3.662 \(\int e^{-\tanh ^{-1}(a x)} (c-\frac{c}{a^2 x^2})^4 \, dx\)

Optimal. Leaf size=169 \[ -\frac{c^4 (6-7 a x) \left (1-a^2 x^2\right )^{7/2}}{42 a^8 x^7}+\frac{c^4 (24-35 a x) \left (1-a^2 x^2\right )^{5/2}}{120 a^6 x^5}-\frac{c^4 (16-35 a x) \left (1-a^2 x^2\right )^{3/2}}{48 a^4 x^3}+\frac{c^4 (35 a x+16) \sqrt{1-a^2 x^2}}{16 a^2 x}-\frac{35 c^4 \tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right )}{16 a}+\frac{c^4 \sin ^{-1}(a x)}{a} \]

[Out]

(c^4*(16 + 35*a*x)*Sqrt[1 - a^2*x^2])/(16*a^2*x) - (c^4*(16 - 35*a*x)*(1 - a^2*x^2)^(3/2))/(48*a^4*x^3) + (c^4
*(24 - 35*a*x)*(1 - a^2*x^2)^(5/2))/(120*a^6*x^5) - (c^4*(6 - 7*a*x)*(1 - a^2*x^2)^(7/2))/(42*a^8*x^7) + (c^4*
ArcSin[a*x])/a - (35*c^4*ArcTanh[Sqrt[1 - a^2*x^2]])/(16*a)

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Rubi [A]  time = 0.232407, antiderivative size = 169, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 9, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.409, Rules used = {6157, 6149, 811, 813, 844, 216, 266, 63, 208} \[ -\frac{c^4 (6-7 a x) \left (1-a^2 x^2\right )^{7/2}}{42 a^8 x^7}+\frac{c^4 (24-35 a x) \left (1-a^2 x^2\right )^{5/2}}{120 a^6 x^5}-\frac{c^4 (16-35 a x) \left (1-a^2 x^2\right )^{3/2}}{48 a^4 x^3}+\frac{c^4 (35 a x+16) \sqrt{1-a^2 x^2}}{16 a^2 x}-\frac{35 c^4 \tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right )}{16 a}+\frac{c^4 \sin ^{-1}(a x)}{a} \]

Antiderivative was successfully verified.

[In]

Int[(c - c/(a^2*x^2))^4/E^ArcTanh[a*x],x]

[Out]

(c^4*(16 + 35*a*x)*Sqrt[1 - a^2*x^2])/(16*a^2*x) - (c^4*(16 - 35*a*x)*(1 - a^2*x^2)^(3/2))/(48*a^4*x^3) + (c^4
*(24 - 35*a*x)*(1 - a^2*x^2)^(5/2))/(120*a^6*x^5) - (c^4*(6 - 7*a*x)*(1 - a^2*x^2)^(7/2))/(42*a^8*x^7) + (c^4*
ArcSin[a*x])/a - (35*c^4*ArcTanh[Sqrt[1 - a^2*x^2]])/(16*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 6149

