3.4.85 \(\int \frac {e^{\coth ^{-1}(a x)}}{(c-\frac {c}{a x})^3} \, dx\) [385]

Optimal. Leaf size=138 \[ -\frac {8 \left (a+\frac {1}{x}\right )}{5 a^2 c^3 \left (1-\frac {1}{a^2 x^2}\right )^{5/2}}-\frac {4 \left (5 a+\frac {8}{x}\right )}{15 a^2 c^3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}-\frac {60 a+\frac {79}{x}}{15 a^2 c^3 \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {\sqrt {1-\frac {1}{a^2 x^2}} x}{c^3}+\frac {4 \tanh ^{-1}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )}{a c^3} \]

[Out]

-8/5*(a+1/x)/a^2/c^3/(1-1/a^2/x^2)^(5/2)-4/15*(5*a+8/x)/a^2/c^3/(1-1/a^2/x^2)^(3/2)+4*arctanh((1-1/a^2/x^2)^(1
/2))/a/c^3+1/15*(-60*a-79/x)/a^2/c^3/(1-1/a^2/x^2)^(1/2)+x*(1-1/a^2/x^2)^(1/2)/c^3

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Rubi [A]
time = 0.27, antiderivative size = 138, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.350, Rules used = {6312, 866, 1819, 821, 272, 65, 214} \begin {gather*} -\frac {8 \left (a+\frac {1}{x}\right )}{5 a^2 c^3 \left (1-\frac {1}{a^2 x^2}\right )^{5/2}}-\frac {4 \left (5 a+\frac {8}{x}\right )}{15 a^2 c^3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}-\frac {60 a+\frac {79}{x}}{15 a^2 c^3 \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {x \sqrt {1-\frac {1}{a^2 x^2}}}{c^3}+\frac {4 \tanh ^{-1}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )}{a c^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[E^ArcCoth[a*x]/(c - c/(a*x))^3,x]

[Out]

(-8*(a + x^(-1)))/(5*a^2*c^3*(1 - 1/(a^2*x^2))^(5/2)) - (4*(5*a + 8/x))/(15*a^2*c^3*(1 - 1/(a^2*x^2))^(3/2)) -
(60*a + 79/x)/(15*a^2*c^3*Sqrt[1 - 1/(a^2*x^2)]) + (Sqrt[1 - 1/(a^2*x^2)]*x)/c^3 + (4*ArcTanh[Sqrt[1 - 1/(a^2
*x^2)]])/(a*c^3)

Rule 65

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 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 272

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 821

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 866

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 1819

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 6312

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]

