3.9.28 \(\int \frac {e^{-3 \coth ^{-1}(a x)}}{(c-\frac {c}{a^2 x^2})^3} \, dx\) [828]

Optimal. Leaf size=253 \[ -\frac {2}{a c^3 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{7/2}}+\frac {11 \sqrt {1-\frac {1}{a x}}}{7 a c^3 \left (1+\frac {1}{a x}\right )^{7/2}}+\frac {54 \sqrt {1-\frac {1}{a x}}}{35 a c^3 \left (1+\frac {1}{a x}\right )^{5/2}}+\frac {71 \sqrt {1-\frac {1}{a x}}}{35 a c^3 \left (1+\frac {1}{a x}\right )^{3/2}}+\frac {176 \sqrt {1-\frac {1}{a x}}}{35 a c^3 \sqrt {1+\frac {1}{a x}}}+\frac {x}{c^3 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{7/2}}-\frac {3 \tanh ^{-1}\left (\sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}\right )}{a c^3} \]

[Out]

-3*arctanh((1-1/a/x)^(1/2)*(1+1/a/x)^(1/2))/a/c^3-2/a/c^3/(1+1/a/x)^(7/2)/(1-1/a/x)^(1/2)+x/c^3/(1+1/a/x)^(7/2
)/(1-1/a/x)^(1/2)+11/7*(1-1/a/x)^(1/2)/a/c^3/(1+1/a/x)^(7/2)+54/35*(1-1/a/x)^(1/2)/a/c^3/(1+1/a/x)^(5/2)+71/35
*(1-1/a/x)^(1/2)/a/c^3/(1+1/a/x)^(3/2)+176/35*(1-1/a/x)^(1/2)/a/c^3/(1+1/a/x)^(1/2)

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Rubi [A]
time = 0.12, antiderivative size = 253, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 6, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {6329, 105, 157, 12, 94, 214} \begin {gather*} \frac {x}{c^3 \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{7/2}}+\frac {176 \sqrt {1-\frac {1}{a x}}}{35 a c^3 \sqrt {\frac {1}{a x}+1}}+\frac {71 \sqrt {1-\frac {1}{a x}}}{35 a c^3 \left (\frac {1}{a x}+1\right )^{3/2}}+\frac {54 \sqrt {1-\frac {1}{a x}}}{35 a c^3 \left (\frac {1}{a x}+1\right )^{5/2}}+\frac {11 \sqrt {1-\frac {1}{a x}}}{7 a c^3 \left (\frac {1}{a x}+1\right )^{7/2}}-\frac {2}{a c^3 \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{7/2}}-\frac {3 \tanh ^{-1}\left (\sqrt {1-\frac {1}{a x}} \sqrt {\frac {1}{a x}+1}\right )}{a c^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-2/(a*c^3*Sqrt[1 - 1/(a*x)]*(1 + 1/(a*x))^(7/2)) + (11*Sqrt[1 - 1/(a*x)])/(7*a*c^3*(1 + 1/(a*x))^(7/2)) + (54*
Sqrt[1 - 1/(a*x)])/(35*a*c^3*(1 + 1/(a*x))^(5/2)) + (71*Sqrt[1 - 1/(a*x)])/(35*a*c^3*(1 + 1/(a*x))^(3/2)) + (1
76*Sqrt[1 - 1/(a*x)])/(35*a*c^3*Sqrt[1 + 1/(a*x)]) + x/(c^3*Sqrt[1 - 1/(a*x)]*(1 + 1/(a*x))^(7/2)) - (3*ArcTan
h[Sqrt[1 - 1/(a*x)]*Sqrt[1 + 1/(a*x)]])/(a*c^3)

