Integrand size = 24, antiderivative size = 170 \[ \int \frac {e^{3 \coth ^{-1}(a x)}}{\sqrt {c-\frac {c}{a x}}} \, dx=-\frac {3 c \sqrt {1-\frac {1}{a^2 x^2}}}{a \left (c-\frac {c}{a x}\right )^{3/2}}+\frac {c \sqrt {1-\frac {1}{a^2 x^2}} x}{\left (c-\frac {c}{a x}\right )^{3/2}}+\frac {7 \text {arctanh}\left (\frac {\sqrt {c} \sqrt {1-\frac {1}{a^2 x^2}}}{\sqrt {c-\frac {c}{a x}}}\right )}{a \sqrt {c}}-\frac {5 \sqrt {2} \text {arctanh}\left (\frac {\sqrt {c} \sqrt {1-\frac {1}{a^2 x^2}}}{\sqrt {2} \sqrt {c-\frac {c}{a x}}}\right )}{a \sqrt {c}} \] Output:
-3*c*(1-1/a^2/x^2)^(1/2)/a/(c-c/a/x)^(3/2)+c*(1-1/a^2/x^2)^(1/2)*x/(c-c/a/ x)^(3/2)+7*arctanh(c^(1/2)*(1-1/a^2/x^2)^(1/2)/(c-c/a/x)^(1/2))/a/c^(1/2)- 5*2^(1/2)*arctanh(1/2*c^(1/2)*(1-1/a^2/x^2)^(1/2)*2^(1/2)/(c-c/a/x)^(1/2)) /a/c^(1/2)
Time = 0.11 (sec) , antiderivative size = 115, normalized size of antiderivative = 0.68 \[ \int \frac {e^{3 \coth ^{-1}(a x)}}{\sqrt {c-\frac {c}{a x}}} \, dx=\frac {\sqrt {1-\frac {1}{a x}} \left (a \sqrt {1+\frac {1}{a x}} x (-3+a x)+7 (-1+a x) \text {arctanh}\left (\sqrt {1+\frac {1}{a x}}\right )-5 \sqrt {2} (-1+a x) \text {arctanh}\left (\frac {\sqrt {1+\frac {1}{a x}}}{\sqrt {2}}\right )\right )}{a \sqrt {c-\frac {c}{a x}} (-1+a x)} \] Input:
Integrate[E^(3*ArcCoth[a*x])/Sqrt[c - c/(a*x)],x]
Output:
(Sqrt[1 - 1/(a*x)]*(a*Sqrt[1 + 1/(a*x)]*x*(-3 + a*x) + 7*(-1 + a*x)*ArcTan h[Sqrt[1 + 1/(a*x)]] - 5*Sqrt[2]*(-1 + a*x)*ArcTanh[Sqrt[1 + 1/(a*x)]/Sqrt [2]]))/(a*Sqrt[c - c/(a*x)]*(-1 + a*x))
Time = 0.66 (sec) , antiderivative size = 135, normalized size of antiderivative = 0.79, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.458, Rules used = {6731, 585, 27, 109, 25, 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.
