3.813 \(\int \frac{e^{-\coth ^{-1}(a x)}}{(c-\frac{c}{a^2 x^2})^4} \, dx\)

Optimal. Leaf size=329 \[ \frac{x}{c^4 \left (1-\frac{1}{a x}\right )^{5/2} \left (\frac{1}{a x}+1\right )^{7/2}}+\frac{128 \sqrt{1-\frac{1}{a x}}}{35 a c^4 \sqrt{\frac{1}{a x}+1}}+\frac{93 \sqrt{1-\frac{1}{a x}}}{35 a c^4 \left (\frac{1}{a x}+1\right )^{3/2}}+\frac{122 \sqrt{1-\frac{1}{a x}}}{35 a c^4 \left (\frac{1}{a x}+1\right )^{5/2}}+\frac{115 \sqrt{1-\frac{1}{a x}}}{21 a c^4 \left (\frac{1}{a x}+1\right )^{7/2}}-\frac{28}{3 a c^4 \sqrt{1-\frac{1}{a x}} \left (\frac{1}{a x}+1\right )^{7/2}}-\frac{31}{15 a c^4 \left (1-\frac{1}{a x}\right )^{3/2} \left (\frac{1}{a x}+1\right )^{7/2}}-\frac{6}{5 a c^4 \left (1-\frac{1}{a x}\right )^{5/2} \left (\frac{1}{a x}+1\right )^{7/2}}-\frac{\tanh ^{-1}\left (\sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}\right )}{a c^4} \]

[Out]

-6/(5*a*c^4*(1 - 1/(a*x))^(5/2)*(1 + 1/(a*x))^(7/2)) - 31/(15*a*c^4*(1 - 1/(a*x))^(3/2)*(1 + 1/(a*x))^(7/2)) -
28/(3*a*c^4*Sqrt[1 - 1/(a*x)]*(1 + 1/(a*x))^(7/2)) + (115*Sqrt[1 - 1/(a*x)])/(21*a*c^4*(1 + 1/(a*x))^(7/2)) +
(122*Sqrt[1 - 1/(a*x)])/(35*a*c^4*(1 + 1/(a*x))^(5/2)) + (93*Sqrt[1 - 1/(a*x)])/(35*a*c^4*(1 + 1/(a*x))^(3/2)
) + (128*Sqrt[1 - 1/(a*x)])/(35*a*c^4*Sqrt[1 + 1/(a*x)]) + x/(c^4*(1 - 1/(a*x))^(5/2)*(1 + 1/(a*x))^(7/2)) - A
rcTanh[Sqrt[1 - 1/(a*x)]*Sqrt[1 + 1/(a*x)]]/(a*c^4)

