Integrand size = 24, antiderivative size = 251 \[ \int \frac {e^{-3 \text {arctanh}(a x)}}{\left (c-\frac {c}{a x}\right )^{7/2}} \, dx=\frac {(1-a x)^{5/2}}{2 a^2 \left (c-\frac {c}{a x}\right )^{7/2} x \sqrt {1+a x}}-\frac {(1-a x)^{7/2}}{4 a^3 \left (c-\frac {c}{a x}\right )^{7/2} x^2 \sqrt {1+a x}}+\frac {7 (1-a x)^{7/2} \sqrt {1+a x}}{4 a^4 \left (c-\frac {c}{a x}\right )^{7/2} x^3}+\frac {(1-a x)^{7/2} \text {arcsinh}\left (\sqrt {a} \sqrt {x}\right )}{a^{9/2} \left (c-\frac {c}{a x}\right )^{7/2} x^{7/2}}-\frac {11 (1-a x)^{7/2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {x}}{\sqrt {1+a x}}\right )}{4 \sqrt {2} a^{9/2} \left (c-\frac {c}{a x}\right )^{7/2} x^{7/2}} \] Output:
1/2*(-a*x+1)^(5/2)/a^2/(c-c/a/x)^(7/2)/x/(a*x+1)^(1/2)-1/4*(-a*x+1)^(7/2)/ a^3/(c-c/a/x)^(7/2)/x^2/(a*x+1)^(1/2)+7/4*(-a*x+1)^(7/2)*(a*x+1)^(1/2)/a^4 /(c-c/a/x)^(7/2)/x^3+(-a*x+1)^(7/2)*arcsinh(a^(1/2)*x^(1/2))/a^(9/2)/(c-c/ a/x)^(7/2)/x^(7/2)-11/8*(-a*x+1)^(7/2)*arctanh(2^(1/2)*a^(1/2)*x^(1/2)/(a* x+1)^(1/2))*2^(1/2)/a^(9/2)/(c-c/a/x)^(7/2)/x^(7/2)
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.63 (sec) , antiderivative size = 234, normalized size of antiderivative = 0.93 \[ \int \frac {e^{-3 \text {arctanh}(a x)}}{\left (c-\frac {c}{a x}\right )^{7/2}} \, dx=\frac {35 \left (\sqrt {a} \sqrt {x} \left (-25+2 a x+13 a^2 x^2+2 a^3 x^3\right )+19 (-1+a x) \sqrt {1+a x} \text {arcsinh}\left (\sqrt {a} \sqrt {x}\right )-22 (-1+a x) \sqrt {2+2 a x} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {x}}{\sqrt {1+a x}}\right )\right )-56 a^{5/2} x^{5/2} (-1+a x) \sqrt {1+a x} \operatorname {Hypergeometric2F1}\left (\frac {3}{2},\frac {5}{2},\frac {7}{2},-a x\right )-40 a^{7/2} x^{7/2} (-1+a x) \sqrt {1+a x} \operatorname {Hypergeometric2F1}\left (\frac {3}{2},\frac {7}{2},\frac {9}{2},-a x\right )}{560 a^{3/2} c^3 \sqrt {c-\frac {c}{a x}} \sqrt {x} \sqrt {1-a^2 x^2}} \] Input:
Integrate[1/(E^(3*ArcTanh[a*x])*(c - c/(a*x))^(7/2)),x]
Output:
(35*(Sqrt[a]*Sqrt[x]*(-25 + 2*a*x + 13*a^2*x^2 + 2*a^3*x^3) + 19*(-1 + a*x )*Sqrt[1 + a*x]*ArcSinh[Sqrt[a]*Sqrt[x]] - 22*(-1 + a*x)*Sqrt[2 + 2*a*x]*A rcTanh[(Sqrt[2]*Sqrt[a]*Sqrt[x])/Sqrt[1 + a*x]]) - 56*a^(5/2)*x^(5/2)*(-1 + a*x)*Sqrt[1 + a*x]*Hypergeometric2F1[3/2, 5/2, 7/2, -(a*x)] - 40*a^(7/2) *x^(7/2)*(-1 + a*x)*Sqrt[1 + a*x]*Hypergeometric2F1[3/2, 7/2, 9/2, -(a*x)] )/(560*a^(3/2)*c^3*Sqrt[c - c/(a*x)]*Sqrt[x]*Sqrt[1 - a^2*x^2])
Time = 0.53 (sec) , antiderivative size = 177, normalized size of antiderivative = 0.71, 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, 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.
