3.403 \(\int \frac{e^{3 \coth ^{-1}(a x)}}{(c-\frac{c}{a x})^4} \, dx\)

Optimal. Leaf size=204 \[ \frac{16 \left (9 a-\frac{5}{x}\right )}{63 a^2 c^4 \left (1-\frac{1}{a^2 x^2}\right )^{7/2}}-\frac{64 \left (a+\frac{1}{x}\right )}{9 a^2 c^4 \left (1-\frac{1}{a^2 x^2}\right )^{9/2}}-\frac{8 \left (21 a+\frac{41}{x}\right )}{105 a^2 c^4 \left (1-\frac{1}{a^2 x^2}\right )^{5/2}}-\frac{735 a+\frac{1417}{x}}{315 a^2 c^4 \left (1-\frac{1}{a^2 x^2}\right )^{3/2}}-\frac{2205 a+\frac{3149}{x}}{315 a^2 c^4 \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{x \sqrt{1-\frac{1}{a^2 x^2}}}{c^4}+\frac{7 \tanh ^{-1}\left (\sqrt{1-\frac{1}{a^2 x^2}}\right )}{a c^4} \]

[Out]

(16*(9*a - 5/x))/(63*a^2*c^4*(1 - 1/(a^2*x^2))^(7/2)) - (64*(a + x^(-1)))/(9*a^2*c^4*(1 - 1/(a^2*x^2))^(9/2))
- (8*(21*a + 41/x))/(105*a^2*c^4*(1 - 1/(a^2*x^2))^(5/2)) - (735*a + 1417/x)/(315*a^2*c^4*(1 - 1/(a^2*x^2))^(3
/2)) - (2205*a + 3149/x)/(315*a^2*c^4*Sqrt[1 - 1/(a^2*x^2)]) + (Sqrt[1 - 1/(a^2*x^2)]*x)/c^4 + (7*ArcTanh[Sqrt
[1 - 1/(a^2*x^2)]])/(a*c^4)

________________________________________________________________________________________

Rubi [A]  time = 0.639914, antiderivative size = 204, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 7, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.318, Rules used = {6177, 852, 1805, 807, 266, 63, 208} \[ \frac{16 \left (9 a-\frac{5}{x}\right )}{63 a^2 c^4 \left (1-\frac{1}{a^2 x^2}\right )^{7/2}}-\frac{64 \left (a+\frac{1}{x}\right )}{9 a^2 c^4 \left (1-\frac{1}{a^2 x^2}\right )^{9/2}}-\frac{8 \left (21 a+\frac{41}{x}\right )}{105 a^2 c^4 \left (1-\frac{1}{a^2 x^2}\right )^{5/2}}-\frac{735 a+\frac{1417}{x}}{315 a^2 c^4 \left (1-\frac{1}{a^2 x^2}\right )^{3/2}}-\frac{2205 a+\frac{3149}{x}}{315 a^2 c^4 \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{x \sqrt{1-\frac{1}{a^2 x^2}}}{c^4}+\frac{7 \tanh ^{-1}\left (\sqrt{1-\frac{1}{a^2 x^2}}\right )}{a c^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(16*(9*a - 5/x))/(63*a^2*c^4*(1 - 1/(a^2*x^2))^(7/2)) - (64*(a + x^(-1)))/(9*a^2*c^4*(1 - 1/(a^2*x^2))^(9/2))
- (8*(21*a + 41/x))/(105*a^2*c^4*(1 - 1/(a^2*x^2))^(5/2)) - (735*a + 1417/x)/(315*a^2*c^4*(1 - 1/(a^2*x^2))^(3
/2)) - (2205*a + 3149/x)/(315*a^2*c^4*Sqrt[1 - 1/(a^2*x^2)]) + (Sqrt[1 - 1/(a^2*x^2)]*x)/c^4 + (7*ArcTanh[Sqrt
[1 - 1/(a^2*x^2)]])/(a*c^4)

