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

Optimal. Leaf size=190 \[ \frac{(a x+1)^7}{9 a c^4 \left (1-a^2 x^2\right )^{9/2}}-\frac{34 (a x+1)^6}{63 a c^4 \left (1-a^2 x^2\right )^{7/2}}+\frac{344 (a x+1)^5}{315 a c^4 \left (1-a^2 x^2\right )^{5/2}}-\frac{4 (a x+1)^4}{3 a c^4 \left (1-a^2 x^2\right )^{3/2}}+\frac{14 (a x+1)^2}{3 a c^4 \sqrt{1-a^2 x^2}}+\frac{7 \sqrt{1-a^2 x^2}}{a c^4}-\frac{7 \sin ^{-1}(a x)}{a c^4} \]

[Out]

(1 + a*x)^7/(9*a*c^4*(1 - a^2*x^2)^(9/2)) - (34*(1 + a*x)^6)/(63*a*c^4*(1 - a^2*x^2)^(7/2)) + (344*(1 + a*x)^5
)/(315*a*c^4*(1 - a^2*x^2)^(5/2)) - (4*(1 + a*x)^4)/(3*a*c^4*(1 - a^2*x^2)^(3/2)) + (14*(1 + a*x)^2)/(3*a*c^4*
Sqrt[1 - a^2*x^2]) + (7*Sqrt[1 - a^2*x^2])/(a*c^4) - (7*ArcSin[a*x])/(a*c^4)

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Rubi [A]  time = 0.443257, antiderivative size = 190, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 8, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.364, Rules used = {6131, 6128, 852, 1635, 789, 669, 641, 216} \[ \frac{(a x+1)^7}{9 a c^4 \left (1-a^2 x^2\right )^{9/2}}-\frac{34 (a x+1)^6}{63 a c^4 \left (1-a^2 x^2\right )^{7/2}}+\frac{344 (a x+1)^5}{315 a c^4 \left (1-a^2 x^2\right )^{5/2}}-\frac{4 (a x+1)^4}{3 a c^4 \left (1-a^2 x^2\right )^{3/2}}+\frac{14 (a x+1)^2}{3 a c^4 \sqrt{1-a^2 x^2}}+\frac{7 \sqrt{1-a^2 x^2}}{a c^4}-\frac{7 \sin ^{-1}(a x)}{a c^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(1 + a*x)^7/(9*a*c^4*(1 - a^2*x^2)^(9/2)) - (34*(1 + a*x)^6)/(63*a*c^4*(1 - a^2*x^2)^(7/2)) + (344*(1 + a*x)^5
)/(315*a*c^4*(1 - a^2*x^2)^(5/2)) - (4*(1 + a*x)^4)/(3*a*c^4*(1 - a^2*x^2)^(3/2)) + (14*(1 + a*x)^2)/(3*a*c^4*
Sqrt[1 - a^2*x^2]) + (7*Sqrt[1 - a^2*x^2])/(a*c^4) - (7*ArcSin[a*x])/(a*c^4)

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 852

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 1635

Int[(Pq_)*((d_) + (e_.)*(x_))^(m_.)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuotient[Pq,
a*e + c*d*x, x], f = PolynomialRemainder[Pq, a*e + c*d*x, x]}, -Simp[(d*f*(d + e*x)^m*(a + c*x^2)^(p + 1))/(2*
a*e*(p + 1)), x] + Dist[d/(2*a*(p + 1)), Int[(d + e*x)^(m - 1)*(a + c*x^2)^(p + 1)*ExpandToSum[2*a*e*(p + 1)*Q
+ f*(m + 2*p + 2), x], x], x]] /; FreeQ[{a, c, d, e}, x] && PolyQ[Pq, x] && EqQ[c*d^2 + a*e^2, 0] && ILtQ[p +
1/2, 0] && GtQ[m, 0]

Rule 789

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

Rule 669

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

Rule 641

Int[((d_) + (e_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(e*(a + c*x^2)^(p + 1))/(2*c*(p + 1)),
x] + Dist[d, Int[(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, p}, x] && NeQ[p, -1]

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]