| \(\displaystyle \int \frac {e^{-3 \text {arctanh}(a x)}}{\left (c-\frac {c}{a x}\right )^{7/2}} \, dx\) |
| \(\Big \downarrow \) 6684 |
| \(\displaystyle \frac {(1-a x)^{7/2} \int \frac {e^{-3 \text {arctanh}(a x)} x^{7/2}}{(1-a x)^{7/2}}dx}{x^{7/2} \left (c-\frac {c}{a x}\right )^{7/2}}\) |
| \(\Big \downarrow \) 6679 |
| \(\displaystyle \frac {(1-a x)^{7/2} \int \frac {x^{7/2}}{(1-a x)^2 (a x+1)^{3/2}}dx}{x^{7/2} \left (c-\frac {c}{a x}\right )^{7/2}}\) |
| \(\Big \downarrow \) 109 |
| \(\displaystyle \frac {(1-a x)^{7/2} \left (\frac {x^{5/2}}{2 a^2 (1-a x) \sqrt {a x+1}}-\frac {\int \frac {x^{3/2} (6 a x+5)}{2 (1-a x) (a x+1)^{3/2}}dx}{2 a^2}\right )}{x^{7/2} \left (c-\frac {c}{a x}\right )^{7/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {(1-a x)^{7/2} \left (\frac {x^{5/2}}{2 a^2 (1-a x) \sqrt {a x+1}}-\frac {\int \frac {x^{3/2} (6 a x+5)}{(1-a x) (a x+1)^{3/2}}dx}{4 a^2}\right )}{x^{7/2} \left (c-\frac {c}{a x}\right )^{7/2}}\) |
| \(\Big \downarrow \) 167 |
| \(\displaystyle \frac {(1-a x)^{7/2} \left (\frac {x^{5/2}}{2 a^2 (1-a x) \sqrt {a x+1}}-\frac {\frac {\int -\frac {a \sqrt {x} (3-14 a x)}{2 (1-a x) \sqrt {a x+1}}dx}{a^2}+\frac {x^{3/2}}{a \sqrt {a x+1}}}{4 a^2}\right )}{x^{7/2} \left (c-\frac {c}{a x}\right )^{7/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {(1-a x)^{7/2} \left (\frac {x^{5/2}}{2 a^2 (1-a x) \sqrt {a x+1}}-\frac {\frac {x^{3/2}}{a \sqrt {a x+1}}-\frac {\int \frac {\sqrt {x} (3-14 a x)}{(1-a x) \sqrt {a x+1}}dx}{2 a}}{4 a^2}\right )}{x^{7/2} \left (c-\frac {c}{a x}\right )^{7/2}}\) |
| \(\Big \downarrow \) 171 |
| \(\displaystyle \frac {(1-a x)^{7/2} \left (\frac {x^{5/2}}{2 a^2 (1-a x) \sqrt {a x+1}}-\frac {\frac {x^{3/2}}{a \sqrt {a x+1}}-\frac {\frac {14 \sqrt {x} \sqrt {a x+1}}{a}-\frac {\int \frac {a (4 a x+7)}{\sqrt {x} (1-a x) \sqrt {a x+1}}dx}{a^2}}{2 a}}{4 a^2}\right )}{x^{7/2} \left (c-\frac {c}{a x}\right )^{7/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {(1-a x)^{7/2} \left (\frac {x^{5/2}}{2 a^2 (1-a x) \sqrt {a x+1}}-\frac {\frac {x^{3/2}}{a \sqrt {a x+1}}-\frac {\frac {14 \sqrt {x} \sqrt {a x+1}}{a}-\frac {\int \frac {4 a x+7}{\sqrt {x} (1-a x) \sqrt {a x+1}}dx}{a}}{2 a}}{4 a^2}\right )}{x^{7/2} \left (c-\frac {c}{a x}\right )^{7/2}}\) |
| \(\Big \downarrow \) 175 |
| \(\displaystyle \frac {(1-a x)^{7/2} \left (\frac {x^{5/2}}{2 a^2 (1-a x) \sqrt {a x+1}}-\frac {\frac {x^{3/2}}{a \sqrt {a x+1}}-\frac {\frac {14 \sqrt {x} \sqrt {a x+1}}{a}-\frac {11 \int \frac {1}{\sqrt {x} (1-a x) \sqrt {a x+1}}dx-4 \int \frac {1}{\sqrt {x} \sqrt {a x+1}}dx}{a}}{2 a}}{4 a^2}\right )}{x^{7/2} \left (c-\frac {c}{a x}\right )^{7/2}}\) |
| \(\Big \downarrow \) 63 |