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Rubi [A]  time = 0.225071, 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} \[ \frac{x}{c^4 \left (1-\frac{1}{a x}\right )^{5/2} \left (\frac{1}{a x}+1\right )^{7/2}}+\frac{128 \sqrt{1-\frac{1}{a x}}}{35 a c^4 \sqrt{\frac{1}{a x}+1}}+\frac{93 \sqrt{1-\frac{1}{a x}}}{35 a c^4 \left (\frac{1}{a x}+1\right )^{3/2}}+\frac{122 \sqrt{1-\frac{1}{a x}}}{35 a c^4 \left (\frac{1}{a x}+1\right )^{5/2}}+\frac{115 \sqrt{1-\frac{1}{a x}}}{21 a c^4 \left (\frac{1}{a x}+1\right )^{7/2}}-\frac{28}{3 a c^4 \sqrt{1-\frac{1}{a x}} \left (\frac{1}{a x}+1\right )^{7/2}}-\frac{31}{15 a c^4 \left (1-\frac{1}{a x}\right )^{3/2} \left (\frac{1}{a x}+1\right )^{7/2}}-\frac{6}{5 a c^4 \left (1-\frac{1}{a x}\right )^{5/2} \left (\frac{1}{a x}+1\right )^{7/2}}-\frac{\tanh ^{-1}\left (\sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}\right )}{a c^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-6/(5*a*c^4*(1 - 1/(a*x))^(5/2)*(1 + 1/(a*x))^(7/2)) - 31/(15*a*c^4*(1 - 1/(a*x))^(3/2)*(1 + 1/(a*x))^(7/2)) -
28/(3*a*c^4*Sqrt[1 - 1/(a*x)]*(1 + 1/(a*x))^(7/2)) + (115*Sqrt[1 - 1/(a*x)])/(21*a*c^4*(1 + 1/(a*x))^(7/2)) +
(122*Sqrt[1 - 1/(a*x)])/(35*a*c^4*(1 + 1/(a*x))^(5/2)) + (93*Sqrt[1 - 1/(a*x)])/(35*a*c^4*(1 + 1/(a*x))^(3/2)
) + (128*Sqrt[1 - 1/(a*x)])/(35*a*c^4*Sqrt[1 + 1/(a*x)]) + x/(c^4*(1 - 1/(a*x))^(5/2)*(1 + 1/(a*x))^(7/2)) - A
rcTanh[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^{-\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 )^{7/2} \left (1+\frac{x}{a}\right )^{9/2}} \, dx,x,\frac{1}{x}\right )}{c^4}\\ &=\frac{x}{c^4 \left (1-\frac{1}{a x}\right )^{5/2} \left (1+\frac{1}{a x}\right )^{7/2}}+\frac{\operatorname{Subst}\left (\int \frac{\frac{1}{a}-\frac{7 x}{a^2}}{x \left (1-\frac{x}{a}\right )^{7/2} \left (1+\frac{x}{a}\right )^{9/2}} \, dx,x,\frac{1}{x}\right )}{c^4}\\ &=-\frac{6}{5 a c^4 \left (1-\frac{1}{a x}\right )^{5/2} \left (1+\frac{1}{a x}\right )^{7/2}}+\frac{x}{c^4 \left (1-\frac{1}{a x}\right )^{5/2} \left (1+\frac{1}{a x}\right )^{7/2}}-\frac{a \operatorname{Subst}\left (\int \frac{-\frac{5}{a^2}+\frac{36 x}{a^3}}{x \left (1-\frac{x}{a}\right )^{5/2} \left (1+\frac{x}{a}\right )^{9/2}} \, dx,x,\frac{1}{x}\right )}{5 c^4}\\ &=-\frac{6}{5 a c^4 \left (1-\frac{1}{a x}\right )^{5/2} \left (1+\frac{1}{a x}\right )^{7/2}}-\frac{31}{15 a c^4 \left (1-\frac{1}{a x}\right )^{3/2} \left (1+\frac{1}{a x}\right )^{7/2}}+\frac{x}{c^4 \left (1-\frac{1}{a x}\right )^{5/2} \left (1+\frac{1}{a x}\right )^{7/2}}+\frac{a^2 \operatorname{Subst}\left (\int \frac{\frac{15}{a^3}-\frac{155 x}{a^4}}{x \left (1-\frac{x}{a}\right )^{3/2} \left (1+\frac{x}{a}\right )^{9/2}} \, dx,x,\frac{1}{x}\right )}{15 c^4}\\ &=-\frac{6}{5 a c^4 \left (1-\frac{1}{a x}\right )^{5/2} \left (1+\frac{1}{a x}\right )^{7/2}}-\frac{31}{15 a c^4 \left (1-\frac{1}{a x}\right )^{3/2} \left (1+\frac{1}{a x}\right )^{7/2}}-\frac{28}{3 a c^4 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{7/2}}+\frac{x}{c^4 \left (1-\frac{1}{a x}\right )^{5/2} \left (1+\frac{1}{a x}\right )^{7/2}}-\frac{a^3 \operatorname{Subst}\left (\int \frac{-\frac{15}{a^4}+\frac{560 x}{a^5}}{x \sqrt{1-\frac{x}{a}} \left (1+\frac{x}{a}\right )^{9/2}} \, dx,x,\frac{1}{x}\right )}{15 c^4}\\ &=-\frac{6}{5 a c^4 \left (1-\frac{1}{a x}\right )^{5/2} \left (1+\frac{1}{a x}\right )^{7/2}}-\frac{31}{15 a c^4 \left (1-\frac{1}{a x}\right )^{3/2} \left (1+\frac{1}{a x}\right )^{7/2}}-\frac{28}{3 a c^4 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{7/2}}+\frac{115 \sqrt{1-\frac{1}{a x}}}{21 a c^4 \left (1+\frac{1}{a x}\right )^{7/2}}+\frac{x}{c^4 \left (1-\frac{1}{a x}\right )^{5/2} \left (1+\frac{1}{a x}\right )^{7/2}}-\frac{a^4 \operatorname{Subst}\left (\int \frac{-\frac{105}{a^5}+\frac{1725 x}{a^6}}{x \sqrt{1-\frac{x}{a}} \left (1+\frac{x}{a}\right )^{7/2}} \, dx,x,\frac{1}{x}\right )}{105 c^4}\\ &=-\frac{6}{5 a c^4 \left (1-\frac{1}{a x}\right )^{5/2} \left (1+\frac{1}{a x}\right )^{7/2}}-\frac{31}{15 a c^4 \left (1-\frac{1}{a x}\right )^{3/2} \left (1+\frac{1}{a x}\right )^{7/2}}-\frac{28}{3 a c^4 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{7/2}}+\frac{115 \sqrt{1-\frac{1}{a x}}}{21 a c^4 \left (1+\frac{1}{a x}\right )^{7/2}}+\frac{122 \sqrt{1-\frac{1}{a x}}}{35 a c^4 \left (1+\frac{1}{a x}\right )^{5/2}}+\frac{x}{c^4 \left (1-\frac{1}{a x}\right )^{5/2} \left (1+\frac{1}{a x}\right )^{7/2}}-\frac{a^5 \operatorname{Subst}\left (\int \frac{-\frac{525}{a^6}+\frac{3660 x}{a^7}}{x \sqrt{1-\frac{x}{a}} \left (1+\frac{x}{a}\right )^{5/2}} \, dx,x,\frac{1}{x}\right )}{525 c^4}\\ &=-\frac{6}{5 a c^4 \left (1-\frac{1}{a x}\right )^{5/2} \left (1+\frac{1}{a x}\right )^{7/2}}-\frac{31}{15 a c^4 \left (1-\frac{1}{a x}\right )^{3/2} \left (1+\frac{1}{a x}\right )^{7/2}}-\frac{28}{3 a c^4 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{7/2}}+\frac{115 \sqrt{1-\frac{1}{a x}}}{21 a c^4 \left (1+\frac{1}{a x}\right )^{7/2}}+\frac{122 \sqrt{1-\frac{1}{a x}}}{35 a c^4 \left (1+\frac{1}{a x}\right )^{5/2}}+\frac{93 \sqrt{1-\frac{1}{a x}}}{35 a c^4 \left (1+\frac{1}{a x}\right )^{3/2}}+\frac{x}{c^4 \left (1-\frac{1}{a x}\right )^{5/2} \left (1+\frac{1}{a x}\right )^{7/2}}-\frac{a^6 \operatorname{Subst}\left (\int \frac{-\frac{1575}{a^7}+\frac{4185 x}{a^8}}{x \sqrt{1-\frac{x}{a}} \left (1+\frac{x}{a}\right )^{3/2}} \, dx,x,\frac{1}{x}\right )}{1575 c^4}\\ &=-\frac{6}{5 a c^4 \left (1-\frac{1}{a x}\right )^{5/2} \left (1+\frac{1}{a x}\right )^{7/2}}-\frac{31}{15 a c^4 \left (1-\frac{1}{a x}\right )^{3/2} \left (1+\frac{1}{a x}\right )^{7/2}}-\frac{28}{3 a c^4 