3.778 \(\int \frac{e^{\coth ^{-1}(a x)}}{(c-\frac{c}{a^2 x^2})^3} \, dx\)

Optimal. Leaf size=254 \[ \frac{x}{c^3 \left (1-\frac{1}{a x}\right )^{5/2} \left (\frac{1}{a x}+1\right )^{3/2}}+\frac{16 \sqrt{1-\frac{1}{a x}}}{5 a c^3 \sqrt{\frac{1}{a x}+1}}+\frac{21 \sqrt{1-\frac{1}{a x}}}{5 a c^3 \left (\frac{1}{a x}+1\right )^{3/2}}-\frac{34}{5 a c^3 \sqrt{1-\frac{1}{a x}} \left (\frac{1}{a x}+1\right )^{3/2}}-\frac{29}{15 a c^3 \left (1-\frac{1}{a x}\right )^{3/2} \left (\frac{1}{a x}+1\right )^{3/2}}-\frac{6}{5 a c^3 \left (1-\frac{1}{a x}\right )^{5/2} \left (\frac{1}{a x}+1\right )^{3/2}}+\frac{\tanh ^{-1}\left (\sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}\right )}{a c^3} \]

[Out]

-6/(5*a*c^3*(1 - 1/(a*x))^(5/2)*(1 + 1/(a*x))^(3/2)) - 29/(15*a*c^3*(1 - 1/(a*x))^(3/2)*(1 + 1/(a*x))^(3/2)) -
34/(5*a*c^3*Sqrt[1 - 1/(a*x)]*(1 + 1/(a*x))^(3/2)) + (21*Sqrt[1 - 1/(a*x)])/(5*a*c^3*(1 + 1/(a*x))^(3/2)) + (
16*Sqrt[1 - 1/(a*x)])/(5*a*c^3*Sqrt[1 + 1/(a*x)]) + x/(c^3*(1 - 1/(a*x))^(5/2)*(1 + 1/(a*x))^(3/2)) + ArcTanh[
Sqrt[1 - 1/(a*x)]*Sqrt[1 + 1/(a*x)]]/(a*c^3)

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Rubi [A]  time = 0.165665, antiderivative size = 254, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 6, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {6194, 103, 152, 12, 92, 208} \[ \frac{x}{c^3 \left (1-\frac{1}{a x}\right )^{5/2} \left (\frac{1}{a x}+1\right )^{3/2}}+\frac{16 \sqrt{1-\frac{1}{a x}}}{5 a c^3 \sqrt{\frac{1}{a x}+1}}+\frac{21 \sqrt{1-\frac{1}{a x}}}{5 a c^3 \left (\frac{1}{a x}+1\right )^{3/2}}-\frac{34}{5 a c^3 \sqrt{1-\frac{1}{a x}} \left (\frac{1}{a x}+1\right )^{3/2}}-\frac{29}{15 a c^3 \left (1-\frac{1}{a x}\right )^{3/2} \left (\frac{1}{a x}+1\right )^{3/2}}-\frac{6}{5 a c^3 \left (1-\frac{1}{a x}\right )^{5/2} \left (\frac{1}{a x}+1\right )^{3/2}}+\frac{\tanh ^{-1}\left (\sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}\right )}{a c^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-6/(5*a*c^3*(1 - 1/(a*x))^(5/2)*(1 + 1/(a*x))^(3/2)) - 29/(15*a*c^3*(1 - 1/(a*x))^(3/2)*(1 + 1/(a*x))^(3/2)) -
34/(5*a*c^3*Sqrt[1 - 1/(a*x)]*(1 + 1/(a*x))^(3/2)) + (21*Sqrt[1 - 1/(a*x)])/(5*a*c^3*(1 + 1/(a*x))^(3/2)) + (
16*Sqrt[1 - 1/(a*x)])/(5*a*c^3*Sqrt[1 + 1/(a*x)]) + x/(c^3*(1 - 1/(a*x))^(5/2)*(1 + 1/(a*x))^(3/2)) + ArcTanh[
Sqrt[1 - 1/(a*x)]*Sqrt[1 + 1/(a*x)]]/(a*c^3)