| \(\displaystyle \frac {(1-a x)^{7/2} \left (\frac {x^{5/2}}{2 a^2 (1-a x) \sqrt {a x+1}}-\frac {\frac {x^{3/2}}{a \sqrt {a x+1}}-\frac {\frac {14 \sqrt {x} \sqrt {a x+1}}{a}-\frac {11 \int \frac {1}{\sqrt {x} (1-a x) \sqrt {a x+1}}dx-8 \int \frac {1}{\sqrt {a x+1}}d\sqrt {x}}{a}}{2 a}}{4 a^2}\right )}{x^{7/2} \left (c-\frac {c}{a x}\right )^{7/2}}\) |
| \(\Big \downarrow \) 104 |
| \(\displaystyle \frac {(1-a x)^{7/2} \left (\frac {x^{5/2}}{2 a^2 (1-a x) \sqrt {a x+1}}-\frac {\frac {x^{3/2}}{a \sqrt {a x+1}}-\frac {\frac {14 \sqrt {x} \sqrt {a x+1}}{a}-\frac {22 \int \frac {1}{1-\frac {2 a x}{a x+1}}d\frac {\sqrt {x}}{\sqrt {a x+1}}-8 \int \frac {1}{\sqrt {a x+1}}d\sqrt {x}}{a}}{2 a}}{4 a^2}\right )}{x^{7/2} \left (c-\frac {c}{a x}\right )^{7/2}}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {(1-a x)^{7/2} \left (\frac {x^{5/2}}{2 a^2 (1-a x) \sqrt {a x+1}}-\frac {\frac {x^{3/2}}{a \sqrt {a x+1}}-\frac {\frac {14 \sqrt {x} \sqrt {a x+1}}{a}-\frac {\frac {11 \sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {x}}{\sqrt {a x+1}}\right )}{\sqrt {a}}-8 \int \frac {1}{\sqrt {a x+1}}d\sqrt {x}}{a}}{2 a}}{4 a^2}\right )}{x^{7/2} \left (c-\frac {c}{a x}\right )^{7/2}}\) |
| \(\Big \downarrow \) 222 |
| \(\displaystyle \frac {(1-a x)^{7/2} \left (\frac {x^{5/2}}{2 a^2 (1-a x) \sqrt {a x+1}}-\frac {\frac {x^{3/2}}{a \sqrt {a x+1}}-\frac {\frac {14 \sqrt {x} \sqrt {a x+1}}{a}-\frac {\frac {11 \sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {x}}{\sqrt {a x+1}}\right )}{\sqrt {a}}-\frac {8 \text {arcsinh}\left (\sqrt {a} \sqrt {x}\right )}{\sqrt {a}}}{a}}{2 a}}{4 a^2}\right )}{x^{7/2} \left (c-\frac {c}{a x}\right )^{7/2}}\) |
Input:
Int[1/(E^(3*ArcTanh[a*x])*(c - c/(a*x))^(7/2)),x]
Output:
((1 - a*x)^(7/2)*(x^(5/2)/(2*a^2*(1 - a*x)*Sqrt[1 + a*x]) - (x^(3/2)/(a*Sq rt[1 + a*x]) - ((14*Sqrt[x]*Sqrt[1 + a*x])/a - ((-8*ArcSinh[Sqrt[a]*Sqrt[x ]])/Sqrt[a] + (11*Sqrt[2]*ArcTanh[(Sqrt[2]*Sqrt[a]*Sqrt[x])/Sqrt[1 + a*x]] )/Sqrt[a])/a)/(2*a))/(4*a^2)))/((c - c/(a*x))^(7/2)*x^(7/2))
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[(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*((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[((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 = 0.26 (sec) , antiderivative size = 315, normalized size of antiderivative = 1.25
| method | result | size |
