### 3.796 $$\int \frac{e^{3 \coth ^{-1}(a x)}}{(c-\frac{c}{a^2 x^2})^4} \, dx$$

Optimal. Leaf size=329 $\frac{x}{c^4 \left (1-\frac{1}{a x}\right )^{9/2} \left (\frac{1}{a x}+1\right )^{3/2}}+\frac{1664 \sqrt{1-\frac{1}{a x}}}{315 a c^4 \sqrt{\frac{1}{a x}+1}}+\frac{2609 \sqrt{1-\frac{1}{a x}}}{315 a c^4 \left (\frac{1}{a x}+1\right )^{3/2}}-\frac{1462}{105 a c^4 \sqrt{1-\frac{1}{a x}} \left (\frac{1}{a x}+1\right )^{3/2}}-\frac{1147}{315 a c^4 \left (1-\frac{1}{a x}\right )^{3/2} \left (\frac{1}{a x}+1\right )^{3/2}}-\frac{208}{105 a c^4 \left (1-\frac{1}{a x}\right )^{5/2} \left (\frac{1}{a x}+1\right )^{3/2}}-\frac{29}{21 a c^4 \left (1-\frac{1}{a x}\right )^{7/2} \left (\frac{1}{a x}+1\right )^{3/2}}-\frac{10}{9 a c^4 \left (1-\frac{1}{a x}\right )^{9/2} \left (\frac{1}{a x}+1\right )^{3/2}}+\frac{3 \tanh ^{-1}\left (\sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}\right )}{a c^4}$

[Out]

-10/(9*a*c^4*(1 - 1/(a*x))^(9/2)*(1 + 1/(a*x))^(3/2)) - 29/(21*a*c^4*(1 - 1/(a*x))^(7/2)*(1 + 1/(a*x))^(3/2))
- 208/(105*a*c^4*(1 - 1/(a*x))^(5/2)*(1 + 1/(a*x))^(3/2)) - 1147/(315*a*c^4*(1 - 1/(a*x))^(3/2)*(1 + 1/(a*x))^
(3/2)) - 1462/(105*a*c^4*Sqrt[1 - 1/(a*x)]*(1 + 1/(a*x))^(3/2)) + (2609*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))^(9/2)*(1 + 1/(a*x
))^(3/2)) + (3*ArcTanh[Sqrt[1 - 1/(a*x)]*Sqrt[1 + 1/(a*x)]])/(a*c^4)

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Rubi [A]  time = 0.236387, antiderivative size = 329, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 6, integrand size = 22, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.273, Rules used = {6194, 103, 152, 12, 92, 208} $\frac{x}{c^4 \left (1-\frac{1}{a x}\right )^{9/2} \left (\frac{1}{a x}+1\right )^{3/2}}+\frac{1664 \sqrt{1-\frac{1}{a x}}}{315 a c^4 \sqrt{\frac{1}{a x}+1}}+\frac{2609 \sqrt{1-\frac{1}{a x}}}{315 a c^4 \left (\frac{1}{a x}+1\right )^{3/2}}-\frac{1462}{105 a c^4 \sqrt{1-\frac{1}{a x}} \left (\frac{1}{a x}+1\right )^{3/2}}-\frac{1147}{315 a c^4 \left (1-\frac{1}{a x}\right )^{3/2} \left (\frac{1}{a x}+1\right )^{3/2}}-\frac{208}{105 a c^4 \left (1-\frac{1}{a x}\right )^{5/2} \left (\frac{1}{a x}+1\right )^{3/2}}-\frac{29}{21 a c^4 \left (1-\frac{1}{a x}\right )^{7/2} \left (\frac{1}{a x}+1\right )^{3/2}}-\frac{10}{9 a c^4 \left (1-\frac{1}{a x}\right )^{9/2} \left (\frac{1}{a x}+1\right )^{3/2}}+\frac{3 \tanh ^{-1}\left (\sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}\right )}{a c^4}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