| \(\displaystyle \int \frac {e^{3 \coth ^{-1}(a x)}}{\sqrt {c-\frac {c}{a x}}} \, dx\) |
| \(\Big \downarrow \) 6731 |
| \(\displaystyle -c^3 \int \frac {\left (1-\frac {1}{a^2 x^2}\right )^{3/2} x^2}{\left (c-\frac {c}{a x}\right )^{7/2}}d\frac {1}{x}\) |
| \(\Big \downarrow \) 585 |
| \(\displaystyle -\frac {\sqrt {1-\frac {1}{a x}} \int \frac {a^2 \left (1+\frac {1}{a x}\right )^{3/2} x^2}{\left (a-\frac {1}{x}\right )^2}d\frac {1}{x}}{\sqrt {c-\frac {c}{a x}}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {a^2 \sqrt {1-\frac {1}{a x}} \int \frac {\left (1+\frac {1}{a x}\right )^{3/2} x^2}{\left (a-\frac {1}{x}\right )^2}d\frac {1}{x}}{\sqrt {c-\frac {c}{a x}}}\) |
| \(\Big \downarrow \) 109 |
| \(\displaystyle -\frac {a^2 \sqrt {1-\frac {1}{a x}} \left (\frac {2 x \sqrt {\frac {1}{a x}+1}}{a \left (a-\frac {1}{x}\right )}-\frac {\int -\frac {\left (3 a+\frac {2}{x}\right ) x^2}{a \left (a-\frac {1}{x}\right ) \sqrt {1+\frac {1}{a x}}}d\frac {1}{x}}{a}\right )}{\sqrt {c-\frac {c}{a x}}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {a^2 \sqrt {1-\frac {1}{a x}} \left (\frac {\int \frac {\left (3 a+\frac {2}{x}\right ) x^2}{a \left (a-\frac {1}{x}\right ) \sqrt {1+\frac {1}{a x}}}d\frac {1}{x}}{a}+\frac {2 x \sqrt {\frac {1}{a x}+1}}{a \left (a-\frac {1}{x}\right )}\right )}{\sqrt {c-\frac {c}{a x}}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {a^2 \sqrt {1-\frac {1}{a x}} \left (\frac {\int \frac {\left (3 a+\frac {2}{x}\right ) x^2}{\left (a-\frac {1}{x}\right ) \sqrt {1+\frac {1}{a x}}}d\frac {1}{x}}{a^2}+\frac {2 x \sqrt {\frac {1}{a x}+1}}{a \left (a-\frac {1}{x}\right )}\right )}{\sqrt {c-\frac {c}{a x}}}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle -\frac {a^2 \sqrt {1-\frac {1}{a x}} \left (\frac {-\frac {\int -\frac {\left (7 a+\frac {3}{x}\right ) x}{2 \left (a-\frac {1}{x}\right ) \sqrt {1+\frac {1}{a x}}}d\frac {1}{x}}{a}-3 x \sqrt {\frac {1}{a x}+1}}{a^2}+\frac {2 x \sqrt {\frac {1}{a x}+1}}{a \left (a-\frac {1}{x}\right )}\right )}{\sqrt {c-\frac {c}{a x}}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {a^2 \sqrt {1-\frac {1}{a x}} \left (\frac {\frac {\int \frac {\left (7 a+\frac {3}{x}\right ) x}{\left (a-\frac {1}{x}\right ) \sqrt {1+\frac {1}{a x}}}d\frac {1}{x}}{2 a}-3 x \sqrt {\frac {1}{a x}+1}}{a^2}+\frac {2 x \sqrt {\frac {1}{a x}+1}}{a \left (a-\frac {1}{x}\right )}\right )}{\sqrt {c-\frac {c}{a x}}}\) |
| \(\Big \downarrow \) 174 |
