3.466 \(\int e^{3 \tanh ^{-1}(a x)} (c-\frac{c}{a x})^4 \, dx\)

Optimal. Leaf size=103 \[ -\frac{c^4 (2-3 a x) \left (1-a^2 x^2\right )^{3/2}}{6 a^4 x^3}+\frac{c^4 (3 a x+2) \sqrt{1-a^2 x^2}}{2 a^2 x}-\frac{3 c^4 \tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right )}{2 a}+\frac{c^4 \sin ^{-1}(a x)}{a} \]

[Out]

(c^4*(2 + 3*a*x)*Sqrt[1 - a^2*x^2])/(2*a^2*x) - (c^4*(2 - 3*a*x)*(1 - a^2*x^2)^(3/2))/(6*a^4*x^3) + (c^4*ArcSi
n[a*x])/a - (3*c^4*ArcTanh[Sqrt[1 - a^2*x^2]])/(2*a)

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Rubi [A]  time = 0.167968, antiderivative size = 103, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 9, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.409, Rules used = {6131, 6128, 811, 813, 844, 216, 266, 63, 208} \[ -\frac{c^4 (2-3 a x) \left (1-a^2 x^2\right )^{3/2}}{6 a^4 x^3}+\frac{c^4 (3 a x+2) \sqrt{1-a^2 x^2}}{2 a^2 x}-\frac{3 c^4 \tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right )}{2 a}+\frac{c^4 \sin ^{-1}(a x)}{a} \]

Antiderivative was successfully verified.

[In]

Int[E^(3*ArcTanh[a*x])*(c - c/(a*x))^4,x]

[Out]

(c^4*(2 + 3*a*x)*Sqrt[1 - a^2*x^2])/(2*a^2*x) - (c^4*(2 - 3*a*x)*(1 - a^2*x^2)^(3/2))/(6*a^4*x^3) + (c^4*ArcSi
n[a*x])/a - (3*c^4*ArcTanh[Sqrt[1 - a^2*x^2]])/(2*a)

Rule 6131

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

Rule 6128

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

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^{3 \tanh ^{-1}(a x)} \left (c-\frac{c}{a x}\right )^4 \, dx &=\frac{c^4 \int \frac{e^{3 \tanh ^{-1}(a x)} (1-a x)^4}{x^4} \, dx}{a^4}\\ &=\frac{c^4 \int \frac{(1-a x) \left (1-a^2 x^2\right )^{3/2}}{x^4} \, dx}{a^4}\\ &=-\frac{c^4 (2-3 a x) \left (1-a^2 x^2\right )^{3/2}}{6 a^4 x^3}-\frac{c^4 \int \frac{\left (4 a^2-6 a^3 x\right ) \sqrt{1-a^2 x^2}}{x^2} \, dx}{4 a^4}\\ &=\frac{c^4 (2+3 a x) \sqrt{1-a^2 x^2}}{2 a^2 x}-\frac{c^4 (2-3 a x) \left (1-a^2 x^2\right )^{3/2}}{6 a^4 x^3}+\frac{c^4 \int \frac{12 a^3+8 a^4 x}{x \sqrt{1-a^2 x^2}} \, dx}{8 a^4}\\ &=\frac{c^4 (2+3 a x) \sqrt{1-a^2 x^2}}{2 a^2 x}-\frac{c^4 (2-3 a x) \left (1-a^2 x^2\right )^{3/2}}{6 a^4 x^3}+c^4 \int \frac{1}{\sqrt{1-a^2 x^2}} \, dx+\frac{\left (3 c^4\right ) \int \frac{1}{x \sqrt{1-a^2 x^2}} \, dx}{2 a}\\ &=\frac{c^4 (2+3 a x) \sqrt{1-a^2 x^2}}{2 a^2 x}-\frac{c^4 (2-3 a x) \left (1-a^2 x^2\right )^{3/2}}{6 a^4 x^3}+\frac{c^4 \sin ^{-1}(a x)}{a}+\frac{\left (3 c^4\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{1-a^2 x}} \, dx,x,x^2\right )}{4 a}\\ &=\frac{c^4 (2+3 a x) \sqrt{1-a^2 x^2}}{2 a^2 x}-\frac{c^4 (2-3 a x) \left (1-a^2 x^2\right )^{3/2}}{6 a^4 x^3}+\frac{c^4 \sin ^{-1}(a x)}{a}-\frac{\left (3 c^4\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 )}{2 a^3}\\ &=\frac{c^4 (2+3 a x) \sqrt{1-a^2 x^2}}{2 a^2 x}-\frac{c^4 (2-3 a x) \left (1-a^2 x^2\right )^{3/2}}{6 a^4 x^3}+\frac{c^4 \sin ^{-1}(a x)}{a}-\frac{3 c^4 \tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right )}{2 a}\\ \end{align*}

