Integrand size = 23, antiderivative size = 275 \[ \int \frac {e^{3 \coth ^{-1}(a x)} \sqrt {c-a c x}}{x^4} \, dx=\frac {a \left (1+\frac {1}{a x}\right )^{3/2} \sqrt {c-a c x}}{3 \sqrt {1-\frac {1}{a x}} x^2}+\frac {13 a^2 \sqrt {1+\frac {1}{a x}} \sqrt {c-a c x}}{8 \sqrt {1-\frac {1}{a x}} x}+\frac {3 a^2 \left (1+\frac {1}{a x}\right )^{3/2} \sqrt {c-a c x}}{4 \sqrt {1-\frac {1}{a x}} x}+\frac {45 a^3 \sqrt {\frac {1}{a x}} \sqrt {c-a c x} \text {arcsinh}\left (\sqrt {\frac {1}{a x}}\right )}{8 \sqrt {1-\frac {1}{a x}}}-\frac {4 \sqrt {2} a^3 \sqrt {\frac {1}{a x}} \sqrt {c-a c x} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {\frac {1}{a x}}}{\sqrt {1+\frac {1}{a x}}}\right )}{\sqrt {1-\frac {1}{a x}}} \] Output:
1/3*a*(1+1/a/x)^(3/2)*(-a*c*x+c)^(1/2)/(1-1/a/x)^(1/2)/x^2+13/8*a^2*(1+1/a /x)^(1/2)*(-a*c*x+c)^(1/2)/(1-1/a/x)^(1/2)/x+3/4*a^2*(1+1/a/x)^(3/2)*(-a*c *x+c)^(1/2)/(1-1/a/x)^(1/2)/x+45/8*a^3*(1/a/x)^(1/2)*(-a*c*x+c)^(1/2)*arcs inh((1/a/x)^(1/2))/(1-1/a/x)^(1/2)-4*2^(1/2)*a^3*(1/a/x)^(1/2)*(-a*c*x+c)^ (1/2)*arctanh(2^(1/2)*(1/a/x)^(1/2)/(1+1/a/x)^(1/2))/(1-1/a/x)^(1/2)
Time = 0.30 (sec) , antiderivative size = 140, normalized size of antiderivative = 0.51 \[ \int \frac {e^{3 \coth ^{-1}(a x)} \sqrt {c-a c x}}{x^4} \, dx=\frac {\sqrt {c-a c x} \left (\sqrt {1+\frac {1}{a x}} \left (8+26 a x+57 a^2 x^2\right )+\frac {135 a^{5/2} \text {arcsinh}\left (\frac {\sqrt {\frac {1}{x}}}{\sqrt {a}}\right )}{\left (\frac {1}{x}\right )^{5/2}}-\frac {96 \sqrt {2} a^{5/2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {\frac {1}{x}}}{\sqrt {a} \sqrt {1+\frac {1}{a x}}}\right )}{\left (\frac {1}{x}\right )^{5/2}}\right )}{24 \sqrt {1-\frac {1}{a x}} x^3} \] Input:
Integrate[(E^(3*ArcCoth[a*x])*Sqrt[c - a*c*x])/x^4,x]
Output:
(Sqrt[c - a*c*x]*(Sqrt[1 + 1/(a*x)]*(8 + 26*a*x + 57*a^2*x^2) + (135*a^(5/ 2)*ArcSinh[Sqrt[x^(-1)]/Sqrt[a]])/(x^(-1))^(5/2) - (96*Sqrt[2]*a^(5/2)*Arc Tanh[(Sqrt[2]*Sqrt[x^(-1)])/(Sqrt[a]*Sqrt[1 + 1/(a*x)])])/(x^(-1))^(5/2))) /(24*Sqrt[1 - 1/(a*x)]*x^3)
Time = 0.66 (sec) , antiderivative size = 188, normalized size of antiderivative = 0.68, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.565, Rules used = {6730, 27, 112, 27, 171, 27, 171, 27, 175, 63, 104, 219, 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 |