Rubi steps

\begin{align*} \int \frac{e^{3 \tanh ^{-1}(a x)}}{\left (c-\frac{c}{a x}\right )^4} \, dx &=\frac{a^4 \int \frac{e^{3 \tanh ^{-1}(a x)} x^4}{(1-a x)^4} \, dx}{c^4}\\ &=\frac{a^4 \int \frac{x^4 \left (1-a^2 x^2\right )^{3/2}}{(1-a x)^7} \, dx}{c^4}\\ &=\frac{a^4 \int \frac{x^4 (1+a x)^7}{\left (1-a^2 x^2\right )^{11/2}} \, dx}{c^4}\\ &=\frac{(1+a x)^7}{9 a c^4 \left (1-a^2 x^2\right )^{9/2}}-\frac{a^4 \int \frac{(1+a x)^6 \left (\frac{7}{a^4}+\frac{9 x}{a^3}+\frac{9 x^2}{a^2}+\frac{9 x^3}{a}\right )}{\left (1-a^2 x^2\right )^{9/2}} \, dx}{9 c^4}\\ &=\frac{(1+a x)^7}{9 a c^4 \left (1-a^2 x^2\right )^{9/2}}-\frac{34 (1+a x)^6}{63 a c^4 \left (1-a^2 x^2\right )^{7/2}}+\frac{a^4 \int \frac{(1+a x)^5 \left (\frac{155}{a^4}+\frac{126 x}{a^3}+\frac{63 x^2}{a^2}\right )}{\left (1-a^2 x^2\right )^{7/2}} \, dx}{63 c^4}\\ &=\frac{(1+a x)^7}{9 a c^4 \left (1-a^2 x^2\right )^{9/2}}-\frac{34 (1+a x)^6}{63 a c^4 \left (1-a^2 x^2\right )^{7/2}}+\frac{344 (1+a x)^5}{315 a c^4 \left (1-a^2 x^2\right )^{5/2}}-\frac{a^4 \int \frac{\left (\frac{945}{a^4}+\frac{315 x}{a^3}\right ) (1+a x)^4}{\left (1-a^2 x^2\right )^{5/2}} \, dx}{315 c^4}\\ &=\frac{(1+a x)^7}{9 a c^4 \left (1-a^2 x^2\right )^{9/2}}-\frac{34 (1+a x)^6}{63 a c^4 \left (1-a^2 x^2\right )^{7/2}}+\frac{344 (1+a x)^5}{315 a c^4 \left (1-a^2 x^2\right )^{5/2}}-\frac{4 (1+a x)^4}{3 a c^4 \left (1-a^2 x^2\right )^{3/2}}+\frac{7 \int \frac{(1+a x)^3}{\left (1-a^2 x^2\right )^{3/2}} \, dx}{3 c^4}\\ &=\frac{(1+a x)^7}{9 a c^4 \left (1-a^2 x^2\right )^{9/2}}-\frac{34 (1+a x)^6}{63 a c^4 \left (1-a^2 x^2\right )^{7/2}}+\frac{344 (1+a x)^5}{315 a c^4 \left (1-a^2 x^2\right )^{5/2}}-\frac{4 (1+a x)^4}{3 a c^4 \left (1-a^2 x^2\right )^{3/2}}+\frac{14 (1+a x)^2}{3 a c^4 \sqrt{1-a^2 x^2}}-\frac{7 \int \frac{1+a x}{\sqrt{1-a^2 x^2}} \, dx}{c^4}\\ &=\frac{(1+a x)^7}{9 a c^4 \left (1-a^2 x^2\right )^{9/2}}-\frac{34 (1+a x)^6}{63 a c^4 \left (1-a^2 x^2\right )^{7/2}}+\frac{344 (1+a x)^5}{315 a c^4 \left (1-a^2 x^2\right )^{5/2}}-\frac{4 (1+a x)^4}{3 a c^4 \left (1-a^2 x^2\right )^{3/2}}+\frac{14 (1+a x)^2}{3 a c^4 \sqrt{1-a^2 x^2}}+\frac{7 \sqrt{1-a^2 x^2}}{a c^4}-\frac{7 \int \frac{1}{\sqrt{1-a^2 x^2}} \, dx}{c^4}\\ &=\frac{(1+a x)^7}{9 a c^4 \left (1-a^2 x^2\right )^{9/2}}-\frac{34 (1+a x)^6}{63 a c^4 \left (1-a^2 x^2\right )^{7/2}}+\frac{344 (1+a x)^5}{315 a c^4 \left (1-a^2 x^2\right )^{5/2}}-\frac{4 (1+a x)^4}{3 a c^4 \left (1-a^2 x^2\right )^{3/2}}+\frac{14 (1+a x)^2}{3 a c^4 \sqrt{1-a^2 x^2}}+\frac{7 \sqrt{1-a^2 x^2}}{a c^4}-\frac{7 \sin ^{-1}(a x)}{a c^4}\\ \end{align*}

