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

Optimal. Leaf size=192 \[ \frac{a^2 (1867 a x+1470)}{105 c^4 \sqrt{1-a^2 x^2}}+\frac{a^2 (671 a x+455)}{105 c^4 \left (1-a^2 x^2\right )^{3/2}}+\frac{4 a^2 (31 a x+21)}{35 c^4 \left (1-a^2 x^2\right )^{5/2}}+\frac{16 a^2 (a x+1)}{7 c^4 \left (1-a^2 x^2\right )^{7/2}}-\frac{5 a \sqrt{1-a^2 x^2}}{c^4 x}-\frac{\sqrt{1-a^2 x^2}}{2 c^4 x^2}-\frac{29 a^2 \tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right )}{2 c^4} \]

[Out]

(16*a^2*(1 + a*x))/(7*c^4*(1 - a^2*x^2)^(7/2)) + (4*a^2*(21 + 31*a*x))/(35*c^4*(1 - a^2*x^2)^(5/2)) + (a^2*(45
5 + 671*a*x))/(105*c^4*(1 - a^2*x^2)^(3/2)) + (a^2*(1470 + 1867*a*x))/(105*c^4*Sqrt[1 - a^2*x^2]) - Sqrt[1 - a
^2*x^2]/(2*c^4*x^2) - (5*a*Sqrt[1 - a^2*x^2])/(c^4*x) - (29*a^2*ArcTanh[Sqrt[1 - a^2*x^2]])/(2*c^4)

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Rubi [A]  time = 0.523759, antiderivative size = 192, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 8, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.421, Rules used = {6128, 852, 1805, 1807, 807, 266, 63, 208} \[ \frac{a^2 (1867 a x+1470)}{105 c^4 \sqrt{1-a^2 x^2}}+\frac{a^2 (671 a x+455)}{105 c^4 \left (1-a^2 x^2\right )^{3/2}}+\frac{4 a^2 (31 a x+21)}{35 c^4 \left (1-a^2 x^2\right )^{5/2}}+\frac{16 a^2 (a x+1)}{7 c^4 \left (1-a^2 x^2\right )^{7/2}}-\frac{5 a \sqrt{1-a^2 x^2}}{c^4 x}-\frac{\sqrt{1-a^2 x^2}}{2 c^4 x^2}-\frac{29 a^2 \tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right )}{2 c^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(16*a^2*(1 + a*x))/(7*c^4*(1 - a^2*x^2)^(7/2)) + (4*a^2*(21 + 31*a*x))/(35*c^4*(1 - a^2*x^2)^(5/2)) + (a^2*(45
5 + 671*a*x))/(105*c^4*(1 - a^2*x^2)^(3/2)) + (a^2*(1470 + 1867*a*x))/(105*c^4*Sqrt[1 - a^2*x^2]) - Sqrt[1 - a
^2*x^2]/(2*c^4*x^2) - (5*a*Sqrt[1 - a^2*x^2])/(c^4*x) - (29*a^2*ArcTanh[Sqrt[1 - a^2*x^2]])/(2*c^4)

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 1805

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 1807

Int[(Pq_)*((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuotient[Pq, c*x, x],
R = PolynomialRemainder[Pq, c*x, x]}, Simp[(R*(c*x)^(m + 1)*(a + b*x^2)^(p + 1))/(a*c*(m + 1)), x] + Dist[1/(
a*c*(m + 1)), Int[(c*x)^(m + 1)*(a + b*x^2)^p*ExpandToSum[a*c*(m + 1)*Q - b*R*(m + 2*p + 3)*x, x], x], x]] /;
FreeQ[{a, b, c, p}, x] && PolyQ[Pq, x] && LtQ[m, -1] && (IntegerQ[2*p] || NeQ[Expon[Pq, x], 1])

