3.629 \(\int e^{\tanh ^{-1}(a x)} (c-\frac{c}{a^2 x^2})^3 \, dx\)

Optimal. Leaf size=136 \[ \frac{c^3 (5 a x+4) \left (1-a^2 x^2\right )^{5/2}}{20 a^6 x^5}-\frac{c^3 (15 a x+8) \left (1-a^2 x^2\right )^{3/2}}{24 a^4 x^3}+\frac{c^3 (8-15 a x) \sqrt{1-a^2 x^2}}{8 a^2 x}+\frac{15 c^3 \tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right )}{8 a}+\frac{c^3 \sin ^{-1}(a x)}{a} \]

[Out]

(c^3*(8 - 15*a*x)*Sqrt[1 - a^2*x^2])/(8*a^2*x) - (c^3*(8 + 15*a*x)*(1 - a^2*x^2)^(3/2))/(24*a^4*x^3) + (c^3*(4
+ 5*a*x)*(1 - a^2*x^2)^(5/2))/(20*a^6*x^5) + (c^3*ArcSin[a*x])/a + (15*c^3*ArcTanh[Sqrt[1 - a^2*x^2]])/(8*a)

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Rubi [A]  time = 0.187181, antiderivative size = 136, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 9, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.45, Rules used = {6157, 6148, 811, 813, 844, 216, 266, 63, 208} \[ \frac{c^3 (5 a x+4) \left (1-a^2 x^2\right )^{5/2}}{20 a^6 x^5}-\frac{c^3 (15 a x+8) \left (1-a^2 x^2\right )^{3/2}}{24 a^4 x^3}+\frac{c^3 (8-15 a x) \sqrt{1-a^2 x^2}}{8 a^2 x}+\frac{15 c^3 \tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right )}{8 a}+\frac{c^3 \sin ^{-1}(a x)}{a} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(c^3*(8 - 15*a*x)*Sqrt[1 - a^2*x^2])/(8*a^2*x) - (c^3*(8 + 15*a*x)*(1 - a^2*x^2)^(3/2))/(24*a^4*x^3) + (c^3*(4
+ 5*a*x)*(1 - a^2*x^2)^(5/2))/(20*a^6*x^5) + (c^3*ArcSin[a*x])/a + (15*c^3*ArcTanh[Sqrt[1 - a^2*x^2]])/(8*a)

Rule 6157

Int[E^(ArcTanh[(a_.)*(x_)]*(n_.))*(u_.)*((c_) + (d_.)/(x_)^2)^(p_.), x_Symbol] :> Dist[d^p, Int[(u*(1 - a^2*x^
2)^p*E^(n*ArcTanh[a*x]))/x^(2*p), x], x] /; FreeQ[{a, c, d, n}, x] && EqQ[c + a^2*d, 0] && IntegerQ[p]

Rule 6148

Int[E^(ArcTanh[(a_.)*(x_)]*(n_.))*(x_)^(m_.)*((c_) + (d_.)*(x_)^2)^(p_.), x_Symbol] :> Dist[c^p, Int[x^m*(1 -
a^2*x^2)^(p - n/2)*(1 + a*x)^n, x], x] /; FreeQ[{a, c, d, m, p}, x] && EqQ[a^2*c + d, 0] && (IntegerQ[p] || Gt
Q[c, 0]) && IGtQ[(n + 1)/2, 0] &&  !IntegerQ[p - n/2]

