3.9.12 \(\int \frac {e^{-\coth ^{-1}(a x)}}{(c-\frac {c}{a^2 x^2})^3} \, dx\) [812]

Optimal. Leaf size=255 \[ -\frac {4}{3 a c^3 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{5/2}}-\frac {13}{3 a c^3 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{5/2}}+\frac {14 \sqrt {1-\frac {1}{a x}}}{5 a c^3 \left (1+\frac {1}{a x}\right )^{5/2}}+\frac {11 \sqrt {1-\frac {1}{a x}}}{5 a c^3 \left (1+\frac {1}{a x}\right )^{3/2}}+\frac {16 \sqrt {1-\frac {1}{a x}}}{5 a c^3 \sqrt {1+\frac {1}{a x}}}+\frac {x}{c^3 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{5/2}}-\frac {\tanh ^{-1}\left (\sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}\right )}{a c^3} \]

[Out]

-4/3/a/c^3/(1-1/a/x)^(3/2)/(1+1/a/x)^(5/2)+x/c^3/(1-1/a/x)^(3/2)/(1+1/a/x)^(5/2)-arctanh((1-1/a/x)^(1/2)*(1+1/
a/x)^(1/2))/a/c^3-13/3/a/c^3/(1+1/a/x)^(5/2)/(1-1/a/x)^(1/2)+14/5*(1-1/a/x)^(1/2)/a/c^3/(1+1/a/x)^(5/2)+11/5*(
1-1/a/x)^(1/2)/a/c^3/(1+1/a/x)^(3/2)+16/5*(1-1/a/x)^(1/2)/a/c^3/(1+1/a/x)^(1/2)

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Rubi [A]
time = 0.11, antiderivative size = 255, 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 \left (1-\frac {1}{a x}\right )^{3/2} \left (\frac {1}{a x}+1\right )^{5/2}}+\frac {16 \sqrt {1-\frac {1}{a x}}}{5 a c^3 \sqrt {\frac {1}{a x}+1}}+\frac {11 \sqrt {1-\frac {1}{a x}}}{5 a c^3 \left (\frac {1}{a x}+1\right )^{3/2}}+\frac {14 \sqrt {1-\frac {1}{a x}}}{5 a c^3 \left (\frac {1}{a x}+1\right )^{5/2}}-\frac {13}{3 a c^3 \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{5/2}}-\frac {4}{3 a c^3 \left (1-\frac {1}{a x}\right )^{3/2} \left (\frac {1}{a x}+1\right )^{5/2}}-\frac {\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^ArcCoth[a*x]*(c - c/(a^2*x^2))^3),x]

[Out]