| \(\displaystyle -\frac {\sqrt {\frac {1}{x}} \sqrt {c-a c x} \int \frac {a \left (1+\frac {1}{a x}\right )^{3/2} \left (\frac {1}{x}\right )^{3/2}}{a-\frac {1}{x}}d\frac {1}{x}}{\sqrt {1-\frac {1}{a x}}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {a \sqrt {\frac {1}{x}} \sqrt {c-a c x} \int \frac {\left (1+\frac {1}{a x}\right )^{3/2} \left (\frac {1}{x}\right )^{3/2}}{a-\frac {1}{x}}d\frac {1}{x}}{\sqrt {1-\frac {1}{a x}}}\) |
| \(\Big \downarrow \) 112 |
| \(\displaystyle -\frac {a \sqrt {\frac {1}{x}} \sqrt {c-a c x} \left (\frac {1}{3} \int \frac {3 \left (a+\frac {3}{x}\right ) \sqrt {1+\frac {1}{a x}} \sqrt {\frac {1}{x}}}{2 \left (a-\frac {1}{x}\right )}d\frac {1}{x}-\frac {1}{3} \left (\frac {1}{x}\right )^{3/2} \left (\frac {1}{a x}+1\right )^{3/2}\right )}{\sqrt {1-\frac {1}{a x}}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {a \sqrt {\frac {1}{x}} \sqrt {c-a c x} \left (\frac {1}{2} \int \frac {\left (a+\frac {3}{x}\right ) \sqrt {1+\frac {1}{a x}} \sqrt {\frac {1}{x}}}{a-\frac {1}{x}}d\frac {1}{x}-\frac {1}{3} \left (\frac {1}{x}\right )^{3/2} \left (\frac {1}{a x}+1\right )^{3/2}\right )}{\sqrt {1-\frac {1}{a x}}}\) |
| \(\Big \downarrow \) 171 |
| \(\displaystyle -\frac {a \sqrt {\frac {1}{x}} \sqrt {c-a c x} \left (\frac {1}{2} \left (-\frac {1}{2} a \int -\frac {\left (3 a+\frac {13}{x}\right ) \sqrt {1+\frac {1}{a x}}}{2 \left (a-\frac {1}{x}\right ) \sqrt {\frac {1}{x}}}d\frac {1}{x}-\frac {3}{2} a \sqrt {\frac {1}{x}} \left (\frac {1}{a x}+1\right )^{3/2}\right )-\frac {1}{3} \left (\frac {1}{x}\right )^{3/2} \left (\frac {1}{a x}+1\right )^{3/2}\right )}{\sqrt {1-\frac {1}{a x}}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {a \sqrt {\frac {1}{x}} \sqrt {c-a c x} \left (\frac {1}{2} \left (\frac {1}{4} a \int \frac {\left (3 a+\frac {13}{x}\right ) \sqrt {1+\frac {1}{a x}}}{\left (a-\frac {1}{x}\right ) \sqrt {\frac {1}{x}}}d\frac {1}{x}-\frac {3}{2} a \sqrt {\frac {1}{x}} \left (\frac {1}{a x}+1\right )^{3/2}\right )-\frac {1}{3} \left (\frac {1}{x}\right )^{3/2} \left (\frac {1}{a x}+1\right )^{3/2}\right )}{\sqrt {1-\frac {1}{a x}}}\) |
| \(\Big \downarrow \) 171 |
| \(\displaystyle -\frac {a \sqrt {\frac {1}{x}} \sqrt {c-a c x} \left (\frac {1}{2} \left (\frac {1}{4} a \left (-\int -\frac {19 a+\frac {45}{x}}{2 \left (a-\frac {1}{x}\right ) \sqrt {1+\frac {1}{a x}} \sqrt {\frac {1}{x}}}d\frac {1}{x}-13 \sqrt {\frac {1}{x}} \sqrt {\frac {1}{a x}+1}\right )-\frac {3}{2} a \sqrt {\frac {1}{x}} \left (\frac {1}{a x}+1\right )^{3/2}\right )-\frac {1}{3} \left (\frac {1}{x}\right )^{3/2} \left (\frac {1}{a x}+1\right )^{3/2}\right )}{\sqrt {1-\frac {1}{a