Rule 807

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 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 \frac{e^{\tanh ^{-1}(a x)}}{x^3 (c-a c x)^4} \, dx &=c \int \frac{\sqrt{1-a^2 x^2}}{x^3 (c-a c x)^5} \, dx\\ &=\frac{\int \frac{(c+a c x)^5}{x^3 \left (1-a^2 x^2\right )^{9/2}} \, dx}{c^9}\\ &=\frac{16 a^2 (1+a x)}{7 c^4 \left (1-a^2 x^2\right )^{7/2}}-\frac{\int \frac{-7 c^5-35 a c^5 x-77 a^2 c^5 x^2-89 a^3 c^5 x^3}{x^3 \left (1-a^2 x^2\right )^{7/2}} \, dx}{7 c^9}\\ &=\frac{16 a^2 (1+a x)}{7 c^4 \left (1-a^2 x^2\right )^{7/2}}+\frac{4 a^2 (21+31 a x)}{35 c^4 \left (1-a^2 x^2\right )^{5/2}}+\frac{\int \frac{35 c^5+175 a c^5 x+420 a^2 c^5 x^2+496 a^3 c^5 x^3}{x^3 \left (1-a^2 x^2\right )^{5/2}} \, dx}{35 c^9}\\ &=\frac{16 a^2 (1+a x)}{7 c^4 \left (1-a^2 x^2\right )^{7/2}}+\frac{4 a^2 (21+31 a x)}{35 c^4 \left (1-a^2 x^2\right )^{5/2}}+\frac{a^2 (455+671 a x)}{105 c^4 \left (1-a^2 x^2\right )^{3/2}}-\frac{\int \frac{-105 c^5-525 a c^5 x-1365 a^2 c^5 x^2-1342 a^3 c^5 x^3}{x^3 \left (1-a^2 x^2\right )^{3/2}} \, dx}{105 c^9}\\ &=\frac{16 a^2 (1+a x)}{7 c^4 \left (1-a^2 x^2\right )^{7/2}}+\frac{4 a^2 (21+31 a x)}{35 c^4 \left (1-a^2 x^2\right )^{5/2}}+\frac{a^2 (455+671 a x)}{105 c^4 \left (1-a^2 x^2\right )^{3/2}}+\frac{a^2 (1470+1867 a x)}{105 c^4 \sqrt{1-a^2 x^2}}+\frac{\int \frac{105 c^5+525 a c^5 x+1470 a^2 c^5 x^2}{x^3 \sqrt{1-a^2 x^2}} \, dx}{105 c^9}\\ &=\frac{16 a^2 (1+a x)}{7 c^4 \left (1-a^2 x^2\right )^{7/2}}+\frac{4 a^2 (21+31 a x)}{35 c^4 \left (1-a^2 x^2\right )^{5/2}}+\frac{a^2 (455+671 a x)}{105 c^4 \left (1-a^2 x^2\right )^{3/2}}+\frac{a^2 (1470+1867 a x)}{105 c^4 \sqrt{1-a^2 x^2}}-\frac{\sqrt{1-a^2 x^2}}{2 c^4 x^2}-\frac{\int \frac{-1050 a c^5-3045 a^2 c^5 x}{x^2 \sqrt{1-a^2 x^2}} \, dx}{210 c^9}\\ &=\frac{16 a^2 (1+a x)}{7 c^4 \left (1-a^2 x^2\right )^{7/2}}+\frac{4 a^2 (21+31 a x)}{35 c^4 \left (1-a^2 x^2\right )^{5/2}}+\frac{a^2 (455+671 a x)}{105 c^4 \left (1-a^2 x^2\right )^{3/2}}+\frac{a^2 (1470+1867 a x)}{105 c^4 \sqrt{1-a^2 x^2}}-\frac{\sqrt{1-a^2 x^2}}{2 c^4 x^2}-\frac{5 a \sqrt{1-a^2 x^2}}{c^4 x}+\frac{\left (29 a^2\right ) \int \frac{1}{x \sqrt{1-a^2 x^2}} \, dx}{2 c^4}\\ &=\frac{16 a^2 (1+a x)}{7 c^4 \left (1-a^2 x^2\right )^{7/2}}+\frac{4 a^2 (21+31 a x)}{35 c^4 \left (1-a^2 x^2\right )^{5/2}}+\frac{a^2 (455+671 a x)}{105 c^4 \left (1-a^2 x^2\right )^{3/2}}+\frac{a^2 (1470+1867 a x)}{105 c^4 \sqrt{1-a^2 x^2}}-\frac{\sqrt{1-a^2 x^2}}{2 c^4 x^2}-\frac{5 a \sqrt{1-a^2 x^2}}{c^4 x}+\frac{\left (29 a^2\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{1-a^2 x}} \, dx,x,x^2\right )}{4 c^4}\\ &=\frac{16 a^2 (1+a x)}{7 c^4 \left (1-a^2 x^2\right )^{7/2}}+\frac{4 a^2 (21+31 a x)}{35 c^4 \left (1-a^2 x^2\right )^{5/2}}+\frac{a^2 (455+671 a x)}{105 c^4 \left (1-a^2 x^2\right )^{3/2}}+\frac{a^2 (1470+1867 a x)}{105 c^4 \sqrt{1-a^2 x^2}}-\frac{\sqrt{1-a^2 x^2}}{2 c^4 x^2}-\frac{5 a \sqrt{1-a^2 x^2}}{c^4 x}-\frac{29 \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 c^4}\\ &=\frac{16 a^2 (1+a x)}{7 c^4 \left (1-a^2 x^2\right )^{7/2}}+\frac{4 a^2 (21+31 a x)}{35 c^4 \left (1-a^2 x^2\right )^{5/2}}+\frac{a^2 (455+671 a x)}{105 c^4 \left (1-a^2 x^2\right )^{3/2}}+\frac{a^2 (1470+1867 a x)}{105 c^4 \sqrt{1-a^2 x^2}}-\frac{\sqrt{1-a^2 x^2}}{2 c^4 x^2}-\frac{5 a \sqrt{1-a^2 x^2}}{c^4 x}-\frac{29 a^2 \tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right )}{2 c^4}\\ \end{align*}