-4/(3*a*c^3*(1 - 1/(a*x))^(3/2)*(1 + 1/(a*x))^(5/2)) - 13/(3*a*c^3*Sqrt[1 - 1/(a*x)]*(1 + 1/(a*x))^(5/2)) + (1
4*Sqrt[1 - 1/(a*x)])/(5*a*c^3*(1 + 1/(a*x))^(5/2)) + (11*Sqrt[1 - 1/(a*x)])/(5*a*c^3*(1 + 1/(a*x))^(3/2)) + (1
6*Sqrt[1 - 1/(a*x)])/(5*a*c^3*Sqrt[1 + 1/(a*x)]) + x/(c^3*(1 - 1/(a*x))^(3/2)*(1 + 1/(a*x))^(5/2)) - ArcTanh[S
qrt[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^{-\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 )^{5/2} \left (1+\frac {x}{a}\right )^{7/2}} \, dx,x,\frac {1}{x}\right )}{c^3}\\ &=\frac {x}{c^3 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{5/2}}+\frac {\text {Subst}\left (\int \frac {\frac {1}{a}-\frac {5 x}{a^2}}{x \left (1-\frac {x}{a}\right )^{5/2} \left (1+\frac {x}{a}\right )^{7/2}} \, dx,x,\frac {1}{x}\right )}{c^3}\\ &=-\frac {4}{3 a c^3 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{5/2}}+\frac {x}{c^3 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{5/2}}-\frac {a \text {Subst}\left (\int \frac {-\frac {3}{a^2}+\frac {16 x}{a^3}}{x \left (1-\frac {x}{a}\right )^{3/2} \left (1+\frac {x}{a}\right )^{7/2}} \, dx,x,\frac {1}{x}\right )}{3 c^3}\\ &=-\frac {4}{3 a c^3 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{5/2}}-\frac {13}{3 a c^3 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{5/2}}+\frac {x}{c^3 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{5/2}}+\frac {a^2 \text {Subst}\left (\int \frac {\frac {3}{a^3}-\frac {39 x}{a^4}}{x \sqrt {1-\frac {x}{a}} \left (1+\frac {x}{a}\right )^{7/2}} \, dx,x,\frac {1}{x}\right )}{3 c^3}\\ &=-\frac {4}{3 a c^3 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{5/2}}-\frac {13}{3 a c^3 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{5/2}}+\frac {14 \sqrt {1-\frac {1}{a x}}}{5 a c^3 \left (1+\frac {1}{a x}\right )^{5/2}}+\frac {x}{c^3 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{5/2}}+\frac {a^3 \text {Subst}\left (\int \frac {\frac {15}{a^4}-\frac {84 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 {4}{3 a c^3 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{5/2}}-\frac {13}{3 a c^3 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{5/2}}+\frac {14 \sqrt {1-\frac {1}{a x}}}{5 a c^3 \left (1+\frac {1}{a x}\right )^{5/2}}+\frac {11 \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 )^{3/2} \left (1+\frac {1}{a x}\right )^{5/2}}+\frac {a^4 \text {Subst}\left (\int \frac {\frac {45}{a^5}-\frac {99 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 {4}{3 a c^3 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{5/2}}-\frac {13}{3 a c^3 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{5/2}}+\frac {14 \sqrt {1-\frac {1}{a x}}}{5 a c^3 \left (1+\frac {1}{a x}\right )^{5/2}}+\frac {11 \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 )^{3/2} \left (1+\frac {1}{a x}\right )^{5/2}}+\frac {a^5 \text {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 {4}{3 a c^3 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{5/2}}-\frac {13}{3 a c^3 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{5/2}}+\frac {14 \sqrt {1-\frac {1}{a x}}}{5 a c^3 \left (1+\frac {1}{a x}\right )^{5/2}}+\frac {11 \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 )^{3/2} \left (1+\frac {1}{a x}\right )^{5/2}}+\frac {\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 {4}{3 a c^3 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{5/2}}-\frac {13}{3 a c^3 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{5/2}}+\frac {14 \sqrt {1-\frac {1}{a x}}}{5 a c^3 \left (1+\frac {1}{a x}\right )^{5/2}}+\frac {11 \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 )^{3/2} \left (1+\frac {1}{a x}\right )^{5/2}}-\frac {\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 {4}{3 a c^3 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{5/2}}-\frac {13}{3 a c^3 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{5/2}}+\frac {14 \sqrt {1-\frac {1}{a x}}}{5 a c^3 \left (1+\frac {1}{a x}\right )^{5/2}}+\frac {11 \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 )^{3/2} \left (1+\frac {1}{a x}\right )^{5/2}}-\frac {\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 (48+33 a x-87 a^2 x^2-52 a^3 x^3+38 a^4 x^4+15 a^5 x^5\right )}{15 (-1+a x)^2 (1+a x)^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^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)^
2*(1 + a*x)^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 = 0.19, size = 714, normalized size = 2.80