Mathematica [A]  time = 0.182409, size = 77, normalized size = 0.41 \[ \frac{\frac{\sqrt{1-a^2 x^2} \left (315 a^5 x^5-6539 a^4 x^4+19780 a^3 x^3-25347 a^2 x^2+15115 a x-3464\right )}{(a x-1)^5}-2205 \sin ^{-1}(a x)}{315 a c^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((Sqrt[1 - a^2*x^2]*(-3464 + 15115*a*x - 25347*a^2*x^2 + 19780*a^3*x^3 - 6539*a^4*x^4 + 315*a^5*x^5))/(-1 + a*
x)^5 - 2205*ArcSin[a*x])/(315*a*c^4)

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Maple [A]  time = 0.059, size = 300, normalized size = 1.6 \begin{align*} -{\frac{a{x}^{2}}{{c}^{4}}{\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}}+27\,{\frac{1}{a{c}^{4}\sqrt{-{a}^{2}{x}^{2}+1}}}+70\,{\frac{x}{{c}^{4}\sqrt{-{a}^{2}{x}^{2}+1}}}-7\,{\frac{1}{{c}^{4}\sqrt{{a}^{2}}}\arctan \left ({\frac{\sqrt{{a}^{2}}x}{\sqrt{-{a}^{2}{x}^{2}+1}}} \right ) }+{\frac{8}{9\,{a}^{5}{c}^{4}} \left ( x-{a}^{-1} \right ) ^{-4}{\frac{1}{\sqrt{-{a}^{2} \left ( x-{a}^{-1} \right ) ^{2}-2\,a \left ( x-{a}^{-1} \right ) }}}}+{\frac{356}{63\,{a}^{4}{c}^{4}} \left ( x-{a}^{-1} \right ) ^{-3}{\frac{1}{\sqrt{-{a}^{2} \left ( x-{a}^{-1} \right ) ^{2}-2\,a \left ( x-{a}^{-1} \right ) }}}}+{\frac{5002}{315\,{a}^{3}{c}^{4}} \left ( x-{a}^{-1} \right ) ^{-2}{\frac{1}{\sqrt{-{a}^{2} \left ( x-{a}^{-1} \right ) ^{2}-2\,a \left ( x-{a}^{-1} \right ) }}}}+{\frac{8543}{315\,{a}^{2}{c}^{4}} \left ( x-{a}^{-1} \right ) ^{-1}{\frac{1}{\sqrt{-{a}^{2} \left ( x-{a}^{-1} \right ) ^{2}-2\,a \left ( x-{a}^{-1} \right ) }}}}-{\frac{17086\,x}{315\,{c}^{4}}{\frac{1}{\sqrt{-{a}^{2} \left ( x-{a}^{-1} \right ) ^{2}-2\,a \left ( x-{a}^{-1} \right ) }}}} \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]

-a/c^4*x^2/(-a^2*x^2+1)^(1/2)+27/a/c^4/(-a^2*x^2+1)^(1/2)+70*x/c^4/(-a^2*x^2+1)^(1/2)-7/c^4/(a^2)^(1/2)*arctan
((a^2)^(1/2)*x/(-a^2*x^2+1)^(1/2))+8/9/a^5/c^4/(x-1/a)^4/(-a^2*(x-1/a)^2-2*a*(x-1/a))^(1/2)+356/63/a^4/c^4/(x-
1/a)^3/(-a^2*(x-1/a)^2-2*a*(x-1/a))^(1/2)+5002/315/a^3/c^4/(x-1/a)^2/(-a^2*(x-1/a)^2-2*a*(x-1/a))^(1/2)+8543/3
15/a^2/c^4/(x-1/a)/(-a^2*(x-1/a)^2-2*a*(x-1/a))^(1/2)-17086/315/c^4/(-a^2*(x-1/a)^2-2*a*(x-1/a))^(1/2)*x