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{7/2}}+\frac{115 \sqrt{1-\frac{1}{a x}}}{21 a c^4 \left (1+\frac{1}{a x}\right )^{7/2}}+\frac{122 \sqrt{1-\frac{1}{a x}}}{35 a c^4 \left (1+\frac{1}{a x}\right )^{5/2}}+\frac{93 \sqrt{1-\frac{1}{a x}}}{35 a c^4 \left (1+\frac{1}{a x}\right )^{3/2}}+\frac{128 \sqrt{1-\frac{1}{a x}}}{35 a c^4 \sqrt{1+\frac{1}{a x}}}+\frac{x}{c^4 \left (1-\frac{1}{a x}\right )^{5/2} \left (1+\frac{1}{a x}\right )^{7/2}}-\frac{a^7 \operatorname{Subst}\left (\int -\frac{1575}{a^8 x \sqrt{1-\frac{x}{a}} \sqrt{1+\frac{x}{a}}} \, dx,x,\frac{1}{x}\right )}{1575 c^4}\\ &=-\frac{6}{5 a c^4 \left (1-\frac{1}{a x}\right )^{5/2} \left (1+\frac{1}{a x}\right )^{7/2}}-\frac{31}{15 a c^4 \left (1-\frac{1}{a x}\right )^{3/2} \left (1+\frac{1}{a x}\right )^{7/2}}-\frac{28}{3 a c^4 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{7/2}}+\frac{115 \sqrt{1-\frac{1}{a x}}}{21 a c^4 \left (1+\frac{1}{a x}\right )^{7/2}}+\frac{122 \sqrt{1-\frac{1}{a x}}}{35 a c^4 \left (1+\frac{1}{a x}\right )^{5/2}}+\frac{93 \sqrt{1-\frac{1}{a x}}}{35 a c^4 \left (1+\frac{1}{a x}\right )^{3/2}}+\frac{128 \sqrt{1-\frac{1}{a x}}}{35 a c^4 \sqrt{1+\frac{1}{a x}}}+\frac{x}{c^4 \left (1-\frac{1}{a x}\right )^{5/2} \left (1+\frac{1}{a x}\right )^{7/2}}+\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^4}\\ &=-\frac{6}{5 a c^4 \left (1-\frac{1}{a x}\right )^{5/2} \left (1+\frac{1}{a x}\right )^{7/2}}-\frac{31}{15 a c^4 \left (1-\frac{1}{a x}\right )^{3/2} \left (1+\frac{1}{a x}\right )^{7/2}}-\frac{28}{3 a c^4 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{7/2}}+\frac{115 \sqrt{1-\frac{1}{a x}}}{21 a c^4 \left (1+\frac{1}{a x}\right )^{7/2}}+\frac{122 \sqrt{1-\frac{1}{a x}}}{35 a c^4 \left (1+\frac{1}{a x}\right )^{5/2}}+\frac{93 \sqrt{1-\frac{1}{a x}}}{35 a c^4 \left (1+\frac{1}{a x}\right )^{3/2}}+\frac{128 \sqrt{1-\frac{1}{a x}}}{35 a c^4 \sqrt{1+\frac{1}{a x}}}+\frac{x}{c^4 \left (1-\frac{1}{a x}\right )^{5/2} \left (1+\frac{1}{a x}\right )^{7/2}}-\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^4}\\ &=-\frac{6}{5 a c^4 \left (1-\frac{1}{a x}\right )^{5/2} \left (1+\frac{1}{a x}\right )^{7/2}}-\frac{31}{15 a c^4 \left (1-\frac{1}{a x}\right )^{3/2} \left (1+\frac{1}{a x}\right )^{7/2}}-\frac{28}{3 a c^4 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{7/2}}+\frac{115 \sqrt{1-\frac{1}{a x}}}{21 a c^4 \left (1+\frac{1}{a x}\right )^{7/2}}+\frac{122 \sqrt{1-\frac{1}{a x}}}{35 a c^4 \left (1+\frac{1}{a x}\right )^{5/2}}+\frac{93 \sqrt{1-\frac{1}{a x}}}{35 a c^4 \left (1+\frac{1}{a x}\right )^{3/2}}+\frac{128 \sqrt{1-\frac{1}{a x}}}{35 a c^4 \sqrt{1+\frac{1}{a x}}}+\frac{x}{c^4 \left (1-\frac{1}{a x}\right )^{5/2} \left (1+\frac{1}{a x}\right )^{7/2}}-\frac{\tanh ^{-1}\left (\sqrt{1-\frac{1}{a x}} \sqrt{1+\frac{1}{a x}}\right )}{a c^4}\\ \end{align*}