| default | \(-\frac {\sqrt {\frac {c \left (a x -1\right )}{a x}}\, x \sqrt {2}\, \left (8 \sqrt {-x \left (a x +1\right )}\, a^{\frac {7}{2}} \sqrt {2}\, \sqrt {-\frac {1}{a}}\, x^{2}+2 \sqrt {-x \left (a x +1\right )}\, a^{\frac {5}{2}} \sqrt {2}\, \sqrt {-\frac {1}{a}}\, x -4 \arctan \left (\frac {2 a x +1}{2 \sqrt {a}\, \sqrt {-x \left (a x +1\right )}}\right ) \sqrt {2}\, \sqrt {-\frac {1}{a}}\, a^{3} x^{2}+11 a^{\frac {5}{2}} \ln \left (\frac {2 \sqrt {2}\, \sqrt {-\frac {1}{a}}\, \sqrt {-x \left (a x +1\right )}\, a -3 a x -1}{a x -1}\right ) x^{2}-14 \sqrt {-x \left (a x +1\right )}\, a^{\frac {3}{2}} \sqrt {2}\, \sqrt {-\frac {1}{a}}+4 \arctan \left (\frac {2 a x +1}{2 \sqrt {a}\, \sqrt {-x \left (a x +1\right )}}\right ) a \sqrt {2}\, \sqrt {-\frac {1}{a}}-11 \ln \left (\frac {2 \sqrt {2}\, \sqrt {-\frac {1}{a}}\, \sqrt {-x \left (a x +1\right )}\, a -3 a x -1}{a x -1}\right ) \sqrt {a}\right ) \sqrt {-a^{2} x^{2}+1}}{16 a^{\frac {3}{2}} c^{4} \left (a x +1\right ) \sqrt {-\frac {1}{a}}\, \sqrt {-x \left (a x +1\right )}\, \left (a x -1\right )^{2}}\) | \(315\) |
| risch | \(\frac {\left (a x +1\right ) \sqrt {\frac {c a x \left (-a^{2} x^{2}+1\right )}{a x -1}}\, \left (a x -1\right )}{a \sqrt {-\left (a x +1\right ) a c x}\, \sqrt {\frac {c \left (a x -1\right )}{a x}}\, \sqrt {-a^{2} x^{2}+1}\, c^{3}}+\frac {\left (\frac {\arctan \left (\frac {\sqrt {a^{2} c}\, \left (x +\frac {1}{2 a}\right )}{\sqrt {-a^{2} c \,x^{2}-a c x}}\right )}{2 a^{4} \sqrt {a^{2} c}}+\frac {\sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2} c -3 \left (x -\frac {1}{a}\right ) a c -2 c}}{4 a^{6} c \left (x -\frac {1}{a}\right )}-\frac {11 \ln \left (\frac {-4 c -3 \left (x -\frac {1}{a}\right ) a c +2 \sqrt {-2 c}\, \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2} c -3 \left (x -\frac {1}{a}\right ) a c -2 c}}{x -\frac {1}{a}}\right )}{8 a^{5} \sqrt {-2 c}}-\frac {\sqrt {-\left (x +\frac {1}{a}\right )^{2} a^{2} c +\left (x +\frac {1}{a}\right ) a c}}{2 a^{6} c \left (x +\frac {1}{a}\right )}\right ) a^{3} \sqrt {\frac {c a x \left (-a^{2} x^{2}+1\right )}{a x -1}}\, \left (a x -1\right )}{x \sqrt {\frac {c \left (a x -1\right )}{a x}}\, \sqrt {-a^{2} x^{2}+1}\, c^{3}}\) | \(355\) |
Input:
int(1/(a*x+1)^3*(-a^2*x^2+1)^(3/2)/(c-c/a/x)^(7/2),x,method=_RETURNVERBOSE )
Output:
-1/16*(c*(a*x-1)/a/x)^(1/2)*x*2^(1/2)*(8*(-x*(a*x+1))^(1/2)*a^(7/2)*2^(1/2 )*(-1/a)^(1/2)*x^2+2*(-x*(a*x+1))^(1/2)*a^(5/2)*2^(1/2)*(-1/a)^(1/2)*x-4*a rctan(1/2/a^(1/2)*(2*a*x+1)/(-x*(a*x+1))^(1/2))*2^(1/2)*(-1/a)^(1/2)*a^3*x ^2+11*a^(5/2)*ln((2*2^(1/2)*(-1/a)^(1/2)*(-x*(a*x+1))^(1/2)*a-3*a*x-1)/(a* x-1))*x^2-14*(-x*(a*x+1))^(1/2)*a^(3/2)*2^(1/2)*(-1/a)^(1/2)+4*arctan(1/2/ a^(1/2)*(2*a*x+1)/(-x*(a*x+1))^(1/2))*a*2^(1/2)*(-1/a)^(1/2)-11*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^2*x^2 +1)^(1/2)/a^(3/2)/c^4/(a*x+1)/(-1/a)^(1/2)/(-x*(a*x+1))^(1/2)/(a*x-1)^2
