Integrand size = 24, antiderivative size = 305 \[ \int e^{-3 \text {arctanh}(a x)} \left (c-\frac {c}{a x}\right )^{9/2} \, dx=\frac {3480 a^4 \left (c-\frac {c}{a x}\right )^{9/2} x^5}{7 (1-a x)^{9/2} \sqrt {1+a x}}+\frac {70 a^3 \left (c-\frac {c}{a x}\right )^{9/2} x^4}{(1-a x)^{5/2} \sqrt {1+a x}}-\frac {50 a^2 \left (c-\frac {c}{a x}\right )^{9/2} x^3}{7 (1-a x)^{3/2} \sqrt {1+a x}}+\frac {10 a \left (c-\frac {c}{a x}\right )^{9/2} x^2}{7 \sqrt {1-a x} \sqrt {1+a x}}-\frac {2 \left (c-\frac {c}{a x}\right )^{9/2} x \sqrt {1-a x}}{7 \sqrt {1+a x}}-\frac {545 a^4 \left (c-\frac {c}{a x}\right )^{9/2} x^5 \sqrt {1+a x}}{7 (1-a x)^{9/2}}-\frac {15 a^{7/2} \left (c-\frac {c}{a x}\right )^{9/2} x^{9/2} \text {arcsinh}\left (\sqrt {a} \sqrt {x}\right )}{(1-a x)^{9/2}} \] Output:
3480/7*a^4*(c-c/a/x)^(9/2)*x^5/(-a*x+1)^(9/2)/(a*x+1)^(1/2)+70*a^3*(c-c/a/ x)^(9/2)*x^4/(-a*x+1)^(5/2)/(a*x+1)^(1/2)-50/7*a^2*(c-c/a/x)^(9/2)*x^3/(-a *x+1)^(3/2)/(a*x+1)^(1/2)+10/7*a*(c-c/a/x)^(9/2)*x^2/(-a*x+1)^(1/2)/(a*x+1 )^(1/2)-2/7*(c-c/a/x)^(9/2)*x*(-a*x+1)^(1/2)/(a*x+1)^(1/2)-545/7*a^4*(c-c/ a/x)^(9/2)*x^5*(a*x+1)^(1/2)/(-a*x+1)^(9/2)-15*a^(7/2)*(c-c/a/x)^(9/2)*x^( 9/2)*arcsinh(a^(1/2)*x^(1/2))/(-a*x+1)^(9/2)
Result contains complex when optimal does not.
Time = 0.33 (sec) , antiderivative size = 152, normalized size of antiderivative = 0.50 \[ \int e^{-3 \text {arctanh}(a x)} \left (c-\frac {c}{a x}\right )^{9/2} \, dx=\frac {c^4 \sqrt {c-\frac {c}{a x}} \left (-2+20 a x-110 a^2 x^2+720 a^3 x^3+1755 a^4 x^4+7 a^5 x^5\right )}{7 a^4 x^3 \sqrt {1-a^2 x^2}}-\frac {15 i c^{9/2} \log \left (-i \sqrt {c} (1+2 a x)+\frac {2 a \sqrt {c-\frac {c}{a x}} x \sqrt {1-a^2 x^2}}{-1+a x}\right )}{2 a} \] Input:
Integrate[(c - c/(a*x))^(9/2)/E^(3*ArcTanh[a*x]),x]
Output:
(c^4*Sqrt[c - c/(a*x)]*(-2 + 20*a*x - 110*a^2*x^2 + 720*a^3*x^3 + 1755*a^4 *x^4 + 7*a^5*x^5))/(7*a^4*x^3*Sqrt[1 - a^2*x^2]) - (((15*I)/2)*c^(9/2)*Log [(-I)*Sqrt[c]*(1 + 2*a*x) + (2*a*Sqrt[c - c/(a*x)]*x*Sqrt[1 - a^2*x^2])/(- 1 + a*x)])/a
Time = 0.52 (sec) , antiderivative size = 188, normalized size of antiderivative = 0.62, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.542, Rules used = {6684, 6679, 109, 27, 167, 27, 167, 27, 167, 27, 160, 63, 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.