| \(\displaystyle -\frac {a^2 \sqrt {1-\frac {1}{a x}} \left (\frac {\frac {10 \int \frac {1}{\left (a-\frac {1}{x}\right ) \sqrt {1+\frac {1}{a x}}}d\frac {1}{x}+7 \int \frac {x}{\sqrt {1+\frac {1}{a x}}}d\frac {1}{x}}{2 a}-3 x \sqrt {\frac {1}{a x}+1}}{a^2}+\frac {2 x \sqrt {\frac {1}{a x}+1}}{a \left (a-\frac {1}{x}\right )}\right )}{\sqrt {c-\frac {c}{a x}}}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle -\frac {a^2 \sqrt {1-\frac {1}{a x}} \left (\frac {\frac {20 a \int \frac {1}{2 a-\frac {a}{x^2}}d\sqrt {1+\frac {1}{a x}}+14 a \int \frac {1}{\frac {a}{x^2}-a}d\sqrt {1+\frac {1}{a x}}}{2 a}-3 x \sqrt {\frac {1}{a x}+1}}{a^2}+\frac {2 x \sqrt {\frac {1}{a x}+1}}{a \left (a-\frac {1}{x}\right )}\right )}{\sqrt {c-\frac {c}{a x}}}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle -\frac {a^2 \sqrt {1-\frac {1}{a x}} \left (\frac {\frac {10 \sqrt {2} \text {arctanh}\left (\frac {\sqrt {\frac {1}{a x}+1}}{\sqrt {2}}\right )-14 \text {arctanh}\left (\sqrt {\frac {1}{a x}+1}\right )}{2 a}-3 x \sqrt {\frac {1}{a x}+1}}{a^2}+\frac {2 x \sqrt {\frac {1}{a x}+1}}{a \left (a-\frac {1}{x}\right )}\right )}{\sqrt {c-\frac {c}{a x}}}\) |
Input:
Int[E^(3*ArcCoth[a*x])/Sqrt[c - c/(a*x)],x]
Output:
-((a^2*Sqrt[1 - 1/(a*x)]*((2*Sqrt[1 + 1/(a*x)]*x)/(a*(a - x^(-1))) + (-3*S qrt[1 + 1/(a*x)]*x + (-14*ArcTanh[Sqrt[1 + 1/(a*x)]] + 10*Sqrt[2]*ArcTanh[ Sqrt[1 + 1/(a*x)]/Sqrt[2]])/(2*a))/a^2))/Sqrt[c - c/(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[((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[((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_Symbol] :> S imp[-c^n Subst[Int[(c + d*x)^(p - n)*((1 - x^2/a^2)^(n/2)/x^2), x], x, 1/ x], x] /; FreeQ[{a, c, d, p}, x] && EqQ[c + a*d, 0] && IntegerQ[(n - 1)/2] && IntegerQ[2*p]
Time = 0.19 (sec) , antiderivative size = 259, normalized size of antiderivative = 1.52
| method | result | size |
| default | \(\frac {\sqrt {\frac {c \left (a x -1\right )}{a x}}\, x \left (2 a^{\frac {5}{2}} \sqrt {\frac {1}{a}}\, \sqrt {x \left (a x +1\right )}\, x -5 a^{\frac {3}{2}} \sqrt {2}\, \ln \left (\frac {2 \sqrt {2}\, \sqrt {\frac {1}{a}}\, \sqrt {x \left (a x +1\right )}\, a +3 a x +1}{a x -1}\right ) x +7 \ln \left (\frac {2 \sqrt {x \left (a x +1\right )}\, \sqrt {a}+2 a x +1}{2 \sqrt {a}}\right ) a^{2} \sqrt {\frac {1}{a}}\, x -6 \sqrt {x \left (a x +1\right )}\, a^{\frac {3}{2}} \sqrt {\frac {1}{a}}-7 \ln \left (\frac {2 \sqrt {x \left (a x +1\right )}\, \sqrt {a}+2 a x +1}{2 \sqrt {a}}\right ) a \sqrt {\frac {1}{a}}+5 \sqrt {2}\, \ln \left (\frac {2 \sqrt {2}\, \sqrt {\frac {1}{a}}\, \sqrt {x \left (a x +1\right )}\, a +3 a x +1}{a x -1}\right ) \sqrt {a}\right )}{2 \left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}} \left (a x +1\right ) a^{\frac {3}{2}} c \sqrt {x \left (a x +1\right )}\, \sqrt {\frac {1}{a}}}\) | \(259\) |