method	result	size
risch	\(\frac {\left (a x +1\right ) \sqrt {\frac {a x -1}{a x +1}}}{a \,c^{3}}+\frac {\left (-\frac {\ln \left (\frac {a^{2} x}{\sqrt {a^{2}}}+\sqrt {a^{2} x^{2}-1}\right )}{a^{6} \sqrt {a^{2}}}-\frac {\sqrt {a^{2} \left (x -\frac {1}{a}\right )^{2}+2 a \left (x -\frac {1}{a}\right )}}{24 a^{9} \left (x -\frac {1}{a}\right )^{2}}-\frac {25 \sqrt {a^{2} \left (x -\frac {1}{a}\right )^{2}+2 a \left (x -\frac {1}{a}\right )}}{48 a^{8} \left (x -\frac {1}{a}\right )}+\frac {493 \sqrt {a^{2} \left (x +\frac {1}{a}\right )^{2}-2 a \left (x +\frac {1}{a}\right )}}{240 a^{8} \left (x +\frac {1}{a}\right )}+\frac {\sqrt {a^{2} \left (x +\frac {1}{a}\right )^{2}-2 a \left (x +\frac {1}{a}\right )}}{20 a^{10} \left (x +\frac {1}{a}\right )^{3}}-\frac {23 \sqrt {a^{2} \left (x +\frac {1}{a}\right )^{2}-2 a \left (x +\frac {1}{a}\right )}}{60 a^{9} \left (x +\frac {1}{a}\right )^{2}}\right ) a^{6} \sqrt {\frac {a x -1}{a x +1}}\, \sqrt {\left (a x +1\right ) \left (a x -1\right )}}{c^{3} \left (a x -1\right )}\)	\(287\)
default	\(-\frac {\left (-525 \sqrt {\left (a x +1\right ) \left (a x -1\right )}\, \sqrt {a^{2}}\, a^{7} x^{7}+240 \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}+285 \left (\left (a x +1\right ) \left (a x -1\right )\right )^{\frac {3}{2}} \sqrt {a^{2}}\, a^{5} x^{5}-525 \sqrt {\left (a x +1\right ) \left (a x -1\right )}\, \sqrt {a^{2}}\, a^{6} x^{6}+240 \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}-83 \left (\left (a x +1\right ) \left (a x -1\right )\right )^{\frac {3}{2}} \sqrt {a^{2}}\, a^{4} x^{4}+1575 \sqrt {\left (a x +1\right ) \left (a x -1\right )}\, \sqrt {a^{2}}\, a^{5} x^{5}-720 \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}-218 \left (\left (a x +1\right ) \left (a x -1\right )\right )^{\frac {3}{2}} \sqrt {a^{2}}\, a^{3} x^{3}+1575 \sqrt {a^{2}}\, \sqrt {\left (a x +1\right ) \left (a x -1\right )}\, a^{4} x^{4}-720 \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}+342 \sqrt {a^{2}}\, \left (\left (a x +1\right ) \left (a x -1\right )\right )^{\frac {3}{2}} a^{2} x^{2}-1575 \sqrt {a^{2}}\, \sqrt {\left (a x +1\right ) \left (a x -1\right )}\, a^{3} x^{3}+720 \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}-3 \sqrt {a^{2}}\, \left (\left (a x +1\right ) \left (a x -1\right )\right )^{\frac {3}{2}} a x -1575 \sqrt {\left (a x +1\right ) \left (a x -1\right )}\, \sqrt {a^{2}}\, a^{2} x^{2}+720 \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}-243 \left (\left (a x +1\right ) \left (a x -1\right )\right )^{\frac {3}{2}} \sqrt {a^{2}}+525 \sqrt {\left (a x +1\right ) \left (a x -1\right )}\, \sqrt {a^{2}}\, a x -240 \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 +525 \sqrt {a^{2}}\, \sqrt {\left (a x +1\right ) \left (a x -1\right )}-240 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 ) \sqrt {\frac {a x -1}{a x +1}}}{240 a \left (a x -1\right )^{3} \sqrt {a^{2}}\, \left (a x +1\right )^{3} c^{3} \sqrt {\left (a x +1\right ) \left (a x -1\right )}}\)	\(714\)