Rule 811

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> -Simp[((d + e*x)^
(m + 1)*(a + c*x^2)^p*((d*g - e*f*(m + 2))*(c*d^2 + a*e^2) - 2*c*d^2*p*(e*f - d*g) - e*(g*(m + 1)*(c*d^2 + a*e
^2) + 2*c*d*p*(e*f - d*g))*x))/(e^2*(m + 1)*(m + 2)*(c*d^2 + a*e^2)), x] - Dist[p/(e^2*(m + 1)*(m + 2)*(c*d^2
+ a*e^2)), Int[(d + e*x)^(m + 2)*(a + c*x^2)^(p - 1)*Simp[2*a*c*e*(e*f - d*g)*(m + 2) - c*(2*c*d*(d*g*(2*p + 1
) - e*f*(m + 2*p + 2)) - 2*a*e^2*g*(m + 1))*x, x], x], x] /; FreeQ[{a, c, d, e, f, g}, x] && NeQ[c*d^2 + a*e^2
, 0] && GtQ[p, 0] && LtQ[m, -2] && LtQ[m + 2*p, 0] &&  !ILtQ[m + 2*p + 3, 0]

Rule 813

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[((d + e*x)^(
m + 1)*(e*f*(m + 2*p + 2) - d*g*(2*p + 1) + e*g*(m + 1)*x)*(a + c*x^2)^p)/(e^2*(m + 1)*(m + 2*p + 2)), x] + Di
st[p/(e^2*(m + 1)*(m + 2*p + 2)), Int[(d + e*x)^(m + 1)*(a + c*x^2)^(p - 1)*Simp[g*(2*a*e + 2*a*e*m) + (g*(2*c
*d + 4*c*d*p) - 2*c*e*f*(m + 2*p + 2))*x, x], x], x] /; FreeQ[{a, c, d, e, f, g, m}, x] && NeQ[c*d^2 + a*e^2,
0] && RationalQ[p] && p > 0 && (LtQ[m, -1] || EqQ[p, 1] || (IntegerQ[p] &&  !RationalQ[m])) && NeQ[m, -1] &&
!ILtQ[m + 2*p + 1, 0] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])

Rule 844

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Dist[g/e, Int[(d
+ e*x)^(m + 1)*(a + c*x^2)^p, x], x] + Dist[(e*f - d*g)/e, Int[(d + e*x)^m*(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] &&  !IGtQ[m, 0]