| risch | \(\frac {a x -1}{a \sqrt {\frac {a x -1}{a x +1}}\, \sqrt {\frac {c \left (a x -1\right )}{a x}}}+\frac {\left (\frac {7 \ln \left (\frac {\frac {1}{2} a c +a^{2} c x}{\sqrt {a^{2} c}}+\sqrt {a^{2} c \,x^{2}+a c x}\right )}{2 a \sqrt {a^{2} c}}-\frac {2 \sqrt {\left (x -\frac {1}{a}\right )^{2} a^{2} c +3 \left (x -\frac {1}{a}\right ) a c +2 c}}{a^{3} c \left (x -\frac {1}{a}\right )}-\frac {5 \sqrt {2}\, \ln \left (\frac {4 c +3 \left (x -\frac {1}{a}\right ) a c +2 \sqrt {2}\, \sqrt {c}\, \sqrt {\left (x -\frac {1}{a}\right )^{2} a^{2} c +3 \left (x -\frac {1}{a}\right ) a c +2 c}}{x -\frac {1}{a}}\right )}{2 a^{2} \sqrt {c}}\right ) \sqrt {\left (a x +1\right ) a c x}\, \left (a x -1\right )}{\sqrt {\frac {a x -1}{a x +1}}\, \left (a x +1\right ) \sqrt {\frac {c \left (a x -1\right )}{a x}}\, x}\) | \(271\) |
Input:
int(1/((a*x-1)/(a*x+1))^(3/2)/(c-c/a/x)^(1/2),x,method=_RETURNVERBOSE)
Output:
1/2/((a*x-1)/(a*x+1))^(3/2)/(a*x+1)*(c*(a*x-1)/a/x)^(1/2)*x*(2*a^(5/2)*(1/ a)^(1/2)*(x*(a*x+1))^(1/2)*x-5*a^(3/2)*2^(1/2)*ln((2*2^(1/2)*(1/a)^(1/2)*( x*(a*x+1))^(1/2)*a+3*a*x+1)/(a*x-1))*x+7*ln(1/2*(2*(x*(a*x+1))^(1/2)*a^(1/ 2)+2*a*x+1)/a^(1/2))*a^2*(1/a)^(1/2)*x-6*(x*(a*x+1))^(1/2)*a^(3/2)*(1/a)^( 1/2)-7*ln(1/2*(2*(x*(a*x+1))^(1/2)*a^(1/2)+2*a*x+1)/a^(1/2))*a*(1/a)^(1/2) +5*2^(1/2)*ln((2*2^(1/2)*(1/a)^(1/2)*(x*(a*x+1))^(1/2)*a+3*a*x+1)/(a*x-1)) *a^(1/2))/a^(3/2)/c/(x*(a*x+1))^(1/2)/(1/a)^(1/2)
Time = 0.18 (sec) , antiderivative size = 581, normalized size of antiderivative = 3.42 \[ \int \frac {e^{3 \coth ^{-1}(a x)}}{\sqrt {c-\frac {c}{a x}}} \, dx=\left [\frac {7 \, {\left (a^{2} x^{2} - 2 \, a x + 1\right )} \sqrt {c} \log \left (-\frac {8 \, a^{3} c x^{3} - 7 \, a c x + 4 \, {\left (2 \, a^{3} x^{3} + 3 \, a^{2} x^{2} + a x\right )} \sqrt {c} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}} - c}{a x - 1}\right ) + 4 \, {\left (a^{3} x^{3} - 2 \, a^{2} x^{2} - 3 \, a x\right )} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}} + \frac {5 \, \sqrt {2} {\left (a^{2} c x^{2} - 2 \, a c x + c\right )} \log \left (-\frac {17 \, a^{3} x^{3} - 3 \, a^{2} x^{2} - 13 \, a x - \frac {4 \, \sqrt {2} {\left (3 \, a^{3} x^{3} + 4 \, a^{2} x^{2} + a x\right )} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}}}{\sqrt {c}} - 1}{a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1}\right )}{\sqrt {c}}}{4 \, {\left (a^{3} c x^{2} - 2 \, a^{2} c x + a c\right )}}, \frac {5 \, \sqrt {2} {\left (a^{2} c x^{2} - 2 \, a c x + c\right )} \sqrt {-\frac {1}{c}} \arctan \left (\frac {2 \, \sqrt {2} {\left (a^{2} x^{2} + a x\right )} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {-\frac {1}{c}} \sqrt {\frac {a c x - c}{a x}}}{3 \, a^{2} x^{2} - 2 \, a x - 1}\right ) - 7 \, {\left (a^{2} x^{2} - 2 \, a x + 1\right )} \sqrt {-c} \arctan \left (\frac {2 \, {\left (a^{2} x^{2} + a x\right )} \sqrt {-c} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}}}{2 \, a^{2} c x^{2} - a c x - c}\right ) + 2 \, {\left (a^{3} x^{3} - 2 \, a^{2} x^{2} - 3 \, a x\right )} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}}}{2 \, {\left (a^{3} c x^{2} - 2 \, a^{2} c x + a c\right )}}\right ] \] Input:
integrate(1/((a*x-1)/(a*x+1))^(3/2)/(c-c/a/x)^(1/2),x, algorithm="fricas")
Output:
[1/4*(7*(a^2*x^2 - 2*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*(a^3*x^3 - 2*a^2*x^2 - 3*a*x)*sqrt((a*x - 1) /(a*x + 1))*sqrt((a*c*x - c)/(a*x)) + 5*sqrt(2)*(a^2*c*x^2 - 2*a*c*x + c)* log(-(17*a^3*x^3 - 3*a^2*x^2 - 13*a*x - 4*sqrt(2)*(3*a^3*x^3 + 4*a^2*x^2 + a*x)*sqrt((a*x - 1)/(a*x + 1))*sqrt((a*c*x - c)/(a*x))/sqrt(c) - 1)/(a^3* x^3 - 3*a^2*x^2 + 3*a*x - 1))/sqrt(c))/(a^3*c*x^2 - 2*a^2*c*x + a*c), 1/2* (5*sqrt(2)*(a^2*c*x^2 - 2*a*c*x + c)*sqrt(-1/c)*arctan(2*sqrt(2)*(a^2*x^2 + a*x)*sqrt((a*x - 1)/(a*x + 1))*sqrt(-1/c)*sqrt((a*c*x - c)/(a*x))/(3*a^2 *x^2 - 2*a*x - 1)) - 7*(a^2*x^2 - 2*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*(a^3*x^3 - 2*a^2*x^2 - 3*a*x)*sqrt((a*x - 1)/(a*x + 1 ))*sqrt((a*c*x - c)/(a*x)))/(a^3*c*x^2 - 2*a^2*c*x + a*c)]
Timed out. \[ \int \frac {e^{3 \coth ^{-1}(a x)}}{\sqrt {c-\frac {c}{a x}}} \, dx=\text {Timed out} \] Input:
integrate(1/((a*x-1)/(a*x+1))**(3/2)/(c-c/a/x)**(1/2),x)
Output:
Timed out
\[ \int \frac {e^{3 \coth ^{-1}(a x)}}{\sqrt {c-\frac {c}{a x}}} \, dx=\int { \frac {1}{\sqrt {c - \frac {c}{a x}} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}}} \,d x } \] Input:
integrate(1/((a*x-1)/(a*x+1))^(3/2)/(c-c/a/x)^(1/2),x, algorithm="maxima")
Output:
integrate(1/(sqrt(c - c/(a*x))*((a*x - 1)/(a*x + 1))^(3/2)), x)
Exception generated. \[ \int \frac {e^{3 \coth ^{-1}(a x)}}{\sqrt {c-\frac {c}{a x}}} \, dx=\text {Exception raised: RuntimeError} \] Input:
integrate(1/((a*x-1)/(a*x+1))^(3/2)/(c-c/a/x)^(1/2),x, algorithm="giac")
