3.8.90 \(\int e^{3 \coth ^{-1}(a x)} (c-\frac {c}{a^2 x^2})^3 \, dx\) [790]

Optimal. Leaf size=269 \[ -\frac {27 c^3 \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}{8 a}-\frac {17 c^3 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/2}}{8 a}-\frac {29 c^3 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{5/2}}{20 a}-\frac {21 c^3 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{7/2}}{20 a}+\frac {6 c^3 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{9/2}}{5 a}+c^3 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{9/2} x+\frac {3 c^3 \csc ^{-1}(a x)}{8 a}+\frac {3 c^3 \tanh ^{-1}\left (\sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}\right )}{a} \]

[Out]

c^3*(1-1/a/x)^(3/2)*(1+1/a/x)^(9/2)*x+3/8*c^3*arccsc(a*x)/a+3*c^3*arctanh((1-1/a/x)^(1/2)*(1+1/a/x)^(1/2))/a-1
7/8*c^3*(1+1/a/x)^(3/2)*(1-1/a/x)^(1/2)/a-29/20*c^3*(1+1/a/x)^(5/2)*(1-1/a/x)^(1/2)/a-21/20*c^3*(1+1/a/x)^(7/2
)*(1-1/a/x)^(1/2)/a+6/5*c^3*(1+1/a/x)^(9/2)*(1-1/a/x)^(1/2)/a-27/8*c^3*(1-1/a/x)^(1/2)*(1+1/a/x)^(1/2)/a

________________________________________________________________________________________

Rubi [A]
time = 0.13, antiderivative size = 269, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 8, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {6329, 99, 159, 163, 41, 222, 94, 214} \begin {gather*} c^3 x \left (1-\frac {1}{a x}\right )^{3/2} \left (\frac {1}{a x}+1\right )^{9/2}+\frac {6 c^3 \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{9/2}}{5 a}-\frac {21 c^3 \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{7/2}}{20 a}-\frac {29 c^3 \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{5/2}}{20 a}-\frac {17 c^3 \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{3/2}}{8 a}-\frac {27 c^3 \sqrt {1-\frac {1}{a x}} \sqrt {\frac {1}{a x}+1}}{8 a}+\frac {3 c^3 \csc ^{-1}(a x)}{8 a}+\frac {3 c^3 \tanh ^{-1}\left (\sqrt {1-\frac {1}{a x}} \sqrt {\frac {1}{a x}+1}\right )}{a} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-27*c^3*Sqrt[1 - 1/(a*x)]*Sqrt[1 + 1/(a*x)])/(8*a) - (17*c^3*Sqrt[1 - 1/(a*x)]*(1 + 1/(a*x))^(3/2))/(8*a) - (
29*c^3*Sqrt[1 - 1/(a*x)]*(1 + 1/(a*x))^(5/2))/(20*a) - (21*c^3*Sqrt[1 - 1/(a*x)]*(1 + 1/(a*x))^(7/2))/(20*a) +
(6*c^3*Sqrt[1 - 1/(a*x)]*(1 + 1/(a*x))^(9/2))/(5*a) + c^3*(1 - 1/(a*x))^(3/2)*(1 + 1/(a*x))^(9/2)*x + (3*c^3*
ArcCsc[a*x])/(8*a) + (3*c^3*ArcTanh[Sqrt[1 - 1/(a*x)]*Sqrt[1 + 1/(a*x)]])/a

Rule 41

Int[((a_) + (b_.)*(x_))^(m_.)*((c_) + (d_.)*(x_))^(m_.), x_Symbol] :> Int[(a*c + b*d*x^2)^m, x] /; FreeQ[{a, b
, c, d, m}, x] && EqQ[b*c + a*d, 0] && (IntegerQ[m] || (GtQ[a, 0] && GtQ[c, 0]))

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 99

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(a + b*
x)^(m + 1)*(c + d*x)^n*((e + f*x)^p/(b*(m + 1))), x] - Dist[1/(b*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^(n
- 1)*(e + f*x)^(p - 1)*Simp[d*e*n + c*f*p + d*f*(n + p)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && LtQ[m
, -1] && GtQ[n, 0] && GtQ[p, 0] && (IntegersQ[2*m, 2*n, 2*p] || IntegersQ[m, n + p] || IntegersQ[p, m + n])

