Integrand size = 22, antiderivative size = 249 \[ \int \frac {e^{\text {arctanh}(a x)}}{\left (c-\frac {c}{a x}\right )^{5/2}} \, dx=\frac {\sqrt {1-a x} \sqrt {1+a x}}{2 a \left (c-\frac {c}{a x}\right )^{5/2}}-\frac {11 (1-a x)^{3/2} \sqrt {1+a x}}{8 a^2 \left (c-\frac {c}{a x}\right )^{5/2} x}-\frac {23 (1-a x)^{5/2} \sqrt {1+a x}}{8 a^3 \left (c-\frac {c}{a x}\right )^{5/2} x^2}-\frac {7 (1-a x)^{5/2} \text {arcsinh}\left (\sqrt {a} \sqrt {x}\right )}{a^{7/2} \left (c-\frac {c}{a x}\right )^{5/2} x^{5/2}}+\frac {79 (1-a x)^{5/2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {x}}{\sqrt {1+a x}}\right )}{8 \sqrt {2} a^{7/2} \left (c-\frac {c}{a x}\right )^{5/2} x^{5/2}} \] Output:
1/2*(-a*x+1)^(1/2)*(a*x+1)^(1/2)/a/(c-c/a/x)^(5/2)-11/8*(-a*x+1)^(3/2)*(a* x+1)^(1/2)/a^2/(c-c/a/x)^(5/2)/x-23/8*(-a*x+1)^(5/2)*(a*x+1)^(1/2)/a^3/(c- c/a/x)^(5/2)/x^2-7*(-a*x+1)^(5/2)*arcsinh(a^(1/2)*x^(1/2))/a^(7/2)/(c-c/a/ x)^(5/2)/x^(5/2)+79/16*(-a*x+1)^(5/2)*arctanh(2^(1/2)*a^(1/2)*x^(1/2)/(a*x +1)^(1/2))*2^(1/2)/a^(7/2)/(c-c/a/x)^(5/2)/x^(5/2)
Time = 0.19 (sec) , antiderivative size = 139, normalized size of antiderivative = 0.56 \[ \int \frac {e^{\text {arctanh}(a x)}}{\left (c-\frac {c}{a x}\right )^{5/2}} \, dx=\frac {-2 \sqrt {a} \sqrt {x} \sqrt {1+a x} \left (23-35 a x+8 a^2 x^2\right )-112 (-1+a x)^2 \text {arcsinh}\left (\sqrt {a} \sqrt {x}\right )+79 \sqrt {2} (-1+a x)^2 \text {arctanh}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {x}}{\sqrt {1+a x}}\right )}{16 a^{3/2} c^2 \sqrt {c-\frac {c}{a x}} \sqrt {x} (1-a x)^{3/2}} \] Input:
Integrate[E^ArcTanh[a*x]/(c - c/(a*x))^(5/2),x]
Output:
(-2*Sqrt[a]*Sqrt[x]*Sqrt[1 + a*x]*(23 - 35*a*x + 8*a^2*x^2) - 112*(-1 + a* x)^2*ArcSinh[Sqrt[a]*Sqrt[x]] + 79*Sqrt[2]*(-1 + a*x)^2*ArcTanh[(Sqrt[2]*S qrt[a]*Sqrt[x])/Sqrt[1 + a*x]])/(16*a^(3/2)*c^2*Sqrt[c - c/(a*x)]*Sqrt[x]* (1 - a*x)^(3/2))
Time = 0.85 (sec) , antiderivative size = 187, normalized size of antiderivative = 0.75, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.636, Rules used = {6684, 6679, 108, 27, 166, 27, 171, 25, 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.