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)^3,x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.27, size = 197, normalized size = 0.77 \begin {gather*} \frac {1}{240} \, a {\left (\frac {5 \, {\left (\frac {23 \, {\left (a x - 1\right )}}{a x + 1} - \frac {120 \, {\left (a x - 1\right )}^{2}}{{\left (a x + 1\right )}^{2}} + 1\right )}}{a^{2} c^{3} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {5}{2}} - a^{2} c^{3} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}}} + \frac {3 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {5}{2}} + 40 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}} + 450 \, \sqrt {\frac {a x - 1}{a x + 1}}}{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 {gather*}

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)^3,x, algorithm="maxima")

[Out]

1/240*a*(5*(23*(a*x - 1)/(a*x + 1) - 120*(a*x - 1)^2/(a*x + 1)^2 + 1)/(a^2*c^3*((a*x - 1)/(a*x + 1))^(5/2) - a
^2*c^3*((a*x - 1)/(a*x + 1))^(3/2)) + (3*((a*x - 1)/(a*x + 1))^(5/2) + 40*((a*x - 1)/(a*x + 1))^(3/2) + 450*sq
rt((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 = 0.36, size = 161, normalized size = 0.63 \begin {gather*} -\frac {15 \, {\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right ) - 15 \, {\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 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^{3} c^{3} x^{2} + a c^{3}\right )}} \end {gather*}

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)^3,x, algorithm="fricas")

[Out]

-1/15*(15*(a^4*x^4 - 2*a^2*x^2 + 1)*log(sqrt((a*x - 1)/(a*x + 1)) + 1) - 15*(a^4*x^4 - 2*a^2*x^2 + 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^3*c^3*x^2 + a*c^3)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \frac {a^{6} \int \frac {x^{6} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a^{6} x^{6} - 3 a^{4} x^{4} + 3 a^{2} x^{2} - 1}\, dx}{c^{3}} \end {gather*}

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)**3,x)

[Out]

a**6*Integral(x**6*sqrt(a*x/(a*x + 1) - 1/(a*x + 1))/(a**6*x**6 - 3*a**4*x**4 + 3*a**2*x**2 - 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))^(1/2)/(c-c/a^2/x^2)^3,x, algorithm="giac")

[Out]

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

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Mupad [B]
time = 0.06, size = 178, normalized size = 0.70 \begin {gather*} \frac {15\,\sqrt {\frac {a\,x-1}{a\,x+1}}}{8\,a\,c^3}-\frac {\frac {23\,\left (a\,x-1\right )}{3\,\left (a\,x+1\right )}-\frac {40\,{\left (a\,x-1\right )}^2}{{\left (a\,x+1\right )}^2}+\frac {1}{3}}{16\,a\,c^3\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/2}-16\,a\,c^3\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{5/2}}+\frac {{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/2}}{6\,a\,c^3}+\frac {{\left (\frac {a\,x-1}{a\,x+1}\right )}^{5/2}}{80\,a\,c^3}+\frac {\mathrm {atan}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\,1{}\mathrm {i}\right )\,2{}\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))^(1/2)/(c - c/(a^2*x^2))^3,x)

[Out]

(15*((a*x - 1)/(a*x + 1))^(1/2))/(8*a*c^3) - ((23*(a*x - 1))/(3*(a*x + 1)) - (40*(a*x - 1)^2)/(a*x + 1)^2 + 1/
3)/(16*a*c^3*((a*x - 1)/(a*x + 1))^(3/2) - 16*a*c^3*((a*x - 1)/(a*x + 1))^(5/2)) + ((a*x - 1)/(a*x + 1))^(3/2)
/(6*a*c^3) + ((a*x - 1)/(a*x + 1))^(5/2)/(80*a*c^3) + (atan(((a*x - 1)/(a*x + 1))^(1/2)*1i)*2i)/(a*c^3)

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