Rubi steps

\begin {align*} \int \frac {e^{\coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^3} \, dx &=-\left (c \text {Subst}\left (\int \frac {\sqrt {1-\frac {x^2}{a^2}}}{x^2 \left (c-\frac {c x}{a}\right )^4} \, dx,x,\frac {1}{x}\right )\right )\\ &=-\frac {\text {Subst}\left (\int \frac {\left (c+\frac {c x}{a}\right )^4}{x^2 \left (1-\frac {x^2}{a^2}\right )^{7/2}} \, dx,x,\frac {1}{x}\right )}{c^7}\\ &=-\frac {8 \left (a+\frac {1}{x}\right )}{5 a^2 c^3 \left (1-\frac {1}{a^2 x^2}\right )^{5/2}}+\frac {\text {Subst}\left (\int \frac {-5 c^4-\frac {20 c^4 x}{a}-\frac {27 c^4 x^2}{a^2}}{x^2 \left (1-\frac {x^2}{a^2}\right )^{5/2}} \, dx,x,\frac {1}{x}\right )}{5 c^7}\\ &=-\frac {8 \left (a+\frac {1}{x}\right )}{5 a^2 c^3 \left (1-\frac {1}{a^2 x^2}\right )^{5/2}}-\frac {4 \left (5 a+\frac {8}{x}\right )}{15 a^2 c^3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}-\frac {\text {Subst}\left (\int \frac {15 c^4+\frac {60 c^4 x}{a}+\frac {64 c^4 x^2}{a^2}}{x^2 \left (1-\frac {x^2}{a^2}\right )^{3/2}} \, dx,x,\frac {1}{x}\right )}{15 c^7}\\ &=-\frac {8 \left (a+\frac {1}{x}\right )}{5 a^2 c^3 \left (1-\frac {1}{a^2 x^2}\right )^{5/2}}-\frac {4 \left (5 a+\frac {8}{x}\right )}{15 a^2 c^3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}-\frac {60 a+\frac {79}{x}}{15 a^2 c^3 \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {\text {Subst}\left (\int \frac {-15 c^4-\frac {60 c^4 x}{a}}{x^2 \sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )}{15 c^7}\\ &=-\frac {8 \left (a+\frac {1}{x}\right )}{5 a^2 c^3 \left (1-\frac {1}{a^2 x^2}\right )^{5/2}}-\frac {4 \left (5 a+\frac {8}{x}\right )}{15 a^2 c^3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}-\frac {60 a+\frac {79}{x}}{15 a^2 c^3 \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {\sqrt {1-\frac {1}{a^2 x^2}} x}{c^3}-\frac {4 \text {Subst}\left (\int \frac {1}{x \sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )}{a c^3}\\ &=-\frac {8 \left (a+\frac {1}{x}\right )}{5 a^2 c^3 \left (1-\frac {1}{a^2 x^2}\right )^{5/2}}-\frac {4 \left (5 a+\frac {8}{x}\right )}{15 a^2 c^3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}-\frac {60 a+\frac {79}{x}}{15 a^2 c^3 \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {\sqrt {1-\frac {1}{a^2 x^2}} x}{c^3}-\frac {2 \text {Subst}\left (\int \frac {1}{x \sqrt {1-\frac {x}{a^2}}} \, dx,x,\frac {1}{x^2}\right )}{a c^3}\\ &=-\frac {8 \left (a+\frac {1}{x}\right )}{5 a^2 c^3 \left (1-\frac {1}{a^2 x^2}\right )^{5/2}}-\frac {4 \left (5 a+\frac {8}{x}\right )}{15 a^2 c^3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}-\frac {60 a+\frac {79}{x}}{15 a^2 c^3 \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {\sqrt {1-\frac {1}{a^2 x^2}} x}{c^3}+\frac {(4 a) \text {Subst}\left (\int \frac {1}{a^2-a^2 x^2} \, dx,x,\sqrt {1-\frac {1}{a^2 x^2}}\right )}{c^3}\\ &=-\frac {8 \left (a+\frac {1}{x}\right )}{5 a^2 c^3 \left (1-\frac {1}{a^2 x^2}\right )^{5/2}}-\frac {4 \left (5 a+\frac {8}{x}\right )}{15 a^2 c^3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}-\frac {60 a+\frac {79}{x}}{15 a^2 c^3 \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {\sqrt {1-\frac {1}{a^2 x^2}} x}{c^3}+\frac {4 \tanh ^{-1}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )}{a c^3}\\ \end {align*}

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Mathematica [A]
time = 0.05, size = 104, normalized size = 0.75 \begin {gather*} \frac {-94+128 a x+73 a^2 x^2-134 a^3 x^3+15 a^4 x^4+60 a \sqrt {1-\frac {1}{a^2 x^2}} x (-1+a x)^2 \tanh ^{-1}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )}{15 a^2 c^3 \sqrt {1-\frac {1}{a^2 x^2}} x (-1+a x)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-94 + 128*a*x + 73*a^2*x^2 - 134*a^3*x^3 + 15*a^4*x^4 + 60*a*Sqrt[1 - 1/(a^2*x^2)]*x*(-1 + a*x)^2*ArcTanh[Sqr
t[1 - 1/(a^2*x^2)]])/(15*a^2*c^3*Sqrt[1 - 1/(a^2*x^2)]*x*(-1 + a*x)^2)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(430\) vs. \(2(122)=244\).
time = 0.12, size = 431, normalized size = 3.12