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*} \int \frac {e^{-3 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^3} \, dx &=-\frac {\text {Subst}\left (\int \frac {1}{x^2 \left (1-\frac {x}{a}\right )^{3/2} \left (1+\frac {x}{a}\right )^{9/2}} \, dx,x,\frac {1}{x}\right )}{c^3}\\ &=\frac {x}{c^3 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{7/2}}+\frac {\text {Subst}\left (\int \frac {\frac {3}{a}-\frac {5 x}{a^2}}{x \left (1-\frac {x}{a}\right )^{3/2} \left (1+\frac {x}{a}\right )^{9/2}} \, dx,x,\frac {1}{x}\right )}{c^3}\\ &=-\frac {2}{a c^3 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{7/2}}+\frac {x}{c^3 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{7/2}}-\frac {a \text {Subst}\left (\int \frac {-\frac {3}{a^2}+\frac {8 x}{a^3}}{x \sqrt {1-\frac {x}{a}} \left (1+\frac {x}{a}\right )^{9/2}} \, dx,x,\frac {1}{x}\right )}{c^3}\\ &=-\frac {2}{a c^3 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{7/2}}+\frac {11 \sqrt {1-\frac {1}{a x}}}{7 a c^3 \left (1+\frac {1}{a x}\right )^{7/2}}+\frac {x}{c^3 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{7/2}}-\frac {a^2 \text {Subst}\left (\int \frac {-\frac {21}{a^3}+\frac {33 x}{a^4}}{x \sqrt {1-\frac {x}{a}} \left (1+\frac {x}{a}\right )^{7/2}} \, dx,x,\frac {1}{x}\right )}{7 c^3}\\ &=-\frac {2}{a c^3 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{7/2}}+\frac {11 \sqrt {1-\frac {1}{a x}}}{7 a c^3 \left (1+\frac {1}{a x}\right )^{7/2}}+\frac {54 \sqrt {1-\frac {1}{a x}}}{35 a c^3 \left (1+\frac {1}{a x}\right )^{5/2}}+\frac {x}{c^3 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{7/2}}-\frac {a^3 \text {Subst}\left (\int \frac {-\frac {105}{a^4}+\frac {108 x}{a^5}}{x \sqrt {1-\frac {x}{a}} \left (1+\frac {x}{a}\right )^{5/2}} \, dx,x,\frac {1}{x}\right )}{35 c^3}\\ &=-\frac {2}{a c^3 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{7/2}}+\frac {11 \sqrt {1-\frac {1}{a x}}}{7 a c^3 \left (1+\frac {1}{a x}\right )^{7/2}}+\frac {54 \sqrt {1-\frac {1}{a x}}}{35 a c^3 \left (1+\frac {1}{a x}\right )^{5/2}}+\frac {71 \sqrt {1-\frac {1}{a x}}}{35 a c^3 \left (1+\frac {1}{a x}\right )^{3/2}}+\frac {x}{c^3 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{7/2}}-\frac {a^4 \text {Subst}\left (\int \frac {-\frac {315}{a^5}+\frac {213 x}{a^6}}{x \sqrt {1-\frac {x}{a}} \left (1+\frac {x}{a}\right )^{3/2}} \, dx,x,\frac {1}{x}\right )}{105 c^3}\\ &=-\frac {2}{a c^3 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{7/2}}+\frac {11 \sqrt {1-\frac {1}{a x}}}{7 a c^3 \left (1+\frac {1}{a x}\right )^{7/2}}+\frac {54 \sqrt {1-\frac {1}{a x}}}{35 a c^3 \left (1+\frac {1}{a x}\right )^{5/2}}+\frac {71 \sqrt {1-\frac {1}{a x}}}{35 a c^3 \left (1+\frac {1}{a x}\right )^{3/2}}+\frac {176 \sqrt {1-\frac {1}{a x}}}{35 a c^3 \sqrt {1+\frac {1}{a x}}}+\frac {x}{c^3 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{7/2}}-\frac {a^5 \text {Subst}\left (\int -\frac {315}{a^6 x \sqrt {1-\frac {x}{a}} \sqrt {1+\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )}{105 c^3}\\ &=-\frac {2}{a c^3 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{7/2}}+\frac {11 \sqrt {1-\frac {1}{a x}}}{7 a c^3 \left (1+\frac {1}{a x}\right )^{7/2}}+\frac {54 \sqrt {1-\frac {1}{a x}}}{35 a c^3 \left (1+\frac {1}{a x}\right )^{5/2}}+\frac {71 \sqrt {1-\frac {1}{a x}}}{35 a c^3 \left (1+\frac {1}{a x}\right )^{3/2}}+\frac {176 \sqrt {1-\frac {1}{a x}}}{35 a c^3 \sqrt {1+\frac {1}{a x}}}+\frac {x}{c^3 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{7/2}}+\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^3}\\ &=-\frac {2}{a c^3 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{7/2}}+\frac {11 \sqrt {1-\frac {1}{a x}}}{7 a c^3 \left (1+\frac {1}{a x}\right )^{7/2}}+\frac {54 \sqrt {1-\frac {1}{a x}}}{35 a c^3 \left (1+\frac {1}{a x}\right )^{5/2}}+\frac {71 \sqrt {1-\frac {1}{a x}}}{35 a c^3 \left (1+\frac {1}{a x}\right )^{3/2}}+\frac {176 \sqrt {1-\frac {1}{a x}}}{35 a c^3 \sqrt {1+\frac {1}{a x}}}+\frac {x}{c^3 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{7/2}}-\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^3}\\ &=-\frac {2}{a c^3 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{7/2}}+\frac {11 \sqrt {1-\frac {1}{a x}}}{7 a c^3 \left (1+\frac {1}{a x}\right )^{7/2}}+\frac {54 \sqrt {1-\frac {1}{a x}}}{35 a c^3 \left (1+\frac {1}{a x}\right )^{5/2}}+\frac {71 \sqrt {1-\frac {1}{a x}}}{35 a c^3 \left (1+\frac {1}{a x}\right )^{3/2}}+\frac {176 \sqrt {1-\frac {1}{a x}}}{35 a c^3 \sqrt {1+\frac {1}{a x}}}+\frac {x}{c^3 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{7/2}}-\frac {3 \tanh ^{-1}\left (\sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}\right )}{a c^3}\\ \end {align*}