Time = 0.15 (sec) , antiderivative size = 596, normalized size of antiderivative = 2.37 \[ \int \frac {e^{-3 \text {arctanh}(a x)}}{\left (c-\frac {c}{a x}\right )^{7/2}} \, dx=\left [-\frac {11 \, \sqrt {2} {\left (a^{3} x^{3} - a^{2} x^{2} - a x + 1\right )} \sqrt {-c} \log \left (-\frac {17 \, a^{3} c x^{3} - 3 \, a^{2} c x^{2} - 13 \, a c x + 4 \, \sqrt {2} {\left (3 \, a^{2} x^{2} + a x\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {-c} \sqrt {\frac {a c x - c}{a x}} - c}{a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1}\right ) + 8 \, {\left (a^{3} x^{3} - a^{2} x^{2} - a x + 1\right )} \sqrt {-c} \log \left (-\frac {8 \, a^{3} c x^{3} - 7 \, a c x - 4 \, {\left (2 \, a^{2} x^{2} + a x\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {-c} \sqrt {\frac {a c x - c}{a x}} - c}{a x - 1}\right ) + 8 \, {\left (4 \, a^{3} x^{3} + a^{2} x^{2} - 7 \, a x\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {\frac {a c x - c}{a x}}}{32 \, {\left (a^{4} c^{4} x^{3} - a^{3} c^{4} x^{2} - a^{2} c^{4} x + a c^{4}\right )}}, \frac {11 \, \sqrt {2} {\left (a^{3} x^{3} - a^{2} x^{2} - a x + 1\right )} \sqrt {c} \arctan \left (\frac {2 \, \sqrt {2} \sqrt {-a^{2} x^{2} + 1} a \sqrt {c} x \sqrt {\frac {a c x - c}{a x}}}{3 \, a^{2} c x^{2} - 2 \, a c x - c}\right ) - 8 \, {\left (a^{3} x^{3} - a^{2} x^{2} - a x + 1\right )} \sqrt {c} \arctan \left (\frac {2 \, \sqrt {-a^{2} x^{2} + 1} a \sqrt {c} x \sqrt {\frac {a c x - c}{a x}}}{2 \, a^{2} c x^{2} - a c x - c}\right ) - 4 \, {\left (4 \, a^{3} x^{3} + a^{2} x^{2} - 7 \, a x\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {\frac {a c x - c}{a x}}}{16 \, {\left (a^{4} c^{4} x^{3} - a^{3} c^{4} x^{2} - a^{2} c^{4} x + a c^{4}\right )}}\right ] \] Input:
integrate(1/(a*x+1)^3*(-a^2*x^2+1)^(3/2)/(c-c/a/x)^(7/2),x, algorithm="fri cas")
Output:
[-1/32*(11*sqrt(2)*(a^3*x^3 - a^2*x^2 - 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) ) + 8*(a^3*x^3 - a^2*x^2 - 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*(4*a^3*x^3 + a^2*x^2 - 7*a*x)*sqrt(-a^2*x^2 + 1)*sqrt(( a*c*x - c)/(a*x)))/(a^4*c^4*x^3 - a^3*c^4*x^2 - a^2*c^4*x + a*c^4), 1/16*( 11*sqrt(2)*(a^3*x^3 - a^2*x^2 - a*x + 1)*sqrt(c)*arctan(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) ) - 8*(a^3*x^3 - a^2*x^2 - a*x + 1)*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)) - 4*(4*a^3*x^ 3 + a^2*x^2 - 7*a*x)*sqrt(-a^2*x^2 + 1)*sqrt((a*c*x - c)/(a*x)))/(a^4*c^4* x^3 - a^3*c^4*x^2 - a^2*c^4*x + a*c^4)]
\[ \int \frac {e^{-3 \text {arctanh}(a x)}}{\left (c-\frac {c}{a x}\right )^{7/2}} \, dx=\int \frac {\left (- \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {3}{2}}}{\left (- c \left (-1 + \frac {1}{a x}\right )\right )^{\frac {7}{2}} \left (a x + 1\right )^{3}}\, dx \] Input:
integrate(1/(a*x+1)**3*(-a**2*x**2+1)**(3/2)/(c-c/a/x)**(7/2),x)