Mathematica [A]  time = 0.243841, size = 85, normalized size = 0.83 \[ \frac{c^4 \left (\frac{\sqrt{1-a^2 x^2} \left (6 a^3 x^3+8 a^2 x^2+3 a x-2\right )}{a^3 x^3}-9 \log \left (\sqrt{1-a^2 x^2}+1\right )+9 \log (a x)+6 \sin ^{-1}(a x)\right )}{6 a} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[E^(3*ArcTanh[a*x])*(c - c/(a*x))^4,x]

[Out]

(c^4*((Sqrt[1 - a^2*x^2]*(-2 + 3*a*x + 8*a^2*x^2 + 6*a^3*x^3))/(a^3*x^3) + 6*ArcSin[a*x] + 9*Log[a*x] - 9*Log[
1 + Sqrt[1 - a^2*x^2]]))/(6*a)

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Maple [A]  time = 0.052, size = 180, normalized size = 1.8 \begin{align*} -{{c}^{4}a{x}^{2}{\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}}+{\frac{{c}^{4}}{2\,a}{\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}}-{\frac{4\,{c}^{4}x}{3}{\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}}+{{c}^{4}\arctan \left ({x\sqrt{{a}^{2}}{\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}} \right ){\frac{1}{\sqrt{{a}^{2}}}}}+{\frac{5\,{c}^{4}}{3\,{a}^{2}x}{\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}}-{\frac{3\,{c}^{4}}{2\,a}{\it Artanh} \left ({\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}} \right ) }+{\frac{{c}^{4}}{2\,{x}^{2}{a}^{3}}{\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}}-{\frac{{c}^{4}}{3\,{a}^{4}{x}^{3}}{\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a*x+1)^3/(-a^2*x^2+1)^(3/2)*(c-c/a/x)^4,x)

[Out]

-c^4*a*x^2/(-a^2*x^2+1)^(1/2)+1/2*c^4/a/(-a^2*x^2+1)^(1/2)-4/3*c^4*x/(-a^2*x^2+1)^(1/2)+c^4/(a^2)^(1/2)*arctan
((a^2)^(1/2)*x/(-a^2*x^2+1)^(1/2))+5/3*c^4/a^2/x/(-a^2*x^2+1)^(1/2)-3/2*c^4/a*arctanh(1/(-a^2*x^2+1)^(1/2))+1/
2*c^4/a^3/x^2/(-a^2*x^2+1)^(1/2)-1/3*c^4/a^4/x^3/(-a^2*x^2+1)^(1/2)

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Maxima [B]  time = 1.47497, size = 487, normalized size = 4.73 \begin{align*} -a^{3} c^{4}{\left (\frac{x^{2}}{\sqrt{-a^{2} x^{2} + 1} a^{2}} - \frac{2}{\sqrt{-a^{2} x^{2} + 1} a^{4}}\right )} - a^{2} c^{4}{\left (\frac{x}{\sqrt{-a^{2} x^{2} + 1} a^{2}} - \frac{\arcsin \left (\frac{a^{2} x}{\sqrt{a^{2}}}\right )}{\sqrt{a^{2}} a^{2}}\right )} + \frac{3 \, c^{4} x}{\sqrt{-a^{2} x^{2} + 1}} + \frac{3 \, c^{4}{\left (\frac{1}{\sqrt{-a^{2} x^{2} + 1}} - \log \left (\frac{2 \, \sqrt{-a^{2} x^{2} + 1}}{{\left | x \right |}} + \frac{2}{{\left | x \right |}}\right )\right )}}{a} - \frac{3 \,{\left (\frac{2 \, a^{2} x}{\sqrt{-a^{2} x^{2} + 1}} - \frac{1}{\sqrt{-a^{2} x^{2} + 1} x}\right )} c^{4}}{a^{2}} - \frac{3 \, c^{4}}{\sqrt{-a^{2} x^{2} + 1} a} + \frac{{\left (3 \, a^{2} \log \left (\frac{2 \, \sqrt{-a^{2} x^{2} + 1}}{{\left | x \right |}} + \frac{2}{{\left | x \right |}}\right ) - \frac{3 \, a^{2}}{\sqrt{-a^{2} x^{2} + 1}} + \frac{1}{\sqrt{-a^{2} x^{2} + 1} x^{2}}\right )} c^{4}}{2 \, a^{3}} + \frac{{\left (\frac{8 \, a^{4} x}{\sqrt{-a^{2} x^{2} + 1}} - \frac{4 \, a^{2}}{\sqrt{-a^{2} x^{2} + 1} x} - \frac{1}{\sqrt{-a^{2} x^{2} + 1} x^{3}}\right )} c^{4}}{3 \, a^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)^3/(-a^2*x^2+1)^(3/2)*(c-c/a/x)^4,x, algorithm="maxima")

