\(\int \frac {e^{-3 \coth ^{-1}(a x)}}{(c-\frac {c}{a^2 x^2})^4} \, dx\) [829]

Optimal result

Integrand size = 22, antiderivative size = 327 \[ \int \frac {e^{-3 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^4} \, dx=-\frac {4}{3 a c^4 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{9/2}}-\frac {5}{a c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{9/2}}+\frac {28 \sqrt {1-\frac {1}{a x}}}{9 a c^4 \left (1+\frac {1}{a x}\right )^{9/2}}+\frac {139 \sqrt {1-\frac {1}{a x}}}{63 a c^4 \left (1+\frac {1}{a x}\right )^{7/2}}+\frac {202 \sqrt {1-\frac {1}{a x}}}{105 a c^4 \left (1+\frac {1}{a x}\right )^{5/2}}+\frac {719 \sqrt {1-\frac {1}{a x}}}{315 a c^4 \left (1+\frac {1}{a x}\right )^{3/2}}+\frac {1664 \sqrt {1-\frac {1}{a x}}}{315 a c^4 \sqrt {1+\frac {1}{a x}}}+\frac {x}{c^4 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{9/2}}-\frac {3 \text {arctanh}\left (\sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}\right )}{a c^4} \]

[Out]

-4/3/a/c^4/(1-1/a/x)^(3/2)/(1+1/a/x)^(9/2)+x/c^4/(1-1/a/x)^(3/2)/(1+1/a/x)^(9/2)-3*arctanh((1-1/a/x)^(1/2)*(1+
1/a/x)^(1/2))/a/c^4-5/a/c^4/(1+1/a/x)^(9/2)/(1-1/a/x)^(1/2)+28/9*(1-1/a/x)^(1/2)/a/c^4/(1+1/a/x)^(9/2)+139/63*
(1-1/a/x)^(1/2)/a/c^4/(1+1/a/x)^(7/2)+202/105*(1-1/a/x)^(1/2)/a/c^4/(1+1/a/x)^(5/2)+719/315*(1-1/a/x)^(1/2)/a/
c^4/(1+1/a/x)^(3/2)+1664/315*(1-1/a/x)^(1/2)/a/c^4/(1+1/a/x)^(1/2)

Rubi [A] (verified)

Time = 0.17 (sec) , antiderivative size = 327, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {6329, 105, 157, 12, 94, 214} \[ \int \frac {e^{-3 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^4} \, dx=-\frac {3 \text {arctanh}\left (\sqrt {1-\frac {1}{a x}} \sqrt {\frac {1}{a x}+1}\right )}{a c^4}+\frac {x}{c^4 \left (1-\frac {1}{a x}\right )^{3/2} \left (\frac {1}{a x}+1\right )^{9/2}}+\frac {1664 \sqrt {1-\frac {1}{a x}}}{315 a c^4 \sqrt {\frac {1}{a x}+1}}+\frac {719 \sqrt {1-\frac {1}{a x}}}{315 a c^4 \left (\frac {1}{a x}+1\right )^{3/2}}+\frac {202 \sqrt {1-\frac {1}{a x}}}{105 a c^4 \left (\frac {1}{a x}+1\right )^{5/2}}+\frac {139 \sqrt {1-\frac {1}{a x}}}{63 a c^4 \left (\frac {1}{a x}+1\right )^{7/2}}+\frac {28 \sqrt {1-\frac {1}{a x}}}{9 a c^4 \left (\frac {1}{a x}+1\right )^{9/2}}-\frac {5}{a c^4 \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{9/2}}-\frac {4}{3 a c^4 \left (1-\frac {1}{a x}\right )^{3/2} \left (\frac {1}{a x}+1\right )^{9/2}} \]

[In]

Int[1/(E^(3*ArcCoth[a*x])*(c - c/(a^2*x^2))^4),x]

[Out]