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] || G
tQ[c, 0]) && ILtQ[(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 )^4 \, dx &=\frac{c^4 \int \frac{e^{-\tanh ^{-1}(a x)} \left (1-a^2 x^2\right )^4}{x^8} \, dx}{a^8}\\ &=\frac{c^4 \int \frac{(1-a x) \left (1-a^2 x^2\right )^{7/2}}{x^8} \, dx}{a^8}\\ &=-\frac{c^4 (6-7 a x) \left (1-a^2 x^2\right )^{7/2}}{42 a^8 x^7}-\frac{c^4 \int \frac{\left (12 a^2-14 a^3 x\right ) \left (1-a^2 x^2\right )^{5/2}}{x^6} \, dx}{12 a^8}\\ &=\frac{c^4 (24-35 a x) \left (1-a^2 x^2\right )^{5/2}}{120 a^6 x^5}-\frac{c^4 (6-7 a x) \left (1-a^2 x^2\right )^{7/2}}{42 a^8 x^7}+\frac{c^4 \int \frac{\left (96 a^4-140 a^5 x\right ) \left (1-a^2 x^2\right )^{3/2}}{x^4} \, dx}{96 a^8}\\ &=-\frac{c^4 (16-35 a x) \left (1-a^2 x^2\right )^{3/2}}{48 a^4 x^3}+\frac{c^4 (24-35 a x) \left (1-a^2 x^2\right )^{5/2}}{120 a^6 x^5}-\frac{c^4 (6-7 a x) \left (1-a^2 x^2\right )^{7/2}}{42 a^8 x^7}-\frac{c^4 \int \frac{\left (384 a^6-840 a^7 x\right ) \sqrt{1-a^2 x^2}}{x^2} \, dx}{384 a^8}\\ &=\frac{c^4 (16+35 a x) \sqrt{1-a^2 x^2}}{16 a^2 x}-\frac{c^4 (16-35 a x) \left (1-a^2 x^2\right )^{3/2}}{48 a^4 x^3}+\frac{c^4 (24-35 a x) \left (1-a^2 x^2\right )^{5/2}}{120 a^6 x^5}-\frac{c^4 (6-7 a x) \left (1-a^2 x^2\right )^{7/2}}{42 a^8 x^7}+\frac{c^4 \int \frac{1680 a^7+768 a^8 x}{x \sqrt{1-a^2 x^2}} \, dx}{768 a^8}\\ &=\frac{c^4 (16+35 a x) \sqrt{1-a^2 x^2}}{16 a^2 x}-\frac{c^4 (16-35 a x) \left (1-a^2 x^2\right )^{3/2}}{48 a^4 x^3}+\frac{c^4 (24-35 a x) \left (1-a^2 x^2\right )^{5/2}}{120 a^6 x^5}-\frac{c^4 (6-7 a x) \left (1-a^2 x^2\right )^{7/2}}{42 a^8 x^7}+c^4 \int \frac{1}{\sqrt{1-a^2 x^2}} \, dx+\frac{\left (35 c^4\right ) \int \frac{1}{x \sqrt{1-a^2 x^2}} \, dx}{16 a}\\ &=\frac{c^4 (16+35 a x) \sqrt{1-a^2 x^2}}{16 a^2 x}-\frac{c^4 (16-35 a x) \left (1-a^2 x^2\right )^{3/2}}{48 a^4 x^3}+\frac{c^4 (24-35 a x) \left (1-a^2 x^2\right )^{5/2}}{120 a^6 x^5}-\frac{c^4 (6-7 a x) \left (1-a^2 x^2\right )^{7/2}}{42 a^8 x^7}+\frac{c^4 \sin ^{-1}(a x)}{a}+\frac{\left (35 c^4\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{1-a^2 x}} \, dx,x,x^2\right )}{32 a}\\ &=\frac{c^4 (16+35 a x) \sqrt{1-a^2 x^2}}{16 a^2 x}-\frac{c^4 (16-35 a x) \left (1-a^2 x^2\right )^{3/2}}{48 a^4 x^3}+\frac{c^4 (24-35 a x) \left (1-a^2 x^2\right )^{5/2}}{120 a^6 x^5}-\frac{c^4 (6-7 a x) \left (1-a^2 x^2\right )^{7/2}}{42 a^8 x^7}+\frac{c^4 \sin ^{-1}(a x)}{a}-\frac{\left (35 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 )}{16 a^3}\\ &=\frac{c^4 (16+35 a x) \sqrt{1-a^2 x^2}}{16 a^2 x}-\frac{c^4 (16-35 a x) \left (1-a^2 x^2\right )^{3/2}}{48 a^4 x^3}+\frac{c^4 (24-35 a x) \left (1-a^2 x^2\right )^{5/2}}{120 a^6 x^5}-\frac{c^4 (6-7 a x) \left (1-a^2 x^2\right )^{7/2}}{42 a^8 x^7}+\frac{c^4 \sin ^{-1}(a x)}{a}-\frac{35 c^4 \tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right )}{16 a}\\ \end{align*}

Mathematica [C]  time = 0.0276227, size = 70, normalized size = 0.41 \[ \frac{c^4 \left (7 a^7 \left (1-a^2 x^2\right )^{9/2} \text{Hypergeometric2F1}\left (4,\frac{9}{2},\frac{11}{2},1-a^2 x^2\right )-\frac{9 \text{Hypergeometric2F1}\left (-\frac{7}{2},-\frac{7}{2},-\frac{5}{2},a^2 x^2\right )}{x^7}\right )}{63 a^8} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[(c - c/(a^2*x^2))^4/E^ArcTanh[a*x],x]

[Out]

(c^4*((-9*Hypergeometric2F1[-7/2, -7/2, -5/2, a^2*x^2])/x^7 + 7*a^7*(1 - a^2*x^2)^(9/2)*Hypergeometric2F1[4, 9
/2, 11/2, 1 - a^2*x^2]))/(63*a^8)

