Integrand size = 27, antiderivative size = 237 \[ \int \frac {e^{3 \text {arctanh}(a x)} \sqrt {c-\frac {c}{a x}}}{x^4} \, dx=-\frac {104 a^3 \sqrt {c-\frac {c}{a x}} \sqrt {1+a x}}{21 \sqrt {1-a x}}-\frac {2 \sqrt {c-\frac {c}{a x}} \sqrt {1+a x}}{7 x^3 \sqrt {1-a x}}-\frac {6 a \sqrt {c-\frac {c}{a x}} \sqrt {1+a x}}{7 x^2 \sqrt {1-a x}}-\frac {32 a^2 \sqrt {c-\frac {c}{a x}} \sqrt {1+a x}}{21 x \sqrt {1-a x}}+\frac {4 \sqrt {2} a^{7/2} \sqrt {c-\frac {c}{a x}} \sqrt {x} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {x}}{\sqrt {1+a x}}\right )}{\sqrt {1-a x}} \]
4*a^(7/2)*arctanh(2^(1/2)*a^(1/2)*x^(1/2)/(a*x+1)^(1/2))*2^(1/2)*(c-c/a/x) ^(1/2)*x^(1/2)/(-a*x+1)^(1/2)-104/21*a^3*(c-c/a/x)^(1/2)*(a*x+1)^(1/2)/(-a *x+1)^(1/2)-2/7*(c-c/a/x)^(1/2)*(a*x+1)^(1/2)/x^3/(-a*x+1)^(1/2)-6/7*a*(c- c/a/x)^(1/2)*(a*x+1)^(1/2)/x^2/(-a*x+1)^(1/2)-32/21*a^2*(c-c/a/x)^(1/2)*(a *x+1)^(1/2)/x/(-a*x+1)^(1/2)
Time = 0.05 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.46 \[ \int \frac {e^{3 \text {arctanh}(a x)} \sqrt {c-\frac {c}{a x}}}{x^4} \, dx=\frac {2 \sqrt {c-\frac {c}{a x}} \left (-\sqrt {1+a x} \left (3+9 a x+16 a^2 x^2+52 a^3 x^3\right )+42 \sqrt {2} a^{7/2} x^{7/2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {x}}{\sqrt {1+a x}}\right )\right )}{21 x^3 \sqrt {1-a x}} \]
(2*Sqrt[c - c/(a*x)]*(-(Sqrt[1 + a*x]*(3 + 9*a*x + 16*a^2*x^2 + 52*a^3*x^3 )) + 42*Sqrt[2]*a^(7/2)*x^(7/2)*ArcTanh[(Sqrt[2]*Sqrt[a]*Sqrt[x])/Sqrt[1 + a*x]]))/(21*x^3*Sqrt[1 - a*x])
Time = 0.54 (sec) , antiderivative size = 154, normalized size of antiderivative = 0.65, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.444, Rules used = {6684, 6679, 109, 27, 169, 27, 169, 27, 169, 27, 104, 219}
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.
| \(\displaystyle \int \frac {e^{3 \text {arctanh}(a x)} \sqrt {c-\frac {c}{a x}}}{x^4} \, dx\) |
| \(\Big \downarrow \) 6684 |
| \(\displaystyle \frac {\sqrt {x} \sqrt {c-\frac {c}{a x}} \int \frac {e^{3 \text {arctanh}(a x)} \sqrt {1-a x}}{x^{9/2}}dx}{\sqrt {1-a x}}\) |
| \(\Big \downarrow \) 6679 |
| \(\displaystyle \frac {\sqrt {x} \sqrt {c-\frac {c}{a x}} \int \frac {(a x+1)^{3/2}}{x^{9/2} (1-a x)}dx}{\sqrt {1-a x}}\) |
| \(\Big \downarrow \) 109 |
| \(\displaystyle \frac {\sqrt {x} \sqrt {c-\frac {c}{a x}} \left (-\frac {2}{7} \int -\frac {a (13 a x+15)}{2 x^{7/2} (1-a x) \sqrt {a x+1}}dx-\frac {2 \sqrt {a x+1}}{7 x^{7/2}}\right )}{\sqrt {1-a x}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sqrt {x} \sqrt {c-\frac {c}{a x}} \left (\frac {1}{7} a \int \frac {13 a x+15}{x^{7/2} (1-a x) \sqrt {a x+1}}dx-\frac {2 \sqrt {a x+1}}{7 x^{7/2}}\right )}{\sqrt {1-a x}}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle \frac {\sqrt {x} \sqrt {c-\frac {c}{a x}} \left (\frac {1}{7} a \left (-\frac {2}{5} \int -\frac {10 a (3 a x+4)}{x^{5/2} (1-a x) \sqrt {a x+1}}dx-\frac {6 \sqrt {a x+1}}{x^{5/2}}\right )-\frac {2 \sqrt {a x+1}}{7 x^{7/2}}\right )}{\sqrt {1-a x}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sqrt {x} \sqrt {c-\frac {c}{a x}} \left (\frac {1}{7} a \left (4 a \int \frac {3 a x+4}{x^{5/2} (1-a x) \sqrt {a x+1}}dx-\frac {6 \sqrt {a x+1}}{x^{5/2}}\right )-\frac {2 \sqrt {a x+1}}{7 x^{7/2}}\right )}{\sqrt {1-a x}}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle \frac {\sqrt {x} \sqrt {c-\frac {c}{a x}} \left (\frac {1}{7} a \left (4 a \left (-\frac {2}{3} \int -\frac {a (8 a x+13)}{2 x^{3/2} (1-a x) \sqrt {a x+1}}dx-\frac {8 \sqrt {a x+1}}{3 x^{3/2}}\right )-\frac {6 \sqrt {a x+1}}{x^{5/2}}\right )-\frac {2 \sqrt {a x+1}}{7 x^{7/2}}\right )}{\sqrt {1-a x}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sqrt {x} \sqrt {c-\frac {c}{a x}} \left (\frac {1}{7} a \left (4 a \left (\frac {1}{3} a \int \frac {8 a x+13}{x^{3/2} (1-a x) \sqrt {a x+1}}dx-\frac {8 \sqrt {a x+1}}{3 x^{3/2}}\right )-\frac {6 \sqrt {a x+1}}{x^{5/2}}\right )-\frac {2 \sqrt {a x+1}}{7 x^{7/2}}\right )}{\sqrt {1-a x}}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle \frac {\sqrt {x} \sqrt {c-\frac {c}{a x}} \left (\frac {1}{7} a \left (4 a \left (\frac {1}{3} a \left (-2 \int -\frac {21 a}{2 \sqrt {x} (1-a x) \sqrt {a x+1}}dx-\frac {26 \sqrt {a x+1}}{\sqrt {x}}\right )-\frac {8 \sqrt {a x+1}}{3 x^{3/2}}\right )-\frac {6 \sqrt {a x+1}}{x^{5/2}}\right )-\frac {2 \sqrt {a x+1}}{7 x^{7/2}}\right )}{\sqrt {1-a x}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sqrt {x} \sqrt {c-\frac {c}{a x}} \left (\frac {1}{7} a \left (4 a \left (\frac {1}{3} a \left (21 a \int \frac {1}{\sqrt {x} (1-a x) \sqrt {a x+1}}dx-\frac {26 \sqrt {a x+1}}{\sqrt {x}}\right )-\frac {8 \sqrt {a x+1}}{3 x^{3/2}}\right )-\frac {6 \sqrt {a x+1}}{x^{5/2}}\right )-\frac {2 \sqrt {a x+1}}{7 x^{7/2}}\right )}{\sqrt {1-a x}}\) |
| \(\Big \downarrow \) 104 |
| \(\displaystyle \frac {\sqrt {x} \sqrt {c-\frac {c}{a x}} \left (\frac {1}{7} a \left (4 a \left (\frac {1}{3} a \left (42 a \int \frac {1}{1-\frac {2 a x}{a x+1}}d\frac {\sqrt {x}}{\sqrt {a x+1}}-\frac {26 \sqrt {a x+1}}{\sqrt {x}}\right )-\frac {8 \sqrt {a x+1}}{3 x^{3/2}}\right )-\frac {6 \sqrt {a x+1}}{x^{5/2}}\right )-\frac {2 \sqrt {a x+1}}{7 x^{7/2}}\right )}{\sqrt {1-a x}}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {\sqrt {x} \left (\frac {1}{7} a \left (4 a \left (\frac {1}{3} a \left (21 \sqrt {2} \sqrt {a} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {x}}{\sqrt {a x+1}}\right )-\frac {26 \sqrt {a x+1}}{\sqrt {x}}\right )-\frac {8 \sqrt {a x+1}}{3 x^{3/2}}\right )-\frac {6 \sqrt {a x+1}}{x^{5/2}}\right )-\frac {2 \sqrt {a x+1}}{7 x^{7/2}}\right ) \sqrt {c-\frac {c}{a x}}}{\sqrt {1-a x}}\) |