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^{\coth ^{-1}(a x)}}{\left (c-\frac{c}{a^2 x^2}\right )^3} \, dx &=-\frac{\operatorname{Subst}\left (\int \frac{1}{x^2 \left (1-\frac{x}{a}\right )^{7/2} \left (1+\frac{x}{a}\right )^{5/2}} \, dx,x,\frac{1}{x}\right )}{c^3}\\ &=\frac{x}{c^3 \left (1-\frac{1}{a x}\right )^{5/2} \left (1+\frac{1}{a x}\right )^{3/2}}+\frac{\operatorname{Subst}\left (\int \frac{-\frac{1}{a}-\frac{5 x}{a^2}}{x \left (1-\frac{x}{a}\right )^{7/2} \left (1+\frac{x}{a}\right )^{5/2}} \, dx,x,\frac{1}{x}\right )}{c^3}\\ &=-\frac{6}{5 a c^3 \left (1-\frac{1}{a x}\right )^{5/2} \left (1+\frac{1}{a x}\right )^{3/2}}+\frac{x}{c^3 \left (1-\frac{1}{a x}\right )^{5/2} \left (1+\frac{1}{a x}\right )^{3/2}}-\frac{a \operatorname{Subst}\left (\int \frac{\frac{5}{a^2}+\frac{24 x}{a^3}}{x \left (1-\frac{x}{a}\right )^{5/2} \left (1+\frac{x}{a}\right )^{5/2}} \, dx,x,\frac{1}{x}\right )}{5 c^3}\\ &=-\frac{6}{5 a c^3 \left (1-\frac{1}{a x}\right )^{5/2} \left (1+\frac{1}{a x}\right )^{3/2}}-\frac{29}{15 a c^3 \left (1-\frac{1}{a x}\right )^{3/2} \left (1+\frac{1}{a x}\right )^{3/2}}+\frac{x}{c^3 \left (1-\frac{1}{a x}\right )^{5/2} \left (1+\frac{1}{a x}\right )^{3/2}}+\frac{a^2 \operatorname{Subst}\left (\int \frac{-\frac{15}{a^3}-\frac{87 x}{a^4}}{x \left (1-\frac{x}{a}\right )^{3/2} \left (1+\frac{x}{a}\right )^{5/2}} \, dx,x,\frac{1}{x}\right )}{15 c^3}\\ &=-\frac{6}{5 a c^3 \left (1-\frac{1}{a x}\right )^{5/2} \left (1+\frac{1}{a x}\right )^{3/2}}-\frac{29}{15 a c^3 \left (1-\frac{1}{a x}\right )^{3/2} \left (1+\frac{1}{a x}\right )^{3/2}}-\frac{34}{5 a c^3 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{3/2}}+\frac{x}{c^3 \left (1-\frac{1}{a x}\right )^{5/2} \left (1+\frac{1}{a x}\right )^{3/2}}-\frac{a^3 \operatorname{Subst}\left (\int \frac{\frac{15}{a^4}+\frac{204 x}{a^5}}{x \sqrt{1-\frac{x}{a}} \left (1+\frac{x}{a}\right )^{5/2}} \, dx,x,\frac{1}{x}\right )}{15 c^3}\\ &=-\frac{6}{5 a c^3 \left (1-\frac{1}{a x}\right )^{5/2} \left (1+\frac{1}{a x}\right )^{3/2}}-\frac{29}{15 a c^3 \left (1-\frac{1}{a x}\right )^{3/2} \left (1+\frac{1}{a x}\right )^{3/2}}-\frac{34}{5 a c^3 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{3/2}}+\frac{21 \sqrt{1-\frac{1}{a x}}}{5 a c^3 \left (1+\frac{1}{a x}\right )^{3/2}}+\frac{x}{c^3 \left (1-\frac{1}{a x}\right )^{5/2} \left (1+\frac{1}{a x}\right )^{3/2}}-\frac{a^4 \operatorname{Subst}\left (\int \frac{\frac{45}{a^5}+\frac{189 x}{a^6}}{x \sqrt{1-\frac{x}{a}} \left (1+\frac{x}{a}\right )^{3/2}} \, dx,x,\frac{1}{x}\right )}{45 c^3}\\ &=-\frac{6}{5 a c^3 \left (1-\frac{1}{a x}\right )^{5/2} \left (1+\frac{1}{a x}\right )^{3/2}}-\frac{29}{15 a c^3 \left (1-\frac{1}{a x}\right )^{3/2} \left (1+\frac{1}{a x}\right )^{3/2}}-\frac{34}{5 a c^3 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{3/2}}+\frac{21 \sqrt{1-\frac{1}{a x}}}{5 a c^3 \left (1+\frac{1}{a x}\right )^{3/2}}+\frac{16 \sqrt{1-\frac{1}{a x}}}{5 a c^3 \sqrt{1+\frac{1}{a x}}}+\frac{x}{c^3 \left (1-\frac{1}{a x}\right )^{5/2} \left (1+\frac{1}{a x}\right )^{3/2}}-\frac{a^5 \operatorname{Subst}\left (\int \frac{45}{a^6 x \sqrt{1-\frac{x}{a}} \sqrt{1+\frac{x}{a}}} \, dx,x,\frac{1}{x}\right )}{45 c^3}\\ &=-\frac{6}{5 a c^3 \left (1-\frac{1}{a x}\right )^{5/2} \left (1+\frac{1}{a x}\right )^{3/2}}-\frac{29}{15 a c^3 \left (1-\frac{1}{a x}\right )^{3/2} \left (1+\frac{1}{a x}\right )^{3/2}}-\frac{34}{5 a c^3 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{3/2}}+\frac{21 \sqrt{1-\frac{1}{a x}}}{5 a c^3 \left (1+\frac{1}{a x}\right )^{3/2}}+\frac{16 \sqrt{1-\frac{1}{a x}}}{5 a c^3 \sqrt{1+\frac{1}{a x}}}+\frac{x}{c^3 \left (1-\frac{1}{a x}\right )^{5/2} \left (1+\frac{1}{a x}\right )^{3/2}}-\frac{\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^3}\\ &=-\frac{6}{5 a c^3 \left (1-\frac{1}{a x}\right )^{5/2} \left (1+\frac{1}{a x}\right )^{3/2}}-\frac{29}{15 a c^3 \left (1-\frac{1}{a x}\right )^{3/2} \left (1+\frac{1}{a x}\right )^{3/2}}-\frac{34}{5 a c^3 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{3/2}}+\frac{21 \sqrt{1-\frac{1}{a x}}}{5 a c^3 \left (1+\frac{1}{a x}\right )^{3/2}}+\frac{16 \sqrt{1-\frac{1}{a x}}}{5 a c^3 \sqrt{1+\frac{1}{a x}}}+\frac{x}{c^3 \left (1-\frac{1}{a x}\right )^{5/2} \left (1+\frac{1}{a x}\right )^{3/2}}+\frac{\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^3}\\ &=-\frac{6}{5 a c^3 \left (1-\frac{1}{a x}\right )^{5/2} \left (1+\frac{1}{a x}\right )^{3/2}}-\frac{29}{15 a c^3 \left (1-\frac{1}{a x}\right )^{3/2} \left (1+\frac{1}{a x}\right )^{3/2}}-\frac{34}{5 a c^3 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{3/2}}+\frac{21 \sqrt{1-\frac{1}{a x}}}{5 a c^3 \left (1+\frac{1}{a x}\right )^{3/2}}+\frac{16 \sqrt{1-\frac{1}{a x}}}{5 a c^3 \sqrt{1+\frac{1}{a x}}}+\frac{x}{c^3 \left (1-\frac{1}{a x}\right )^{5/2} \left (1+\frac{1}{a x}\right )^{3/2}}+\frac{\tanh ^{-1}\left (\sqrt{1-\frac{1}{a x}} \sqrt{1+\frac{1}{a x}}\right )}{a c^3}\\ \end{align*}