Rule 6177

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

Rule 852

Int[((d_) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Dist[d^(2*m)/a
^m, Int[((f + g*x)^n*(a + c*x^2)^(m + p))/(d - e*x)^m, x], x] /; FreeQ[{a, c, d, e, f, g, n, p}, x] && NeQ[e*f
- d*g, 0] && EqQ[c*d^2 + a*e^2, 0] &&  !IntegerQ[p] && EqQ[f, 0] && ILtQ[m, -1] &&  !(IGtQ[n, 0] && ILtQ[m +
n, 0] &&  !GtQ[p, 1])

Rule 1805

Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuotient[(c*x)^m*Pq,
a + b*x^2, x], f = Coeff[PolynomialRemainder[(c*x)^m*Pq, a + b*x^2, x], x, 0], g = Coeff[PolynomialRemainder[
(c*x)^m*Pq, a + b*x^2, x], x, 1]}, Simp[((a*g - b*f*x)*(a + b*x^2)^(p + 1))/(2*a*b*(p + 1)), x] + Dist[1/(2*a*
(p + 1)), Int[(c*x)^m*(a + b*x^2)^(p + 1)*ExpandToSum[(2*a*(p + 1)*Q)/(c*x)^m + (f*(2*p + 3))/(c*x)^m, x], x],
x]] /; FreeQ[{a, b, c}, x] && PolyQ[Pq, x] && LtQ[p, -1] && ILtQ[m, 0]

Rule 807

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> -Simp[((e*f - d*g
)*(d + e*x)^(m + 1)*(a + c*x^2)^(p + 1))/(2*(p + 1)*(c*d^2 + a*e^2)), x] + Dist[(c*d*f + a*e*g)/(c*d^2 + a*e^2
), Int[(d + e*x)^(m + 1)*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g, m, p}, x] && NeQ[c*d^2 + a*e^2, 0]
&& EqQ[Simplify[m + 2*p + 3], 0]