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Mathematica [A]
time = 0.09, size = 101, normalized size = 0.40 \begin {gather*} \frac {\frac {a \sqrt {1-\frac {1}{a^2 x^2}} x \left (-176-423 a x-125 a^2 x^2+368 a^3 x^3+286 a^4 x^4+35 a^5 x^5\right )}{35 (-1+a x) (1+a x)^4}-3 \log \left (\left (1+\sqrt {1-\frac {1}{a^2 x^2}}\right ) x\right )}{a c^3} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

((a*Sqrt[1 - 1/(a^2*x^2)]*x*(-176 - 423*a*x - 125*a^2*x^2 + 368*a^3*x^3 + 286*a^4*x^4 + 35*a^5*x^5))/(35*(-1 +
a*x)*(1 + a*x)^4) - 3*Log[(1 + Sqrt[1 - 1/(a^2*x^2)])*x])/(a*c^3)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(713\) vs. \(2(217)=434\).
time = 0.16, size = 714, normalized size = 2.82

method	result	size
risch	\(\frac {\left (a x +1\right ) \sqrt {\frac {a x -1}{a x +1}}}{a \,c^{3}}+\frac {\left (-\frac {3 \ln \left (\frac {a^{2} x}{\sqrt {a^{2}}}+\sqrt {a^{2} x^{2}-1}\right )}{a^{6} \sqrt {a^{2}}}-\frac {477 \sqrt {a^{2} \left (x +\frac {1}{a}\right )^{2}-2 a \left (x +\frac {1}{a}\right )}}{280 a^{9} \left (x +\frac {1}{a}\right )^{2}}+\frac {2931 \sqrt {a^{2} \left (x +\frac {1}{a}\right )^{2}-2 a \left (x +\frac {1}{a}\right )}}{560 a^{8} \left (x +\frac {1}{a}\right )}-\frac {\sqrt {a^{2} \left (x -\frac {1}{a}\right )^{2}+2 a \left (x -\frac {1}{a}\right )}}{16 a^{8} \left (x -\frac {1}{a}\right )}-\frac {\sqrt {a^{2} \left (x +\frac {1}{a}\right )^{2}-2 a \left (x +\frac {1}{a}\right )}}{14 a^{11} \left (x +\frac {1}{a}\right )^{4}}+\frac {71 \sqrt {a^{2} \left (x +\frac {1}{a}\right )^{2}-2 a \left (x +\frac {1}{a}\right )}}{140 a^{10} \left (x +\frac {1}{a}\right )^{3}}\right ) a^{6} \sqrt {\frac {a x -1}{a x +1}}\, \sqrt {\left (a x +1\right ) \left (a x -1\right )}}{c^{3} \left (a x -1\right )}\)	\(281\)
default	\(-\frac {\left (-3675 \sqrt {\left (a x +1\right ) \left (a x -1\right )}\, \sqrt {a^{2}}\, a^{7} x^{7}+3360 \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^{8} x^{7}+2555 \left (\left (a x +1\right ) \left (a x -1\right )\right )^{\frac {3}{2}} \sqrt {a^{2}}\, a^{5} x^{5}-11025 \sqrt {\left (a x +1\right ) \left (a x -1\right )}\, \sqrt {a^{2}}\, a^{6} x^{6}+10080 \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}+1873 \left (\left (a x +1\right ) \left (a x -1\right )\right )^{\frac {3}{2}} \sqrt {a^{2}}\, a^{4} x^{4}-3675 \sqrt {\left (a x +1\right ) \left (a x -1\right )}\, \sqrt {a^{2}}\, a^{5} x^{5}+3360 \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}-4426 \left (\left (a x +1\right ) \left (a x -1\right )\right )^{\frac {3}{2}} \sqrt {a^{2}}\, a^{3} x^{3}+18375 \sqrt {a^{2}}\, \sqrt {\left (a x +1\right ) \left (a x -1\right )}\, a^{4} x^{4}-16800 \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}-3350 \sqrt {a^{2}}\, \left (\left (a x +1\right ) \left (a x -1\right )\right )^{\frac {3}{2}} a^{2} x^{2}+18375 \sqrt {a^{2}}\, \sqrt {\left (a x +1\right ) \left (a x -1\right )}\, a^{3} x^{3}-16800 \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}+2511 \sqrt {a^{2}}\, \left (\left (a x +1\right ) \left (a x -1\right )\right )^{\frac {3}{2}} a x -3675 \sqrt {\left (a x +1\right ) \left (a x -1\right )}\, \sqrt {a^{2}}\, a^{2} x^{2}+3360 \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^{3} x^{2}+1957 \left (\left (a x +1\right ) \left (a x -1\right )\right )^{\frac {3}{2}} \sqrt {a^{2}}-11025 \sqrt {\left (a x +1\right ) \left (a x -1\right )}\, \sqrt {a^{2}}\, a x +10080 \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 -3675 \sqrt {a^{2}}\, \sqrt {\left (a x +1\right ) \left (a x -1\right )}+3360 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}}}{1120 a \sqrt {a^{2}}\, \left (a x +1\right )^{3} c^{3} \sqrt {\left (a x +1\right ) \left (a x -1\right )}\, \left (a x -1\right )^{3}}\)	\(714\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/1120*(-3675*((a*x+1)*(a*x-1))^(1/2)*(a^2)^(1/2)*a^7*x^7+3360*ln((a^2*x+(a^2)^(1/2)*((a*x+1)*(a*x-1))^(1/2))
/(a^2)^(1/2))*a^8*x^7+2555*((a*x+1)*(a*x-1))^(3/2)*(a^2)^(1/2)*a^5*x^5-11025*((a*x+1)*(a*x-1))^(1/2)*(a^2)^(1/
2)*a^6*x^6+10080*ln((a^2*x+(a^2)^(1/2)*((a*x+1)*(a*x-1))^(1/2))/(a^2)^(1/2))*a^7*x^6+1873*((a*x+1)*(a*x-1))^(3
/2)*(a^2)^(1/2)*a^4*x^4-3675*((a*x+1)*(a*x-1))^(1/2)*(a^2)^(1/2)*a^5*x^5+3360*ln((a^2*x+(a^2)^(1/2)*((a*x+1)*(
a*x-1))^(1/2))/(a^2)^(1/2))*a^6*x^5-4426*((a*x+1)*(a*x-1))^(3/2)*(a^2)^(1/2)*a^3*x^3+18375*(a^2)^(1/2)*((a*x+1
)*(a*x-1))^(1/2)*a^4*x^4-16800*ln((a^2*x+(a^2)^(1/2)*((a*x+1)*(a*x-1))^(1/2))/(a^2)^(1/2))*a^5*x^4-3350*(a^2)^
(1/2)*((a*x+1)*(a*x-1))^(3/2)*a^2*x^2+18375*(a^2)^(1/2)*((a*x+1)*(a*x-1))^(1/2)*a^3*x^3-16800*ln((a^2*x+(a^2)^
(1/2)*((a*x+1)*(a*x-1))^(1/2))/(a^2)^(1/2))*a^4*x^3+2511*(a^2)^(1/2)*((a*x+1)*(a*x-1))^(3/2)*a*x-3675*((a*x+1)
*(a*x-1))^(1/2)*(a^2)^(1/2)*a^2*x^2+3360*ln((a^2*x+(a^2)^(1/2)*((a*x+1)*(a*x-1))^(1/2))/(a^2)^(1/2))*a^3*x^2+1
957*((a*x+1)*(a*x-1))^(3/2)*(a^2)^(1/2)-11025*((a*x+1)*(a*x-1))^(1/2)*(a^2)^(1/2)*a*x+10080*ln((a^2*x+(a^2)^(1
/2)*((a*x+1)*(a*x-1))^(1/2))/(a^2)^(1/2))*a^2*x-3675*(a^2)^(1/2)*((a*x+1)*(a*x-1))^(1/2)+3360*a*ln((a^2*x+(a^2
)^(1/2)*((a*x+1)*(a*x-1))^(1/2))/(a^2)^(1/2)))/a*((a*x-1)/(a*x+1))^(3/2)/(a^2)^(1/2)/(a*x+1)^3/c^3/((a*x+1)*(a
*x-1))^(1/2)/(a*x-1)^3