| \(\displaystyle \int e^{-3 \text {arctanh}(a x)} \left (c-\frac {c}{a x}\right )^{9/2} \, dx\) |
| \(\Big \downarrow \) 6684 |
| \(\displaystyle \frac {x^{9/2} \left (c-\frac {c}{a x}\right )^{9/2} \int \frac {e^{-3 \text {arctanh}(a x)} (1-a x)^{9/2}}{x^{9/2}}dx}{(1-a x)^{9/2}}\) |
| \(\Big \downarrow \) 6679 |
| \(\displaystyle \frac {x^{9/2} \left (c-\frac {c}{a x}\right )^{9/2} \int \frac {(1-a x)^6}{x^{9/2} (a x+1)^{3/2}}dx}{(1-a x)^{9/2}}\) |
| \(\Big \downarrow \) 109 |
| \(\displaystyle \frac {x^{9/2} \left (c-\frac {c}{a x}\right )^{9/2} \left (-\frac {2}{7} \int \frac {5 a (1-a x)^4 (5-a x)}{2 x^{7/2} (a x+1)^{3/2}}dx-\frac {2 (1-a x)^5}{7 x^{7/2} \sqrt {a x+1}}\right )}{(1-a x)^{9/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {x^{9/2} \left (c-\frac {c}{a x}\right )^{9/2} \left (-\frac {5}{7} a \int \frac {(1-a x)^4 (5-a x)}{x^{7/2} (a x+1)^{3/2}}dx-\frac {2 (1-a x)^5}{7 x^{7/2} \sqrt {a x+1}}\right )}{(1-a x)^{9/2}}\) |
| \(\Big \downarrow \) 167 |
| \(\displaystyle \frac {x^{9/2} \left (c-\frac {c}{a x}\right )^{9/2} \left (-\frac {5}{7} a \left (\frac {2}{5} \int -\frac {5 a (1-a x)^3 (a x+15)}{2 x^{5/2} (a x+1)^{3/2}}dx-\frac {2 (1-a x)^4}{x^{5/2} \sqrt {a x+1}}\right )-\frac {2 (1-a x)^5}{7 x^{7/2} \sqrt {a x+1}}\right )}{(1-a x)^{9/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {x^{9/2} \left (c-\frac {c}{a x}\right )^{9/2} \left (-\frac {5}{7} a \left (-a \int \frac {(1-a x)^3 (a x+15)}{x^{5/2} (a x+1)^{3/2}}dx-\frac {2 (1-a x)^4}{x^{5/2} \sqrt {a x+1}}\right )-\frac {2 (1-a x)^5}{7 x^{7/2} \sqrt {a x+1}}\right )}{(1-a x)^{9/2}}\) |
| \(\Big \downarrow \) 167 |
| \(\displaystyle \frac {x^{9/2} \left (c-\frac {c}{a x}\right )^{9/2} \left (-\frac {5}{7} a \left (-a \left (\frac {2}{3} \int -\frac {3 a (1-a x)^2 (11 a x+49)}{2 x^{3/2} (a x+1)^{3/2}}dx-\frac {10 (1-a x)^3}{x^{3/2} \sqrt {a x+1}}\right )-\frac {2 (1-a x)^4}{x^{5/2} \sqrt {a x+1}}\right )-\frac {2 (1-a x)^5}{7 x^{7/2} \sqrt {a x+1}}\right )}{(1-a x)^{9/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {x^{9/2} \left (c-\frac {c}{a x}\right )^{9/2} \left (-\frac {5}{7} a \left (-a \left (-a \int \frac {(1-a x)^2 (11 a x+49)}{x^{3/2} (a x+1)^{3/2}}dx-\frac {10 (1-a x)^3}{x^{3/2} \sqrt {a x+1}}\right )-\frac {2 (1-a x)^4}{x^{5/2} \sqrt {a x+1}}\right )-\frac {2 (1-a x)^5}{7 x^{7/2} \sqrt {a x+1}}\right )}{(1-a x)^{9/2}}\) |
| \(\Big \downarrow \) 167 |