[Out]

-a^3*c^4*(x^2/(sqrt(-a^2*x^2 + 1)*a^2) - 2/(sqrt(-a^2*x^2 + 1)*a^4)) - a^2*c^4*(x/(sqrt(-a^2*x^2 + 1)*a^2) - a
rcsin(a^2*x/sqrt(a^2))/(sqrt(a^2)*a^2)) + 3*c^4*x/sqrt(-a^2*x^2 + 1) + 3*c^4*(1/sqrt(-a^2*x^2 + 1) - log(2*sqr
t(-a^2*x^2 + 1)/abs(x) + 2/abs(x)))/a - 3*(2*a^2*x/sqrt(-a^2*x^2 + 1) - 1/(sqrt(-a^2*x^2 + 1)*x))*c^4/a^2 - 3*
c^4/(sqrt(-a^2*x^2 + 1)*a) + 1/2*(3*a^2*log(2*sqrt(-a^2*x^2 + 1)/abs(x) + 2/abs(x)) - 3*a^2/sqrt(-a^2*x^2 + 1)
+ 1/(sqrt(-a^2*x^2 + 1)*x^2))*c^4/a^3 + 1/3*(8*a^4*x/sqrt(-a^2*x^2 + 1) - 4*a^2/(sqrt(-a^2*x^2 + 1)*x) - 1/(s
qrt(-a^2*x^2 + 1)*x^3))*c^4/a^4

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Fricas [A]  time = 2.1517, size = 282, normalized size = 2.74 \begin{align*} -\frac{12 \, a^{3} c^{4} x^{3} \arctan \left (\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right ) - 9 \, a^{3} c^{4} x^{3} \log \left (\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{x}\right ) - 6 \, a^{3} c^{4} x^{3} -{\left (6 \, a^{3} c^{4} x^{3} + 8 \, a^{2} c^{4} x^{2} + 3 \, a c^{4} x - 2 \, c^{4}\right )} \sqrt{-a^{2} x^{2} + 1}}{6 \, a^{4} x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)^3/(-a^2*x^2+1)^(3/2)*(c-c/a/x)^4,x, algorithm="fricas")

[Out]

-1/6*(12*a^3*c^4*x^3*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) - 9*a^3*c^4*x^3*log((sqrt(-a^2*x^2 + 1) - 1)/x) -
6*a^3*c^4*x^3 - (6*a^3*c^4*x^3 + 8*a^2*c^4*x^2 + 3*a*c^4*x - 2*c^4)*sqrt(-a^2*x^2 + 1))/(a^4*x^3)