-10/(9*a*c^4*(1 - 1/(a*x))^(9/2)*(1 + 1/(a*x))^(3/2)) - 29/(21*a*c^4*(1 - 1/(a*x))^(7/2)*(1 + 1/(a*x))^(3/2))
- 208/(105*a*c^4*(1 - 1/(a*x))^(5/2)*(1 + 1/(a*x))^(3/2)) - 1147/(315*a*c^4*(1 - 1/(a*x))^(3/2)*(1 + 1/(a*x))^
(3/2)) - 1462/(105*a*c^4*Sqrt[1 - 1/(a*x)]*(1 + 1/(a*x))^(3/2)) + (2609*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))^(9/2)*(1 + 1/(a*x
))^(3/2)) + (3*ArcTanh[Sqrt[1 - 1/(a*x)]*Sqrt[1 + 1/(a*x)]])/(a*c^4)

Rule 6194

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]

Rule 103

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] && LtQ[m, -1] &&
IntegerQ[m] && (IntegerQ[n] || IntegersQ[2*n, 2*p])

Rule 152

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 12

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

Rule 92

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 208

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-(a/b), 2]*ArcTanh[x/Rt[-(a/b), 2]])/a, x] /; FreeQ[{a,
b}, x] && NegQ[a/b]

Rubi steps

\begin{align*} \int \frac{e^{3 \coth ^{-1}(a x)}}{\left (c-\frac{c}{a^2 x^2}\right )^4} \, dx &=-\frac{\operatorname{Subst}\left (\int \frac{1}{x^2 \left (1-\frac{x}{a}\right )^{11/2} \left (1+\frac{x}{a}\right )^{5/2}} \, dx,x,\frac{1}{x}\right )}{c^4}\\ &=\frac{x}{c^4 \left (1-\frac{1}{a x}\right )^{9/2} \left (1+\frac{1}{a x}\right )^{3/2}}+\frac{\operatorname{Subst}\left (\int \frac{-\frac{3}{a}-\frac{7 x}{a^2}}{x \left (1-\frac{x}{a}\right )^{11/2} \left (1+\frac{x}{a}\right )^{5/2}} \, dx,x,\frac{1}{x}\right )}{c^4}\\ &=-\frac{10}{9 a c^4 \left (1-\frac{1}{a x}\right )^{9/2} \left (1+\frac{1}{a x}\right )^{3/2}}+\frac{x}{c^4 \left (1-\frac{1}{a x}\right )^{9/2} \left (1+\frac{1}{a x}\right )^{3/2}}-\frac{a \operatorname{Subst}\left (\int \frac{\frac{27}{a^2}+\frac{60 x}{a^3}}{x \left (1-\frac{x}{a}\right )^{9/2} \left (1+\frac{x}{a}\right )^{5/2}} \, dx,x,\frac{1}{x}\right )}{9 c^4}\\ &=-\frac{10}{9 a c^4 \left (1-\frac{1}{a x}\right )^{9/2} \left (1+\frac{1}{a x}\right )^{3/2}}-\frac{29}{21 a c^4 \left (1-\frac{1}{a x}\right )^{7/2} \left (1+\frac{1}{a x}\right )^{3/2}}+\frac{x}{c^4 \left (1-\frac{1}{a x}\right )^{9/2} \left (1+\frac{1}{a x}\right )^{3/2}}+\frac{a^2 \operatorname{Subst}\left (\int \frac{-\frac{189}{a^3}-\frac{435 x}{a^4}}{x \left (1-\frac{x}{a}\right )^{7/2} \left (1+\frac{x}{a}\right )^{5/2}} \, dx,x,\frac{1}{x}\right )}{63 c^4}\\ &=-\frac{10}{9 a c^4 \left (1-\frac{1}{a x}\right )^{9/2} \left (1+\frac{1}{a x}\right )^{3/2}}-\frac{29}{21 a c^4 \left (1-\frac{1}{a x}\right )^{7/2} \left (1+\frac{1}{a x}\right )^{3/2}}-\frac{208}{105 a c^4 \left (1-\frac{1}{a x}\right )^{5/2} \left (1+\frac{1}{a x}\right )^{3/2}}+\frac{x}{c^4 \left (1-\frac{1}{a x}\right )^{9/2} \left (1+\frac{1}{a x}\right )^{3/2}}-\frac{a^3 \operatorname{Subst}\left (\int \frac{\frac{945}{a^4}+\frac{2496 x}{a^5}}{x \left (1-\frac{x}{a}\right )^{5/2} \left (1+\frac{x}{a}\right )^{5/2}} \, dx,x,\frac{1}{x}\right )}{315 c^4}\\ &=-\frac{10}{9 a c^4 \left (1-\frac{1}{a x}\right )^{9/2} \left (1+\frac{1}{a x}\right )^{3/2}}-\frac{29}{21 a c^4 \left (1-\frac{1}{a x}\right )^{7/2} \left (1+\frac{1}{a x}\right )^{3/2}}-\frac{208}{105 a c^4 \left (1-\frac{1}{a x}\right )^{5/2} \left (1+\frac{1}{a x}\right )^{3/2}}-\frac{1147}{315 a c^4 \left (1-\frac{1}{a x}\right )^{3/2} \left (1+\frac{1}{a x}\right )^{3/2}}+\frac{x}{c^4 \left (1-\frac{1}{a x}\right )^{9/2} \left (1+\frac{1}{a x}\right )^{3/2}}+\frac{a^4 \operatorname{Subst}\left (\int \frac{-\frac{2835}{a^5}-\frac{10323 x}{a^6}}{x \left (1-\frac{x}{a}\right )^{3/2} \left (1+\frac{x}{a}\right )^{5/2}} \, dx,x,\frac{1}{x}\right )}{945 c^4}\\ &=-\frac{10}{9 a c^4 \left (1-\frac{1}{a x}\right )^{9/2} \left (1+\frac{1}{a x}\right )^{3/2}}-\frac{29}{21 a c^4 \left (1-\frac{1}{a x}\right )^{7/2} \left (1+\frac{1}{a x}\right )^{3/2}}-\frac{208}{105 a c^4 \left (1-\frac{1}{a x}\right )^{5/2} \left (1+\frac{1}{a x}\right )^{3/2}}-\frac{1147}{315 a c^4 \left (1-\frac{1}{a x}\right )^{3/2} \left (1+\frac{1}{a x}\right )^{3/2}}-\frac{1462}{105 a c^4 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{3/2}}+\frac{x}{c^4 \left (1-\frac{1}{a x}\right )^{9/2} \left (1+\frac{1}{a x}\right )^{3/2}}-\frac{a^5 \operatorname{Subst}\left (\int \frac{\frac{2835}{a^6}+\frac{26316 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{10}{9 a c^4 \left (1-\frac{1}{a x}\right )^{9/2} \left (1+\frac{1}{a x}\right )^{3/2}}-\frac{29}{21 a c^4 \left (1-\frac{1}{a x}\right )^{7/2} \left (1+\frac{1}{a x}\right )^{3/2}}-\frac{208}{105 a c^4 \left (1-\frac{1}{a x}\right )^{5/2} \left (1+\frac{1}{a x}\right )^{3/2}}-\frac{1147}{315 a c^4 \left (1-\frac{1}{a x}\right )^{3/2} \left (1+\frac{1}{a x}\right )^{3/2}}-\frac{1462}{105 a c^4 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{3/2}}+\frac{2609 \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 )^{9/2} \left (1+\frac{1}{a x}\right )^{3/2}}-\frac{a^6 \operatorname{Subst}\left (\int \frac{\frac{8505}{a^7}+\frac{23481 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{10}{9 a c^4 \left (1-\frac{1}{a x}\right )^{9/2} \left (1+\frac{1}{a x}\right )^{3/2}}-\frac{29}{21 a c^4 \left (1-\frac{1}{a x}\right )^{7/2} \left (1+\frac{1}{a x}\right )^{3/2}}-\frac{208}{105 a c^4 \left (1-\frac{1}{a x}\right )^{5/2} \left (1+\frac{1}{a x}\right )^{3/2}}-\frac{1147}{315 a c^4 \left (1-\frac{1}{a x}\right )^{3/2} \left (1+\frac{1}{a x}\right )^{3/2}}-\frac{1462}{105 a c^4 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{3/2}}+\frac{2609 \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 )^{9/2} \left (1+\frac{1}{a x}\right )^{3/2}}-\frac{a^7 \operatorname{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{10}{9 a c^4 \left (1-\frac{1}{a x}\right )^{9/2} \left (1+\frac{1}{a x}\right )^{3/2}}-\frac{29}{21 a c^4 \left (1-\frac{1}{a x}\right )^{7/2} \left (1+\frac{1}{a x}\right )^{3/2}}-\frac{208}{105 a c^4 \left (1-\frac{1}{a x}\right )^{5/2} \left (1+\frac{1}{a x}\right )^{3/2}}-\frac{1147}{315 a c^4 \left (1-\frac{1}{a x}\right )^{3/2} \left (1+\frac{1}{a x}\right )^{3/2}}-\frac{1462}{105 a c^4 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{3/2}}+\frac{2609 \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 )^{9/2} \left (1+\frac{1}{a x}\right )^{3/2}}-\frac{3 \operatorname{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{10}{9 a c^4 \left (1-\frac{1}{a x}\right )^{9/2} \left (1+\frac{1}{a x}\right )^{3/2}}-\frac{29}{21 a c^4 \left (1-\frac{1}{a x}\right )^{7/2} \left (1+\frac{1}{a x}\right )^{3/2}}-\frac{208}{105 a c^4 \left (1-\frac{1}{a x}\right )^{5/2} \left (1+\frac{1}{a x}\right )^{3/2}}-\frac{1147}{315 a c^4 \left (1-\frac{1}{a x}\right )^{3/2} \left (1+\frac{1}{a x}\right )^{3/2}}-\frac{1462}{105 a c^4 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{3/2}}+\frac{2609 \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 )^{9/2} \left (1+\frac{1}{a x}\right )^{3/2}}+\frac{3 \operatorname{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{10}{9 a c^4 \left (1-\frac{1}{a x}\right )^{9/2} \left (1+\frac{1}{a x}\right )^{3/2}}-\frac{29}{21 a c^4 \left (1-\frac{1}{a x}\right )^{7/2} \left (1+\frac{1}{a x}\right )^{3/2}}-\frac{208}{105 a c^4 \left (1-\frac{1}{a x}\right )^{5/2} \left (1+\frac{1}{a x}\right )^{3/2}}-\frac{1147}{315 a c^4 \left (1-\frac{1}{a x}\right )^{3/2} \left (1+\frac{1}{a x}\right )^{3/2}}-\frac{1462}{105 a c^4 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{3/2}}+\frac{2609 \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 )^{9/2} \left (1+\frac{1}{a x}\right )^{3/2}}+\frac{3 \tanh ^{-1}\left (\sqrt{1-\frac{1}{a x}} \sqrt{1+\frac{1}{a x}}\right )}{a c^4}\\ \end{align*}