| \(\displaystyle \frac {x^{9/2} \left (c-\frac {c}{a x}\right )^{9/2} \left (-\frac {5}{7} a \left (-a \left (-a \left (2 \int -\frac {a (1-a x) (109 a x+283)}{2 \sqrt {x} (a x+1)^{3/2}}dx-\frac {98 (1-a x)^2}{\sqrt {x} \sqrt {a x+1}}\right )-\frac {10 (1-a x)^3}{x^{3/2} \sqrt {a x+1}}\right )-\frac {2 (1-a x)^4}{x^{5/2} \sqrt {a x+1}}\right )-\frac {2 (1-a x)^5}{7 x^{7/2} \sqrt {a x+1}}\right )}{(1-a x)^{9/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {x^{9/2} \left (c-\frac {c}{a x}\right )^{9/2} \left (-\frac {5}{7} a \left (-a \left (-a \left (-a \int \frac {(1-a x) (109 a x+283)}{\sqrt {x} (a x+1)^{3/2}}dx-\frac {98 (1-a x)^2}{\sqrt {x} \sqrt {a x+1}}\right )-\frac {10 (1-a x)^3}{x^{3/2} \sqrt {a x+1}}\right )-\frac {2 (1-a x)^4}{x^{5/2} \sqrt {a x+1}}\right )-\frac {2 (1-a x)^5}{7 x^{7/2} \sqrt {a x+1}}\right )}{(1-a x)^{9/2}}\) |
| \(\Big \downarrow \) 160 |
| \(\displaystyle \frac {x^{9/2} \left (c-\frac {c}{a x}\right )^{9/2} \left (-\frac {5}{7} a \left (-a \left (-a \left (-a \left (\frac {\sqrt {x} (587-109 a x)}{\sqrt {a x+1}}-\frac {21}{2} \int \frac {1}{\sqrt {x} \sqrt {a x+1}}dx\right )-\frac {98 (1-a x)^2}{\sqrt {x} \sqrt {a x+1}}\right )-\frac {10 (1-a x)^3}{x^{3/2} \sqrt {a x+1}}\right )-\frac {2 (1-a x)^4}{x^{5/2} \sqrt {a x+1}}\right )-\frac {2 (1-a x)^5}{7 x^{7/2} \sqrt {a x+1}}\right )}{(1-a x)^{9/2}}\) |
| \(\Big \downarrow \) 63 |
| \(\displaystyle \frac {x^{9/2} \left (c-\frac {c}{a x}\right )^{9/2} \left (-\frac {5}{7} a \left (-a \left (-a \left (-a \left (\frac {\sqrt {x} (587-109 a x)}{\sqrt {a x+1}}-21 \int \frac {1}{\sqrt {a x+1}}d\sqrt {x}\right )-\frac {98 (1-a x)^2}{\sqrt {x} \sqrt {a x+1}}\right )-\frac {10 (1-a x)^3}{x^{3/2} \sqrt {a x+1}}\right )-\frac {2 (1-a x)^4}{x^{5/2} \sqrt {a x+1}}\right )-\frac {2 (1-a x)^5}{7 x^{7/2} \sqrt {a x+1}}\right )}{(1-a x)^{9/2}}\) |
| \(\Big \downarrow \) 222 |
| \(\displaystyle \frac {x^{9/2} \left (-\frac {5}{7} a \left (-a \left (-a \left (-a \left (\frac {\sqrt {x} (587-109 a x)}{\sqrt {a x+1}}-\frac {21 \text {arcsinh}\left (\sqrt {a} \sqrt {x}\right )}{\sqrt {a}}\right )-\frac {98 (1-a x)^2}{\sqrt {x} \sqrt {a x+1}}\right )-\frac {10 (1-a x)^3}{x^{3/2} \sqrt {a x+1}}\right )-\frac {2 (1-a x)^4}{x^{5/2} \sqrt {a x+1}}\right )-\frac {2 (1-a x)^5}{7 x^{7/2} \sqrt {a x+1}}\right ) \left (c-\frac {c}{a x}\right )^{9/2}}{(1-a x)^{9/2}}\) |
Input:
Int[(c - c/(a*x))^(9/2)/E^(3*ArcTanh[a*x]),x]
Output:
((c - c/(a*x))^(9/2)*x^(9/2)*((-2*(1 - a*x)^5)/(7*x^(7/2)*Sqrt[1 + a*x]) - (5*a*((-2*(1 - a*x)^4)/(x^(5/2)*Sqrt[1 + a*x]) - a*((-10*(1 - a*x)^3)/(x^ (3/2)*Sqrt[1 + a*x]) - a*((-98*(1 - a*x)^2)/(Sqrt[x]*Sqrt[1 + a*x]) - a*(( Sqrt[x]*(587 - 109*a*x))/Sqrt[1 + a*x] - (21*ArcSinh[Sqrt[a]*Sqrt[x]])/Sqr t[a])))))/7))/(1 - a*x)^(9/2)