Rule 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

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 x}\right )^4} \, dx &=-\left (c^3 \operatorname{Subst}\left (\int \frac{\left (1-\frac{x^2}{a^2}\right )^{3/2}}{x^2 \left (c-\frac{c x}{a}\right )^7} \, dx,x,\frac{1}{x}\right )\right )\\ &=-\frac{\operatorname{Subst}\left (\int \frac{\left (c+\frac{c x}{a}\right )^7}{x^2 \left (1-\frac{x^2}{a^2}\right )^{11/2}} \, dx,x,\frac{1}{x}\right )}{c^{11}}\\ &=-\frac{64 \left (a+\frac{1}{x}\right )}{9 a^2 c^4 \left (1-\frac{1}{a^2 x^2}\right )^{9/2}}+\frac{\operatorname{Subst}\left (\int \frac{-9 c^7-\frac{63 c^7 x}{a}-\frac{134 c^7 x^2}{a^2}+\frac{198 c^7 x^3}{a^3}+\frac{63 c^7 x^4}{a^4}+\frac{9 c^7 x^5}{a^5}}{x^2 \left (1-\frac{x^2}{a^2}\right )^{9/2}} \, dx,x,\frac{1}{x}\right )}{9 c^{11}}\\ &=\frac{16 \left (9 a-\frac{5}{x}\right )}{63 a^2 c^4 \left (1-\frac{1}{a^2 x^2}\right )^{7/2}}-\frac{64 \left (a+\frac{1}{x}\right )}{9 a^2 c^4 \left (1-\frac{1}{a^2 x^2}\right )^{9/2}}-\frac{\operatorname{Subst}\left (\int \frac{63 c^7+\frac{441 c^7 x}{a}+\frac{921 c^7 x^2}{a^2}+\frac{63 c^7 x^3}{a^3}}{x^2 \left (1-\frac{x^2}{a^2}\right )^{7/2}} \, dx,x,\frac{1}{x}\right )}{63 c^{11}}\\ &=\frac{16 \left (9 a-\frac{5}{x}\right )}{63 a^2 c^4 \left (1-\frac{1}{a^2 x^2}\right )^{7/2}}-\frac{64 \left (a+\frac{1}{x}\right )}{9 a^2 c^4 \left (1-\frac{1}{a^2 x^2}\right )^{9/2}}-\frac{8 \left (21 a+\frac{41}{x}\right )}{105 a^2 c^4 \left (1-\frac{1}{a^2 x^2}\right )^{5/2}}+\frac{\operatorname{Subst}\left (\int \frac{-315 c^7-\frac{2205 c^7 x}{a}-\frac{3936 c^7 x^2}{a^2}}{x^2 \left (1-\frac{x^2}{a^2}\right )^{5/2}} \, dx,x,\frac{1}{x}\right )}{315 c^{11}}\\ &=\frac{16 \left (9 a-\frac{5}{x}\right )}{63 a^2 c^4 \left (1-\frac{1}{a^2 x^2}\right )^{7/2}}-\frac{64 \left (a+\frac{1}{x}\right )}{9 a^2 c^4 \left (1-\frac{1}{a^2 x^2}\right )^{9/2}}-\frac{8 \left (21 a+\frac{41}{x}\right )}{105 a^2 c^4 \left (1-\frac{1}{a^2 x^2}\right )^{5/2}}-\frac{735 a+\frac{1417}{x}}{315 a^2 c^4 \left (1-\frac{1}{a^2 x^2}\right )^{3/2}}-\frac{\operatorname{Subst}\left (\int \frac{945 c^7+\frac{6615 c^7 x}{a}+\frac{8502 c^7 x^2}{a^2}}{x^2 \left (1-\frac{x^2}{a^2}\right )^{3/2}} \, dx,x,\frac{1}{x}\right )}{945 c^{11}}\\ &=\frac{16 \left (9 a-\frac{5}{x}\right )}{63 a^2 c^4 \left (1-\frac{1}{a^2 x^2}\right )^{7/2}}-\frac{64 \left (a+\frac{1}{x}\right )}{9 a^2 c^4 \left (1-\frac{1}{a^2 x^2}\right )^{9/2}}-\frac{8 \left (21 a+\frac{41}{x}\right )}{105 a^2 c^4 \left (1-\frac{1}{a^2 x^2}\right )^{5/2}}-\frac{735 a+\frac{1417}{x}}{315 a^2 c^4 \left (1-\frac{1}{a^2 x^2}\right )^{3/2}}-\frac{2205 a+\frac{3149}{x}}{315 