Output:
Integral((-(a*x - 1)*(a*x + 1))**(3/2)/((-c*(-1 + 1/(a*x)))**(7/2)*(a*x + 1)**3), x)
\[ \int \frac {e^{-3 \text {arctanh}(a x)}}{\left (c-\frac {c}{a x}\right )^{7/2}} \, dx=\int { \frac {{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}}}{{\left (a x + 1\right )}^{3} {\left (c - \frac {c}{a x}\right )}^{\frac {7}{2}}} \,d x } \] Input:
integrate(1/(a*x+1)^3*(-a^2*x^2+1)^(3/2)/(c-c/a/x)^(7/2),x, algorithm="max ima")
Output:
integrate((-a^2*x^2 + 1)^(3/2)/((a*x + 1)^3*(c - c/(a*x))^(7/2)), x)
\[ \int \frac {e^{-3 \text {arctanh}(a x)}}{\left (c-\frac {c}{a x}\right )^{7/2}} \, dx=\int { \frac {{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}}}{{\left (a x + 1\right )}^{3} {\left (c - \frac {c}{a x}\right )}^{\frac {7}{2}}} \,d x } \] Input:
integrate(1/(a*x+1)^3*(-a^2*x^2+1)^(3/2)/(c-c/a/x)^(7/2),x, algorithm="gia c")
Output:
integrate((-a^2*x^2 + 1)^(3/2)/((a*x + 1)^3*(c - c/(a*x))^(7/2)), x)
Timed out. \[ \int \frac {e^{-3 \text {arctanh}(a x)}}{\left (c-\frac {c}{a x}\right )^{7/2}} \, dx=\int \frac {{\left (1-a^2\,x^2\right )}^{3/2}}{{\left (c-\frac {c}{a\,x}\right )}^{7/2}\,{\left (a\,x+1\right )}^3} \,d x \] Input:
int((1 - a^2*x^2)^(3/2)/((c - c/(a*x))^(7/2)*(a*x + 1)^3),x)
Output:
int((1 - a^2*x^2)^(3/2)/((c - c/(a*x))^(7/2)*(a*x + 1)^3), x)
Time = 0.18 (sec) , antiderivative size = 177, normalized size of antiderivative = 0.71 \[ \int \frac {e^{-3 \text {arctanh}(a x)}}{\left (c-\frac {c}{a x}\right )^{7/2}} \, dx=\frac {\sqrt {c}\, \left (11 \sqrt {2}\, \mathit {atan} \left (\frac {\sqrt {x}\, \sqrt {a}\, \sqrt {a x +1}\, \sqrt {2}\, i}{a x +1}\right ) a^{2} x^{2}-11 \sqrt {2}\, \mathit {atan} \left (\frac {\sqrt {x}\, \sqrt {a}\, \sqrt {a x +1}\, \sqrt {2}\, i}{a x +1}\right )-8 \mathit {atan} \left (\frac {\sqrt {x}\, \sqrt {a}\, \sqrt {a x +1}\, i}{a x +1}\right ) a^{2} x^{2}+8 \mathit {atan} \left (\frac {\sqrt {x}\, \sqrt {a}\, \sqrt {a x +1}\, i}{a x +1}\right )-8 \sqrt {x}\, \sqrt {a}\, \sqrt {a x +1}\, a^{2} i \,x^{2}-2 \sqrt {x}\, \sqrt {a}\, \sqrt {a x +1}\, a i x +14 \sqrt {x}\, \sqrt {a}\, \sqrt {a x +1}\, i \right )}{8 a \,c^{4} \left (a^{2} x^{2}-1\right )} \] Input:
int(1/(a*x+1)^3*(-a^2*x^2+1)^(3/2)/(c-c/a/x)^(7/2),x)
Output:
(sqrt(c)*(11*sqrt(2)*atan((sqrt(x)*sqrt(a)*sqrt(a*x + 1)*sqrt(2)*i)/(a*x + 1))*a**2*x**2 - 11*sqrt(2)*atan((sqrt(x)*sqrt(a)*sqrt(a*x + 1)*sqrt(2)*i) /(a*x + 1)) - 8*atan((sqrt(x)*sqrt(a)*sqrt(a*x + 1)*i)/(a*x + 1))*a**2*x** 2 + 8*atan((sqrt(x)*sqrt(a)*sqrt(a*x + 1)*i)/(a*x + 1)) - 8*sqrt(x)*sqrt(a )*sqrt(a*x + 1)*a**2*i*x**2 - 2*sqrt(x)*sqrt(a)*sqrt(a*x + 1)*a*i*x + 14*s qrt(x)*sqrt(a)*sqrt(a*x + 1)*i))/(8*a*c**4*(a**2*x**2 - 1))