(Sqrt[c - c/(a*x)]*Sqrt[x]*((-2*Sqrt[1 + a*x])/(7*x^(7/2)) + (a*((-6*Sqrt[ 1 + a*x])/x^(5/2) + 4*a*((-8*Sqrt[1 + a*x])/(3*x^(3/2)) + (a*((-26*Sqrt[1 + a*x])/Sqrt[x] + 21*Sqrt[2]*Sqrt[a]*ArcTanh[(Sqrt[2]*Sqrt[a]*Sqrt[x])/Sqr t[1 + a*x]]))/3)))/7))/Sqrt[1 - a*x]
3.6.89.3.1 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_))/((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[(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] && LtQ[m, -1] && IntegersQ[ 2*m, 2*n, 2*p]
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[E^(ArcTanh[(a_.)*(x_)]*(n_.))*(u_.)*((c_) + (d_.)*(x_))^(p_.), x_Symbol ] :> Simp[c^p Int[u*(1 + d*(x/c))^p*((1 + a*x)^(n/2)/(1 - a*x)^(n/2)), x] , x] /; FreeQ[{a, c, d, n, p}, x] && EqQ[a^2*c^2 - d^2, 0] && (IntegerQ[p] || GtQ[c, 0])
Int[E^(ArcTanh[(a_.)*(x_)]*(n_.))*(u_.)*((c_) + (d_.)/(x_))^(p_), x_Symbol] :> Simp[x^p*((c + d/x)^p/(1 + c*(x/d))^p) Int[u*(1 + c*(x/d))^p*(E^(n*Ar cTanh[a*x])/x^p), x], x] /; FreeQ[{a, c, d, n, p}, x] && EqQ[c^2 - a^2*d^2, 0] && !IntegerQ[p]
Time = 0.12 (sec) , antiderivative size = 209, normalized size of antiderivative = 0.88
| method | result | size |
| default | \(\frac {\sqrt {\frac {c \left (a x -1\right )}{a x}}\, \sqrt {-a^{2} x^{2}+1}\, \left (52 a^{3} \sqrt {2}\, \sqrt {-\frac {1}{a}}\, x^{3} \sqrt {-\left (a x +1\right ) x}+42 a^{3} \ln \left (\frac {2 \sqrt {2}\, \sqrt {-\frac {1}{a}}\, \sqrt {-\left (a x +1\right ) x}\, a -3 a x -1}{a x -1}\right ) x^{4}+16 a^{2} \sqrt {2}\, \sqrt {-\frac {1}{a}}\, x^{2} \sqrt {-\left (a x +1\right ) x}+9 a \sqrt {2}\, \sqrt {-\frac {1}{a}}\, x \sqrt {-\left (a x +1\right ) x}+3 \sqrt {2}\, \sqrt {-\frac {1}{a}}\, \sqrt {-\left (a x +1\right ) x}\right ) \sqrt {2}}{21 x^{3} \left (a x -1\right ) \sqrt {-\left (a x +1\right ) x}\, \sqrt {-\frac {1}{a}}}\) | \(209\) |
| risch | \(-\frac {2 \left (52 a^{4} x^{4}+68 a^{3} x^{3}+25 a^{2} x^{2}+12 a x +3\right ) \sqrt {\frac {c \left (a x -1\right )}{a x}}\, \sqrt {\frac {c a x \left (-a^{2} x^{2}+1\right )}{a x -1}}}{21 x^{3} \sqrt {-\left (a x +1\right ) a c x}\, \sqrt {-a^{2} x^{2}+1}}+\frac {4 a^{3} \ln \left (\frac {-4 c -3 \left (x -\frac {1}{a}\right ) a c +2 \sqrt {-2 c}\, \sqrt {-a^{2} c \left (x -\frac {1}{a}\right )^{2}-3 \left (x -\frac {1}{a}\right ) a c -2 c}}{x -\frac {1}{a}}\right ) \sqrt {\frac {c \left (a x -1\right )}{a x}}\, \sqrt {\frac {c a x \left (-a^{2} x^{2}+1\right )}{a x -1}}}{\sqrt {-2 c}\, \sqrt {-a^{2} x^{2}+1}}\) | \(224\) |
1/21*(c*(a*x-1)/a/x)^(1/2)/x^3*(-a^2*x^2+1)^(1/2)*(52*a^3*2^(1/2)*(-1/a)^( 1/2)*x^3*(-(a*x+1)*x)^(1/2)+42*a^3*ln((2*2^(1/2)*(-1/a)^(1/2)*(-(a*x+1)*x) ^(1/2)*a-3*a*x-1)/(a*x-1))*x^4+16*a^2*2^(1/2)*(-1/a)^(1/2)*x^2*(-(a*x+1)*x )^(1/2)+9*a*2^(1/2)*(-1/a)^(1/2)*x*(-(a*x+1)*x)^(1/2)+3*2^(1/2)*(-1/a)^(1/ 2)*(-(a*x+1)*x)^(1/2))*2^(1/2)/(a*x-1)/(-(a*x+1)*x)^(1/2)/(-1/a)^(1/2)