a^2 c^4 \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{\operatorname{Subst}\left (\int \frac{-945 c^7-\frac{6615 c^7 x}{a}}{x^2 \sqrt{1-\frac{x^2}{a^2}}} \, dx,x,\frac{1}{x}\right )}{945 c^{11}}\\ &=\frac{16 \left (9 a-\frac{5}{x}\right )}{63 a^2 c^4 \left (1-\frac{1}{a^2 x^2}\right )^{7/2}}-\frac{64 \left (a+\frac{1}{x}\right )}{9 a^2 c^4 \left (1-\frac{1}{a^2 x^2}\right )^{9/2}}-\frac{8 \left (21 a+\frac{41}{x}\right )}{105 a^2 c^4 \left (1-\frac{1}{a^2 x^2}\right )^{5/2}}-\frac{735 a+\frac{1417}{x}}{315 a^2 c^4 \left (1-\frac{1}{a^2 x^2}\right )^{3/2}}-\frac{2205 a+\frac{3149}{x}}{315 a^2 c^4 \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{\sqrt{1-\frac{1}{a^2 x^2}} x}{c^4}-\frac{7 \operatorname{Subst}\left (\int \frac{1}{x \sqrt{1-\frac{x^2}{a^2}}} \, dx,x,\frac{1}{x}\right )}{a c^4}\\ &=\frac{16 \left (9 a-\frac{5}{x}\right )}{63 a^2 c^4 \left (1-\frac{1}{a^2 x^2}\right )^{7/2}}-\frac{64 \left (a+\frac{1}{x}\right )}{9 a^2 c^4 \left (1-\frac{1}{a^2 x^2}\right )^{9/2}}-\frac{8 \left (21 a+\frac{41}{x}\right )}{105 a^2 c^4 \left (1-\frac{1}{a^2 x^2}\right )^{5/2}}-\frac{735 a+\frac{1417}{x}}{315 a^2 c^4 \left (1-\frac{1}{a^2 x^2}\right )^{3/2}}-\frac{2205 a+\frac{3149}{x}}{315 a^2 c^4 \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{\sqrt{1-\frac{1}{a^2 x^2}} x}{c^4}-\frac{7 \operatorname{Subst}\left (\int \frac{1}{x \sqrt{1-\frac{x}{a^2}}} \, dx,x,\frac{1}{x^2}\right )}{2 a c^4}\\ &=\frac{16 \left (9 a-\frac{5}{x}\right )}{63 a^2 c^4 \left (1-\frac{1}{a^2 x^2}\right )^{7/2}}-\frac{64 \left (a+\frac{1}{x}\right )}{9 a^2 c^4 \left (1-\frac{1}{a^2 x^2}\right )^{9/2}}-\frac{8 \left (21 a+\frac{41}{x}\right )}{105 a^2 c^4 \left (1-\frac{1}{a^2 x^2}\right )^{5/2}}-\frac{735 a+\frac{1417}{x}}{315 a^2 c^4 \left (1-\frac{1}{a^2 x^2}\right )^{3/2}}-\frac{2205 a+\frac{3149}{x}}{315 a^2 c^4 \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{\sqrt{1-\frac{1}{a^2 x^2}} x}{c^4}+\frac{(7 a) \operatorname{Subst}\left (\int \frac{1}{a^2-a^2 x^2} \, dx,x,\sqrt{1-\frac{1}{a^2 x^2}}\right )}{c^4}\\ &=\frac{16 \left (9 a-\frac{5}{x}\right )}{63 a^2 c^4 \left (1-\frac{1}{a^2 x^2}\right )^{7/2}}-\frac{64 \left (a+\frac{1}{x}\right )}{9 a^2 c^4 \left (1-\frac{1}{a^2 x^2}\right )^{9/2}}-\frac{8 \left (21 a+\frac{41}{x}\right )}{105 a^2 c^4 \left (1-\frac{1}{a^2 x^2}\right )^{5/2}}-\frac{735 a+\frac{1417}{x}}{315 a^2 c^4 \left (1-\frac{1}{a^2 x^2}\right )^{3/2}}-\frac{2205 a+\frac{3149}{x}}{315 a^2 c^4 \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{\sqrt{1-\frac{1}{a^2 x^2}} x}{c^4}+\frac{7 \tanh ^{-1}\left (\sqrt{1-\frac{1}{a^2 x^2}}\right )}{a c^4}\\ \end{align*}