| \(\displaystyle \int \frac {e^{\text {arctanh}(a x)}}{\left (c-\frac {c}{a x}\right )^{5/2}} \, dx\) |
| \(\Big \downarrow \) 6684 |
| \(\displaystyle \frac {(1-a x)^{5/2} \int \frac {e^{\text {arctanh}(a x)} x^{5/2}}{(1-a x)^{5/2}}dx}{x^{5/2} \left (c-\frac {c}{a x}\right )^{5/2}}\) |
| \(\Big \downarrow \) 6679 |
| \(\displaystyle \frac {(1-a x)^{5/2} \int \frac {x^{5/2} \sqrt {a x+1}}{(1-a x)^3}dx}{x^{5/2} \left (c-\frac {c}{a x}\right )^{5/2}}\) |
| \(\Big \downarrow \) 108 |
| \(\displaystyle \frac {(1-a x)^{5/2} \left (\frac {x^{5/2} \sqrt {a x+1}}{2 a (1-a x)^2}-\frac {\int \frac {x^{3/2} (6 a x+5)}{2 (1-a x)^2 \sqrt {a x+1}}dx}{2 a}\right )}{x^{5/2} \left (c-\frac {c}{a x}\right )^{5/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {(1-a x)^{5/2} \left (\frac {x^{5/2} \sqrt {a x+1}}{2 a (1-a x)^2}-\frac {\int \frac {x^{3/2} (6 a x+5)}{(1-a x)^2 \sqrt {a x+1}}dx}{4 a}\right )}{x^{5/2} \left (c-\frac {c}{a x}\right )^{5/2}}\) |
| \(\Big \downarrow \) 166 |
| \(\displaystyle \frac {(1-a x)^{5/2} \left (\frac {x^{5/2} \sqrt {a x+1}}{2 a (1-a x)^2}-\frac {\frac {\int -\frac {a \sqrt {x} (46 a x+33)}{2 (1-a x) \sqrt {a x+1}}dx}{2 a^2}+\frac {11 x^{3/2} \sqrt {a x+1}}{2 a (1-a x)}}{4 a}\right )}{x^{5/2} \left (c-\frac {c}{a x}\right )^{5/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {(1-a x)^{5/2} \left (\frac {x^{5/2} \sqrt {a x+1}}{2 a (1-a x)^2}-\frac {\frac {11 x^{3/2} \sqrt {a x+1}}{2 a (1-a x)}-\frac {\int \frac {\sqrt {x} (46 a x+33)}{(1-a x) \sqrt {a x+1}}dx}{4 a}}{4 a}\right )}{x^{5/2} \left (c-\frac {c}{a x}\right )^{5/2}}\) |
| \(\Big \downarrow \) 171 |
| \(\displaystyle \frac {(1-a x)^{5/2} \left (\frac {x^{5/2} \sqrt {a x+1}}{2 a (1-a x)^2}-\frac {\frac {11 x^{3/2} \sqrt {a x+1}}{2 a (1-a x)}-\frac {-\frac {\int -\frac {a (56 a x+23)}{\sqrt {x} (1-a x) \sqrt {a x+1}}dx}{a^2}-\frac {46 \sqrt {x} \sqrt {a x+1}}{a}}{4 a}}{4 a}\right )}{x^{5/2} \left (c-\frac {c}{a x}\right )^{5/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {(1-a x)^{5/2} \left (\frac {x^{5/2} \sqrt {a x+1}}{2 a (1-a x)^2}-\frac {\frac {11 x^{3/2} \sqrt {a x+1}}{2 a (1-a x)}-\frac {\frac {\int \frac {a (56 a x+23)}{\sqrt {x} (1-a x) \sqrt {a x+1}}dx}{a^2}-\frac {46 \sqrt {x} \sqrt {a x+1}}{a}}{4 a}}{4 a}\right )}{x^{5/2} \left (c-\frac {c}{a x}\right )^{5/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {(1-a x)^{5/2} \left (\frac {x^{5/2} \sqrt {a x+1}}{2 a (1-a x)^2}-\frac {\frac {11 x^{3/2} \sqrt {a x+1}}{2 a (1-a x)}-\frac {\frac {\int \frac {56 a x+23}{\sqrt {x} (1-a x) \sqrt {a x+1}}dx}{a}-\frac {46 \sqrt {x} \sqrt {a x+1}}{a}}{4 a}}{4 a}\right )}{x^{5/2} \left (c-\frac {c}{a x}\right )^{5/2}}\) |