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_) )^(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_) )*((g_.) + (h_.)*(x_)), x_] :> Simp[(b^2*d*e*g - a^2*d*f*h*m - a*b*(d*(f*g + e*h) - c*f*h*(m + 1)) + b*f*h*(b*c - a*d)*(m + 1)*x)*(a + b*x)^(m + 1)*(( c + d*x)^(n + 1)/(b^2*d*(b*c - a*d)*(m + 1))), x] + Simp[(a*d*f*h*m + b*(d* (f*g + e*h) - c*f*h*(m + 2)))/(b^2*d) Int[(a + b*x)^(m + 1)*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, e, f, g, h, m, n}, x] && EqQ[m + n + 2, 0] && NeQ[m, -1] && (SumSimplerQ[m, 1] || !SumSimplerQ[n, 1])
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*((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 - 1)*(e + f*x)^p*Simp[b* c*(f*g - e*h)*(m + 1) + (b*g - a*h)*(d*e*n + c*f*(p + 1)) + d*(b*(f*g - e*h )*(m + 1) + f*(b*g - a*h)*(n + p + 1))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, p}, x] && LtQ[m, -1] && GtQ[n, 0] && IntegersQ[2*m, 2*n, 2*p]
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^(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 = 0.22 (sec) , antiderivative size = 227, normalized size of antiderivative = 0.74
| method | result | size |
| default | \(-\frac {\sqrt {\frac {c \left (a x -1\right )}{a x}}\, c^{4} \left (14 \sqrt {-x \left (a x +1\right )}\, a^{\frac {11}{2}} x^{5}+105 \arctan \left (\frac {2 a x +1}{2 \sqrt {a}\, \sqrt {-x \left (a x +1\right )}}\right ) a^{5} x^{5}+3510 a^{\frac {9}{2}} \sqrt {-x \left (a x +1\right )}\, x^{4}+105 \arctan \left (\frac {2 a x +1}{2 \sqrt {a}\, \sqrt {-x \left (a x +1\right )}}\right ) a^{4} x^{4}+1440 a^{\frac {7}{2}} x^{3} \sqrt {-x \left (a x +1\right )}-220 a^{\frac {5}{2}} x^{2} \sqrt {-x \left (a x +1\right )}+40 a^{\frac {3}{2}} x \sqrt {-x \left (a x +1\right )}-4 \sqrt {a}\, \sqrt {-x \left (a x +1\right )}\right ) \sqrt {-a^{2} x^{2}+1}}{14 x^{3} a^{\frac {9}{2}} \left (a x +1\right ) \sqrt {-x \left (a x +1\right )}\, \left (a x -1\right )}\) | \(227\) |
| risch | \(\frac {\left (7 a^{5} x^{5}+859 a^{4} x^{4}+720 a^{3} x^{3}-110 a^{2} x^{2}+20 a x -2\right ) c^{4} \sqrt {\frac {c \left (a x -1\right )}{a x}}\, \sqrt {\frac {c a x \left (-a^{2} x^{2}+1\right )}{a x -1}}}{7 x^{3} \sqrt {-\left (a x +1\right ) a c x}\, \sqrt {-a^{2} x^{2}+1}\, a^{4}}+\frac {\left (-\frac {15 a^{4} \arctan \left (\frac {\sqrt {a^{2} c}\, \left (x +\frac {1}{2 a}\right )}{\sqrt {-a^{2} c \,x^{2}-a c x}}\right )}{2 \sqrt {a^{2} c}}-\frac {128 a^{2} \sqrt {-\left (x +\frac {1}{a}\right )^{2} a^{2} c +\left (x +\frac {1}{a}\right ) a c}}{c \left (x +\frac {1}{a}\right )}\right ) c^{4} \sqrt {\frac {c \left (a x -1\right )}{a x}}\, \sqrt {\frac {c a x \left (-a^{2} x^{2}+1\right )}{a x -1}}}{\sqrt {-a^{2} x^{2}+1}\, a^{4}}\) | \(255\) |