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Maxima [A]
time = 0.27, size = 199, normalized size = 0.79 \begin {gather*} -\frac {1}{560} \, a {\left (\frac {35 \, {\left (\frac {33 \, {\left (a x - 1\right )}}{a x + 1} - 1\right )}}{a^{2} c^{3} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}} - a^{2} c^{3} \sqrt {\frac {a x - 1}{a x + 1}}} - \frac {5 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {7}{2}} + 56 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {5}{2}} + 350 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}} + 2520 \, \sqrt {\frac {a x - 1}{a x + 1}}}{a^{2} c^{3}} + \frac {1680 \, \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right )}{a^{2} c^{3}} - \frac {1680 \, \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right )}{a^{2} c^{3}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/560*a*(35*(33*(a*x - 1)/(a*x + 1) - 1)/(a^2*c^3*((a*x - 1)/(a*x + 1))^(3/2) - a^2*c^3*sqrt((a*x - 1)/(a*x +
1))) - (5*((a*x - 1)/(a*x + 1))^(7/2) + 56*((a*x - 1)/(a*x + 1))^(5/2) + 350*((a*x - 1)/(a*x + 1))^(3/2) + 25
20*sqrt((a*x - 1)/(a*x + 1)))/(a^2*c^3) + 1680*log(sqrt((a*x - 1)/(a*x + 1)) + 1)/(a^2*c^3) - 1680*log(sqrt((a
*x - 1)/(a*x + 1)) - 1)/(a^2*c^3))