Mathematica [A]  time = 0.0654024, size = 121, normalized size = 0.63 \[ \frac{4784 a^6 x^6-11307 a^5 x^5+2825 a^4 x^4+10512 a^3 x^3-7774 a^2 x^2-3045 a^2 x^2 (a x-1)^3 \sqrt{1-a^2 x^2} \tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right )+735 a x+105}{210 c^4 x^2 (a x-1)^3 \sqrt{1-a^2 x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(105 + 735*a*x - 7774*a^2*x^2 + 10512*a^3*x^3 + 2825*a^4*x^4 - 11307*a^5*x^5 + 4784*a^6*x^6 - 3045*a^2*x^2*(-1
+ a*x)^3*Sqrt[1 - a^2*x^2]*ArcTanh[Sqrt[1 - a^2*x^2]])/(210*c^4*x^2*(-1 + a*x)^3*Sqrt[1 - a^2*x^2])

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Maple [B]  time = 0.052, size = 397, normalized size = 2.1 \begin{align*}{\frac{1}{{c}^{4}} \left ( -{\frac{1}{a}\sqrt{-{a}^{2} \left ( x-{a}^{-1} \right ) ^{2}-2\,a \left ( x-{a}^{-1} \right ) } \left ( x-{a}^{-1} \right ) ^{-3}}+11\,a \left ( 1/3\,{\frac{1}{a}\sqrt{-{a}^{2} \left ( x-{a}^{-1} \right ) ^{2}-2\,a \left ( x-{a}^{-1} \right ) } \left ( x-{a}^{-1} \right ) ^{-2}}-1/3\,{\sqrt{-{a}^{2} \left ( x-{a}^{-1} \right ) ^{2}-2\,a \left ( x-{a}^{-1} \right ) } \left ( x-{a}^{-1} \right ) ^{-1}} \right ) -5\,{\frac{a\sqrt{-{a}^{2}{x}^{2}+1}}{x}}+2\,{\frac{1}{a} \left ( 1/7\,{\frac{1}{a}\sqrt{-{a}^{2} \left ( x-{a}^{-1} \right ) ^{2}-2\,a \left ( x-{a}^{-1} \right ) } \left ( x-{a}^{-1} \right ) ^{-4}}-3/7\,a \left ( 1/5\,{\frac{1}{a}\sqrt{-{a}^{2} \left ( x-{a}^{-1} \right ) ^{2}-2\,a \left ( x-{a}^{-1} \right ) } \left ( x-{a}^{-1} \right ) ^{-3}}-2/5\,a \left ( 1/3\,{\frac{1}{a}\sqrt{-{a}^{2} \left ( x-{a}^{-1} \right ) ^{2}-2\,a \left ( x-{a}^{-1} \right ) } \left ( x-{a}^{-1} \right ) ^{-2}}-1/3\,{\sqrt{-{a}^{2} \left ( x-{a}^{-1} \right ) ^{2}-2\,a \left ( x-{a}^{-1} \right ) } \left ( x-{a}^{-1} \right ) ^{-1}} \right ) \right ) \right ) }-{\frac{29\,{a}^{2}}{2}{\it Artanh} \left ({\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}} \right ) }-14\,{a\sqrt{-{a}^{2} \left ( x-{a}^{-1} \right ) ^{2}-2\,a \left ( x-{a}^{-1} \right ) } \left ( x-{a}^{-1} \right ) ^{-1}}-{\frac{1}{2\,{x}^{2}}\sqrt{-{a}^{2}{x}^{2}+1}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/c^4*(-1/a/(x-1/a)^3*(-a^2*(x-1/a)^2-2*a*(x-1/a))^(1/2)+11*a*(1/3/a/(x-1/a)^2*(-a^2*(x-1/a)^2-2*a*(x-1/a))^(1
/2)-1/3/(x-1/a)*(-a^2*(x-1/a)^2-2*a*(x-1/a))^(1/2))-5*a*(-a^2*x^2+1)^(1/2)/x+2/a*(1/7/a/(x-1/a)^4*(-a^2*(x-1/a
)^2-2*a*(x-1/a))^(1/2)-3/7*a*(1/5/a/(x-1/a)^3*(-a^2*(x-1/a)^2-2*a*(x-1/a))^(1/2)-2/5*a*(1/3/a/(x-1/a)^2*(-a^2*
(x-1/a)^2-2*a*(x-1/a))^(1/2)-1/3/(x-1/a)*(-a^2*(x-1/a)^2-2*a*(x-1/a))^(1/2))))-29/2*a^2*arctanh(1/(-a^2*x^2+1)
^(1/2))-14*a/(x-1/a)*(-a^2*(x-1/a)^2-2*a*(x-1/a))^(1/2)-1/2*(-a^2*x^2+1)^(1/2)/x^2)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.97885, size = 471, normalized size = 2.45 \begin{align*} \frac{4834 \, a^{6} x^{6} - 19336 \, a^{5} x^{5} + 29004 \, a^{4} x^{4} - 19336 \, a^{3} x^{3} + 4834 \, a^{2} x^{2} + 3045 \,{\left (a^{6} x^{6} - 4 \, a^{5} x^{5} + 6 \, a^{4} x^{4} - 4 \, a^{3} x^{3} + a^{2} x^{2}\right )} \log \left (\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{x}\right ) -{\left (4784 \, a^{5} x^{5} - 16091 \, a^{4} x^{4} + 18916 \, a^{3} x^{3} - 8404 \, a^{2} x^{2} + 630 \, a x + 105\right )} \sqrt{-a^{2} x^{2} + 1}}{210 \,{\left (a^{4} c^{4} x^{6} - 4 \, a^{3} c^{4} x^{5} + 6 \, a^{2} c^{4} x^{4} - 4 \, a c^{4} x^{3} + c^{4} x^{2}\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)/x^3/(-a*c*x+c)^4,x, algorithm="fricas")