-4/(3*a*c^4*(1 - 1/(a*x))^(3/2)*(1 + 1/(a*x))^(9/2)) - 5/(a*c^4*Sqrt[1 - 1/(a*x)]*(1 + 1/(a*x))^(9/2)) + (28*S
qrt[1 - 1/(a*x)])/(9*a*c^4*(1 + 1/(a*x))^(9/2)) + (139*Sqrt[1 - 1/(a*x)])/(63*a*c^4*(1 + 1/(a*x))^(7/2)) + (20
2*Sqrt[1 - 1/(a*x)])/(105*a*c^4*(1 + 1/(a*x))^(5/2)) + (719*Sqrt[1 - 1/(a*x)])/(315*a*c^4*(1 + 1/(a*x))^(3/2))
+ (1664*Sqrt[1 - 1/(a*x)])/(315*a*c^4*Sqrt[1 + 1/(a*x)]) + x/(c^4*(1 - 1/(a*x))^(3/2)*(1 + 1/(a*x))^(9/2)) -
(3*ArcTanh[Sqrt[1 - 1/(a*x)]*Sqrt[1 + 1/(a*x)]])/(a*c^4)

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 94

Int[1/(Sqrt[(a_.) + (b_.)*(x_)]*Sqrt[(c_.) + (d_.)*(x_)]*((e_.) + (f_.)*(x_))), x_Symbol] :> Dist[b*f, Subst[I
nt[1/(d*(b*e - a*f)^2 + b*f^2*x^2), x], x, Sqrt[a + b*x]*Sqrt[c + d*x]], x] /; FreeQ[{a, b, c, d, e, f}, x] &&
EqQ[2*b*d*e - f*(b*c + a*d), 0]

Rule 105

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[b*(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] + Dist[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*(m + 1) - b*(d*e*(m + n + 2) +
c*f*(m + p + 2)) - b*d*f*(m + n + p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && ILtQ[m, -1] &
& (IntegerQ[n] || IntegersQ[2*n, 2*p] || ILtQ[m + n + p + 3, 0])

Rule 157

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)), x_Symb
ol] :> 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] + Dist[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]

Rule 214

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]

Rule 6329

Int[E^(ArcCoth[(a_.)*(x_)]*(n_.))*((c_) + (d_.)/(x_)^2)^(p_.), x_Symbol] :> Dist[-c^p, Subst[Int[(1 - x/a)^(p
- n/2)*((1 + x/a)^(p + n/2)/x^2), x], x, 1/x], x] /; FreeQ[{a, c, d, n, p}, x] && EqQ[c + a^2*d, 0] &&  !Integ
erQ[n/2] && (IntegerQ[p] || GtQ[c, 0]) &&  !IntegersQ[2*p, p + n/2]