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Sympy [A]  time = 13.3834, size = 354, normalized size = 3.44 \begin{align*} - a c^{4} \left (\begin{cases} \frac{x^{2}}{2} & \text{for}\: a^{2} = 0 \\- \frac{\sqrt{- a^{2} x^{2} + 1}}{a^{2}} & \text{otherwise} \end{cases}\right ) + c^{4} \left (\begin{cases} \sqrt{\frac{1}{a^{2}}} \operatorname{asin}{\left (x \sqrt{a^{2}} \right )} & \text{for}\: a^{2} > 0 \\\sqrt{- \frac{1}{a^{2}}} \operatorname{asinh}{\left (x \sqrt{- a^{2}} \right )} & \text{for}\: a^{2} < 0 \end{cases}\right ) + \frac{2 c^{4} \left (\begin{cases} - \operatorname{acosh}{\left (\frac{1}{a x} \right )} & \text{for}\: \frac{1}{\left |{a^{2} x^{2}}\right |} > 1 \\i \operatorname{asin}{\left (\frac{1}{a x} \right )} & \text{otherwise} \end{cases}\right )}{a} - \frac{2 c^{4} \left (\begin{cases} - \frac{i \sqrt{a^{2} x^{2} - 1}}{x} & \text{for}\: \left |{a^{2} x^{2}}\right | > 1 \\- \frac{\sqrt{- a^{2} x^{2} + 1}}{x} & \text{otherwise} \end{cases}\right )}{a^{2}} - \frac{c^{4} \left (\begin{cases} - \frac{a^{2} \operatorname{acosh}{\left (\frac{1}{a x} \right )}}{2} - \frac{a \sqrt{-1 + \frac{1}{a^{2} x^{2}}}}{2 x} & \text{for}\: \frac{1}{\left |{a^{2} x^{2}}\right |} > 1 \\\frac{i a^{2} \operatorname{asin}{\left (\frac{1}{a x} \right )}}{2} - \frac{i a}{2 x \sqrt{1 - \frac{1}{a^{2} x^{2}}}} + \frac{i}{2 a x^{3} \sqrt{1 - \frac{1}{a^{2} x^{2}}}} & \text{otherwise} \end{cases}\right )}{a^{3}} + \frac{c^{4} \left (\begin{cases} - \frac{2 i a^{2} \sqrt{a^{2} x^{2} - 1}}{3 x} - \frac{i \sqrt{a^{2} x^{2} - 1}}{3 x^{3}} & \text{for}\: \left |{a^{2} x^{2}}\right | > 1 \\- \frac{2 a^{2} \sqrt{- a^{2} x^{2} + 1}}{3 x} - \frac{\sqrt{- a^{2} x^{2} + 1}}{3 x^{3}} & \text{otherwise} \end{cases}\right )}{a^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)**3/(-a**2*x**2+1)**(3/2)*(c-c/a/x)**4,x)

[Out]

-a*c**4*Piecewise((x**2/2, Eq(a**2, 0)), (-sqrt(-a**2*x**2 + 1)/a**2, True)) + c**4*Piecewise((sqrt(a**(-2))*a
sin(x*sqrt(a**2)), a**2 > 0), (sqrt(-1/a**2)*asinh(x*sqrt(-a**2)), a**2 < 0)) + 2*c**4*Piecewise((-acosh(1/(a*
x)), 1/Abs(a**2*x**2) > 1), (I*asin(1/(a*x)), True))/a - 2*c**4*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 - c**4*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 + c**4*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

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Giac [B]  time = 1.19015, size = 354, normalized size = 3.44 \begin{align*} \frac{{\left (c^{4} - \frac{3 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )} c^{4}}{a^{2} x} - \frac{15 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{2} c^{4}}{a^{4} x^{2}}\right )} a^{6} x^{3}}{24 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{3}{\left | a \right |}} + \frac{c^{4} \arcsin \left (a x\right ) \mathrm{sgn}\left (a\right )}{{\left | a \right |}} - \frac{3 \, c^{4} \log \left (\frac{{\left | -2 \, \sqrt{-a^{2} x^{2} + 1}{\left | a \right |} - 2 \, a \right |}}{2 \, a^{2}{\left | x \right |}}\right )}{2 \,{\left | a \right |}} + \frac{\sqrt{-a^{2} x^{2} + 1} c^{4}}{a} + \frac{\frac{15 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )} c^{4}}{x} + \frac{3 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{2} c^{4}}{a^{2} x^{2}} - \frac{{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{3} c^{4}}{a^{4} x^{3}}}{24 \, a^{2}{\left | a \right |}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/24*(c^4 - 3*(sqrt(-a^2*x^2 + 1)*abs(a) + a)*c^4/(a^2*x) - 15*(sqrt(-a^2*x^2 + 1)*abs(a) + a)^2*c^4/(a^4*x^2)
)*a^6*x^3/((sqrt(-a^2*x^2 + 1)*abs(a) + a)^3*abs(a)) + c^4*arcsin(a*x)*sgn(a)/abs(a) - 3/2*c^4*log(1/2*abs(-2*
sqrt(-a^2*x^2 + 1)*abs(a) - 2*a)/(a^2*abs(x)))/abs(a) + sqrt(-a^2*x^2 + 1)*c^4/a + 1/24*(15*(sqrt(-a^2*x^2 + 1
)*abs(a) + a)*c^4/x + 3*(sqrt(-a^2*x^2 + 1)*abs(a) + a)^2*c^4/(a^2*x^2) - (sqrt(-a^2*x^2 + 1)*abs(a) + a)^3*c^
4/(a^4*x^3))/(a^2*abs(a))