Rule 216

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 266

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 63

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 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 e^{\tanh ^{-1}(a x)} \left (c-\frac{c}{a^2 x^2}\right )^3 \, dx &=-\frac{c^3 \int \frac{e^{\tanh ^{-1}(a x)} \left (1-a^2 x^2\right )^3}{x^6} \, dx}{a^6}\\ &=-\frac{c^3 \int \frac{(1+a x) \left (1-a^2 x^2\right )^{5/2}}{x^6} \, dx}{a^6}\\ &=\frac{c^3 (4+5 a x) \left (1-a^2 x^2\right )^{5/2}}{20 a^6 x^5}+\frac{c^3 \int \frac{\left (8 a^2+10 a^3 x\right ) \left (1-a^2 x^2\right )^{3/2}}{x^4} \, dx}{8 a^6}\\ &=-\frac{c^3 (8+15 a x) \left (1-a^2 x^2\right )^{3/2}}{24 a^4 x^3}+\frac{c^3 (4+5 a x) \left (1-a^2 x^2\right )^{5/2}}{20 a^6 x^5}-\frac{c^3 \int \frac{\left (32 a^4+60 a^5 x\right ) \sqrt{1-a^2 x^2}}{x^2} \, dx}{32 a^6}\\ &=\frac{c^3 (8-15 a x) \sqrt{1-a^2 x^2}}{8 a^2 x}-\frac{c^3 (8+15 a x) \left (1-a^2 x^2\right )^{3/2}}{24 a^4 x^3}+\frac{c^3 (4+5 a x) \left (1-a^2 x^2\right )^{5/2}}{20 a^6 x^5}+\frac{c^3 \int \frac{-120 a^5+64 a^6 x}{x \sqrt{1-a^2 x^2}} \, dx}{64 a^6}\\ &=\frac{c^3 (8-15 a x) \sqrt{1-a^2 x^2}}{8 a^2 x}-\frac{c^3 (8+15 a x) \left (1-a^2 x^2\right )^{3/2}}{24 a^4 x^3}+\frac{c^3 (4+5 a x) \left (1-a^2 x^2\right )^{5/2}}{20 a^6 x^5}+c^3 \int \frac{1}{\sqrt{1-a^2 x^2}} \, dx-\frac{\left (15 c^3\right ) \int \frac{1}{x \sqrt{1-a^2 x^2}} \, dx}{8 a}\\ &=\frac{c^3 (8-15 a x) \sqrt{1-a^2 x^2}}{8 a^2 x}-\frac{c^3 (8+15 a x) \left (1-a^2 x^2\right )^{3/2}}{24 a^4 x^3}+\frac{c^3 (4+5 a x) \left (1-a^2 x^2\right )^{5/2}}{20 a^6 x^5}+\frac{c^3 \sin ^{-1}(a x)}{a}-\frac{\left (15 c^3\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{1-a^2 x}} \, dx,x,x^2\right )}{16 a}\\ &=\frac{c^3 (8-15 a x) \sqrt{1-a^2 x^2}}{8 a^2 x}-\frac{c^3 (8+15 a x) \left (1-a^2 x^2\right )^{3/2}}{24 a^4 x^3}+\frac{c^3 (4+5 a x) \left (1-a^2 x^2\right )^{5/2}}{20 a^6 x^5}+\frac{c^3 \sin ^{-1}(a x)}{a}+\frac{\left (15 c^3\right ) \operatorname{Subst}\left (\int \frac{1}{\frac{1}{a^2}-\frac{x^2}{a^2}} \, dx,x,\sqrt{1-a^2 x^2}\right )}{8 a^3}\\ &=\frac{c^3 (8-15 a x) \sqrt{1-a^2 x^2}}{8 a^2 x}-\frac{c^3 (8+15 a x) \left (1-a^2 x^2\right )^{3/2}}{24 a^4 x^3}+\frac{c^3 (4+5 a x) \left (1-a^2 x^2\right )^{5/2}}{20 a^6 x^5}+\frac{c^3 \sin ^{-1}(a x)}{a}+\frac{15 c^3 \tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right )}{8 a}\\ \end{align*}

Mathematica [C]  time = 0.0263298, size = 70, normalized size = 0.51 \[ \frac{c^3 \left (5 a^5 \left (1-a^2 x^2\right )^{7/2} \text{Hypergeometric2F1}\left (3,\frac{7}{2},\frac{9}{2},1-a^2 x^2\right )+\frac{7 \text{Hypergeometric2F1}\left (-\frac{5}{2},-\frac{5}{2},-\frac{3}{2},a^2 x^2\right )}{x^5}\right )}{35 a^6} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(c^3*((7*Hypergeometric2F1[-5/2, -5/2, -3/2, a^2*x^2])/x^5 + 5*a^5*(1 - a^2*x^2)^(7/2)*Hypergeometric2F1[3, 7/
2, 9/2, 1 - a^2*x^2]))/(35*a^6)