Rubi steps \begin{align*} \text {integral}& = -\frac {\text {Subst}\left (\int \frac {1}{x^2 \left (1-\frac {x}{a}\right )^{5/2} \left (1+\frac {x}{a}\right )^{11/2}} \, dx,x,\frac {1}{x}\right )}{c^4} \\ & = \frac {x}{c^4 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{9/2}}+\frac {\text {Subst}\left (\int \frac {\frac {3}{a}-\frac {7 x}{a^2}}{x \left (1-\frac {x}{a}\right )^{5/2} \left (1+\frac {x}{a}\right )^{11/2}} \, dx,x,\frac {1}{x}\right )}{c^4} \\ & = -\frac {4}{3 a c^4 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{9/2}}+\frac {x}{c^4 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{9/2}}-\frac {a \text {Subst}\left (\int \frac {-\frac {9}{a^2}+\frac {24 x}{a^3}}{x \left (1-\frac {x}{a}\right )^{3/2} \left (1+\frac {x}{a}\right )^{11/2}} \, dx,x,\frac {1}{x}\right )}{3 c^4} \\ & = -\frac {4}{3 a c^4 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{9/2}}-\frac {5}{a c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{9/2}}+\frac {x}{c^4 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{9/2}}+\frac {a^2 \text {Subst}\left (\int \frac {\frac {9}{a^3}-\frac {75 x}{a^4}}{x \sqrt {1-\frac {x}{a}} \left (1+\frac {x}{a}\right )^{11/2}} \, dx,x,\frac {1}{x}\right )}{3 c^4} \\ & = -\frac {4}{3 a c^4 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{9/2}}-\frac {5}{a c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{9/2}}+\frac {28 \sqrt {1-\frac {1}{a x}}}{9 a c^4 \left (1+\frac {1}{a x}\right )^{9/2}}+\frac {x}{c^4 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{9/2}}+\frac {a^3 \text {Subst}\left (\int \frac {\frac {81}{a^4}-\frac {336 x}{a^5}}{x \sqrt {1-\frac {x}{a}} \left (1+\frac {x}{a}\right )^{9/2}} \, dx,x,\frac {1}{x}\right )}{27 c^4} \\ & = -\frac {4}{3 a c^4 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{9/2}}-\frac {5}{a c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{9/2}}+\frac {28 \sqrt {1-\frac {1}{a x}}}{9 a c^4 \left (1+\frac {1}{a x}\right )^{9/2}}+\frac {139 \sqrt {1-\frac {1}{a x}}}{63 a c^4 \left (1+\frac {1}{a x}\right )^{7/2}}+\frac {x}{c^4 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{9/2}}+\frac {a^4 \text {Subst}\left (\int \frac {\frac {567}{a^5}-\frac {1251 x}{a^6}}{x \sqrt {1-\frac {x}{a}} \left (1+\frac {x}{a}\right )^{7/2}} \, dx,x,\frac {1}{x}\right )}{189 c^4} \\ & = -\frac {4}{3 a c^4 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{9/2}}-\frac {5}{a c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{9/2}}+\frac {28 \sqrt {1-\frac {1}{a x}}}{9 a c^4 \left (1+\frac {1}{a x}\right )^{9/2}}+\frac {139 \sqrt {1-\frac {1}{a x}}}{63 a c^4 \left (1+\frac {1}{a x}\right )^{7/2}}+\frac {202 \sqrt {1-\frac {1}{a x}}}{105 a c^4 \left (1+\frac {1}{a x}\right )^{5/2}}+\frac {x}{c^4 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{9/2}}+\frac {a^5 \text {Subst}\left (\int \frac {\frac {2835}{a^6}-\frac {3636 x}{a^7}}{x \sqrt {1-\frac {x}{a}} \left (1+\frac {x}{a}\right )^{5/2}} \, dx,x,\frac {1}{x}\right )}{945 c^4} \\ & = -\frac {4}{3 a c^4 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{9/2}}-\frac {5}{a c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{9/2}}+\frac {28 \sqrt {1-\frac {1}{a x}}}{9 a c^4 \left (1+\frac {1}{a x}\right )^{9/2}}+\frac {139 \sqrt {1-\frac {1}{a x}}}{63 a c^4 \left (1+\frac {1}{a x}\right )^{7/2}}+\frac {202 \sqrt {1-\frac {1}{a x}}}{105 a c^4 \left (1+\frac {1}{a x}\right )^{5/2}}+\frac {719 \sqrt {1-\frac {1}{a x}}}{315 a c^4 \left (1+\frac {1}{a x}\right )^{3/2}}+\frac {x}{c^4 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{9/2}}+\frac {a^6 \text {Subst}\left (\int \frac {\frac {8505}{a^7}-\frac {6471 x}{a^8}}{x \sqrt {1-\frac {x}{a}} \left (1+\frac {x}{a}\right )^{3/2}} \, dx,x,\frac {1}{x}\right )}{2835 c^4} \\ & = -\frac {4}{3 a c^4 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{9/2}}-\frac {5}{a c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{9/2}}+\frac {28 \sqrt {1-\frac {1}{a x}}}{9 a c^4 \left (1+\frac {1}{a x}\right )^{9/2}}+\frac {139 \sqrt {1-\frac {1}{a x}}}{63 a c^4 \left (1+\frac {1}{a x}\right )^{7/2}}+\frac {202 \sqrt {1-\frac {1}{a x}}}{105 a c^4 \left (1+\frac {1}{a x}\right )^{5/2}}+\frac {719 \sqrt {1-\frac {1}{a x}}}{315 a c^4 \left (1+\frac {1}{a x}\right )^{3/2}}+\frac {1664 \sqrt {1-\frac {1}{a x}}}{315 a c^4 \sqrt {1+\frac {1}{a x}}}+\frac {x}{c^4 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{9/2}}+\frac {a^7 \text {Subst}\left (\int \frac {8505}{a^8 x \sqrt {1-\frac {x}{a}} \sqrt {1+\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )}{2835 c^4} \\ & = -\frac {4}{3 a c^4 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{9/2}}-\frac {5}{a c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{9/2}}+\frac {28 \sqrt {1-\frac {1}{a x}}}{9 a c^4 \left (1+\frac {1}{a x}\right )^{9/2}}+\frac {139 \sqrt {1-\frac {1}{a x}}}{63 a c^4 \left (1+\frac {1}{a x}\right )^{7/2}}+\frac {202 \sqrt {1-\frac {1}{a x}}}{105 a c^4 \left (1+\frac {1}{a x}\right )^{5/2}}+\frac {719 \sqrt {1-\frac {1}{a x}}}{315 a c^4 \left (1+\frac {1}{a x}\right )^{3/2}}+\frac {1664 \sqrt {1-\frac {1}{a x}}}{315 a c^4 \sqrt {1+\frac {1}{a x}}}+\frac {x}{c^4 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{9/2}}+\frac {3 \text {Subst}\left (\int \frac {1}{x \sqrt {1-\frac {x}{a}} \sqrt {1+\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )}{a c^4} \\ & = -\frac {4}{3 a c^4 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{9/2}}-\frac {5}{a c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{9/2}}+\frac {28 \sqrt {1-\frac {1}{a x}}}{9 a c^4 \left (1+\frac {1}{a x}\right )^{9/2}}+\frac {139 \sqrt {1-\frac {1}{a x}}}{63 a c^4 \left (1+\frac {1}{a x}\right )^{7/2}}+\frac {202 \sqrt {1-\frac {1}{a x}}}{105 a c^4 \left (1+\frac {1}{a x}\right )^{5/2}}+\frac {719 \sqrt {1-\frac {1}{a x}}}{315 a c^4 \left (1+\frac {1}{a x}\right )^{3/2}}+\frac {1664 \sqrt {1-\frac {1}{a x}}}{315 a c^4 \sqrt {1+\frac {1}{a x}}}+\frac {x}{c^4 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{9/2}}-\frac {3 \text {Subst}\left (\int \frac {1}{\frac {1}{a}-\frac {x^2}{a}} \, dx,x,\sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}\right )}{a^2 c^4} \\ & = -\frac {4}{3 a c^4 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{9/2}}-\frac {5}{a c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{9/2}}+\frac {28 \sqrt {1-\frac {1}{a x}}}{9 a c^4 \left (1+\frac {1}{a x}\right )^{9/2}}+\frac {139 \sqrt {1-\frac {1}{a x}}}{63 a c^4 \left (1+\frac {1}{a x}\right )^{7/2}}+\frac {202 \sqrt {1-\frac {1}{a x}}}{105 a c^4 \left (1+\frac {1}{a x}\right )^{5/2}}+\frac {719 \sqrt {1-\frac {1}{a x}}}{315 a c^4 \left (1+\frac {1}{a x}\right )^{3/2}}+\frac {1664 \sqrt {1-\frac {1}{a x}}}{315 a c^4 \sqrt {1+\frac {1}{a x}}}+\frac {x}{c^4 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{9/2}}-\frac {3 \text {arctanh}\left (\sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}\right )}{a c^4} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.53 (sec) , antiderivative size = 117, normalized size of antiderivative = 0.36 \[ \int \frac {e^{-3 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^4} \, dx=\frac {\frac {a \sqrt {1-\frac {1}{a^2 x^2}} x \left (1664+4047 a x-339 a^2 x^2-7399 a^3 x^3-4029 a^4 x^4+2967 a^5 x^5+2669 a^6 x^6+315 a^7 x^7\right )}{315 (-1+a x)^2 (1+a x)^5}-3 \log \left (\left (1+\sqrt {1-\frac {1}{a^2 x^2}}\right ) x\right )}{a c^4} \]