Input:
int((c-c/a/x)^(9/2)/(a*x+1)^3*(-a^2*x^2+1)^(3/2),x,method=_RETURNVERBOSE)
Output:
-1/14*(c*(a*x-1)/a/x)^(1/2)/x^3*c^4/a^(9/2)*(14*(-x*(a*x+1))^(1/2)*a^(11/2 )*x^5+105*arctan(1/2/a^(1/2)*(2*a*x+1)/(-x*(a*x+1))^(1/2))*a^5*x^5+3510*a^ (9/2)*(-x*(a*x+1))^(1/2)*x^4+105*arctan(1/2/a^(1/2)*(2*a*x+1)/(-x*(a*x+1)) ^(1/2))*a^4*x^4+1440*a^(7/2)*x^3*(-x*(a*x+1))^(1/2)-220*a^(5/2)*x^2*(-x*(a *x+1))^(1/2)+40*a^(3/2)*x*(-x*(a*x+1))^(1/2)-4*a^(1/2)*(-x*(a*x+1))^(1/2)) *(-a^2*x^2+1)^(1/2)/(a*x+1)/(-x*(a*x+1))^(1/2)/(a*x-1)
Time = 0.13 (sec) , antiderivative size = 408, normalized size of antiderivative = 1.34 \[ \int e^{-3 \text {arctanh}(a x)} \left (c-\frac {c}{a x}\right )^{9/2} \, dx=\left [\frac {105 \, {\left (a^{5} c^{4} x^{5} - a^{3} c^{4} x^{3}\right )} \sqrt {-c} \log \left (-\frac {8 \, a^{3} c x^{3} - 7 \, a c x - 4 \, {\left (2 \, a^{2} x^{2} + a x\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {-c} \sqrt {\frac {a c x - c}{a x}} - c}{a x - 1}\right ) - 4 \, {\left (7 \, a^{5} c^{4} x^{5} + 1755 \, a^{4} c^{4} x^{4} + 720 \, a^{3} c^{4} x^{3} - 110 \, a^{2} c^{4} x^{2} + 20 \, a c^{4} x - 2 \, c^{4}\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {\frac {a c x - c}{a x}}}{28 \, {\left (a^{6} x^{5} - a^{4} x^{3}\right )}}, \frac {105 \, {\left (a^{5} c^{4} x^{5} - a^{3} c^{4} x^{3}\right )} \sqrt {c} \arctan \left (\frac {2 \, \sqrt {-a^{2} x^{2} + 1} a \sqrt {c} x \sqrt {\frac {a c x - c}{a x}}}{2 \, a^{2} c x^{2} - a c x - c}\right ) - 2 \, {\left (7 \, a^{5} c^{4} x^{5} + 1755 \, a^{4} c^{4} x^{4} + 720 \, a^{3} c^{4} x^{3} - 110 \, a^{2} c^{4} x^{2} + 20 \, a c^{4} x - 2 \, c^{4}\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {\frac {a c x - c}{a x}}}{14 \, {\left (a^{6} x^{5} - a^{4} x^{3}\right )}}\right ] \] Input:
integrate((c-c/a/x)^(9/2)/(a*x+1)^3*(-a^2*x^2+1)^(3/2),x, algorithm="frica s")
Output:
[1/28*(105*(a^5*c^4*x^5 - a^3*c^4*x^3)*sqrt(-c)*log(-(8*a^3*c*x^3 - 7*a*c* x - 4*(2*a^2*x^2 + a*x)*sqrt(-a^2*x^2 + 1)*sqrt(-c)*sqrt((a*c*x - c)/(a*x) ) - c)/(a*x - 1)) - 4*(7*a^5*c^4*x^5 + 1755*a^4*c^4*x^4 + 720*a^3*c^4*x^3 - 110*a^2*c^4*x^2 + 20*a*c^4*x - 2*c^4)*sqrt(-a^2*x^2 + 1)*sqrt((a*c*x - c )/(a*x)))/(a^6*x^5 - a^4*x^3), 1/14*(105*(a^5*c^4*x^5 - a^3*c^4*x^3)*sqrt( c)*arctan(2*sqrt(-a^2*x^2 + 1)*a*sqrt(c)*x*sqrt((a*c*x - c)/(a*x))/(2*a^2* c*x^2 - a*c*x - c)) - 2*(7*a^5*c^4*x^5 + 1755*a^4*c^4*x^4 + 720*a^3*c^4*x^ 3 - 110*a^2*c^4*x^2 + 20*a*c^4*x - 2*c^4)*sqrt(-a^2*x^2 + 1)*sqrt((a*c*x - c)/(a*x)))/(a^6*x^5 - a^4*x^3)]