| \(\Big \downarrow \) 175 |
| \(\displaystyle \frac {(1-a x)^{5/2} \left (\frac {x^{5/2} \sqrt {a x+1}}{2 a (1-a x)^2}-\frac {\frac {11 x^{3/2} \sqrt {a x+1}}{2 a (1-a x)}-\frac {\frac {79 \int \frac {1}{\sqrt {x} (1-a x) \sqrt {a x+1}}dx-56 \int \frac {1}{\sqrt {x} \sqrt {a x+1}}dx}{a}-\frac {46 \sqrt {x} \sqrt {a x+1}}{a}}{4 a}}{4 a}\right )}{x^{5/2} \left (c-\frac {c}{a x}\right )^{5/2}}\) |
| \(\Big \downarrow \) 63 |
| \(\displaystyle \frac {(1-a x)^{5/2} \left (\frac {x^{5/2} \sqrt {a x+1}}{2 a (1-a x)^2}-\frac {\frac {11 x^{3/2} \sqrt {a x+1}}{2 a (1-a x)}-\frac {\frac {79 \int \frac {1}{\sqrt {x} (1-a x) \sqrt {a x+1}}dx-112 \int \frac {1}{\sqrt {a x+1}}d\sqrt {x}}{a}-\frac {46 \sqrt {x} \sqrt {a x+1}}{a}}{4 a}}{4 a}\right )}{x^{5/2} \left (c-\frac {c}{a x}\right )^{5/2}}\) |
| \(\Big \downarrow \) 104 |
| \(\displaystyle \frac {(1-a x)^{5/2} \left (\frac {x^{5/2} \sqrt {a x+1}}{2 a (1-a x)^2}-\frac {\frac {11 x^{3/2} \sqrt {a x+1}}{2 a (1-a x)}-\frac {\frac {158 \int \frac {1}{1-\frac {2 a x}{a x+1}}d\frac {\sqrt {x}}{\sqrt {a x+1}}-112 \int \frac {1}{\sqrt {a x+1}}d\sqrt {x}}{a}-\frac {46 \sqrt {x} \sqrt {a x+1}}{a}}{4 a}}{4 a}\right )}{x^{5/2} \left (c-\frac {c}{a x}\right )^{5/2}}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {(1-a x)^{5/2} \left (\frac {x^{5/2} \sqrt {a x+1}}{2 a (1-a x)^2}-\frac {\frac {11 x^{3/2} \sqrt {a x+1}}{2 a (1-a x)}-\frac {\frac {\frac {79 \sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {x}}{\sqrt {a x+1}}\right )}{\sqrt {a}}-112 \int \frac {1}{\sqrt {a x+1}}d\sqrt {x}}{a}-\frac {46 \sqrt {x} \sqrt {a x+1}}{a}}{4 a}}{4 a}\right )}{x^{5/2} \left (c-\frac {c}{a x}\right )^{5/2}}\) |
| \(\Big \downarrow \) 222 |
| \(\displaystyle \frac {(1-a x)^{5/2} \left (\frac {x^{5/2} \sqrt {a x+1}}{2 a (1-a x)^2}-\frac {\frac {11 x^{3/2} \sqrt {a x+1}}{2 a (1-a x)}-\frac {\frac {\frac {79 \sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {x}}{\sqrt {a x+1}}\right )}{\sqrt {a}}-\frac {112 \text {arcsinh}\left (\sqrt {a} \sqrt {x}\right )}{\sqrt {a}}}{a}-\frac {46 \sqrt {x} \sqrt {a x+1}}{a}}{4 a}}{4 a}\right )}{x^{5/2} \left (c-\frac {c}{a x}\right )^{5/2}}\) |
Input:
Int[E^ArcTanh[a*x]/(c - c/(a*x))^(5/2),x]
Output:
((1 - a*x)^(5/2)*((x^(5/2)*Sqrt[1 + a*x])/(2*a*(1 - a*x)^2) - ((11*x^(3/2) *Sqrt[1 + a*x])/(2*a*(1 - a*x)) - ((-46*Sqrt[x]*Sqrt[1 + a*x])/a + ((-112* ArcSinh[Sqrt[a]*Sqrt[x]])/Sqrt[a] + (79*Sqrt[2]*ArcTanh[(Sqrt[2]*Sqrt[a]*S qrt[x])/Sqrt[1 + a*x]])/Sqrt[a])/a)/(4*a))/(4*a)))/((c - c/(a*x))^(5/2)*x^ (5/2))
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 _)), 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 + 1)*(c + d*x)^n*((e + f*x)^p/(b*(m + 1))) , x] - Simp[1/(b*(m + 1)) Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f* x)^(p - 1)*Simp[d*e*n + c*f*p + d*f*(n + p)*x, x], x], x] /; FreeQ[{a, b, c , d, e, f}, x] && LtQ[m, -1] && GtQ[n, 0] && GtQ[p, 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[(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] && ILtQ[m, -1] && GtQ[n, 0]
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^(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 = 0.17 (sec) , antiderivative size = 365, normalized size of antiderivative = 1.47
| method | result | size |
| risch | \(\frac {\left (a x +1\right ) \sqrt {\frac {c a x \left (-a^{2} x^{2}+1\right )}{a x -1}}\, \left (a x -1\right )}{a \sqrt {-\left (a x +1\right ) a c x}\, \sqrt {\frac {c \left (a x -1\right )}{a x}}\, \sqrt {-a^{2} x^{2}+1}\, c^{2}}+\frac {\left (\frac {7 \arctan \left (\frac {\sqrt {a^{2} c}\, \left (x +\frac {1}{2 a}\right )}{\sqrt {-a^{2} c \,x^{2}-a c x}}\right )}{2 a^{3} \sqrt {a^{2} c}}+\frac {\sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2} c -3 \left (x -\frac {1}{a}\right ) a c -2 c}}{2 a^{6} c \left (x -\frac {1}{a}\right )^{2}}+\frac {19 \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2} c -3 \left (x -\frac {1}{a}\right ) a c -2 c}}{8 a^{5} c \left (x -\frac {1}{a}\right )}-\frac {79 \ln \left (\frac {-4 c -3 \left (x -\frac {1}{a}\right ) a c +2 \sqrt {-2 c}\, \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2} c -3 \left (x -\frac {1}{a}\right ) a c -2 c}}{x -\frac {1}{a}}\right )}{16 a^{4} \sqrt {-2 c}}\right ) a^{2} \sqrt {\frac {c a x \left (-a^{2} x^{2}+1\right )}{a x -1}}\, \left (a x -1\right )}{x \sqrt {\frac {c \left (a x -1\right )}{a x}}\, \sqrt {-a^{2} x^{2}+1}\, c^{2}}\) | \(365\) |
| default | \(-\frac {\sqrt {\frac {c \left (a x -1\right )}{a x}}\, x \sqrt {-a^{2} x^{2}+1}\, \left (16 \sqrt {-x \left (a x +1\right )}\, a^{\frac {7}{2}} \sqrt {2}\, \sqrt {-\frac {1}{a}}\, x^{2}-56 a^{3} \arctan \left (\frac {2 a x +1}{2 \sqrt {a}\, \sqrt {-x \left (a x +1\right )}}\right ) \sqrt {2}\, \sqrt {-\frac {1}{a}}\, x^{2}-70 \sqrt {-x \left (a x +1\right )}\, a^{\frac {5}{2}} \sqrt {2}\, \sqrt {-\frac {1}{a}}\, x +112 a^{2} \arctan \left (\frac {2 a x +1}{2 \sqrt {a}\, \sqrt {-x \left (a x +1\right )}}\right ) \sqrt {2}\, \sqrt {-\frac {1}{a}}\, x +79 a^{\frac {5}{2}} \ln \left (\frac {2 \sqrt {2}\, \sqrt {-\frac {1}{a}}\, \sqrt {-x \left (a x +1\right )}\, a -3 a x -1}{a x -1}\right ) x^{2}+46 \sqrt {-x \left (a x +1\right )}\, a^{\frac {3}{2}} \sqrt {2}\, \sqrt {-\frac {1}{a}}-56 \arctan \left (\frac {2 a x +1}{2 \sqrt {a}\, \sqrt {-x \left (a x +1\right )}}\right ) a \sqrt {2}\, \sqrt {-\frac {1}{a}}-158 a^{\frac {3}{2}} \ln \left (\frac {2 \sqrt {2}\, \sqrt {-\frac {1}{a}}\, \sqrt {-x \left (a x +1\right )}\, a -3 a x -1}{a x -1}\right ) x +79 \ln \left (\frac {2 \sqrt {2}\, \sqrt {-\frac {1}{a}}\, \sqrt {-x \left (a x +1\right )}\, a -3 a x -1}{a x -1}\right ) \sqrt {a}\right ) \sqrt {2}}{32 a^{\frac {3}{2}} c^{3} \left (a x -1\right )^{3} \sqrt {-x \left (a x +1\right )}\, \sqrt {-\frac {1}{a}}}\) | \(390\) |
Input:
int((a*x+1)/(-a^2*x^2+1)^(1/2)/(c-c/a/x)^(5/2),x,method=_RETURNVERBOSE)
Output:
1/a*(a*x+1)/(-(a*x+1)*a*c*x)^(1/2)/(c*(a*x-1)/a/x)^(1/2)*(c/(a*x-1)*a*x*(- a^2*x^2+1))^(1/2)/(-a^2*x^2+1)^(1/2)*(a*x-1)/c^2+(7/2/a^3/(a^2*c)^(1/2)*ar ctan((a^2*c)^(1/2)*(x+1/2/a)/(-a^2*c*x^2-a*c*x)^(1/2))+1/2/a^6/c/(x-1/a)^2 *(-(x-1/a)^2*a^2*c-3*(x-1/a)*a*c-2*c)^(1/2)+19/8/a^5/c/(x-1/a)*(-(x-1/a)^2 *a^2*c-3*(x-1/a)*a*c-2*c)^(1/2)-79/16/a^4/(-2*c)^(1/2)*ln((-4*c-3*(x-1/a)* a*c+2*(-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 ^2/x/(c*(a*x-1)/a/x)^(1/2)*(c/(a*x-1)*a*x*(-a^2*x^2+1))^(1/2)/(-a^2*x^2+1) ^(1/2)*(a*x-1)/c^2
Time = 0.13 (sec) , antiderivative size = 600, normalized size of antiderivative = 2.41 \[ \int \frac {e^{\text {arctanh}(a x)}}{\left (c-\frac {c}{a x}\right )^{5/2}} \, dx=\left [-\frac {79 \, \sqrt {2} {\left (a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1\right )} \sqrt {-c} \log \left (-\frac {17 \, a^{3} c x^{3} - 3 \, a^{2} c x^{2} - 13 \, a c x + 4 \, \sqrt {2} {\left (3 \, a^{2} x^{2} + a x\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {-c} \sqrt {\frac {a c x - c}{a x}} - c}{a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1}\right ) + 112 \, {\left (a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1\right )} \sqrt {-c} \log \left (-\frac {8 \, a^{3} c x^{3} - 7 \, a c x - 4 \, {\left (2 \, a^{2} x^{2} + a x\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {-c} \sqrt {\frac {a c x - c}{a x}} - c}{a x - 1}\right ) + 8 \, {\left (8 \, a^{3} x^{3} - 35 \, a^{2} x^{2} + 23 \, a x\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {\frac {a c x - c}{a x}}}{64 \, {\left (a^{4} c^{3} x^{3} - 3 \, a^{3} c^{3} x^{2} + 3 \, a^{2} c^{3} x - a c^{3}\right )}}, \frac {79 \, \sqrt {2} {\left (a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1\right )} \sqrt {c} \arctan \left (\frac {2 \, \sqrt {2} \sqrt {-a^{2} x^{2} + 1} a \sqrt {c} x \sqrt {\frac {a c x - c}{a x}}}{3 \, a^{2} c x^{2} - 2 \, a c x - c}\right ) - 112 \, {\left (a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1\right )} \sqrt {c} \arctan \left (\frac {2 \, \sqrt {-a^{2} x^{2} + 1} a \sqrt {c} x \sqrt {\frac {a c x - c}{a x}}}{2 \, a^{2} c x^{2} - a c x - c}\right ) - 4 \, {\left (8 \, a^{3} x^{3} - 35 \, a^{2} x^{2} + 23 \, a x\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {\frac {a c x - c}{a x}}}{32 \, {\left (a^{4} c^{3} x^{3} - 3 \, a^{3} c^{3} x^{2} + 3 \, a^{2} c^{3} x - a c^{3}\right )}}\right ] \] Input:
integrate((a*x+1)/(-a^2*x^2+1)^(1/2)/(c-c/a/x)^(5/2),x, algorithm="fricas" )
Output:
[-1/64*(79*sqrt(2)*(a^3*x^3 - 3*a^2*x^2 + 3*a*x - 1)*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)) + 112*(a^3*x^3 - 3*a^2*x^2 + 3*a*x - 1)*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)) + 8*(8*a^3*x^3 - 35*a^2*x^2 + 23*a*x)*sqrt(-a^2*x ^2 + 1)*sqrt((a*c*x - c)/(a*x)))/(a^4*c^3*x^3 - 3*a^3*c^3*x^2 + 3*a^2*c^3* x - a*c^3), 1/32*(79*sqrt(2)*(a^3*x^3 - 3*a^2*x^2 + 3*a*x - 1)*sqrt(c)*arc tan(2*sqrt(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)) - 112*(a^3*x^3 - 3*a^2*x^2 + 3*a*x - 1)*sqrt(c)*ar ctan(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)) - 4*(8*a^3*x^3 - 35*a^2*x^2 + 23*a*x)*sqrt(-a^2*x^2 + 1)*sq rt((a*c*x - c)/(a*x)))/(a^4*c^3*x^3 - 3*a^3*c^3*x^2 + 3*a^2*c^3*x - a*c^3) ]
\[ \int \frac {e^{\text {arctanh}(a x)}}{\left (c-\frac {c}{a x}\right )^{5/2}} \, dx=\int \frac {a x + 1}{\left (- c \left (-1 + \frac {1}{a x}\right )\right )^{\frac {5}{2}} \sqrt {- \left (a x - 1\right ) \left (a x + 1\right )}}\, dx \] Input:
integrate((a*x+1)/(-a**2*x**2+1)**(1/2)/(c-c/a/x)**(5/2),x)
Output:
Integral((a*x + 1)/((-c*(-1 + 1/(a*x)))**(5/2)*sqrt(-(a*x - 1)*(a*x + 1))) , x)
\[ \int \frac {e^{\text {arctanh}(a x)}}{\left (c-\frac {c}{a x}\right )^{5/2}} \, dx=\int { \frac {a x + 1}{\sqrt {-a^{2} x^{2} + 1} {\left (c - \frac {c}{a x}\right )}^{\frac {5}{2}}} \,d x } \] Input:
integrate((a*x+1)/(-a^2*x^2+1)^(1/2)/(c-c/a/x)^(5/2),x, algorithm="maxima" )
Output:
integrate((a*x + 1)/(sqrt(-a^2*x^2 + 1)*(c - c/(a*x))^(5/2)), x)
\[ \int \frac {e^{\text {arctanh}(a x)}}{\left (c-\frac {c}{a x}\right )^{5/2}} \, dx=\int { \frac {a x + 1}{\sqrt {-a^{2} x^{2} + 1} {\left (c - \frac {c}{a x}\right )}^{\frac {5}{2}}} \,d x } \] Input:
integrate((a*x+1)/(-a^2*x^2+1)^(1/2)/(c-c/a/x)^(5/2),x, algorithm="giac")
Output:
integrate((a*x + 1)/(sqrt(-a^2*x^2 + 1)*(c - c/(a*x))^(5/2)), x)
Timed out. \[ \int \frac {e^{\text {arctanh}(a x)}}{\left (c-\frac {c}{a x}\right )^{5/2}} \, dx=\int \frac {a\,x+1}{{\left (c-\frac {c}{a\,x}\right )}^{5/2}\,\sqrt {1-a^2\,x^2}} \,d x \] Input:
int((a*x + 1)/((c - c/(a*x))^(5/2)*(1 - a^2*x^2)^(1/2)),x)
Output:
int((a*x + 1)/((c - c/(a*x))^(5/2)*(1 - a^2*x^2)^(1/2)), x)
Time = 0.20 (sec) , antiderivative size = 233, normalized size of antiderivative = 0.94 \[ \int \frac {e^{\text {arctanh}(a x)}}{\left (c-\frac {c}{a x}\right )^{5/2}} \, dx=\frac {\sqrt {c}\, \left (79 \sqrt {2}\, \mathit {atan} \left (\frac {\sqrt {x}\, \sqrt {a}\, \sqrt {a x +1}\, \sqrt {2}\, i}{a x +1}\right ) a^{2} x^{2}-158 \sqrt {2}\, \mathit {atan} \left (\frac {\sqrt {x}\, \sqrt {a}\, \sqrt {a x +1}\, \sqrt {2}\, i}{a x +1}\right ) a x +79 \sqrt {2}\, \mathit {atan} \left (\frac {\sqrt {x}\, \sqrt {a}\, \sqrt {a x +1}\, \sqrt {2}\, i}{a x +1}\right )-112 \mathit {atan} \left (\frac {\sqrt {x}\, \sqrt {a}\, \sqrt {a x +1}\, i}{a x +1}\right ) a^{2} x^{2}+224 \mathit {atan} \left (\frac {\sqrt {x}\, \sqrt {a}\, \sqrt {a x +1}\, i}{a x +1}\right ) a x -112 \mathit {atan} \left (\frac {\sqrt {x}\, \sqrt {a}\, \sqrt {a x +1}\, i}{a x +1}\right )-16 \sqrt {x}\, \sqrt {a}\, \sqrt {a x +1}\, a^{2} i \,x^{2}+70 \sqrt {x}\, \sqrt {a}\, \sqrt {a x +1}\, a i x -46 \sqrt {x}\, \sqrt {a}\, \sqrt {a x +1}\, i \right )}{16 a \,c^{3} \left (a^{2} x^{2}-2 a x +1\right )} \] Input:
int((a*x+1)/(-a^2*x^2+1)^(1/2)/(c-c/a/x)^(5/2),x)
Output:
(sqrt(c)*(79*sqrt(2)*atan((sqrt(x)*sqrt(a)*sqrt(a*x + 1)*sqrt(2)*i)/(a*x + 1))*a**2*x**2 - 158*sqrt(2)*atan((sqrt(x)*sqrt(a)*sqrt(a*x + 1)*sqrt(2)*i )/(a*x + 1))*a*x + 79*sqrt(2)*atan((sqrt(x)*sqrt(a)*sqrt(a*x + 1)*sqrt(2)* i)/(a*x + 1)) - 112*atan((sqrt(x)*sqrt(a)*sqrt(a*x + 1)*i)/(a*x + 1))*a**2 *x**2 + 224*atan((sqrt(x)*sqrt(a)*sqrt(a*x + 1)*i)/(a*x + 1))*a*x - 112*at an((sqrt(x)*sqrt(a)*sqrt(a*x + 1)*i)/(a*x + 1)) - 16*sqrt(x)*sqrt(a)*sqrt( a*x + 1)*a**2*i*x**2 + 70*sqrt(x)*sqrt(a)*sqrt(a*x + 1)*a*i*x - 46*sqrt(x) *sqrt(a)*sqrt(a*x + 1)*i))/(16*a*c**3*(a**2*x**2 - 2*a*x + 1))