[In]

Integrate[1/(E^(3*ArcCoth[a*x])*(c - c/(a^2*x^2))^4),x]

[Out]

((a*Sqrt[1 - 1/(a^2*x^2)]*x*(1664 + 4047*a*x - 339*a^2*x^2 - 7399*a^3*x^3 - 4029*a^4*x^4 + 2967*a^5*x^5 + 2669
*a^6*x^6 + 315*a^7*x^7))/(315*(-1 + a*x)^2*(1 + a*x)^5) - 3*Log[(1 + Sqrt[1 - 1/(a^2*x^2)])*x])/(a*c^4)

Maple [A] (verified)

Time = 0.25 (sec) , antiderivative size = 355, normalized size of antiderivative = 1.09

method	result	size
risch	\(\frac {\left (a x +1\right ) \sqrt {\frac {a x -1}{a x +1}}}{a \,c^{4}}+\frac {\left (-\frac {3 \ln \left (\frac {a^{2} x}{\sqrt {a^{2}}}+\sqrt {a^{2} x^{2}-1}\right )}{a^{8} \sqrt {a^{2}}}-\frac {59 \sqrt {a^{2} \left (x +\frac {1}{a}\right )^{2}-2 a \left (x +\frac {1}{a}\right )}}{252 a^{13} \left (x +\frac {1}{a}\right )^{4}}+\frac {1507 \sqrt {a^{2} \left (x +\frac {1}{a}\right )^{2}-2 a \left (x +\frac {1}{a}\right )}}{1680 a^{12} \left (x +\frac {1}{a}\right )^{3}}-\frac {691 \sqrt {a^{2} \left (x +\frac {1}{a}\right )^{2}-2 a \left (x +\frac {1}{a}\right )}}{315 a^{11} \left (x +\frac {1}{a}\right )^{2}}+\frac {113591 \sqrt {a^{2} \left (x +\frac {1}{a}\right )^{2}-2 a \left (x +\frac {1}{a}\right )}}{20160 a^{10} \left (x +\frac {1}{a}\right )}+\frac {\sqrt {a^{2} \left (x +\frac {1}{a}\right )^{2}-2 a \left (x +\frac {1}{a}\right )}}{36 a^{14} \left (x +\frac {1}{a}\right )^{5}}-\frac {\sqrt {\left (x -\frac {1}{a}\right )^{2} a^{2}+2 \left (x -\frac {1}{a}\right ) a}}{96 a^{11} \left (x -\frac {1}{a}\right )^{2}}-\frac {31 \sqrt {\left (x -\frac {1}{a}\right )^{2} a^{2}+2 \left (x -\frac {1}{a}\right ) a}}{192 a^{10} \left (x -\frac {1}{a}\right )}\right ) a^{8} \sqrt {\frac {a x -1}{a x +1}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{c^{4} \left (a x -1\right )}\)	\(355\)
default	\(-\frac {\left (-138915 \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}\, a^{9} x^{9}+120960 \ln \left (\frac {a^{2} x +\sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{\sqrt {a^{2}}}\right ) a^{10} x^{9}+98595 \left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} \sqrt {a^{2}}\, a^{7} x^{7}-416745 \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}\, a^{8} x^{8}+362880 \ln \left (\frac {a^{2} x +\sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{\sqrt {a^{2}}}\right ) a^{9} x^{8}+75113 \left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} \sqrt {a^{2}}\, a^{6} x^{6}-240861 \left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} \sqrt {a^{2}}\, a^{5} x^{5}+1111320 \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}\, a^{6} x^{6}-967680 \ln \left (\frac {a^{2} x +\sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{\sqrt {a^{2}}}\right ) a^{7} x^{6}-178863 \left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} \sqrt {a^{2}}\, a^{4} x^{4}+833490 \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}\, a^{5} x^{5}-725760 \ln \left (\frac {a^{2} x +\sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{\sqrt {a^{2}}}\right ) a^{6} x^{5}+252497 \left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} \sqrt {a^{2}}\, a^{3} x^{3}-833490 \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}\, a^{4} x^{4}+725760 \ln \left (\frac {a^{2} x +\sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{\sqrt {a^{2}}}\right ) a^{5} x^{4}+182307 \left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} \sqrt {a^{2}}\, a^{2} x^{2}-1111320 \sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, a^{3} x^{3}+967680 \ln \left (\frac {a^{2} x +\sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{\sqrt {a^{2}}}\right ) a^{4} x^{3}-101271 \sqrt {a^{2}}\, \left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} a x -74077 \left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} \sqrt {a^{2}}+416745 \sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, a x -362880 \ln \left (\frac {a^{2} x +\sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{\sqrt {a^{2}}}\right ) a^{2} x +138915 \sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}-120960 a \ln \left (\frac {a^{2} x +\sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{\sqrt {a^{2}}}\right )\right ) \left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}}}{40320 a \sqrt {a^{2}}\, \left (a x +1\right )^{4} c^{4} \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \left (a x -1\right )^{4}}\)	\(766\)