x}}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {a \sqrt {\frac {1}{x}} \sqrt {c-a c x} \left (\frac {1}{2} \left (\frac {1}{4} a \left (\frac {1}{2} \int \frac {19 a+\frac {45}{x}}{\left (a-\frac {1}{x}\right ) \sqrt {1+\frac {1}{a x}} \sqrt {\frac {1}{x}}}d\frac {1}{x}-13 \sqrt {\frac {1}{x}} \sqrt {\frac {1}{a x}+1}\right )-\frac {3}{2} a \sqrt {\frac {1}{x}} \left (\frac {1}{a x}+1\right )^{3/2}\right )-\frac {1}{3} \left (\frac {1}{x}\right )^{3/2} \left (\frac {1}{a x}+1\right )^{3/2}\right )}{\sqrt {1-\frac {1}{a x}}}\) |
| \(\Big \downarrow \) 175 |
| \(\displaystyle -\frac {a \sqrt {\frac {1}{x}} \sqrt {c-a c x} \left (\frac {1}{2} \left (\frac {1}{4} a \left (\frac {1}{2} \left (64 a \int \frac {1}{\left (a-\frac {1}{x}\right ) \sqrt {1+\frac {1}{a x}} \sqrt {\frac {1}{x}}}d\frac {1}{x}-45 \int \frac {1}{\sqrt {1+\frac {1}{a x}} \sqrt {\frac {1}{x}}}d\frac {1}{x}\right )-13 \sqrt {\frac {1}{x}} \sqrt {\frac {1}{a x}+1}\right )-\frac {3}{2} a \sqrt {\frac {1}{x}} \left (\frac {1}{a x}+1\right )^{3/2}\right )-\frac {1}{3} \left (\frac {1}{x}\right )^{3/2} \left (\frac {1}{a x}+1\right )^{3/2}\right )}{\sqrt {1-\frac {1}{a x}}}\) |
| \(\Big \downarrow \) 63 |
| \(\displaystyle -\frac {a \sqrt {\frac {1}{x}} \sqrt {c-a c x} \left (\frac {1}{2} \left (\frac {1}{4} a \left (\frac {1}{2} \left (64 a \int \frac {1}{\left (a-\frac {1}{x}\right ) \sqrt {1+\frac {1}{a x}} \sqrt {\frac {1}{x}}}d\frac {1}{x}-90 \int \frac {1}{\sqrt {1+\frac {1}{x^2 a}}}d\sqrt {\frac {1}{x}}\right )-13 \sqrt {\frac {1}{x}} \sqrt {\frac {1}{a x}+1}\right )-\frac {3}{2} a \sqrt {\frac {1}{x}} \left (\frac {1}{a x}+1\right )^{3/2}\right )-\frac {1}{3} \left (\frac {1}{x}\right )^{3/2} \left (\frac {1}{a x}+1\right )^{3/2}\right )}{\sqrt {1-\frac {1}{a x}}}\) |
| \(\Big \downarrow \) 104 |
| \(\displaystyle -\frac {a \sqrt {\frac {1}{x}} \sqrt {c-a c x} \left (\frac {1}{2} \left (\frac {1}{4} a \left (\frac {1}{2} \left (128 a \int \frac {1}{a-\frac {2}{x^2}}d\frac {\sqrt {\frac {1}{x}}}{\sqrt {1+\frac {1}{a x}}}-90 \int \frac {1}{\sqrt {1+\frac {1}{x^2 a}}}d\sqrt {\frac {1}{x}}\right )-13 \sqrt {\frac {1}{x}} \sqrt {\frac {1}{a x}+1}\right )-\frac {3}{2} a \sqrt {\frac {1}{x}} \left (\frac {1}{a x}+1\right )^{3/2}\right )-\frac {1}{3} \left (\frac {1}{x}\right )^{3/2} \left (\frac {1}{a x}+1\right )^{3/2}\right )}{\sqrt {1-\frac {1}{a x}}}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle -\frac {a \sqrt {\frac {1}{x}} \sqrt {c-a c x} \left (\frac {1}{2} \left (\frac {1}{4} a \left (\frac {1}{2} \left (64 \sqrt {2} \sqrt {a} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {\frac {1}{x}}}{\sqrt {a} \sqrt {\frac {1}{a x}+1}}\right )-90 \int \frac {1}{\sqrt {1+\frac {1}{x^2 a}}}d\sqrt {\frac {1}{x}}\right )-13 \sqrt {\frac {1}{x}} \sqrt {\frac {1}{a x}+1}\right )-\frac {3}{2} a \sqrt {\frac {1}{x}} \left (\frac {1}{a x}+1\right )^{3/2}\right )-\frac {1}{3} \left (\frac {1}{x}\right )^{3/2} \left (\frac {1}{a x}+1\right )^{3/2}\right )}{\sqrt {1-\frac {1}{a x}}}\) |
| \(\Big \downarrow \) 222 |
| \(\displaystyle -\frac {a \sqrt {\frac {1}{x}} \sqrt {c-a c x} \left (\frac {1}{2} \left (\frac {1}{4} a \left (\frac {1}{2} \left (64 \sqrt {2} \sqrt {a} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {\frac {1}{x}}}{\sqrt {a} \sqrt {\frac {1}{a x}+1}}\right )-90 \sqrt {a} \text {arcsinh}\left (\frac {\sqrt {\frac {1}{x}}}{\sqrt {a}}\right )\right )-13 \sqrt {\frac {1}{x}} \sqrt {\frac {1}{a x}+1}\right )-\frac {3}{2} a \sqrt {\frac {1}{x}} \left (\frac {1}{a x}+1\right )^{3/2}\right )-\frac {1}{3} \left (\frac {1}{x}\right )^{3/2} \left (\frac {1}{a x}+1\right )^{3/2}\right )}{\sqrt {1-\frac {1}{a x}}}\) |
Input:
Int[(E^(3*ArcCoth[a*x])*Sqrt[c - a*c*x])/x^4,x]
Output:
-((a*Sqrt[x^(-1)]*Sqrt[c - a*c*x]*(-1/3*((1 + 1/(a*x))^(3/2)*(x^(-1))^(3/2 )) + ((-3*a*(1 + 1/(a*x))^(3/2)*Sqrt[x^(-1)])/2 + (a*(-13*Sqrt[1 + 1/(a*x) ]*Sqrt[x^(-1)] + (-90*Sqrt[a]*ArcSinh[Sqrt[x^(-1)]/Sqrt[a]] + 64*Sqrt[2]*S qrt[a]*ArcTanh[(Sqrt[2]*Sqrt[x^(-1)])/(Sqrt[a]*Sqrt[1 + 1/(a*x)])])/2))/4) /2))/Sqrt[1 - 1/(a*x)])
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, 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_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x _)), x_] :> With[{q = Denominator[m]}, Simp[q Subst[Int[x^(q*(m + 1) - 1) /(b*e - a*f - (d*e - c*f)*x^q), x], x, (a + b*x)^(1/q)/(c + d*x)^(1/q)], x] ] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[m + n + 1, 0] && RationalQ[n] && L tQ[-1, m, 0] && SimplerQ[a + b*x, c + d*x]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Simp[(a + b*x)^m*(c + d*x)^n*((e + f*x)^(p + 1)/(f*(m + n + p + 1))), x] - Simp[1/(f*(m + n + p + 1)) Int[(a + b*x)^(m - 1)*(c + d*x) ^(n - 1)*(e + f*x)^p*Simp[c*m*(b*e - a*f) + a*n*(d*e - c*f) + (d*m*(b*e - a *f) + b*n*(d*e - c*f))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && GtQ[m, 0] && GtQ[n, 0] && NeQ[m + n + p + 1, 0] && (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[h*(a + b*x)^m*(c + d*x)^(n + 1)*(( e + f*x)^(p + 1)/(d*f*(m + n + p + 2))), x] + Simp[1/(d*f*(m + n + p + 2)) Int[(a + b*x)^(m - 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*g*(m + n + p + 2 ) - h*(b*c*e*m + a*(d*e*(n + 1) + c*f*(p + 1))) + (b*d*f*g*(m + n + p + 2) + h*(a*d*f*m - b*(d*e*(m + n + 1) + c*f*(m + p + 1))))*x, x], x], x] /; Fre eQ[{a, b, c, d, e, f, g, h, n, p}, x] && GtQ[m, 0] && NeQ[m + n + p + 2, 0] && IntegersQ[2*m, 2*n, 2*p]
Int[(((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_ )))/((a_.) + (b_.)*(x_)), x_] :> Simp[h/b Int[(c + d*x)^n*(e + f*x)^p, x] , x] + Simp[(b*g - a*h)/b Int[(c + d*x)^n*((e + f*x)^p/(a + b*x)), x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
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 = 0.18 (sec) , antiderivative size = 165, normalized size of antiderivative = 0.60
| method | result | size |
| default | \(\frac {\left (a x -1\right ) \sqrt {-c \left (a x -1\right )}\, \left (-96 \sqrt {2}\, \arctan \left (\frac {\sqrt {-c \left (a x +1\right )}\, \sqrt {2}}{2 \sqrt {c}}\right ) a^{3} c \,x^{3}+135 c \arctan \left (\frac {\sqrt {-c \left (a x +1\right )}}{\sqrt {c}}\right ) a^{3} x^{3}+57 a^{2} x^{2} \sqrt {-c \left (a x +1\right )}\, \sqrt {c}+26 a x \sqrt {-c \left (a x +1\right )}\, \sqrt {c}+8 \sqrt {-c \left (a x +1\right )}\, \sqrt {c}\right )}{24 \left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}} \left (a x +1\right ) \sqrt {c}\, \sqrt {-c \left (a x +1\right )}\, x^{3}}\) | \(165\) |
| risch | \(-\frac {\left (57 a^{3} x^{3}+83 a^{2} x^{2}+34 a x +8\right ) c \left (a x -1\right )}{24 x^{3} \sqrt {\frac {a x -1}{a x +1}}\, \left (a x +1\right ) \sqrt {-c \left (a x -1\right )}}-\frac {\left (\frac {4 a^{3} \sqrt {2}\, \arctan \left (\frac {\sqrt {-a c x -c}\, \sqrt {2}}{2 \sqrt {c}}\right )}{\sqrt {c}}-\frac {45 a^{3} \arctan \left (\frac {\sqrt {-a c x -c}}{\sqrt {c}}\right )}{8 \sqrt {c}}\right ) c \sqrt {-c \left (a x +1\right )}\, \left (a x -1\right )}{\sqrt {\frac {a x -1}{a x +1}}\, \left (a x +1\right ) \sqrt {-c \left (a x -1\right )}}\) | \(173\) |
Input:
int(1/((a*x-1)/(a*x+1))^(3/2)*(-a*c*x+c)^(1/2)/x^4,x,method=_RETURNVERBOSE )
Output:
1/24*(a*x-1)*(-c*(a*x-1))^(1/2)*(-96*2^(1/2)*arctan(1/2*(-c*(a*x+1))^(1/2) *2^(1/2)/c^(1/2))*a^3*c*x^3+135*c*arctan((-c*(a*x+1))^(1/2)/c^(1/2))*a^3*x ^3+57*a^2*x^2*(-c*(a*x+1))^(1/2)*c^(1/2)+26*a*x*(-c*(a*x+1))^(1/2)*c^(1/2) +8*(-c*(a*x+1))^(1/2)*c^(1/2))/((a*x-1)/(a*x+1))^(3/2)/(a*x+1)/c^(1/2)/(-c *(a*x+1))^(1/2)/x^3