Mathematica [A]  time = 0.0936146, size = 120, normalized size = 0.59 \[ \frac{315 a^6 x^6-6224 a^5 x^5+13241 a^4 x^4-5567 a^3 x^3-10232 a^2 x^2+2205 a x \sqrt{1-\frac{1}{a^2 x^2}} (a x-1)^4 \tanh ^{-1}\left (\sqrt{1-\frac{1}{a^2 x^2}}\right )+11651 a x-3464}{315 a^2 c^4 x \sqrt{1-\frac{1}{a^2 x^2}} (a x-1)^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-3464 + 11651*a*x - 10232*a^2*x^2 - 5567*a^3*x^3 + 13241*a^4*x^4 - 6224*a^5*x^5 + 315*a^6*x^6 + 2205*a*Sqrt[1
- 1/(a^2*x^2)]*x*(-1 + a*x)^4*ArcTanh[Sqrt[1 - 1/(a^2*x^2)]])/(315*a^2*c^4*Sqrt[1 - 1/(a^2*x^2)]*x*(-1 + a*x)
^4)

________________________________________________________________________________________

Maple [B]  time = 0.18, size = 622, normalized size = 3.1 \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/x)^4,x)

[Out]

1/315*(2205*ln((a^2*x+(a^2)^(1/2)*((a*x-1)*(a*x+1))^(1/2))/(a^2)^(1/2))*x^6*a^7+2205*(a^2)^(1/2)*((a*x-1)*(a*x
+1))^(1/2)*x^6*a^6-13230*ln((a^2*x+(a^2)^(1/2)*((a*x-1)*(a*x+1))^(1/2))/(a^2)^(1/2))*x^5*a^6-1890*(a^2)^(1/2)*
((a*x-1)*(a*x+1))^(3/2)*x^4*a^4-13230*(a^2)^(1/2)*((a*x-1)*(a*x+1))^(1/2)*x^5*a^5+33075*ln((a^2*x+(a^2)^(1/2)*
((a*x-1)*(a*x+1))^(1/2))/(a^2)^(1/2))*x^4*a^5+6376*(a^2)^(1/2)*((a*x-1)*(a*x+1))^(3/2)*x^3*a^3+33075*(a^2)^(1/
2)*((a*x-1)*(a*x+1))^(1/2)*x^4*a^4-44100*ln((a^2*x+(a^2)^(1/2)*((a*x-1)*(a*x+1))^(1/2))/(a^2)^(1/2))*x^3*a^4-8
646*(a^2)^(1/2)*((a*x-1)*(a*x+1))^(3/2)*x^2*a^2-44100*(a^2)^(1/2)*((a*x-1)*(a*x+1))^(1/2)*x^3*a^3+33075*ln((a^
2*x+(a^2)^(1/2)*((a*x-1)*(a*x+1))^(1/2))/(a^2)^(1/2))*x^2*a^3+5349*(a^2)^(1/2)*((a*x-1)*(a*x+1))^(3/2)*x*a+330
75*(a^2)^(1/2)*((a*x-1)*(a*x+1))^(1/2)*x^2*a^2-13230*ln((a^2*x+(a^2)^(1/2)*((a*x-1)*(a*x+1))^(1/2))/(a^2)^(1/2
))*x*a^2-1259*((a*x-1)*(a*x+1))^(3/2)*(a^2)^(1/2)-13230*(a^2)^(1/2)*((a*x-1)*(a*x+1))^(1/2)*x*a+2205*a*ln((a^2
*x+(a^2)^(1/2)*((a*x-1)*(a*x+1))^(1/2))/(a^2)^(1/2))+2205*(a^2)^(1/2)*((a*x-1)*(a*x+1))^(1/2))/a/(a^2)^(1/2)/(
a*x-1)^4/c^4/((a*x-1)*(a*x+1))^(1/2)/(a*x+1)/((a*x-1)/(a*x+1))^(3/2)

________________________________________________________________________________________

Maxima [A]  time = 1.17138, size = 250, normalized size = 1.23 \begin{align*} \frac{1}{1260} \, a{\left (\frac{\frac{235 \,{\left (a x - 1\right )}}{a x + 1} + \frac{801 \,{\left (a x - 1\right )}^{2}}{{\left (a x + 1\right )}^{2}} + \frac{2289 \,{\left (a x - 1\right )}^{3}}{{\left (a x + 1\right )}^{3}} + \frac{11760 \,{\left (a x - 1\right )}^{4}}{{\left (a x + 1\right )}^{4}} - \frac{17640 \,{\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{8820 \, \log \left (\sqrt{\frac{a x - 1}{a x + 1}} + 1\right )}{a^{2} c^{4}} - \frac{8820 \, \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/x)^4,x, algorithm="maxima")

[Out]

1/1260*a*((235*(a*x - 1)/(a*x + 1) + 801*(a*x - 1)^2/(a*x + 1)^2 + 2289*(a*x - 1)^3/(a*x + 1)^3 + 11760*(a*x -
1)^4/(a*x + 1)^4 - 17640*(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)) + 8820*log(sqrt((a*x - 1)/(a*x + 1)) + 1)/(a^2*c^4) - 8820*log(sqrt((a*x - 1)/(a*x + 1)
) - 1)/(a^2*c^4))