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Fricas [A]
time = 0.43, size = 179, normalized size = 0.71 \begin {gather*} -\frac {105 \, {\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 ) - 105 \, {\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 (35 \, a^{5} x^{5} + 286 \, a^{4} x^{4} + 368 \, a^{3} x^{3} - 125 \, a^{2} x^{2} - 423 \, a x - 176\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{35 \, {\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 {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/35*(105*(a^4*x^4 + 2*a^3*x^3 - 2*a*x - 1)*log(sqrt((a*x - 1)/(a*x + 1)) + 1) - 105*(a^4*x^4 + 2*a^3*x^3 - 2
*a*x - 1)*log(sqrt((a*x - 1)/(a*x + 1)) - 1) - (35*a^5*x^5 + 286*a^4*x^4 + 368*a^3*x^3 - 125*a^2*x^2 - 423*a*x
- 176)*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]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \frac {a^{6} \left (\int \left (- \frac {x^{6} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a^{7} x^{7} + a^{6} x^{6} - 3 a^{5} x^{5} - 3 a^{4} x^{4} + 3 a^{3} x^{3} + 3 a^{2} x^{2} - a x - 1}\right )\, dx + \int \frac {a x^{7} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a^{7} x^{7} + a^{6} x^{6} - 3 a^{5} x^{5} - 3 a^{4} x^{4} + 3 a^{3} x^{3} + 3 a^{2} x^{2} - a x - 1}\, dx\right )}{c^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a**6*(Integral(-x**6*sqrt(a*x/(a*x + 1) - 1/(a*x + 1))/(a**7*x**7 + a**6*x**6 - 3*a**5*x**5 - 3*a**4*x**4 + 3*
a**3*x**3 + 3*a**2*x**2 - a*x - 1), x) + Integral(a*x**7*sqrt(a*x/(a*x + 1) - 1/(a*x + 1))/(a**7*x**7 + a**6*x
**6 - 3*a**5*x**5 - 3*a**4*x**4 + 3*a**3*x**3 + 3*a**2*x**2 - a*x - 1), x))/c**3

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [B]
time = 0.05, size = 183, normalized size = 0.72 \begin {gather*} \frac {\frac {33\,\left (a\,x-1\right )}{a\,x+1}-1}{16\,a\,c^3\,\sqrt {\frac {a\,x-1}{a\,x+1}}-16\,a\,c^3\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/2}}+\frac {9\,\sqrt {\frac {a\,x-1}{a\,x+1}}}{2\,a\,c^3}+\frac {5\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/2}}{8\,a\,c^3}+\frac {{\left (\frac {a\,x-1}{a\,x+1}\right )}^{5/2}}{10\,a\,c^3}+\frac {{\left (\frac {a\,x-1}{a\,x+1}\right )}^{7/2}}{112\,a\,c^3}+\frac {\mathrm {atan}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\,1{}\mathrm {i}\right )\,6{}\mathrm {i}}{a\,c^3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((33*(a*x - 1))/(a*x + 1) - 1)/(16*a*c^3*((a*x - 1)/(a*x + 1))^(1/2) - 16*a*c^3*((a*x - 1)/(a*x + 1))^(3/2)) +
(9*((a*x - 1)/(a*x + 1))^(1/2))/(2*a*c^3) + (5*((a*x - 1)/(a*x + 1))^(3/2))/(8*a*c^3) + ((a*x - 1)/(a*x + 1))
^(5/2)/(10*a*c^3) + ((a*x - 1)/(a*x + 1))^(7/2)/(112*a*c^3) + (atan(((a*x - 1)/(a*x + 1))^(1/2)*1i)*6i)/(a*c^3
)

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