method	result	size
risch	\(\frac {a x -1}{a \,c^{3} \sqrt {\frac {a x -1}{a x +1}}}+\frac {\left (\frac {4 \ln \left (\frac {a^{2} x}{\sqrt {a^{2}}}+\sqrt {a^{2} x^{2}-1}\right )}{a^{3} \sqrt {a^{2}}}-\frac {104 \sqrt {a^{2} \left (x -\frac {1}{a}\right )^{2}+2 a \left (x -\frac {1}{a}\right )}}{15 a^{5} \left (x -\frac {1}{a}\right )}-\frac {31 \sqrt {a^{2} \left (x -\frac {1}{a}\right )^{2}+2 a \left (x -\frac {1}{a}\right )}}{15 a^{6} \left (x -\frac {1}{a}\right )^{2}}-\frac {2 \sqrt {a^{2} \left (x -\frac {1}{a}\right )^{2}+2 a \left (x -\frac {1}{a}\right )}}{5 a^{7} \left (x -\frac {1}{a}\right )^{3}}\right ) a^{3} \sqrt {\left (a x +1\right ) \left (a x -1\right )}}{c^{3} \sqrt {\frac {a x -1}{a x +1}}\, \left (a x +1\right )}\)	\(225\)
default	\(-\frac {-60 \sqrt {a^{2}}\, \sqrt {\left (a x +1\right ) \left (a x -1\right )}\, a^{4} x^{4}-60 \ln \left (\frac {a^{2} x +\sqrt {a^{2}}\, \sqrt {\left (a x +1\right ) \left (a x -1\right )}}{\sqrt {a^{2}}}\right ) a^{5} x^{4}+45 \sqrt {a^{2}}\, \left (\left (a x +1\right ) \left (a x -1\right )\right )^{\frac {3}{2}} a^{2} x^{2}+240 \sqrt {a^{2}}\, \sqrt {\left (a x +1\right ) \left (a x -1\right )}\, a^{3} x^{3}+240 \ln \left (\frac {a^{2} x +\sqrt {a^{2}}\, \sqrt {\left (a x +1\right ) \left (a x -1\right )}}{\sqrt {a^{2}}}\right ) a^{4} x^{3}-76 \sqrt {a^{2}}\, \left (\left (a x +1\right ) \left (a x -1\right )\right )^{\frac {3}{2}} a x -360 \sqrt {\left (a x +1\right ) \left (a x -1\right )}\, \sqrt {a^{2}}\, a^{2} x^{2}-360 \ln \left (\frac {a^{2} x +\sqrt {a^{2}}\, \sqrt {\left (a x +1\right ) \left (a x -1\right )}}{\sqrt {a^{2}}}\right ) a^{3} x^{2}+34 \left (\left (a x +1\right ) \left (a x -1\right )\right )^{\frac {3}{2}} \sqrt {a^{2}}+240 \sqrt {\left (a x +1\right ) \left (a x -1\right )}\, \sqrt {a^{2}}\, a x +240 \ln \left (\frac {a^{2} x +\sqrt {a^{2}}\, \sqrt {\left (a x +1\right ) \left (a x -1\right )}}{\sqrt {a^{2}}}\right ) a^{2} x -60 \sqrt {a^{2}}\, \sqrt {\left (a x +1\right ) \left (a x -1\right )}-60 a \ln \left (\frac {a^{2} x +\sqrt {a^{2}}\, \sqrt {\left (a x +1\right ) \left (a x -1\right )}}{\sqrt {a^{2}}}\right )}{15 a \sqrt {a^{2}}\, \left (a x -1\right )^{3} c^{3} \sqrt {\left (a x +1\right ) \left (a x -1\right )}\, \sqrt {\frac {a x -1}{a x +1}}}\)	\(431\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/15*(-60*(a^2)^(1/2)*((a*x+1)*(a*x-1))^(1/2)*a^4*x^4-60*ln((a^2*x+(a^2)^(1/2)*((a*x+1)*(a*x-1))^(1/2))/(a^2)
^(1/2))*a^5*x^4+45*(a^2)^(1/2)*((a*x+1)*(a*x-1))^(3/2)*a^2*x^2+240*(a^2)^(1/2)*((a*x+1)*(a*x-1))^(1/2)*a^3*x^3
+240*ln((a^2*x+(a^2)^(1/2)*((a*x+1)*(a*x-1))^(1/2))/(a^2)^(1/2))*a^4*x^3-76*(a^2)^(1/2)*((a*x+1)*(a*x-1))^(3/2
)*a*x-360*((a*x+1)*(a*x-1))^(1/2)*(a^2)^(1/2)*a^2*x^2-360*ln((a^2*x+(a^2)^(1/2)*((a*x+1)*(a*x-1))^(1/2))/(a^2)
^(1/2))*a^3*x^2+34*((a*x+1)*(a*x-1))^(3/2)*(a^2)^(1/2)+240*((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-60*(a^2)^(1/2)*((a*x+1)*(a*x-1))^(1/2)-60*a*ln((a^2
*x+(a^2)^(1/2)*((a*x+1)*(a*x-1))^(1/2))/(a^2)^(1/2)))/a/(a^2)^(1/2)/(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 = 0.26, size = 153, normalized size = 1.11 \begin {gather*} \frac {1}{30} \, a {\left (\frac {\frac {22 \, {\left (a x - 1\right )}}{a x + 1} + \frac {155 \, {\left (a x - 1\right )}^{2}}{{\left (a x + 1\right )}^{2}} - \frac {240 \, {\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 {120 \, \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right )}{a^{2} c^{3}} - \frac {120 \, \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(1/((a*x-1)/(a*x+1))^(1/2)/(c-c/a/x)^3,x, algorithm="maxima")