Time = 0.13 (sec) , antiderivative size = 455, normalized size of antiderivative = 1.65 \[ \int \frac {e^{3 \coth ^{-1}(a x)} \sqrt {c-a c x}}{x^4} \, dx=\left [\frac {96 \, \sqrt {2} {\left (a^{4} x^{4} - a^{3} x^{3}\right )} \sqrt {-c} \log \left (-\frac {a^{2} c x^{2} + 2 \, a c x + 2 \, \sqrt {2} \sqrt {-a c x + c} {\left (a x + 1\right )} \sqrt {-c} \sqrt {\frac {a x - 1}{a x + 1}} - 3 \, c}{a^{2} x^{2} - 2 \, a x + 1}\right ) + 135 \, {\left (a^{4} x^{4} - a^{3} x^{3}\right )} \sqrt {-c} \log \left (-\frac {a^{2} c x^{2} + a c x - 2 \, \sqrt {-a c x + c} {\left (a x + 1\right )} \sqrt {-c} \sqrt {\frac {a x - 1}{a x + 1}} - 2 \, c}{a x^{2} - x}\right ) + 2 \, {\left (57 \, a^{3} x^{3} + 83 \, a^{2} x^{2} + 34 \, a x + 8\right )} \sqrt {-a c x + c} \sqrt {\frac {a x - 1}{a x + 1}}}{48 \, {\left (a x^{4} - x^{3}\right )}}, -\frac {96 \, \sqrt {2} {\left (a^{4} x^{4} - a^{3} x^{3}\right )} \sqrt {c} \arctan \left (\frac {\sqrt {2} \sqrt {-a c x + c} {\left (a x + 1\right )} \sqrt {c} \sqrt {\frac {a x - 1}{a x + 1}}}{2 \, {\left (a c x - c\right )}}\right ) - 135 \, {\left (a^{4} x^{4} - a^{3} x^{3}\right )} \sqrt {c} \arctan \left (\frac {\sqrt {-a c x + c} {\left (a x + 1\right )} \sqrt {c} \sqrt {\frac {a x - 1}{a x + 1}}}{a c x - c}\right ) - {\left (57 \, a^{3} x^{3} + 83 \, a^{2} x^{2} + 34 \, a x + 8\right )} \sqrt {-a c x + c} \sqrt {\frac {a x - 1}{a x + 1}}}{24 \, {\left (a x^{4} - x^{3}\right )}}\right ] \] Input:
integrate(1/((a*x-1)/(a*x+1))^(3/2)*(-a*c*x+c)^(1/2)/x^4,x, algorithm="fri cas")
Output:
[1/48*(96*sqrt(2)*(a^4*x^4 - a^3*x^3)*sqrt(-c)*log(-(a^2*c*x^2 + 2*a*c*x + 2*sqrt(2)*sqrt(-a*c*x + c)*(a*x + 1)*sqrt(-c)*sqrt((a*x - 1)/(a*x + 1)) - 3*c)/(a^2*x^2 - 2*a*x + 1)) + 135*(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*(57*a^3*x^3 + 83*a^2*x^2 + 34*a*x + 8)*sqrt( -a*c*x + c)*sqrt((a*x - 1)/(a*x + 1)))/(a*x^4 - x^3), -1/24*(96*sqrt(2)*(a ^4*x^4 - a^3*x^3)*sqrt(c)*arctan(1/2*sqrt(2)*sqrt(-a*c*x + c)*(a*x + 1)*sq rt(c)*sqrt((a*x - 1)/(a*x + 1))/(a*c*x - c)) - 135*(a^4*x^4 - a^3*x^3)*sqr t(c)*arctan(sqrt(-a*c*x + c)*(a*x + 1)*sqrt(c)*sqrt((a*x - 1)/(a*x + 1))/( a*c*x - c)) - (57*a^3*x^3 + 83*a^2*x^2 + 34*a*x + 8)*sqrt(-a*c*x + c)*sqrt ((a*x - 1)/(a*x + 1)))/(a*x^4 - x^3)]
Timed out. \[ \int \frac {e^{3 \coth ^{-1}(a x)} \sqrt {c-a c x}}{x^4} \, dx=\text {Timed out} \] Input:
integrate(1/((a*x-1)/(a*x+1))**(3/2)*(-a*c*x+c)**(1/2)/x**4,x)
Output:
Timed out