Mathematica [A]  time = 0.273661, size = 117, normalized size = 0.36 $\frac{\frac{a x \sqrt{1-\frac{1}{a^2 x^2}} \left (315 a^7 x^7-2669 a^6 x^6+2967 a^5 x^5+4029 a^4 x^4-7399 a^3 x^3+339 a^2 x^2+4047 a x-1664\right )}{315 (a x-1)^5 (a x+1)^2}+3 \log \left (x \left (\sqrt{1-\frac{1}{a^2 x^2}}+1\right )\right )}{a c^4}$

Warning: Unable to verify antiderivative.

[In]

Integrate[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 - 266
9*a^6*x^6 + 315*a^7*x^7))/(315*(-1 + a*x)^5*(1 + a*x)^2) + 3*Log[(1 + Sqrt[1 - 1/(a^2*x^2)])*x])/(a*c^4)

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Maple [B]  time = 0.192, size = 766, normalized size = 2.3 \begin{align*} \text{result too large to display} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/40320*(-138915*(a^2)^(1/2)*((a*x-1)*(a*x+1))^(1/2)*x^9*a^9-120960*ln((a^2*x+(a^2)^(1/2)*((a*x-1)*(a*x+1))^(
1/2))/(a^2)^(1/2))*x^9*a^10+98595*(a^2)^(1/2)*((a*x-1)*(a*x+1))^(3/2)*x^7*a^7+416745*(a^2)^(1/2)*((a*x-1)*(a*x
+1))^(1/2)*x^8*a^8+362880*ln((a^2*x+(a^2)^(1/2)*((a*x-1)*(a*x+1))^(1/2))/(a^2)^(1/2))*x^8*a^9-75113*(a^2)^(1/2
)*((a*x-1)*(a*x+1))^(3/2)*x^6*a^6-240861*(a^2)^(1/2)*((a*x-1)*(a*x+1))^(3/2)*x^5*a^5-1111320*(a^2)^(1/2)*((a*x
-1)*(a*x+1))^(1/2)*x^6*a^6-967680*ln((a^2*x+(a^2)^(1/2)*((a*x-1)*(a*x+1))^(1/2))/(a^2)^(1/2))*x^6*a^7+178863*(
a^2)^(1/2)*((a*x-1)*(a*x+1))^(3/2)*x^4*a^4+833490*(a^2)^(1/2)*((a*x-1)*(a*x+1))^(1/2)*x^5*a^5+725760*ln((a^2*x
+(a^2)^(1/2)*((a*x-1)*(a*x+1))^(1/2))/(a^2)^(1/2))*x^5*a^6+252497*(a^2)^(1/2)*((a*x-1)*(a*x+1))^(3/2)*x^3*a^3+
833490*(a^2)^(1/2)*((a*x-1)*(a*x+1))^(1/2)*x^4*a^4+725760*ln((a^2*x+(a^2)^(1/2)*((a*x-1)*(a*x+1))^(1/2))/(a^2)
^(1/2))*x^4*a^5-182307*(a^2)^(1/2)*((a*x-1)*(a*x+1))^(3/2)*x^2*a^2-1111320*(a^2)^(1/2)*((a*x-1)*(a*x+1))^(1/2)
*x^3*a^3-967680*ln((a^2*x+(a^2)^(1/2)*((a*x-1)*(a*x+1))^(1/2))/(a^2)^(1/2))*x^3*a^4-101271*(a^2)^(1/2)*((a*x-1
)*(a*x+1))^(3/2)*x*a+74077*((a*x-1)*(a*x+1))^(3/2)*(a^2)^(1/2)+416745*(a^2)^(1/2)*((a*x-1)*(a*x+1))^(1/2)*x*a+
362880*ln((a^2*x+(a^2)^(1/2)*((a*x-1)*(a*x+1))^(1/2))/(a^2)^(1/2))*x*a^2-138915*(a^2)^(1/2)*((a*x-1)*(a*x+1))^
(1/2)-120960*a*ln((a^2*x+(a^2)^(1/2)*((a*x-1)*(a*x+1))^(1/2))/(a^2)^(1/2)))/a/(a^2)^(1/2)/(a*x-1)^4/c^4/((a*x-
1)*(a*x+1))^(1/2)/(a*x+1)^4/((a*x-1)/(a*x+1))^(3/2)

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Maxima [A]  time = 1.10717, size = 305, normalized size = 0.93 \begin{align*} \frac{1}{20160} \, a{\left (\frac{\frac{415 \,{\left (a x - 1\right )}}{a x + 1} + \frac{2511 \,{\left (a x - 1\right )}^{2}}{{\left (a x + 1\right )}^{2}} + \frac{11739 \,{\left (a x - 1\right )}^{3}}{{\left (a x + 1\right )}^{3}} + \frac{80745 \,{\left (a x - 1\right )}^{4}}{{\left (a x + 1\right )}^{4}} - \frac{135765 \,{\left (a x - 1\right )}^{5}}{{\left (a x + 1\right )}^{5}} + 35}{a^{2} c^{4} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{11}{2}} - a^{2} c^{4} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{9}{2}}} + \frac{105 \,{\left (\left (\frac{a x - 1}{a x + 1}\right )^{\frac{3}{2}} + 30 \, \sqrt{\frac{a x - 1}{a x + 1}}\right )}}{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 )} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/20160*a*((415*(a*x - 1)/(a*x + 1) + 2511*(a*x - 1)^2/(a*x + 1)^2 + 11739*(a*x - 1)^3/(a*x + 1)^3 + 80745*(a*
x - 1)^4/(a*x + 1)^4 - 135765*(a*x - 1)^5/(a*x + 1)^5 + 35)/(a^2*c^4*((a*x - 1)/(a*x + 1))^(11/2) - a^2*c^4*((
a*x - 1)/(a*x + 1))^(9/2)) + 105*(((a*x - 1)/(a*x + 1))^(3/2) + 30*sqrt((a*x - 1)/(a*x + 1)))/(a^2*c^4) + 6048
0*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))