Time = 0.29 (sec) , antiderivative size = 353, normalized size of antiderivative = 1.49 \[ \int \frac {e^{3 \text {arctanh}(a x)} \sqrt {c-\frac {c}{a x}}}{x^4} \, dx=\left [\frac {21 \, \sqrt {2} {\left (a^{4} x^{4} - a^{3} x^{3}\right )} \sqrt {-c} \log \left (-\frac {17 \, a^{3} c x^{3} - 3 \, a^{2} c x^{2} - 13 \, a c x + 4 \, \sqrt {2} {\left (3 \, a^{2} x^{2} + a x\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {-c} \sqrt {\frac {a c x - c}{a x}} - c}{a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1}\right ) + 2 \, {\left (52 \, a^{3} x^{3} + 16 \, a^{2} x^{2} + 9 \, a x + 3\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {\frac {a c x - c}{a x}}}{21 \, {\left (a x^{4} - x^{3}\right )}}, -\frac {2 \, {\left (21 \, \sqrt {2} {\left (a^{4} x^{4} - a^{3} x^{3}\right )} \sqrt {c} \arctan \left (\frac {2 \, \sqrt {2} \sqrt {-a^{2} x^{2} + 1} a \sqrt {c} x \sqrt {\frac {a c x - c}{a x}}}{3 \, a^{2} c x^{2} - 2 \, a c x - c}\right ) - {\left (52 \, a^{3} x^{3} + 16 \, a^{2} x^{2} + 9 \, a x + 3\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {\frac {a c x - c}{a x}}\right )}}{21 \, {\left (a x^{4} - x^{3}\right )}}\right ] \]
[1/21*(21*sqrt(2)*(a^4*x^4 - a^3*x^3)*sqrt(-c)*log(-(17*a^3*c*x^3 - 3*a^2* c*x^2 - 13*a*c*x + 4*sqrt(2)*(3*a^2*x^2 + a*x)*sqrt(-a^2*x^2 + 1)*sqrt(-c) *sqrt((a*c*x - c)/(a*x)) - c)/(a^3*x^3 - 3*a^2*x^2 + 3*a*x - 1)) + 2*(52*a ^3*x^3 + 16*a^2*x^2 + 9*a*x + 3)*sqrt(-a^2*x^2 + 1)*sqrt((a*c*x - c)/(a*x) ))/(a*x^4 - x^3), -2/21*(21*sqrt(2)*(a^4*x^4 - a^3*x^3)*sqrt(c)*arctan(2*s qrt(2)*sqrt(-a^2*x^2 + 1)*a*sqrt(c)*x*sqrt((a*c*x - c)/(a*x))/(3*a^2*c*x^2 - 2*a*c*x - c)) - (52*a^3*x^3 + 16*a^2*x^2 + 9*a*x + 3)*sqrt(-a^2*x^2 + 1 )*sqrt((a*c*x - c)/(a*x)))/(a*x^4 - x^3)]
\[ \int \frac {e^{3 \text {arctanh}(a x)} \sqrt {c-\frac {c}{a x}}}{x^4} \, dx=\int \frac {\sqrt {- c \left (-1 + \frac {1}{a x}\right )} \left (a x + 1\right )^{3}}{x^{4} \left (- \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {3}{2}}}\, dx \]
\[ \int \frac {e^{3 \text {arctanh}(a x)} \sqrt {c-\frac {c}{a x}}}{x^4} \, dx=\int { \frac {{\left (a x + 1\right )}^{3} \sqrt {c - \frac {c}{a x}}}{{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} x^{4}} \,d x } \]
Exception generated. \[ \int \frac {e^{3 \text {arctanh}(a x)} \sqrt {c-\frac {c}{a x}}}{x^4} \, dx=\text {Exception raised: TypeError} \]
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. \[ \int \frac {e^{3 \text {arctanh}(a x)} \sqrt {c-\frac {c}{a x}}}{x^4} \, dx=\int \frac {\sqrt {c-\frac {c}{a\,x}}\,{\left (a\,x+1\right )}^3}{x^4\,{\left (1-a^2\,x^2\right )}^{3/2}} \,d x \]