Mathematica [A]  time = 0.196055, size = 99, normalized size = 0.39 \[ \frac{\frac{a x \sqrt{1-\frac{1}{a^2 x^2}} \left (15 a^5 x^5-38 a^4 x^4-52 a^3 x^3+87 a^2 x^2+33 a x-48\right )}{15 (a x-1)^3 (a x+1)^2}+\log \left (x \left (\sqrt{1-\frac{1}{a^2 x^2}}+1\right )\right )}{a c^3} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

((a*Sqrt[1 - 1/(a^2*x^2)]*x*(-48 + 33*a*x + 87*a^2*x^2 - 52*a^3*x^3 - 38*a^4*x^4 + 15*a^5*x^5))/(15*(-1 + a*x)
^3*(1 + a*x)^2) + Log[(1 + Sqrt[1 - 1/(a^2*x^2)])*x])/(a*c^3)

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Maple [B]  time = 0.164, size = 714, normalized size = 2.8 \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))^(1/2)/(c-c/a^2/x^2)^3,x)

[Out]

1/240*(240*ln((a^2*x+(a^2)^(1/2)*((a*x-1)*(a*x+1))^(1/2))/(a^2)^(1/2))*x^7*a^8+525*(a^2)^(1/2)*((a*x-1)*(a*x+1
))^(1/2)*x^7*a^7-240*ln((a^2*x+(a^2)^(1/2)*((a*x-1)*(a*x+1))^(1/2))/(a^2)^(1/2))*x^6*a^7-285*(a^2)^(1/2)*((a*x
-1)*(a*x+1))^(3/2)*x^5*a^5-525*(a^2)^(1/2)*((a*x-1)*(a*x+1))^(1/2)*x^6*a^6-720*ln((a^2*x+(a^2)^(1/2)*((a*x-1)*
(a*x+1))^(1/2))/(a^2)^(1/2))*x^5*a^6-83*(a^2)^(1/2)*((a*x-1)*(a*x+1))^(3/2)*x^4*a^4-1575*(a^2)^(1/2)*((a*x-1)*
(a*x+1))^(1/2)*x^5*a^5+720*ln((a^2*x+(a^2)^(1/2)*((a*x-1)*(a*x+1))^(1/2))/(a^2)^(1/2))*x^4*a^5+218*(a^2)^(1/2)
*((a*x-1)*(a*x+1))^(3/2)*x^3*a^3+1575*(a^2)^(1/2)*((a*x-1)*(a*x+1))^(1/2)*x^4*a^4+720*ln((a^2*x+(a^2)^(1/2)*((
a*x-1)*(a*x+1))^(1/2))/(a^2)^(1/2))*x^3*a^4+342*(a^2)^(1/2)*((a*x-1)*(a*x+1))^(3/2)*x^2*a^2+1575*(a^2)^(1/2)*(
(a*x-1)*(a*x+1))^(1/2)*x^3*a^3-720*ln((a^2*x+(a^2)^(1/2)*((a*x-1)*(a*x+1))^(1/2))/(a^2)^(1/2))*x^2*a^3+3*(a^2)
^(1/2)*((a*x-1)*(a*x+1))^(3/2)*x*a-1575*(a^2)^(1/2)*((a*x-1)*(a*x+1))^(1/2)*x^2*a^2-240*ln((a^2*x+(a^2)^(1/2)*
((a*x-1)*(a*x+1))^(1/2))/(a^2)^(1/2))*x*a^2-243*((a*x-1)*(a*x+1))^(3/2)*(a^2)^(1/2)-525*(a^2)^(1/2)*((a*x-1)*(
a*x+1))^(1/2)*x*a+240*a*ln((a^2*x+(a^2)^(1/2)*((a*x-1)*(a*x+1))^(1/2))/(a^2)^(1/2))+525*(a^2)^(1/2)*((a*x-1)*(
a*x+1))^(1/2))/a/(a^2)^(1/2)/(a*x-1)^3/(a*x+1)^3/c^3/((a*x-1)*(a*x+1))^(1/2)/((a*x-1)/(a*x+1))^(1/2)

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Maxima [A]  time = 1.03109, size = 262, normalized size = 1.03 \begin{align*} \frac{1}{240} \, a{\left (\frac{\frac{37 \,{\left (a x - 1\right )}}{a x + 1} + \frac{410 \,{\left (a x - 1\right )}^{2}}{{\left (a x + 1\right )}^{2}} - \frac{930 \,{\left (a x - 1\right )}^{3}}{{\left (a x + 1\right )}^{3}} + 3}{a^{2} c^{3} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{7}{2}} - a^{2} c^{3} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{5}{2}}} + \frac{5 \,{\left (\left (\frac{a x - 1}{a x + 1}\right )^{\frac{3}{2}} + 24 \, \sqrt{\frac{a x - 1}{a x + 1}}\right )}}{a^{2} c^{3}} + \frac{240 \, \log \left (\sqrt{\frac{a x - 1}{a x + 1}} + 1\right )}{a^{2} c^{3}} - \frac{240 \, \log \left (\sqrt{\frac{a x - 1}{a x + 1}} - 1\right )}{a^{2} c^{3}}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/240*a*((37*(a*x - 1)/(a*x + 1) + 410*(a*x - 1)^2/(a*x + 1)^2 - 930*(a*x - 1)^3/(a*x + 1)^3 + 3)/(a^2*c^3*((a
*x - 1)/(a*x + 1))^(7/2) - a^2*c^3*((a*x - 1)/(a*x + 1))^(5/2)) + 5*(((a*x - 1)/(a*x + 1))^(3/2) + 24*sqrt((a*
x - 1)/(a*x + 1)))/(a^2*c^3) + 240*log(sqrt((a*x - 1)/(a*x + 1)) + 1)/(a^2*c^3) - 240*log(sqrt((a*x - 1)/(a*x
+ 1)) - 1)/(a^2*c^3))