________________________________________________________________________________________

Fricas [A]  time = 1.52512, size = 567, normalized size = 2.78 \begin{align*} \frac{2205 \,{\left (a^{5} x^{5} - 5 \, a^{4} x^{4} + 10 \, a^{3} x^{3} - 10 \, a^{2} x^{2} + 5 \, a x - 1\right )} \log \left (\sqrt{\frac{a x - 1}{a x + 1}} + 1\right ) - 2205 \,{\left (a^{5} x^{5} - 5 \, a^{4} x^{4} + 10 \, a^{3} x^{3} - 10 \, a^{2} x^{2} + 5 \, a x - 1\right )} \log \left (\sqrt{\frac{a x - 1}{a x + 1}} - 1\right ) +{\left (315 \, a^{6} x^{6} - 6224 \, a^{5} x^{5} + 13241 \, a^{4} x^{4} - 5567 \, a^{3} x^{3} - 10232 \, a^{2} x^{2} + 11651 \, a x - 3464\right )} \sqrt{\frac{a x - 1}{a x + 1}}}{315 \,{\left (a^{6} c^{4} x^{5} - 5 \, a^{5} c^{4} x^{4} + 10 \, a^{4} c^{4} x^{3} - 10 \, a^{3} c^{4} x^{2} + 5 \, 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/x)^4,x, algorithm="fricas")

[Out]

1/315*(2205*(a^5*x^5 - 5*a^4*x^4 + 10*a^3*x^3 - 10*a^2*x^2 + 5*a*x - 1)*log(sqrt((a*x - 1)/(a*x + 1)) + 1) - 2
205*(a^5*x^5 - 5*a^4*x^4 + 10*a^3*x^3 - 10*a^2*x^2 + 5*a*x - 1)*log(sqrt((a*x - 1)/(a*x + 1)) - 1) + (315*a^6*
x^6 - 6224*a^5*x^5 + 13241*a^4*x^4 - 5567*a^3*x^3 - 10232*a^2*x^2 + 11651*a*x - 3464)*sqrt((a*x - 1)/(a*x + 1)
))/(a^6*c^4*x^5 - 5*a^5*c^4*x^4 + 10*a^4*c^4*x^3 - 10*a^3*c^4*x^2 + 5*a^2*c^4*x - a*c^4)

________________________________________________________________________________________

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/x)**4,x)

[Out]

Timed out

________________________________________________________________________________________

Giac [A]  time = 1.41816, size = 267, normalized size = 1.31 \begin{align*} \frac{1}{1260} \, a{\left (\frac{8820 \, \log \left (\sqrt{\frac{a x - 1}{a x + 1}} + 1\right )}{a^{2} c^{4}} - \frac{8820 \, \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{270 \,{\left (a x - 1\right )}}{a x + 1} + \frac{1071 \,{\left (a x - 1\right )}^{2}}{{\left (a x + 1\right )}^{2}} + \frac{3360 \,{\left (a x - 1\right )}^{3}}{{\left (a x + 1\right )}^{3}} + \frac{15120 \,{\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{2520 \, \sqrt{\frac{a x - 1}{a x + 1}}}{a^{2} c^{4}{\left (\frac{a x - 1}{a x + 1} - 1\right )}}\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/x)^4,x, algorithm="giac")

[Out]

1/1260*a*(8820*log(sqrt((a*x - 1)/(a*x + 1)) + 1)/(a^2*c^4) - 8820*log(abs(sqrt((a*x - 1)/(a*x + 1)) - 1))/(a^
2*c^4) - (a*x + 1)^4*(270*(a*x - 1)/(a*x + 1) + 1071*(a*x - 1)^2/(a*x + 1)^2 + 3360*(a*x - 1)^3/(a*x + 1)^3 +
15120*(a*x - 1)^4/(a*x + 1)^4 + 35)/((a*x - 1)^4*a^2*c^4*sqrt((a*x - 1)/(a*x + 1))) - 2520*sqrt((a*x - 1)/(a*x
+ 1))/(a^2*c^4*((a*x - 1)/(a*x + 1) - 1)))