[In]

int(((a*x-1)/(a*x+1))^(3/2)/(c-c/a^2/x^2)^4,x,method=_RETURNVERBOSE)

[Out]

1/a*(a*x+1)/c^4*((a*x-1)/(a*x+1))^(1/2)+(-3/a^8*ln(a^2*x/(a^2)^(1/2)+(a^2*x^2-1)^(1/2))/(a^2)^(1/2)-59/252/a^1
3/(x+1/a)^4*(a^2*(x+1/a)^2-2*a*(x+1/a))^(1/2)+1507/1680/a^12/(x+1/a)^3*(a^2*(x+1/a)^2-2*a*(x+1/a))^(1/2)-691/3
15/a^11/(x+1/a)^2*(a^2*(x+1/a)^2-2*a*(x+1/a))^(1/2)+113591/20160/a^10/(x+1/a)*(a^2*(x+1/a)^2-2*a*(x+1/a))^(1/2
)+1/36/a^14/(x+1/a)^5*(a^2*(x+1/a)^2-2*a*(x+1/a))^(1/2)-1/96/a^11/(x-1/a)^2*((x-1/a)^2*a^2+2*(x-1/a)*a)^(1/2)-
31/192/a^10/(x-1/a)*((x-1/a)^2*a^2+2*(x-1/a)*a)^(1/2))*a^8/c^4*((a*x-1)/(a*x+1))^(1/2)*((a*x-1)*(a*x+1))^(1/2)
/(a*x-1)

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 275, normalized size of antiderivative = 0.84 \[ \int \frac {e^{-3 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^4} \, dx=-\frac {945 \, {\left (a^{6} x^{6} + 2 \, a^{5} x^{5} - a^{4} x^{4} - 4 \, a^{3} x^{3} - a^{2} x^{2} + 2 \, a x + 1\right )} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right ) - 945 \, {\left (a^{6} x^{6} + 2 \, a^{5} x^{5} - a^{4} x^{4} - 4 \, a^{3} x^{3} - a^{2} x^{2} + 2 \, a x + 1\right )} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right ) - {\left (315 \, a^{7} x^{7} + 2669 \, a^{6} x^{6} + 2967 \, a^{5} x^{5} - 4029 \, a^{4} x^{4} - 7399 \, a^{3} x^{3} - 339 \, a^{2} x^{2} + 4047 \, a x + 1664\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{315 \, {\left (a^{7} c^{4} x^{6} + 2 \, a^{6} c^{4} x^{5} - a^{5} c^{4} x^{4} - 4 \, a^{4} c^{4} x^{3} - a^{3} c^{4} x^{2} + 2 \, a^{2} c^{4} x + a c^{4}\right )}} \]

[In]

integrate(((a*x-1)/(a*x+1))^(3/2)/(c-c/a^2/x^2)^4,x, algorithm="fricas")

[Out]

-1/315*(945*(a^6*x^6 + 2*a^5*x^5 - a^4*x^4 - 4*a^3*x^3 - a^2*x^2 + 2*a*x + 1)*log(sqrt((a*x - 1)/(a*x + 1)) +
1) - 945*(a^6*x^6 + 2*a^5*x^5 - a^4*x^4 - 4*a^3*x^3 - a^2*x^2 + 2*a*x + 1)*log(sqrt((a*x - 1)/(a*x + 1)) - 1)
- (315*a^7*x^7 + 2669*a^6*x^6 + 2967*a^5*x^5 - 4029*a^4*x^4 - 7399*a^3*x^3 - 339*a^2*x^2 + 4047*a*x + 1664)*sq
rt((a*x - 1)/(a*x + 1)))/(a^7*c^4*x^6 + 2*a^6*c^4*x^5 - a^5*c^4*x^4 - 4*a^4*c^4*x^3 - a^3*c^4*x^2 + 2*a^2*c^4*
x + a*c^4)

Sympy [F(-1)]

Timed out. \[ \int \frac {e^{-3 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^4} \, dx=\text {Timed out} \]

[In]

integrate(((a*x-1)/(a*x+1))**(3/2)/(c-c/a**2/x**2)**4,x)