Rule 159

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)), x_Symb
ol] :> Simp[h*(a + b*x)^m*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d*f*(m + n + p + 2))), x] + Dist[1/(d*f*(m + n
+ p + 2)), Int[(a + b*x)^(m - 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*g*(m + n + p + 2) - h*(b*c*e*m + a*(d*e*(
n + 1) + c*f*(p + 1))) + (b*d*f*g*(m + n + p + 2) + h*(a*d*f*m - b*(d*e*(m + n + 1) + c*f*(m + p + 1))))*x, x]
, x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && GtQ[m, 0] && NeQ[m + n + p + 2, 0] && IntegersQ[2*m, 2
*n, 2*p]

Rule 163

Int[(((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)))/((a_.) + (b_.)*(x_)), x_Symbol]
:> Dist[h/b, Int[(c + d*x)^n*(e + f*x)^p, x], x] + Dist[(b*g - a*h)/b, Int[(c + d*x)^n*((e + f*x)^p/(a + b*x)
), x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, 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 222

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSin[Rt[-b, 2]*(x/Sqrt[a])]/Rt[-b, 2], x] /; FreeQ[{a, b}
, x] && GtQ[a, 0] && NegQ[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 e^{3 \coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^3 \, dx &=-\left (c^3 \text {Subst}\left (\int \frac {\left (1-\frac {x}{a}\right )^{3/2} \left (1+\frac {x}{a}\right )^{9/2}}{x^2} \, dx,x,\frac {1}{x}\right )\right )\\ &=c^3 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{9/2} x-c^3 \text {Subst}\left (\int \frac {\left (\frac {3}{a}-\frac {6 x}{a^2}\right ) \sqrt {1-\frac {x}{a}} \left (1+\frac {x}{a}\right )^{7/2}}{x} \, dx,x,\frac {1}{x}\right )\\ &=\frac {6 c^3 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{9/2}}{5 a}+c^3 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{9/2} x-\frac {1}{5} \left (a c^3\right ) \text {Subst}\left (\int \frac {\left (\frac {15}{a^2}-\frac {21 x}{a^3}\right ) \left (1+\frac {x}{a}\right )^{7/2}}{x \sqrt {1-\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )\\ &=-\frac {21 c^3 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{7/2}}{20 a}+\frac {6 c^3 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{9/2}}{5 a}+c^3 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{9/2} x+\frac {1}{20} \left (a^2 c^3\right ) \text {Subst}\left (\int \frac {\left (-\frac {60}{a^3}+\frac {87 x}{a^4}\right ) \left (1+\frac {x}{a}\right )^{5/2}}{x \sqrt {1-\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )\\ &=-\frac {29 c^3 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{5/2}}{20 a}-\frac {21 c^3 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{7/2}}{20 a}+\frac {6 c^3 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{9/2}}{5 a}+c^3 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{9/2} x-\frac {1}{60} \left (a^3 c^3\right ) \text {Subst}\left (\int \frac {\left (\frac {180}{a^4}-\frac {255 x}{a^5}\right ) \left (1+\frac {x}{a}\right )^{3/2}}{x \sqrt {1-\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )\\ &=-\frac {17 c^3 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/2}}{8 a}-\frac {29 c^3 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{5/2}}{20 a}-\frac {21 c^3 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{7/2}}{20 a}+\frac {6 c^3 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{9/2}}{5 a}+c^3 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{9/2} x+\frac {1}{120} \left (a^4 c^3\right ) \text {Subst}\left (\int \frac {\left (-\frac {360}{a^5}+\frac {405 x}{a^6}\right ) \sqrt {1+\frac {x}{a}}}{x \sqrt {1-\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )\\ &=-\frac {27 c^3 \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}{8 a}-\frac {17 c^3 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/2}}{8 a}-\frac {29 c^3 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{5/2}}{20 a}-\frac {21 c^3 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{7/2}}{20 a}+\frac {6 c^3 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{9/2}}{5 a}+c^3 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{9/2} x-\frac {1}{120} \left (a^5 c^3\right ) \text {Subst}\left (\int \frac {\frac {360}{a^6}-\frac {45 x}{a^7}}{x \sqrt {1-\frac {x}{a}} \sqrt {1+\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )\\ &=-\frac {27 c^3 \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}{8 a}-\frac {17 c^3 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/2}}{8 a}-\frac {29 c^3 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{5/2}}{20 a}-\frac {21 c^3 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{7/2}}{20 a}+\frac {6 c^3 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{9/2}}{5 a}+c^3 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{9/2} x+\frac {\left (3 c^3\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-\frac {x}{a}} \sqrt {1+\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )}{8 a^2}-\frac {\left (3 c^3\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {1-\frac {x}{a}} \sqrt {1+\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )}{a}\\ &=-\frac {27 c^3 \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}{8 a}-\frac {17 c^3 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/2}}{8 a}-\frac {29 c^3 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{5/2}}{20 a}-\frac {21 c^3 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{7/2}}{20 a}+\frac {6 c^3 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{9/2}}{5 a}+c^3 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{9/2} x+\frac {\left (3 c^3\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )}{8 a^2}+\frac {\left (3 c^3\right ) \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}\\ &=-\frac {27 c^3 \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}{8 a}-\frac {17 c^3 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/2}}{8 a}-\frac {29 c^3 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{5/2}}{20 a}-\frac {21 c^3 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{7/2}}{20 a}+\frac {6 c^3 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{9/2}}{5 a}+c^3 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{9/2} x+\frac {3 c^3 \csc ^{-1}(a x)}{8 a}+\frac {3 c^3 \tanh ^{-1}\left (\sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}\right )}{a}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.11, size = 110, normalized size = 0.41 \begin {gather*} \frac {c^3 \left (\sqrt {1-\frac {1}{a^2 x^2}} \left (8+30 a x+24 a^2 x^2-55 a^3 x^3-152 a^4 x^4+40 a^5 x^5\right )+15 a^4 x^4 \text {ArcSin}\left (\frac {1}{a x}\right )+120 a^4 x^4 \log \left (\left (1+\sqrt {1-\frac {1}{a^2 x^2}}\right ) x\right )\right )}{40 a^5 x^4} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(c^3*(Sqrt[1 - 1/(a^2*x^2)]*(8 + 30*a*x + 24*a^2*x^2 - 55*a^3*x^3 - 152*a^4*x^4 + 40*a^5*x^5) + 15*a^4*x^4*Arc
Sin[1/(a*x)] + 120*a^4*x^4*Log[(1 + Sqrt[1 - 1/(a^2*x^2)])*x]))/(40*a^5*x^4)