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Maple [A]  time = 0.053, size = 187, normalized size = 1.4 \begin{align*} -{\frac{{c}^{3}}{a}\sqrt{-{a}^{2}{x}^{2}+1}}+{{c}^{3}\arctan \left ({x\sqrt{{a}^{2}}{\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}} \right ){\frac{1}{\sqrt{{a}^{2}}}}}+{\frac{{c}^{3}}{4\,{a}^{5}{x}^{4}}\sqrt{-{a}^{2}{x}^{2}+1}}-{\frac{9\,{c}^{3}}{8\,{x}^{2}{a}^{3}}\sqrt{-{a}^{2}{x}^{2}+1}}+{\frac{15\,{c}^{3}}{8\,a}{\it Artanh} \left ({\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}} \right ) }+{\frac{23\,{c}^{3}}{15\,{a}^{2}x}\sqrt{-{a}^{2}{x}^{2}+1}}+{\frac{{c}^{3}}{5\,{a}^{6}{x}^{5}}\sqrt{-{a}^{2}{x}^{2}+1}}-{\frac{11\,{c}^{3}}{15\,{a}^{4}{x}^{3}}\sqrt{-{a}^{2}{x}^{2}+1}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-c^3*(-a^2*x^2+1)^(1/2)/a+c^3/(a^2)^(1/2)*arctan((a^2)^(1/2)*x/(-a^2*x^2+1)^(1/2))+1/4*c^3/a^5/x^4*(-a^2*x^2+1
)^(1/2)-9/8*c^3*(-a^2*x^2+1)^(1/2)/x^2/a^3+15/8*c^3/a*arctanh(1/(-a^2*x^2+1)^(1/2))+23/15*c^3*(-a^2*x^2+1)^(1/
2)/a^2/x+1/5*c^3/a^6/x^5*(-a^2*x^2+1)^(1/2)-11/15*c^3/a^4/x^3*(-a^2*x^2+1)^(1/2)

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Maxima [B]  time = 1.48176, size = 460, normalized size = 3.38 \begin{align*} \frac{c^{3} \arcsin \left (\frac{a^{2} x}{\sqrt{a^{2}}}\right )}{\sqrt{a^{2}}} + \frac{3 \, c^{3} \log \left (\frac{2 \, \sqrt{-a^{2} x^{2} + 1}}{{\left | x \right |}} + \frac{2}{{\left | x \right |}}\right )}{a} - \frac{\sqrt{-a^{2} x^{2} + 1} c^{3}}{a} - \frac{3 \,{\left (a^{2} \log \left (\frac{2 \, \sqrt{-a^{2} x^{2} + 1}}{{\left | x \right |}} + \frac{2}{{\left | x \right |}}\right ) + \frac{\sqrt{-a^{2} x^{2} + 1}}{x^{2}}\right )} c^{3}}{2 \, a^{3}} + \frac{3 \, \sqrt{-a^{2} x^{2} + 1} c^{3}}{a^{2} x} - \frac{{\left (\frac{2 \, \sqrt{-a^{2} x^{2} + 1} a^{2}}{x} + \frac{\sqrt{-a^{2} x^{2} + 1}}{x^{3}}\right )} c^{3}}{a^{4}} + \frac{{\left (3 \, a^{4} \log \left (\frac{2 \, \sqrt{-a^{2} x^{2} + 1}}{{\left | x \right |}} + \frac{2}{{\left | x \right |}}\right ) + \frac{3 \, \sqrt{-a^{2} x^{2} + 1} a^{2}}{x^{2}} + \frac{2 \, \sqrt{-a^{2} x^{2} + 1}}{x^{4}}\right )} c^{3}}{8 \, a^{5}} + \frac{{\left (\frac{8 \, \sqrt{-a^{2} x^{2} + 1} a^{4}}{x} + \frac{4 \, \sqrt{-a^{2} x^{2} + 1} a^{2}}{x^{3}} + \frac{3 \, \sqrt{-a^{2} x^{2} + 1}}{x^{5}}\right )} c^{3}}{15 \, a^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

c^3*arcsin(a^2*x/sqrt(a^2))/sqrt(a^2) + 3*c^3*log(2*sqrt(-a^2*x^2 + 1)/abs(x) + 2/abs(x))/a - sqrt(-a^2*x^2 +
1)*c^3/a - 3/2*(a^2*log(2*sqrt(-a^2*x^2 + 1)/abs(x) + 2/abs(x)) + sqrt(-a^2*x^2 + 1)/x^2)*c^3/a^3 + 3*sqrt(-a^
2*x^2 + 1)*c^3/(a^2*x) - (2*sqrt(-a^2*x^2 + 1)*a^2/x + sqrt(-a^2*x^2 + 1)/x^3)*c^3/a^4 + 1/8*(3*a^4*log(2*sqrt
(-a^2*x^2 + 1)/abs(x) + 2/abs(x)) + 3*sqrt(-a^2*x^2 + 1)*a^2/x^2 + 2*sqrt(-a^2*x^2 + 1)/x^4)*c^3/a^5 + 1/15*(8
*sqrt(-a^2*x^2 + 1)*a^4/x + 4*sqrt(-a^2*x^2 + 1)*a^2/x^3 + 3*sqrt(-a^2*x^2 + 1)/x^5)*c^3/a^6