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (a x + 1\right )}^{3}}{{\left (-a^{2} x^{2} + 1\right )}^{\frac{3}{2}}{\left (c - \frac{c}{a x}\right )}^{4}}\,{d x} \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]

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

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Fricas [A]  time = 2.26225, size = 516, normalized size = 2.72 \begin{align*} \frac{3464 \, a^{5} x^{5} - 17320 \, a^{4} x^{4} + 34640 \, a^{3} x^{3} - 34640 \, a^{2} x^{2} + 17320 \, a x + 4410 \,{\left (a^{5} x^{5} - 5 \, a^{4} x^{4} + 10 \, a^{3} x^{3} - 10 \, a^{2} x^{2} + 5 \, a x - 1\right )} \arctan \left (\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right ) +{\left (315 \, a^{5} x^{5} - 6539 \, a^{4} x^{4} + 19780 \, a^{3} x^{3} - 25347 \, a^{2} x^{2} + 15115 \, a x - 3464\right )} \sqrt{-a^{2} x^{2} + 1} - 3464}{315 \,{\left (a^{6} c^{4} x^{5} - 5 \, a^{5} c^{4} x^{4} + 10 \, a^{4} c^{4} x^{3} - 10 \, a^{3} c^{4} x^{2} + 5 \, a^{2} c^{4} x - a c^{4}\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="fricas")

[Out]

1/315*(3464*a^5*x^5 - 17320*a^4*x^4 + 34640*a^3*x^3 - 34640*a^2*x^2 + 17320*a*x + 4410*(a^5*x^5 - 5*a^4*x^4 +
10*a^3*x^3 - 10*a^2*x^2 + 5*a*x - 1)*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) + (315*a^5*x^5 - 6539*a^4*x^4 + 19
780*a^3*x^3 - 25347*a^2*x^2 + 15115*a*x - 3464)*sqrt(-a^2*x^2 + 1) - 3464)/(a^6*c^4*x^5 - 5*a^5*c^4*x^4 + 10*a
^4*c^4*x^3 - 10*a^3*c^4*x^2 + 5*a^2*c^4*x - a*c^4)

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

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Giac [A]  time = 1.31472, size = 389, normalized size = 2.05 \begin{align*} -\frac{7 \, \arcsin \left (a x\right ) \mathrm{sgn}\left (a\right )}{c^{4}{\left | a \right |}} + \frac{\sqrt{-a^{2} x^{2} + 1}}{a c^{4}} - \frac{2 \,{\left (\frac{26136 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}}{a^{2} x} - \frac{93834 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{2}}{a^{4} x^{2}} + \frac{188706 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{3}}{a^{6} x^{3}} - \frac{229194 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{4}}{a^{8} x^{4}} + \frac{167580 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{5}}{a^{10} x^{5}} - \frac{75810 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{6}}{a^{12} x^{6}} + \frac{19530 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{7}}{a^{14} x^{7}} - \frac{2205 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{8}}{a^{16} x^{8}} - 3149\right )}}{315 \, c^{4}{\left (\frac{\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a}{a^{2} x} - 1\right )}^{9}{\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]

-7*arcsin(a*x)*sgn(a)/(c^4*abs(a)) + sqrt(-a^2*x^2 + 1)/(a*c^4) - 2/315*(26136*(sqrt(-a^2*x^2 + 1)*abs(a) + a)
/(a^2*x) - 93834*(sqrt(-a^2*x^2 + 1)*abs(a) + a)^2/(a^4*x^2) + 188706*(sqrt(-a^2*x^2 + 1)*abs(a) + a)^3/(a^6*x
^3) - 229194*(sqrt(-a^2*x^2 + 1)*abs(a) + a)^4/(a^8*x^4) + 167580*(sqrt(-a^2*x^2 + 1)*abs(a) + a)^5/(a^10*x^5)
- 75810*(sqrt(-a^2*x^2 + 1)*abs(a) + a)^6/(a^12*x^6) + 19530*(sqrt(-a^2*x^2 + 1)*abs(a) + a)^7/(a^14*x^7) - 2
205*(sqrt(-a^2*x^2 + 1)*abs(a) + a)^8/(a^16*x^8) - 3149)/(c^4*((sqrt(-a^2*x^2 + 1)*abs(a) + a)/(a^2*x) - 1)^9*
abs(a))