________________________________________________________________________________________

Maple [A]
time = 0.08, size = 281, normalized size = 1.04

method	result	size
risch	\(-\frac {\left (a x -1\right ) \left (152 a^{4} x^{4}+55 a^{3} x^{3}-24 a^{2} x^{2}-30 a x -8\right ) c^{3}}{40 x^{5} a^{6} \sqrt {\frac {a x -1}{a x +1}}}+\frac {\left (a^{5} \sqrt {\left (a x +1\right ) \left (a x -1\right )}+\frac {3 a^{6} \ln \left (\frac {a^{2} x}{\sqrt {a^{2}}}+\sqrt {a^{2} x^{2}-1}\right )}{\sqrt {a^{2}}}+\frac {3 a^{5} \arctan \left (\frac {1}{\sqrt {a^{2} x^{2}-1}}\right )}{8}\right ) c^{3} \sqrt {\left (a x +1\right ) \left (a x -1\right )}}{a^{6} \left (a x +1\right ) \sqrt {\frac {a x -1}{a x +1}}}\)	\(173\)
default	\(\frac {\left (a x -1\right )^{2} c^{3} \left (-120 \sqrt {a^{2}}\, \sqrt {a^{2} x^{2}-1}\, a^{6} x^{6}+120 \sqrt {a^{2}}\, \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} a^{4} x^{4}+15 \sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}\, a^{5} x^{5}+15 \sqrt {a^{2}}\, \arctan \left (\frac {1}{\sqrt {a^{2} x^{2}-1}}\right ) a^{5} x^{5}+120 \ln \left (\frac {a^{2} x +\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) a^{6} x^{5}+25 \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}\, a^{3} x^{3}-32 \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}\, a^{2} x^{2}-30 \sqrt {a^{2}}\, \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} a x -8 \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}\right )}{40 \left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}} \left (a x +1\right ) \sqrt {\left (a x +1\right ) \left (a x -1\right )}\, a^{6} x^{5} \sqrt {a^{2}}}\)	\(281\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/40*(a*x-1)^2*c^3*(-120*(a^2)^(1/2)*(a^2*x^2-1)^(1/2)*a^6*x^6+120*(a^2)^(1/2)*(a^2*x^2-1)^(3/2)*a^4*x^4+15*(a
^2*x^2-1)^(1/2)*(a^2)^(1/2)*a^5*x^5+15*(a^2)^(1/2)*arctan(1/(a^2*x^2-1)^(1/2))*a^5*x^5+120*ln((a^2*x+(a^2*x^2-
1)^(1/2)*(a^2)^(1/2))/(a^2)^(1/2))*a^6*x^5+25*(a^2*x^2-1)^(3/2)*(a^2)^(1/2)*a^3*x^3-32*(a^2*x^2-1)^(3/2)*(a^2)
^(1/2)*a^2*x^2-30*(a^2)^(1/2)*(a^2*x^2-1)^(3/2)*a*x-8*(a^2*x^2-1)^(3/2)*(a^2)^(1/2))/((a*x-1)/(a*x+1))^(3/2)/(
a*x+1)/((a*x+1)*(a*x-1))^(1/2)/a^6/x^5/(a^2)^(1/2)

________________________________________________________________________________________

Maxima [A]
time = 0.47, size = 302, normalized size = 1.12 \begin {gather*} -\frac {1}{20} \, {\left (\frac {15 \, c^{3} \arctan \left (\sqrt {\frac {a x - 1}{a x + 1}}\right )}{a^{2}} - \frac {60 \, c^{3} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right )}{a^{2}} + \frac {60 \, c^{3} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right )}{a^{2}} - \frac {135 \, c^{3} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {11}{2}} + 575 \, c^{3} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {9}{2}} + 842 \, c^{3} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {7}{2}} + 298 \, c^{3} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {5}{2}} - 465 \, c^{3} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}} - 105 \, c^{3} \sqrt {\frac {a x - 1}{a x + 1}}}{\frac {4 \, {\left (a x - 1\right )} a^{2}}{a x + 1} + \frac {5 \, {\left (a x - 1\right )}^{2} a^{2}}{{\left (a x + 1\right )}^{2}} - \frac {5 \, {\left (a x - 1\right )}^{4} a^{2}}{{\left (a x + 1\right )}^{4}} - \frac {4 \, {\left (a x - 1\right )}^{5} a^{2}}{{\left (a x + 1\right )}^{5}} - \frac {{\left (a x - 1\right )}^{6} a^{2}}{{\left (a x + 1\right )}^{6}} + a^{2}}\right )} a \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/20*(15*c^3*arctan(sqrt((a*x - 1)/(a*x + 1)))/a^2 - 60*c^3*log(sqrt((a*x - 1)/(a*x + 1)) + 1)/a^2 + 60*c^3*l
og(sqrt((a*x - 1)/(a*x + 1)) - 1)/a^2 - (135*c^3*((a*x - 1)/(a*x + 1))^(11/2) + 575*c^3*((a*x - 1)/(a*x + 1))^
(9/2) + 842*c^3*((a*x - 1)/(a*x + 1))^(7/2) + 298*c^3*((a*x - 1)/(a*x + 1))^(5/2) - 465*c^3*((a*x - 1)/(a*x +
1))^(3/2) - 105*c^3*sqrt((a*x - 1)/(a*x + 1)))/(4*(a*x - 1)*a^2/(a*x + 1) + 5*(a*x - 1)^2*a^2/(a*x + 1)^2 - 5*
(a*x - 1)^4*a^2/(a*x + 1)^4 - 4*(a*x - 1)^5*a^2/(a*x + 1)^5 - (a*x - 1)^6*a^2/(a*x + 1)^6 + a^2))*a