[Out]

1/210*(4834*a^6*x^6 - 19336*a^5*x^5 + 29004*a^4*x^4 - 19336*a^3*x^3 + 4834*a^2*x^2 + 3045*(a^6*x^6 - 4*a^5*x^5
+ 6*a^4*x^4 - 4*a^3*x^3 + a^2*x^2)*log((sqrt(-a^2*x^2 + 1) - 1)/x) - (4784*a^5*x^5 - 16091*a^4*x^4 + 18916*a^
3*x^3 - 8404*a^2*x^2 + 630*a*x + 105)*sqrt(-a^2*x^2 + 1))/(a^4*c^4*x^6 - 4*a^3*c^4*x^5 + 6*a^2*c^4*x^4 - 4*a*c
^4*x^3 + c^4*x^2)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B]  time = 1.22932, size = 529, normalized size = 2.76 \begin{align*} -\frac{29 \, a^{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 )}{2 \, c^{4}{\left | a \right |}} - \frac{{\left (105 \, a^{3} + \frac{1365 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )} a}{x} - \frac{51167 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{2}}{a x^{2}} + \frac{260729 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{3}}{a^{3} x^{3}} - \frac{621537 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{4}}{a^{5} x^{4}} + \frac{826175 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{5}}{a^{7} x^{5}} - \frac{642005 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{6}}{a^{9} x^{6}} + \frac{274995 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{7}}{a^{11} x^{7}} - \frac{52500 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{8}}{a^{13} x^{8}}\right )} a^{4} x^{2}}{840 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{2} c^{4}{\left (\frac{\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a}{a^{2} x} - 1\right )}^{7}{\left | a \right |}} - \frac{\frac{20 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )} a c^{4}{\left | a \right |}}{x} + \frac{{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{2} c^{4}{\left | a \right |}}{a x^{2}}}{8 \, a^{2} c^{8}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-29/2*a^3*log(1/2*abs(-2*sqrt(-a^2*x^2 + 1)*abs(a) - 2*a)/(a^2*abs(x)))/(c^4*abs(a)) - 1/840*(105*a^3 + 1365*(
sqrt(-a^2*x^2 + 1)*abs(a) + a)*a/x - 51167*(sqrt(-a^2*x^2 + 1)*abs(a) + a)^2/(a*x^2) + 260729*(sqrt(-a^2*x^2 +
1)*abs(a) + a)^3/(a^3*x^3) - 621537*(sqrt(-a^2*x^2 + 1)*abs(a) + a)^4/(a^5*x^4) + 826175*(sqrt(-a^2*x^2 + 1)*
abs(a) + a)^5/(a^7*x^5) - 642005*(sqrt(-a^2*x^2 + 1)*abs(a) + a)^6/(a^9*x^6) + 274995*(sqrt(-a^2*x^2 + 1)*abs(
a) + a)^7/(a^11*x^7) - 52500*(sqrt(-a^2*x^2 + 1)*abs(a) + a)^8/(a^13*x^8))*a^4*x^2/((sqrt(-a^2*x^2 + 1)*abs(a)
+ a)^2*c^4*((sqrt(-a^2*x^2 + 1)*abs(a) + a)/(a^2*x) - 1)^7*abs(a)) - 1/8*(20*(sqrt(-a^2*x^2 + 1)*abs(a) + a)*
a*c^4*abs(a)/x + (sqrt(-a^2*x^2 + 1)*abs(a) + a)^2*c^4*abs(a)/(a*x^2))/(a^2*c^8)