Output:
Exception raised: RuntimeError >> an error occurred running a Giac command :INPUT:sage2OUTPUT:sym2poly/r2sym(const gen & e,const index_m & i,const ve cteur & l) Error: Bad Argument Value
Timed out. \[ \int \frac {e^{3 \coth ^{-1}(a x)}}{\sqrt {c-\frac {c}{a x}}} \, dx=\int \frac {1}{\sqrt {c-\frac {c}{a\,x}}\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/2}} \,d x \] Input:
int(1/((c - c/(a*x))^(1/2)*((a*x - 1)/(a*x + 1))^(3/2)),x)
Output:
int(1/((c - c/(a*x))^(1/2)*((a*x - 1)/(a*x + 1))^(3/2)), x)
Time = 0.16 (sec) , antiderivative size = 252, normalized size of antiderivative = 1.48 \[ \int \frac {e^{3 \coth ^{-1}(a x)}}{\sqrt {c-\frac {c}{a x}}} \, dx=\frac {\sqrt {c}\, \left (2 \sqrt {x}\, \sqrt {a}\, \sqrt {a x +1}\, a x -6 \sqrt {x}\, \sqrt {a}\, \sqrt {a x +1}+5 \sqrt {2}\, \mathrm {log}\left (\sqrt {a x +1}+\sqrt {x}\, \sqrt {a}-\sqrt {2}-1\right ) a x -5 \sqrt {2}\, \mathrm {log}\left (\sqrt {a x +1}+\sqrt {x}\, \sqrt {a}-\sqrt {2}-1\right )-5 \sqrt {2}\, \mathrm {log}\left (\sqrt {a x +1}+\sqrt {x}\, \sqrt {a}-\sqrt {2}+1\right ) a x +5 \sqrt {2}\, \mathrm {log}\left (\sqrt {a x +1}+\sqrt {x}\, \sqrt {a}-\sqrt {2}+1\right )-5 \sqrt {2}\, \mathrm {log}\left (\sqrt {a x +1}+\sqrt {x}\, \sqrt {a}+\sqrt {2}-1\right ) a x +5 \sqrt {2}\, \mathrm {log}\left (\sqrt {a x +1}+\sqrt {x}\, \sqrt {a}+\sqrt {2}-1\right )+5 \sqrt {2}\, \mathrm {log}\left (\sqrt {a x +1}+\sqrt {x}\, \sqrt {a}+\sqrt {2}+1\right ) a x -5 \sqrt {2}\, \mathrm {log}\left (\sqrt {a x +1}+\sqrt {x}\, \sqrt {a}+\sqrt {2}+1\right )+14 \,\mathrm {log}\left (\sqrt {a x +1}+\sqrt {x}\, \sqrt {a}\right ) a x -14 \,\mathrm {log}\left (\sqrt {a x +1}+\sqrt {x}\, \sqrt {a}\right )\right )}{2 a c \left (a x -1\right )} \] Input:
int(1/((a*x-1)/(a*x+1))^(3/2)/(c-c/a/x)^(1/2),x)
Output:
(sqrt(c)*(2*sqrt(x)*sqrt(a)*sqrt(a*x + 1)*a*x - 6*sqrt(x)*sqrt(a)*sqrt(a*x + 1) + 5*sqrt(2)*log(sqrt(a*x + 1) + sqrt(x)*sqrt(a) - sqrt(2) - 1)*a*x - 5*sqrt(2)*log(sqrt(a*x + 1) + sqrt(x)*sqrt(a) - sqrt(2) - 1) - 5*sqrt(2)* log(sqrt(a*x + 1) + sqrt(x)*sqrt(a) - sqrt(2) + 1)*a*x + 5*sqrt(2)*log(sqr t(a*x + 1) + sqrt(x)*sqrt(a) - sqrt(2) + 1) - 5*sqrt(2)*log(sqrt(a*x + 1) + sqrt(x)*sqrt(a) + sqrt(2) - 1)*a*x + 5*sqrt(2)*log(sqrt(a*x + 1) + sqrt( x)*sqrt(a) + sqrt(2) - 1) + 5*sqrt(2)*log(sqrt(a*x + 1) + sqrt(x)*sqrt(a) + sqrt(2) + 1)*a*x - 5*sqrt(2)*log(sqrt(a*x + 1) + sqrt(x)*sqrt(a) + sqrt( 2) + 1) + 14*log(sqrt(a*x + 1) + sqrt(x)*sqrt(a))*a*x - 14*log(sqrt(a*x + 1) + sqrt(x)*sqrt(a))))/(2*a*c*(a*x - 1))