________________________________________________________________________________________

Fricas [A]
time = 0.35, size = 179, normalized size = 0.67 \begin {gather*} -\frac {30 \, a^{5} c^{3} x^{5} \arctan \left (\sqrt {\frac {a x - 1}{a x + 1}}\right ) - 120 \, a^{5} c^{3} x^{5} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right ) + 120 \, a^{5} c^{3} x^{5} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right ) - {\left (40 \, a^{6} c^{3} x^{6} - 112 \, a^{5} c^{3} x^{5} - 207 \, a^{4} c^{3} x^{4} - 31 \, a^{3} c^{3} x^{3} + 54 \, a^{2} c^{3} x^{2} + 38 \, a c^{3} x + 8 \, c^{3}\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{40 \, a^{6} x^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/40*(30*a^5*c^3*x^5*arctan(sqrt((a*x - 1)/(a*x + 1))) - 120*a^5*c^3*x^5*log(sqrt((a*x - 1)/(a*x + 1)) + 1) +
120*a^5*c^3*x^5*log(sqrt((a*x - 1)/(a*x + 1)) - 1) - (40*a^6*c^3*x^6 - 112*a^5*c^3*x^5 - 207*a^4*c^3*x^4 - 31
*a^3*c^3*x^3 + 54*a^2*c^3*x^2 + 38*a*c^3*x + 8*c^3)*sqrt((a*x - 1)/(a*x + 1)))/(a^6*x^5)

________________________________________________________________________________________

Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \frac {c^{3} \left (\int \frac {3 a^{2}}{\frac {a x^{5} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1} - \frac {x^{4} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1}}\, dx + \int \left (- \frac {3 a^{4}}{\frac {a x^{3} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1} - \frac {x^{2} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1}}\right )\, dx + \int \frac {a^{6}}{\frac {a x \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1} - \frac {\sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1}}\, dx + \int \left (- \frac {1}{\frac {a x^{7} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1} - \frac {x^{6} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1}}\right )\, dx\right )}{a^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Giac [A]
time = 0.42, size = 355, normalized size = 1.32 \begin {gather*} -\frac {3 \, c^{3} \arctan \left (-x {\left | a \right |} + \sqrt {a^{2} x^{2} - 1}\right )}{4 \, a \mathrm {sgn}\left (a x + 1\right )} - \frac {3 \, c^{3} \log \left ({\left | -x {\left | a \right |} + \sqrt {a^{2} x^{2} - 1} \right |}\right )}{{\left | a \right |} \mathrm {sgn}\left (a x + 1\right )} + \frac {\sqrt {a^{2} x^{2} - 1} c^{3}}{a \mathrm {sgn}\left (a x + 1\right )} + \frac {55 \, {\left (x {\left | a \right |} - \sqrt {a^{2} x^{2} - 1}\right )}^{9} c^{3} {\left | a \right |} - 200 \, {\left (x {\left | a \right |} - \sqrt {a^{2} x^{2} - 1}\right )}^{8} a c^{3} - 10 \, {\left (x {\left | a \right |} - \sqrt {a^{2} x^{2} - 1}\right )}^{7} c^{3} {\left | a \right |} - 720 \, {\left (x {\left | a \right |} - \sqrt {a^{2} x^{2} - 1}\right )}^{6} a c^{3} - 800 \, {\left (x {\left | a \right |} - \sqrt {a^{2} x^{2} - 1}\right )}^{4} a c^{3} + 10 \, {\left (x {\left | a \right |} - \sqrt {a^{2} x^{2} - 1}\right )}^{3} c^{3} {\left | a \right |} - 560 \, {\left (x {\left | a \right |} - \sqrt {a^{2} x^{2} - 1}\right )}^{2} a c^{3} - 55 \, {\left (x {\left | a \right |} - \sqrt {a^{2} x^{2} - 1}\right )} c^{3} {\left | a \right |} - 152 \, a c^{3}}{20 \, {\left ({\left (x {\left | a \right |} - \sqrt {a^{2} x^{2} - 1}\right )}^{2} + 1\right )}^{5} a {\left | a \right |} \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))^(3/2)*(c-c/a^2/x^2)^3,x, algorithm="giac")