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Fricas [A]  time = 2.1743, size = 347, normalized size = 2.55 \begin{align*} -\frac{240 \, a^{5} c^{3} x^{5} \arctan \left (\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right ) + 225 \, a^{5} c^{3} x^{5} \log \left (\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{x}\right ) + 120 \, a^{5} c^{3} x^{5} +{\left (120 \, a^{5} c^{3} x^{5} - 184 \, a^{4} c^{3} x^{4} + 135 \, a^{3} c^{3} x^{3} + 88 \, a^{2} c^{3} x^{2} - 30 \, a c^{3} x - 24 \, c^{3}\right )} \sqrt{-a^{2} x^{2} + 1}}{120 \, a^{6} x^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/120*(240*a^5*c^3*x^5*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) + 225*a^5*c^3*x^5*log((sqrt(-a^2*x^2 + 1) - 1)/
x) + 120*a^5*c^3*x^5 + (120*a^5*c^3*x^5 - 184*a^4*c^3*x^4 + 135*a^3*c^3*x^3 + 88*a^2*c^3*x^2 - 30*a*c^3*x - 24
*c^3)*sqrt(-a^2*x^2 + 1))/(a^6*x^5)

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Sympy [A]  time = 54.0499, size = 687, normalized size = 5.05 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a*c**3*Piecewise((x**2/2, Eq(a**2, 0)), (-sqrt(-a**2*x**2 + 1)/a**2, True)) + c**3*Piecewise((sqrt(a**(-2))*as
in(x*sqrt(a**2)), a**2 > 0), (sqrt(-1/a**2)*asinh(x*sqrt(-a**2)), a**2 < 0)) - 3*c**3*Piecewise((-acosh(1/(a*x
)), 1/Abs(a**2*x**2) > 1), (I*asin(1/(a*x)), True))/a - 3*c**3*Piecewise((-I*sqrt(a**2*x**2 - 1)/x, Abs(a**2*x
**2) > 1), (-sqrt(-a**2*x**2 + 1)/x, True))/a**2 + 3*c**3*Piecewise((-a**2*acosh(1/(a*x))/2 - a*sqrt(-1 + 1/(a
**2*x**2))/(2*x), 1/Abs(a**2*x**2) > 1), (I*a**2*asin(1/(a*x))/2 - I*a/(2*x*sqrt(1 - 1/(a**2*x**2))) + I/(2*a*
x**3*sqrt(1 - 1/(a**2*x**2))), True))/a**3 + 3*c**3*Piecewise((-2*I*a**2*sqrt(a**2*x**2 - 1)/(3*x) - I*sqrt(a*
*2*x**2 - 1)/(3*x**3), Abs(a**2*x**2) > 1), (-2*a**2*sqrt(-a**2*x**2 + 1)/(3*x) - sqrt(-a**2*x**2 + 1)/(3*x**3
), True))/a**4 - c**3*Piecewise((-3*a**4*acosh(1/(a*x))/8 + 3*a**3/(8*x*sqrt(-1 + 1/(a**2*x**2))) - a/(8*x**3*
sqrt(-1 + 1/(a**2*x**2))) - 1/(4*a*x**5*sqrt(-1 + 1/(a**2*x**2))), 1/Abs(a**2*x**2) > 1), (3*I*a**4*asin(1/(a*
x))/8 - 3*I*a**3/(8*x*sqrt(1 - 1/(a**2*x**2))) + I*a/(8*x**3*sqrt(1 - 1/(a**2*x**2))) + I/(4*a*x**5*sqrt(1 - 1
/(a**2*x**2))), True))/a**5 - c**3*Piecewise((-8*a**5*sqrt(-1 + 1/(a**2*x**2))/15 - 4*a**3*sqrt(-1 + 1/(a**2*x
**2))/(15*x**2) - a*sqrt(-1 + 1/(a**2*x**2))/(5*x**4), 1/Abs(a**2*x**2) > 1), (-8*I*a**5*sqrt(1 - 1/(a**2*x**2
))/15 - 4*I*a**3*sqrt(1 - 1/(a**2*x**2))/(15*x**2) - I*a*sqrt(1 - 1/(a**2*x**2))/(5*x**4), True))/a**6