\[ \int \frac {e^{3 \coth ^{-1}(a x)} \sqrt {c-a c x}}{x^4} \, dx=\int { \frac {\sqrt {-a c x + c}}{x^{4} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}}} \,d x } \] Input:
integrate(1/((a*x-1)/(a*x+1))^(3/2)*(-a*c*x+c)^(1/2)/x^4,x, algorithm="max ima")
Output:
integrate(sqrt(-a*c*x + c)/(x^4*((a*x - 1)/(a*x + 1))^(3/2)), x)
Exception generated. \[ \int \frac {e^{3 \coth ^{-1}(a x)} \sqrt {c-a c x}}{x^4} \, dx=\text {Exception raised: TypeError} \] Input:
integrate(1/((a*x-1)/(a*x+1))^(3/2)*(-a*c*x+c)^(1/2)/x^4,x, algorithm="gia c")
Output:
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. \[ \int \frac {e^{3 \coth ^{-1}(a x)} \sqrt {c-a c x}}{x^4} \, dx=\int \frac {\sqrt {c-a\,c\,x}}{x^4\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/2}} \,d x \] Input:
int((c - a*c*x)^(1/2)/(x^4*((a*x - 1)/(a*x + 1))^(3/2)),x)
Output:
int((c - a*c*x)^(1/2)/(x^4*((a*x - 1)/(a*x + 1))^(3/2)), x)
Time = 0.17 (sec) , antiderivative size = 198, normalized size of antiderivative = 0.72 \[ \int \frac {e^{3 \coth ^{-1}(a x)} \sqrt {c-a c x}}{x^4} \, dx=\frac {\sqrt {c}\, i \left (-114 \sqrt {a x +1}\, a^{2} x^{2}-52 \sqrt {a x +1}\, a x -16 \sqrt {a x +1}-192 \sqrt {2}\, \mathrm {log}\left (\tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {-a x +1}}{\sqrt {2}}\right )}{2}\right )\right ) a^{3} x^{3}+182 \sqrt {2}\, a^{3} x^{3}-135 \,\mathrm {log}\left (-\sqrt {2}+\tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {-a x +1}}{\sqrt {2}}\right )}{2}\right )-1\right ) a^{3} x^{3}+135 \,\mathrm {log}\left (-\sqrt {2}+\tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {-a x +1}}{\sqrt {2}}\right )}{2}\right )+1\right ) a^{3} x^{3}+135 \,\mathrm {log}\left (\sqrt {2}+\tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {-a x +1}}{\sqrt {2}}\right )}{2}\right )-1\right ) a^{3} x^{3}-135 \,\mathrm {log}\left (\sqrt {2}+\tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {-a x +1}}{\sqrt {2}}\right )}{2}\right )+1\right ) a^{3} x^{3}\right )}{48 x^{3}} \] Input:
int(1/((a*x-1)/(a*x+1))^(3/2)*(-a*c*x+c)^(1/2)/x^4,x)
Output:
(sqrt(c)*i*( - 114*sqrt(a*x + 1)*a**2*x**2 - 52*sqrt(a*x + 1)*a*x - 16*sqr t(a*x + 1) - 192*sqrt(2)*log(tan(asin(sqrt( - a*x + 1)/sqrt(2))/2))*a**3*x **3 + 182*sqrt(2)*a**3*x**3 - 135*log( - sqrt(2) + tan(asin(sqrt( - a*x + 1)/sqrt(2))/2) - 1)*a**3*x**3 + 135*log( - sqrt(2) + tan(asin(sqrt( - a*x + 1)/sqrt(2))/2) + 1)*a**3*x**3 + 135*log(sqrt(2) + tan(asin(sqrt( - a*x + 1)/sqrt(2))/2) - 1)*a**3*x**3 - 135*log(sqrt(2) + tan(asin(sqrt( - a*x + 1)/sqrt(2))/2) + 1)*a**3*x**3))/(48*x**3)