[Out]

-3/4*c^3*arctan(-x*abs(a) + sqrt(a^2*x^2 - 1))/(a*sgn(a*x + 1)) - 3*c^3*log(abs(-x*abs(a) + sqrt(a^2*x^2 - 1))
)/(abs(a)*sgn(a*x + 1)) + sqrt(a^2*x^2 - 1)*c^3/(a*sgn(a*x + 1)) + 1/20*(55*(x*abs(a) - sqrt(a^2*x^2 - 1))^9*c
^3*abs(a) - 200*(x*abs(a) - sqrt(a^2*x^2 - 1))^8*a*c^3 - 10*(x*abs(a) - sqrt(a^2*x^2 - 1))^7*c^3*abs(a) - 720*
(x*abs(a) - sqrt(a^2*x^2 - 1))^6*a*c^3 - 800*(x*abs(a) - sqrt(a^2*x^2 - 1))^4*a*c^3 + 10*(x*abs(a) - sqrt(a^2*
x^2 - 1))^3*c^3*abs(a) - 560*(x*abs(a) - sqrt(a^2*x^2 - 1))^2*a*c^3 - 55*(x*abs(a) - sqrt(a^2*x^2 - 1))*c^3*ab
s(a) - 152*a*c^3)/(((x*abs(a) - sqrt(a^2*x^2 - 1))^2 + 1)^5*a*abs(a)*sgn(a*x + 1))

________________________________________________________________________________________

Mupad [B]
time = 1.41, size = 258, normalized size = 0.96 \begin {gather*} \frac {\frac {149\,c^3\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{5/2}}{10}-\frac {93\,c^3\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/2}}{4}-\frac {21\,c^3\,\sqrt {\frac {a\,x-1}{a\,x+1}}}{4}+\frac {421\,c^3\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{7/2}}{10}+\frac {115\,c^3\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{9/2}}{4}+\frac {27\,c^3\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{11/2}}{4}}{a+\frac {4\,a\,\left (a\,x-1\right )}{a\,x+1}+\frac {5\,a\,{\left (a\,x-1\right )}^2}{{\left (a\,x+1\right )}^2}-\frac {5\,a\,{\left (a\,x-1\right )}^4}{{\left (a\,x+1\right )}^4}-\frac {4\,a\,{\left (a\,x-1\right )}^5}{{\left (a\,x+1\right )}^5}-\frac {a\,{\left (a\,x-1\right )}^6}{{\left (a\,x+1\right )}^6}}-\frac {3\,c^3\,\mathrm {atan}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\right )}{4\,a}+\frac {6\,c^3\,\mathrm {atanh}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\right )}{a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((149*c^3*((a*x - 1)/(a*x + 1))^(5/2))/10 - (93*c^3*((a*x - 1)/(a*x + 1))^(3/2))/4 - (21*c^3*((a*x - 1)/(a*x +
1))^(1/2))/4 + (421*c^3*((a*x - 1)/(a*x + 1))^(7/2))/10 + (115*c^3*((a*x - 1)/(a*x + 1))^(9/2))/4 + (27*c^3*(
(a*x - 1)/(a*x + 1))^(11/2))/4)/(a + (4*a*(a*x - 1))/(a*x + 1) + (5*a*(a*x - 1)^2)/(a*x + 1)^2 - (5*a*(a*x - 1
)^4)/(a*x + 1)^4 - (4*a*(a*x - 1)^5)/(a*x + 1)^5 - (a*(a*x - 1)^6)/(a*x + 1)^6) - (3*c^3*atan(((a*x - 1)/(a*x
+ 1))^(1/2)))/(4*a) + (6*c^3*atanh(((a*x - 1)/(a*x + 1))^(1/2)))/a

________________________________________________________________________________________