Mathematica [A]  time = 0.255458, size = 117, normalized size = 0.36 \[ \frac{\frac{a x \sqrt{1-\frac{1}{a^2 x^2}} \left (105 a^7 x^7+281 a^6 x^6-559 a^5 x^5-965 a^4 x^4+715 a^3 x^3+1065 a^2 x^2-279 a x-384\right )}{105 (a x-1)^3 (a x+1)^4}-\log \left (x \left (\sqrt{1-\frac{1}{a^2 x^2}}+1\right )\right )}{a c^4} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

((a*Sqrt[1 - 1/(a^2*x^2)]*x*(-384 - 279*a*x + 1065*a^2*x^2 + 715*a^3*x^3 - 965*a^4*x^4 - 559*a^5*x^5 + 281*a^6
*x^6 + 105*a^7*x^7))/(105*(-1 + a*x)^3*(1 + a*x)^4) - Log[(1 + Sqrt[1 - 1/(a^2*x^2)])*x])/(a*c^4)

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Maple [B]  time = 0.175, size = 898, normalized size = 2.7 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/13440*(-33075*(a^2)^(1/2)*((a*x-1)*(a*x+1))^(1/2)*x^9*a^9-53760*ln((a^2*x+(a^2)^(1/2)*((a*x-1)*(a*x+1))^(1/
2))/(a^2)^(1/2))*x^3*a^4-53760*ln((a^2*x+(a^2)^(1/2)*((a*x-1)*(a*x+1))^(1/2))/(a^2)^(1/2))*x^7*a^8+132300*(a^2
)^(1/2)*((a*x-1)*(a*x+1))^(1/2)*x^7*a^7-27673*(a^2)^(1/2)*((a*x-1)*(a*x+1))^(3/2)*x^5*a^5-198450*(a^2)^(1/2)*(
(a*x-1)*(a*x+1))^(1/2)*x^5*a^5+7705*(a^2)^(1/2)*((a*x-1)*(a*x+1))^(3/2)*x^3*a^3+132300*(a^2)^(1/2)*((a*x-1)*(a
*x+1))^(1/2)*x^6*a^6+24295*(a^2)^(1/2)*((a*x-1)*(a*x+1))^(3/2)*x^4*a^4+13440*ln((a^2*x+(a^2)^(1/2)*((a*x-1)*(a
*x+1))^(1/2))/(a^2)^(1/2))*x*a^2-37095*(a^2)^(1/2)*((a*x-1)*(a*x+1))^(3/2)*x^2*a^2+2637*(a^2)^(1/2)*((a*x-1)*(
a*x+1))^(3/2)*x*a-53760*ln((a^2*x+(a^2)^(1/2)*((a*x-1)*(a*x+1))^(1/2))/(a^2)^(1/2))*x^2*a^3+80640*ln((a^2*x+(a
^2)^(1/2)*((a*x-1)*(a*x+1))^(1/2))/(a^2)^(1/2))*x^5*a^6+80640*ln((a^2*x+(a^2)^(1/2)*((a*x-1)*(a*x+1))^(1/2))/(
a^2)^(1/2))*x^4*a^5-33075*(a^2)^(1/2)*((a*x-1)*(a*x+1))^(1/2)*x*a+132300*(a^2)^(1/2)*((a*x-1)*(a*x+1))^(1/2)*x
^3*a^3-198450*(a^2)^(1/2)*((a*x-1)*(a*x+1))^(1/2)*x^4*a^4+13440*ln((a^2*x+(a^2)^(1/2)*((a*x-1)*(a*x+1))^(1/2))
/(a^2)^(1/2))*x^8*a^9-53760*ln((a^2*x+(a^2)^(1/2)*((a*x-1)*(a*x+1))^(1/2))/(a^2)^(1/2))*x^6*a^7+132300*(a^2)^(
1/2)*((a*x-1)*(a*x+1))^(1/2)*x^2*a^2+16077*((a*x-1)*(a*x+1))^(3/2)*(a^2)^(1/2)-33075*(a^2)^(1/2)*((a*x-1)*(a*x
+1))^(1/2)+13440*a*ln((a^2*x+(a^2)^(1/2)*((a*x-1)*(a*x+1))^(1/2))/(a^2)^(1/2))+19635*(a^2)^(1/2)*((a*x-1)*(a*x
+1))^(3/2)*x^7*a^7-33075*(a^2)^(1/2)*((a*x-1)*(a*x+1))^(1/2)*x^8*a^8-2893*(a^2)^(1/2)*((a*x-1)*(a*x+1))^(3/2)*
x^6*a^6+13440*ln((a^2*x+(a^2)^(1/2)*((a*x-1)*(a*x+1))^(1/2))/(a^2)^(1/2))*x^9*a^10)/a*((a*x-1)/(a*x+1))^(1/2)/
(a^2)^(1/2)/(a*x-1)^4/(a*x+1)^4/c^4/((a*x-1)*(a*x+1))^(1/2)