[Out]

1/30*a*((22*(a*x - 1)/(a*x + 1) + 155*(a*x - 1)^2/(a*x + 1)^2 - 240*(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)) + 120*log(sqrt((a*x - 1)/(a*x + 1)) + 1)/(a^2*c
^3) - 120*log(sqrt((a*x - 1)/(a*x + 1)) - 1)/(a^2*c^3))

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Fricas [A]
time = 0.34, size = 170, normalized size = 1.23 \begin {gather*} \frac {60 \, {\left (a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1\right )} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right ) - 60 \, {\left (a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1\right )} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right ) + {\left (15 \, a^{4} x^{4} - 134 \, a^{3} x^{3} + 73 \, a^{2} x^{2} + 128 \, a x - 94\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{15 \, {\left (a^{4} c^{3} x^{3} - 3 \, a^{3} c^{3} x^{2} + 3 \, a^{2} c^{3} x - a c^{3}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/15*(60*(a^3*x^3 - 3*a^2*x^2 + 3*a*x - 1)*log(sqrt((a*x - 1)/(a*x + 1)) + 1) - 60*(a^3*x^3 - 3*a^2*x^2 + 3*a*
x - 1)*log(sqrt((a*x - 1)/(a*x + 1)) - 1) + (15*a^4*x^4 - 134*a^3*x^3 + 73*a^2*x^2 + 128*a*x - 94)*sqrt((a*x -
1)/(a*x + 1)))/(a^4*c^3*x^3 - 3*a^3*c^3*x^2 + 3*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^{3} \int \frac {x^{3}}{a^{3} x^{3} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}} - 3 a^{2} x^{2} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}} + 3 a x \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}} - \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}\, dx}{c^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]
time = 0.42, size = 63, normalized size = 0.46 \begin {gather*} -\frac {4 \, \log \left ({\left | -x {\left | a \right |} + \sqrt {a^{2} x^{2} - 1} \right |}\right )}{c^{3} {\left | a \right |} \mathrm {sgn}\left (a x + 1\right )} + \frac {\sqrt {a^{2} x^{2} - 1}}{a c^{3} \mathrm {sgn}\left (a x + 1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [B]
time = 1.22, size = 121, normalized size = 0.88 \begin {gather*} \frac {8\,\mathrm {atanh}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\right )}{a\,c^3}-\frac {\frac {31\,{\left (a\,x-1\right )}^2}{3\,{\left (a\,x+1\right )}^2}-\frac {16\,{\left (a\,x-1\right )}^3}{{\left (a\,x+1\right )}^3}+\frac {22\,\left (a\,x-1\right )}{15\,\left (a\,x+1\right )}+\frac {1}{5}}{2\,a\,c^3\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{5/2}-2\,a\,c^3\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{7/2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(8*atanh(((a*x - 1)/(a*x + 1))^(1/2)))/(a*c^3) - ((31*(a*x - 1)^2)/(3*(a*x + 1)^2) - (16*(a*x - 1)^3)/(a*x + 1
)^3 + (22*(a*x - 1))/(15*(a*x + 1)) + 1/5)/(2*a*c^3*((a*x - 1)/(a*x + 1))^(5/2) - 2*a*c^3*((a*x - 1)/(a*x + 1)
)^(7/2))

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