Timed out. \[ \int e^{-3 \text {arctanh}(a x)} \left (c-\frac {c}{a x}\right )^{9/2} \, dx=\text {Timed out} \] Input:
integrate((c-c/a/x)**(9/2)/(a*x+1)**3*(-a**2*x**2+1)**(3/2),x)
Output:
Timed out
\[ \int e^{-3 \text {arctanh}(a x)} \left (c-\frac {c}{a x}\right )^{9/2} \, dx=\int { \frac {{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} {\left (c - \frac {c}{a x}\right )}^{\frac {9}{2}}}{{\left (a x + 1\right )}^{3}} \,d x } \] Input:
integrate((c-c/a/x)^(9/2)/(a*x+1)^3*(-a^2*x^2+1)^(3/2),x, algorithm="maxim a")
Output:
integrate((-a^2*x^2 + 1)^(3/2)*(c - c/(a*x))^(9/2)/(a*x + 1)^3, x)
Exception generated. \[ \int e^{-3 \text {arctanh}(a x)} \left (c-\frac {c}{a x}\right )^{9/2} \, dx=\text {Exception raised: TypeError} \] Input:
integrate((c-c/a/x)^(9/2)/(a*x+1)^3*(-a^2*x^2+1)^(3/2),x, algorithm="giac" )
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. \[ \int e^{-3 \text {arctanh}(a x)} \left (c-\frac {c}{a x}\right )^{9/2} \, dx=\int \frac {{\left (c-\frac {c}{a\,x}\right )}^{9/2}\,{\left (1-a^2\,x^2\right )}^{3/2}}{{\left (a\,x+1\right )}^3} \,d x \] Input:
int(((c - c/(a*x))^(9/2)*(1 - a^2*x^2)^(3/2))/(a*x + 1)^3,x)
Output:
int(((c - c/(a*x))^(9/2)*(1 - a^2*x^2)^(3/2))/(a*x + 1)^3, x)
Time = 0.16 (sec) , antiderivative size = 129, normalized size of antiderivative = 0.42 \[ \int e^{-3 \text {arctanh}(a x)} \left (c-\frac {c}{a x}\right )^{9/2} \, dx=\frac {\sqrt {c}\, c^{4} i \left (840 \sqrt {a x +1}\, \mathrm {log}\left (\sqrt {a x +1}\, i +\sqrt {x}\, \sqrt {a}\, i \right ) a^{4} x^{4}+14775 \sqrt {a x +1}\, a^{4} x^{4}-56 \sqrt {x}\, \sqrt {a}\, a^{5} x^{5}-14040 \sqrt {x}\, \sqrt {a}\, a^{4} x^{4}-5760 \sqrt {x}\, \sqrt {a}\, a^{3} x^{3}+880 \sqrt {x}\, \sqrt {a}\, a^{2} x^{2}-160 \sqrt {x}\, \sqrt {a}\, a x +16 \sqrt {x}\, \sqrt {a}\right )}{56 \sqrt {a x +1}\, a^{5} x^{4}} \] Input:
int((c-c/a/x)^(9/2)/(a*x+1)^3*(-a^2*x^2+1)^(3/2),x)
Output:
(sqrt(c)*c**4*i*(840*sqrt(a*x + 1)*log(sqrt(a*x + 1)*i + sqrt(x)*sqrt(a)*i )*a**4*x**4 + 14775*sqrt(a*x + 1)*a**4*x**4 - 56*sqrt(x)*sqrt(a)*a**5*x**5 - 14040*sqrt(x)*sqrt(a)*a**4*x**4 - 5760*sqrt(x)*sqrt(a)*a**3*x**3 + 880* sqrt(x)*sqrt(a)*a**2*x**2 - 160*sqrt(x)*sqrt(a)*a*x + 16*sqrt(x)*sqrt(a))) /(56*sqrt(a*x + 1)*a**5*x**4)