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 231, normalized size of antiderivative = 0.71 \[ \int \frac {e^{-3 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^4} \, dx=\frac {1}{20160} \, a {\left (\frac {105 \, {\left (\frac {29 \, {\left (a x - 1\right )}}{a x + 1} - \frac {414 \, {\left (a x - 1\right )}^{2}}{{\left (a x + 1\right )}^{2}} + 1\right )}}{a^{2} c^{4} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {5}{2}} - a^{2} c^{4} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}}} + \frac {35 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {9}{2}} + 450 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {7}{2}} + 2961 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {5}{2}} + 14700 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}} + 95445 \, \sqrt {\frac {a x - 1}{a x + 1}}}{a^{2} c^{4}} - \frac {60480 \, \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right )}{a^{2} c^{4}} + \frac {60480 \, \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right )}{a^{2} c^{4}}\right )} \]

[In]

integrate(((a*x-1)/(a*x+1))^(3/2)/(c-c/a^2/x^2)^4,x, algorithm="maxima")

[Out]

1/20160*a*(105*(29*(a*x - 1)/(a*x + 1) - 414*(a*x - 1)^2/(a*x + 1)^2 + 1)/(a^2*c^4*((a*x - 1)/(a*x + 1))^(5/2)
- a^2*c^4*((a*x - 1)/(a*x + 1))^(3/2)) + (35*((a*x - 1)/(a*x + 1))^(9/2) + 450*((a*x - 1)/(a*x + 1))^(7/2) +
2961*((a*x - 1)/(a*x + 1))^(5/2) + 14700*((a*x - 1)/(a*x + 1))^(3/2) + 95445*sqrt((a*x - 1)/(a*x + 1)))/(a^2*c
^4) - 60480*log(sqrt((a*x - 1)/(a*x + 1)) + 1)/(a^2*c^4) + 60480*log(sqrt((a*x - 1)/(a*x + 1)) - 1)/(a^2*c^4))

Giac [F]

\[ \int \frac {e^{-3 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^4} \, dx=\int { \frac {\left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}}}{{\left (c - \frac {c}{a^{2} x^{2}}\right )}^{4}} \,d x } \]

[In]

integrate(((a*x-1)/(a*x+1))^(3/2)/(c-c/a^2/x^2)^4,x, algorithm="giac")

[Out]

integrate(((a*x - 1)/(a*x + 1))^(3/2)/(c - c/(a^2*x^2))^4, x)

Mupad [B] (verification not implemented)

Time = 0.06 (sec) , antiderivative size = 224, normalized size of antiderivative = 0.69 \[ \int \frac {e^{-3 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^4} \, dx=\frac {303\,\sqrt {\frac {a\,x-1}{a\,x+1}}}{64\,a\,c^4}-\frac {\frac {29\,\left (a\,x-1\right )}{3\,\left (a\,x+1\right )}-\frac {138\,{\left (a\,x-1\right )}^2}{{\left (a\,x+1\right )}^2}+\frac {1}{3}}{64\,a\,c^4\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/2}-64\,a\,c^4\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{5/2}}+\frac {35\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/2}}{48\,a\,c^4}+\frac {47\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{5/2}}{320\,a\,c^4}+\frac {5\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{7/2}}{224\,a\,c^4}+\frac {{\left (\frac {a\,x-1}{a\,x+1}\right )}^{9/2}}{576\,a\,c^4}+\frac {\mathrm {atan}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\,1{}\mathrm {i}\right )\,6{}\mathrm {i}}{a\,c^4} \]

[In]

int(((a*x - 1)/(a*x + 1))^(3/2)/(c - c/(a^2*x^2))^4,x)

[Out]

(303*((a*x - 1)/(a*x + 1))^(1/2))/(64*a*c^4) - ((29*(a*x - 1))/(3*(a*x + 1)) - (138*(a*x - 1)^2)/(a*x + 1)^2 +
1/3)/(64*a*c^4*((a*x - 1)/(a*x + 1))^(3/2) - 64*a*c^4*((a*x - 1)/(a*x + 1))^(5/2)) + (35*((a*x - 1)/(a*x + 1)
)^(3/2))/(48*a*c^4) + (47*((a*x - 1)/(a*x + 1))^(5/2))/(320*a*c^4) + (5*((a*x - 1)/(a*x + 1))^(7/2))/(224*a*c^
4) + ((a*x - 1)/(a*x + 1))^(9/2)/(576*a*c^4) + (atan(((a*x - 1)/(a*x + 1))^(1/2)*1i)*6i)/(a*c^4)