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Fricas [A]  time = 1.45363, size = 405, normalized size = 1.59 \begin{align*} \frac{15 \,{\left (a^{4} x^{4} - 2 \, a^{3} x^{3} + 2 \, a x - 1\right )} \log \left (\sqrt{\frac{a x - 1}{a x + 1}} + 1\right ) - 15 \,{\left (a^{4} x^{4} - 2 \, a^{3} x^{3} + 2 \, a x - 1\right )} \log \left (\sqrt{\frac{a x - 1}{a x + 1}} - 1\right ) +{\left (15 \, a^{5} x^{5} - 38 \, a^{4} x^{4} - 52 \, a^{3} x^{3} + 87 \, a^{2} x^{2} + 33 \, a x - 48\right )} \sqrt{\frac{a x - 1}{a x + 1}}}{15 \,{\left (a^{5} c^{3} x^{4} - 2 \, a^{4} c^{3} x^{3} + 2 \, a^{2} c^{3} x - a c^{3}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/15*(15*(a^4*x^4 - 2*a^3*x^3 + 2*a*x - 1)*log(sqrt((a*x - 1)/(a*x + 1)) + 1) - 15*(a^4*x^4 - 2*a^3*x^3 + 2*a*
x - 1)*log(sqrt((a*x - 1)/(a*x + 1)) - 1) + (15*a^5*x^5 - 38*a^4*x^4 - 52*a^3*x^3 + 87*a^2*x^2 + 33*a*x - 48)*
sqrt((a*x - 1)/(a*x + 1)))/(a^5*c^3*x^4 - 2*a^4*c^3*x^3 + 2*a^2*c^3*x - a*c^3)

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

[Out]

Timed out

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Giac [A]  time = 1.14955, size = 313, normalized size = 1.23 \begin{align*} \frac{1}{240} \, a{\left (\frac{240 \, \log \left (\sqrt{\frac{a x - 1}{a x + 1}} + 1\right )}{a^{2} c^{3}} - \frac{240 \, \log \left ({\left | \sqrt{\frac{a x - 1}{a x + 1}} - 1 \right |}\right )}{a^{2} c^{3}} - \frac{{\left (a x + 1\right )}^{2}{\left (\frac{40 \,{\left (a x - 1\right )}}{a x + 1} + \frac{450 \,{\left (a x - 1\right )}^{2}}{{\left (a x + 1\right )}^{2}} + 3\right )}}{{\left (a x - 1\right )}^{2} a^{2} c^{3} \sqrt{\frac{a x - 1}{a x + 1}}} - \frac{480 \, \sqrt{\frac{a x - 1}{a x + 1}}}{a^{2} c^{3}{\left (\frac{a x - 1}{a x + 1} - 1\right )}} + \frac{5 \,{\left (\frac{{\left (a x - 1\right )} a^{4} c^{6} \sqrt{\frac{a x - 1}{a x + 1}}}{a x + 1} + 24 \, a^{4} c^{6} \sqrt{\frac{a x - 1}{a x + 1}}\right )}}{a^{6} c^{9}}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/240*a*(240*log(sqrt((a*x - 1)/(a*x + 1)) + 1)/(a^2*c^3) - 240*log(abs(sqrt((a*x - 1)/(a*x + 1)) - 1))/(a^2*c
^3) - (a*x + 1)^2*(40*(a*x - 1)/(a*x + 1) + 450*(a*x - 1)^2/(a*x + 1)^2 + 3)/((a*x - 1)^2*a^2*c^3*sqrt((a*x -
1)/(a*x + 1))) - 480*sqrt((a*x - 1)/(a*x + 1))/(a^2*c^3*((a*x - 1)/(a*x + 1) - 1)) + 5*((a*x - 1)*a^4*c^6*sqrt
((a*x - 1)/(a*x + 1))/(a*x + 1) + 24*a^4*c^6*sqrt((a*x - 1)/(a*x + 1)))/(a^6*c^9))