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Giac [B]  time = 1.22412, size = 520, normalized size = 3.82 \begin{align*} -\frac{{\left (6 \, c^{3} + \frac{15 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )} c^{3}}{a^{2} x} - \frac{70 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{2} c^{3}}{a^{4} x^{2}} - \frac{240 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{3} c^{3}}{a^{6} x^{3}} + \frac{660 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{4} c^{3}}{a^{8} x^{4}}\right )} a^{10} x^{5}}{960 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{5}{\left | a \right |}} + \frac{c^{3} \arcsin \left (a x\right ) \mathrm{sgn}\left (a\right )}{{\left | a \right |}} + \frac{15 \, c^{3} \log \left (\frac{{\left | -2 \, \sqrt{-a^{2} x^{2} + 1}{\left | a \right |} - 2 \, a \right |}}{2 \, a^{2}{\left | x \right |}}\right )}{8 \,{\left | a \right |}} - \frac{\sqrt{-a^{2} x^{2} + 1} c^{3}}{a} + \frac{\frac{660 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )} a^{2} c^{3}}{x} - \frac{240 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{2} c^{3}}{x^{2}} - \frac{70 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{3} c^{3}}{a^{2} x^{3}} + \frac{15 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{4} c^{3}}{a^{4} x^{4}} + \frac{6 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{5} c^{3}}{a^{6} x^{5}}}{960 \, a^{4}{\left | a \right |}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/960*(6*c^3 + 15*(sqrt(-a^2*x^2 + 1)*abs(a) + a)*c^3/(a^2*x) - 70*(sqrt(-a^2*x^2 + 1)*abs(a) + a)^2*c^3/(a^4
*x^2) - 240*(sqrt(-a^2*x^2 + 1)*abs(a) + a)^3*c^3/(a^6*x^3) + 660*(sqrt(-a^2*x^2 + 1)*abs(a) + a)^4*c^3/(a^8*x
^4))*a^10*x^5/((sqrt(-a^2*x^2 + 1)*abs(a) + a)^5*abs(a)) + c^3*arcsin(a*x)*sgn(a)/abs(a) + 15/8*c^3*log(1/2*ab
s(-2*sqrt(-a^2*x^2 + 1)*abs(a) - 2*a)/(a^2*abs(x)))/abs(a) - sqrt(-a^2*x^2 + 1)*c^3/a + 1/960*(660*(sqrt(-a^2*
x^2 + 1)*abs(a) + a)*a^2*c^3/x - 240*(sqrt(-a^2*x^2 + 1)*abs(a) + a)^2*c^3/x^2 - 70*(sqrt(-a^2*x^2 + 1)*abs(a)
+ a)^3*c^3/(a^2*x^3) + 15*(sqrt(-a^2*x^2 + 1)*abs(a) + a)^4*c^3/(a^4*x^4) + 6*(sqrt(-a^2*x^2 + 1)*abs(a) + a)
^5*c^3/(a^6*x^5))/(a^4*abs(a))