac {a c x - c}{a x}}}{a c x - c}\right ) + {\left (48 \, a^{4} x^{4} - 136 \, a^{3} x^{3} + 214 \, a^{2} x^{2} - 447 \, a x\right )} \sqrt {\frac {a c x - c}{a x}} - 1089 \, \sqrt {-c} \arctan \left (\frac {a \sqrt {-c} x \sqrt {\frac {a c x - c}{a x}}}{a c x - c}\right )}{192 \, a^{4}}\right ] \] Input:
integrate((c-c/a/x)^(1/2)*x^3*(a*x-1)/(a*x+1),x, algorithm="fricas")
Output:
[1/384*(768*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*(48*a^4*x^4 - 136*a^3*x^3 + 214*a^2*x^2 - 447*a*x)*sqrt((a*c*x - c)/(a*x)) + 1089*sqrt(c)*log(-2*a*c*x - 2*a*sqrt (c)*x*sqrt((a*c*x - c)/(a*x)) + c))/a^4, 1/192*(768*sqrt(2)*sqrt(-c)*arcta n(sqrt(2)*a*sqrt(-c)*x*sqrt((a*c*x - c)/(a*x))/(a*c*x - c)) + (48*a^4*x^4 - 136*a^3*x^3 + 214*a^2*x^2 - 447*a*x)*sqrt((a*c*x - c)/(a*x)) - 1089*sqrt (-c)*arctan(a*sqrt(-c)*x*sqrt((a*c*x - c)/(a*x))/(a*c*x - c)))/a^4]
\[ \int e^{-2 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}} x^3 \, dx=\int \frac {x^{3} \sqrt {- c \left (-1 + \frac {1}{a x}\right )} \left (a x - 1\right )}{a x + 1}\, dx \] Input:
integrate((c-c/a/x)**(1/2)*x**3*(a*x-1)/(a*x+1),x)
Output:
Integral(x**3*sqrt(-c*(-1 + 1/(a*x)))*(a*x - 1)/(a*x + 1), x)
\[ \int e^{-2 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}} x^3 \, dx=\int { \frac {{\left (a x - 1\right )} \sqrt {c - \frac {c}{a x}} x^{3}}{a x + 1} \,d x } \] Input:
integrate((c-c/a/x)^(1/2)*x^3*(a*x-1)/(a*x+1),x, algorithm="maxima")
Output:
integrate((a*x - 1)*sqrt(c - c/(a*x))*x^3/(a*x + 1), x)
Exception generated. \[ \int e^{-2 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}} x^3 \, dx=\text {Exception raised: TypeError} \] Input:
integrate((c-c/a/x)^(1/2)*x^3*(a*x-1)/(a*x+1),x, algorithm="giac")
Output:
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Error: Bad Argument Type
Timed out. \[ \int e^{-2 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}} x^3 \, dx=\int \frac {x^3\,\sqrt {c-\frac {c}{a\,x}}\,\left (a\,x-1\right )}{a\,x+1} \,d x \] Input:
int((x^3*(c - c/(a*x))^(1/2)*(a*x - 1))/(a*x + 1),x)
Output:
int((x^3*(c - c/(a*x))^(1/2)*(a*x - 1))/(a*x + 1), x)
Time = 0.16 (sec) , antiderivative size = 159, normalized size of antiderivative = 0.92 \[ \int e^{-2 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}} x^3 \, dx=\frac {\sqrt {c}\, \left (48 \sqrt {x}\, \sqrt {a}\, \sqrt {a x -1}\, a^{3} x^{3}-136 \sqrt {x}\, \sqrt {a}\, \sqrt {a x -1}\, a^{2} x^{2}+214 \sqrt {x}\, \sqrt {a}\, \sqrt {a x -1}\, a x -447 \sqrt {x}\, \sqrt {a}\, \sqrt {a x -1}-384 \sqrt {2}\, \mathrm {log}\left (\sqrt {a x -1}+\sqrt {x}\, \sqrt {a}-\sqrt {2}\, i +i \right )-384 \sqrt {2}\, \mathrm {log}\left (\sqrt {a x -1}+\sqrt {x}\, \sqrt {a}+\sqrt {2}\, i -i \right )+384 \sqrt {2}\, \mathrm {log}\left (2 \sqrt {x}\, \sqrt {a}\, \sqrt {a x -1}+2 \sqrt {2}+2 a x +2\right )+1089 \,\mathrm {log}\left (\sqrt {a x -1}+\sqrt {x}\, \sqrt {a}\right )\right )}{192 a^{4}} \] Input:
int((c-c/a/x)^(1/2)*x^3*(a*x-1)/(a*x+1),x)
Output:
(sqrt(c)*(48*sqrt(x)*sqrt(a)*sqrt(a*x - 1)*a**3*x**3 - 136*sqrt(x)*sqrt(a) *sqrt(a*x - 1)*a**2*x**2 + 214*sqrt(x)*sqrt(a)*sqrt(a*x - 1)*a*x - 447*sqr t(x)*sqrt(a)*sqrt(a*x - 1) - 384*sqrt(2)*log(sqrt(a*x - 1) + sqrt(x)*sqrt( a) - sqrt(2)*i + i) - 384*sqrt(2)*log(sqrt(a*x - 1) + sqrt(x)*sqrt(a) + sq rt(2)*i - i) + 384*sqrt(2)*log(2*sqrt(x)*sqrt(a)*sqrt(a*x - 1) + 2*sqrt(2) + 2*a*x + 2) + 1089*log(sqrt(a*x - 1) + sqrt(x)*sqrt(a))))/(192*a**4)