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Fricas [A]  time = 1.71006, size = 571, normalized size = 1.74 \begin{align*} \frac{945 \,{\left (a^{6} x^{6} - 4 \, a^{5} x^{5} + 5 \, a^{4} x^{4} - 5 \, a^{2} x^{2} + 4 \, a x - 1\right )} \log \left (\sqrt{\frac{a x - 1}{a x + 1}} + 1\right ) - 945 \,{\left (a^{6} x^{6} - 4 \, a^{5} x^{5} + 5 \, a^{4} x^{4} - 5 \, a^{2} x^{2} + 4 \, 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} - 4 \, a^{6} c^{4} x^{5} + 5 \, a^{5} c^{4} x^{4} - 5 \, a^{3} c^{4} x^{2} + 4 \, a^{2} c^{4} x - a c^{4}\right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/((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 - 4*a^5*x^5 + 5*a^4*x^4 - 5*a^2*x^2 + 4*a*x - 1)*log(sqrt((a*x - 1)/(a*x + 1)) + 1) - 945*
(a^6*x^6 - 4*a^5*x^5 + 5*a^4*x^4 - 5*a^2*x^2 + 4*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)*sqrt((a*x - 1)/(a*x
+ 1)))/(a^7*c^4*x^6 - 4*a^6*c^4*x^5 + 5*a^5*c^4*x^4 - 5*a^3*c^4*x^2 + 4*a^2*c^4*x - a*c^4)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [A]  time = 1.27637, size = 356, normalized size = 1.08 \begin{align*} \frac{1}{20160} \, a{\left (\frac{60480 \, \log \left (\sqrt{\frac{a x - 1}{a x + 1}} + 1\right )}{a^{2} c^{4}} - \frac{60480 \, \log \left ({\left | \sqrt{\frac{a x - 1}{a x + 1}} - 1 \right |}\right )}{a^{2} c^{4}} - \frac{{\left (a x + 1\right )}^{4}{\left (\frac{450 \,{\left (a x - 1\right )}}{a x + 1} + \frac{2961 \,{\left (a x - 1\right )}^{2}}{{\left (a x + 1\right )}^{2}} + \frac{14700 \,{\left (a x - 1\right )}^{3}}{{\left (a x + 1\right )}^{3}} + \frac{95445 \,{\left (a x - 1\right )}^{4}}{{\left (a x + 1\right )}^{4}} + 35\right )}}{{\left (a x - 1\right )}^{4} a^{2} c^{4} \sqrt{\frac{a x - 1}{a x + 1}}} - \frac{40320 \, \sqrt{\frac{a x - 1}{a x + 1}}}{a^{2} c^{4}{\left (\frac{a x - 1}{a x + 1} - 1\right )}} + \frac{105 \,{\left (\frac{{\left (a x - 1\right )} a^{4} c^{8} \sqrt{\frac{a x - 1}{a x + 1}}}{a x + 1} + 30 \, a^{4} c^{8} \sqrt{\frac{a x - 1}{a x + 1}}\right )}}{a^{6} c^{12}}\right )} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/20160*a*(60480*log(sqrt((a*x - 1)/(a*x + 1)) + 1)/(a^2*c^4) - 60480*log(abs(sqrt((a*x - 1)/(a*x + 1)) - 1))/
(a^2*c^4) - (a*x + 1)^4*(450*(a*x - 1)/(a*x + 1) + 2961*(a*x - 1)^2/(a*x + 1)^2 + 14700*(a*x - 1)^3/(a*x + 1)^
3 + 95445*(a*x - 1)^4/(a*x + 1)^4 + 35)/((a*x - 1)^4*a^2*c^4*sqrt((a*x - 1)/(a*x + 1))) - 40320*sqrt((a*x - 1)
/(a*x + 1))/(a^2*c^4*((a*x - 1)/(a*x + 1) - 1)) + 105*((a*x - 1)*a^4*c^8*sqrt((a*x - 1)/(a*x + 1))/(a*x + 1) +
30*a^4*c^8*sqrt((a*x - 1)/(a*x + 1)))/(a^6*c^12))