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Maxima [A]  time = 1.0368, size = 312, normalized size = 0.95 \begin{align*} \frac{1}{6720} \, a{\left (\frac{7 \,{\left (\frac{47 \,{\left (a x - 1\right )}}{a x + 1} + \frac{655 \,{\left (a x - 1\right )}^{2}}{{\left (a x + 1\right )}^{2}} - \frac{2625 \,{\left (a x - 1\right )}^{3}}{{\left (a x + 1\right )}^{3}} + 3\right )}}{a^{2} c^{4} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{7}{2}} - a^{2} c^{4} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{5}{2}}} + \frac{5 \,{\left (3 \, \left (\frac{a x - 1}{a x + 1}\right )^{\frac{7}{2}} + 42 \, \left (\frac{a x - 1}{a x + 1}\right )^{\frac{5}{2}} + 329 \, \left (\frac{a x - 1}{a x + 1}\right )^{\frac{3}{2}} + 2940 \, \sqrt{\frac{a x - 1}{a x + 1}}\right )}}{a^{2} c^{4}} - \frac{6720 \, \log \left (\sqrt{\frac{a x - 1}{a x + 1}} + 1\right )}{a^{2} c^{4}} + \frac{6720 \, \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(((a*x-1)/(a*x+1))^(1/2)/(c-c/a^2/x^2)^4,x, algorithm="maxima")

[Out]

1/6720*a*(7*(47*(a*x - 1)/(a*x + 1) + 655*(a*x - 1)^2/(a*x + 1)^2 - 2625*(a*x - 1)^3/(a*x + 1)^3 + 3)/(a^2*c^4
*((a*x - 1)/(a*x + 1))^(7/2) - a^2*c^4*((a*x - 1)/(a*x + 1))^(5/2)) + 5*(3*((a*x - 1)/(a*x + 1))^(7/2) + 42*((
a*x - 1)/(a*x + 1))^(5/2) + 329*((a*x - 1)/(a*x + 1))^(3/2) + 2940*sqrt((a*x - 1)/(a*x + 1)))/(a^2*c^4) - 6720
*log(sqrt((a*x - 1)/(a*x + 1)) + 1)/(a^2*c^4) + 6720*log(sqrt((a*x - 1)/(a*x + 1)) - 1)/(a^2*c^4))

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Fricas [A]  time = 1.3448, size = 471, normalized size = 1.43 \begin{align*} -\frac{105 \,{\left (a^{6} x^{6} - 3 \, a^{4} x^{4} + 3 \, a^{2} x^{2} - 1\right )} \log \left (\sqrt{\frac{a x - 1}{a x + 1}} + 1\right ) - 105 \,{\left (a^{6} x^{6} - 3 \, a^{4} x^{4} + 3 \, a^{2} x^{2} - 1\right )} \log \left (\sqrt{\frac{a x - 1}{a x + 1}} - 1\right ) -{\left (105 \, a^{7} x^{7} + 281 \, a^{6} x^{6} - 559 \, a^{5} x^{5} - 965 \, a^{4} x^{4} + 715 \, a^{3} x^{3} + 1065 \, a^{2} x^{2} - 279 \, a x - 384\right )} \sqrt{\frac{a x - 1}{a x + 1}}}{105 \,{\left (a^{7} c^{4} x^{6} - 3 \, a^{5} c^{4} x^{4} + 3 \, a^{3} c^{4} x^{2} - a c^{4}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/105*(105*(a^6*x^6 - 3*a^4*x^4 + 3*a^2*x^2 - 1)*log(sqrt((a*x - 1)/(a*x + 1)) + 1) - 105*(a^6*x^6 - 3*a^4*x^
4 + 3*a^2*x^2 - 1)*log(sqrt((a*x - 1)/(a*x + 1)) - 1) - (105*a^7*x^7 + 281*a^6*x^6 - 559*a^5*x^5 - 965*a^4*x^4
+ 715*a^3*x^3 + 1065*a^2*x^2 - 279*a*x - 384)*sqrt((a*x - 1)/(a*x + 1)))/(a^7*c^4*x^6 - 3*a^5*c^4*x^4 + 3*a^3
*c^4*x^2 - 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(((a*x-1)/(a*x+1))**(1/2)/(c-c/a**2/x**2)**4,x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{\frac{a x - 1}{a x + 1}}}{{\left (c - \frac{c}{a^{2} x^{2}}\right )}^{4}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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