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Maple [A]  time = 0.085, size = 249, normalized size = 1.5 \begin{align*} -{\frac{5\,{c}^{4}}{8\,{a}^{5}{x}^{4}} \left ( -{a}^{2}{x}^{2}+1 \right ) ^{{\frac{3}{2}}}}+{\frac{19\,{c}^{4}}{16\,{x}^{2}{a}^{3}} \left ( -{a}^{2}{x}^{2}+1 \right ) ^{{\frac{3}{2}}}}+{\frac{35\,{c}^{4}}{16\,a}\sqrt{-{a}^{2}{x}^{2}+1}}-{\frac{35\,{c}^{4}}{16\,a}{\it Artanh} \left ({\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}} \right ) }+{\frac{{c}^{4}}{{a}^{2}x} \left ( -{a}^{2}{x}^{2}+1 \right ) ^{{\frac{3}{2}}}}+{c}^{4}x\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{{c}^{4}}{7\,{a}^{8}{x}^{7}} \left ( -{a}^{2}{x}^{2}+1 \right ) ^{{\frac{3}{2}}}}+{\frac{17\,{c}^{4}}{35\,{a}^{6}{x}^{5}} \left ( -{a}^{2}{x}^{2}+1 \right ) ^{{\frac{3}{2}}}}-{\frac{71\,{c}^{4}}{105\,{a}^{4}{x}^{3}} \left ( -{a}^{2}{x}^{2}+1 \right ) ^{{\frac{3}{2}}}}+{\frac{{c}^{4}}{6\,{a}^{7}{x}^{6}} \left ( -{a}^{2}{x}^{2}+1 \right ) ^{{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-5/8*c^4/a^5/x^4*(-a^2*x^2+1)^(3/2)+19/16*c^4*(-a^2*x^2+1)^(3/2)/x^2/a^3+35/16*c^4*(-a^2*x^2+1)^(1/2)/a-35/16*
c^4/a*arctanh(1/(-a^2*x^2+1)^(1/2))+c^4/a^2/x*(-a^2*x^2+1)^(3/2)+c^4*x*(-a^2*x^2+1)^(1/2)+c^4/(a^2)^(1/2)*arct
an((a^2)^(1/2)*x/(-a^2*x^2+1)^(1/2))-1/7*c^4/a^8/x^7*(-a^2*x^2+1)^(3/2)+17/35*c^4/a^6/x^5*(-a^2*x^2+1)^(3/2)-7
1/105*c^4*(-a^2*x^2+1)^(3/2)/a^4/x^3+1/6*c^4/a^7/x^6*(-a^2*x^2+1)^(3/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} c^{4}{\left (\frac{\arcsin \left (a x\right )}{a} + \frac{\sqrt{-a^{2} x^{2} + 1}}{a}\right )} - \int \frac{{\left (4 \, a^{6} c^{4} x^{6} - 6 \, a^{4} c^{4} x^{4} + 4 \, a^{2} c^{4} x^{2} - c^{4}\right )} \sqrt{a x + 1} \sqrt{-a x + 1}}{a^{9} x^{9} + a^{8} x^{8}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

c^4*(arcsin(a*x)/a + sqrt(-a^2*x^2 + 1)/a) - integrate((4*a^6*c^4*x^6 - 6*a^4*c^4*x^4 + 4*a^2*c^4*x^2 - c^4)*s
qrt(a*x + 1)*sqrt(-a*x + 1)/(a^9*x^9 + a^8*x^8), x)

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Fricas [A]  time = 1.98607, size = 413, normalized size = 2.44 \begin{align*} -\frac{3360 \, a^{7} c^{4} x^{7} \arctan \left (\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right ) - 3675 \, a^{7} c^{4} x^{7} \log \left (\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{x}\right ) - 1680 \, a^{7} c^{4} x^{7} -{\left (1680 \, a^{7} c^{4} x^{7} + 2816 \, a^{6} c^{4} x^{6} + 3045 \, a^{5} c^{4} x^{5} - 1952 \, a^{4} c^{4} x^{4} - 1330 \, a^{3} c^{4} x^{3} + 1056 \, a^{2} c^{4} x^{2} + 280 \, a c^{4} x - 240 \, c^{4}\right )} \sqrt{-a^{2} x^{2} + 1}}{1680 \, a^{8} x^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/1680*(3360*a^7*c^4*x^7*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) - 3675*a^7*c^4*x^7*log((sqrt(-a^2*x^2 + 1) -
1)/x) - 1680*a^7*c^4*x^7 - (1680*a^7*c^4*x^7 + 2816*a^6*c^4*x^6 + 3045*a^5*c^4*x^5 - 1952*a^4*c^4*x^4 - 1330*a
^3*c^4*x^3 + 1056*a^2*c^4*x^2 + 280*a*c^4*x - 240*c^4)*sqrt(-a^2*x^2 + 1))/(a^8*x^7)

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Sympy [C]  time = 25.6365, size = 1110, normalized size = 6.57 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

c**4*Piecewise((I*sqrt(a**2*x**2 - 1) - log(a*x) + log(a**2*x**2)/2 + I*asin(1/(a*x)), Abs(a**2*x**2) > 1), (s
qrt(-a**2*x**2 + 1) + log(a**2*x**2)/2 - log(sqrt(-a**2*x**2 + 1) + 1), True))/a - c**4*Piecewise((-I*a**2*x/s
qrt(a**2*x**2 - 1) + I*a*acosh(a*x) + I/(x*sqrt(a**2*x**2 - 1)), Abs(a**2*x**2) > 1), (a**2*x/sqrt(-a**2*x**2
+ 1) - a*asin(a*x) - 1/(x*sqrt(-a**2*x**2 + 1)), True))/a**2 - 3*c**4*Piecewise((a**2*acosh(1/(a*x))/2 + a/(2*
x*sqrt(-1 + 1/(a**2*x**2))) - 1/(2*a*x**3*sqrt(-1 + 1/(a**2*x**2))), 1/Abs(a**2*x**2) > 1), (-I*a**2*asin(1/(a
*x))/2 - I*a*sqrt(1 - 1/(a**2*x**2))/(2*x), True))/a**3 + 3*c**4*Piecewise((a**3*sqrt(-1 + 1/(a**2*x**2))/3 -
a*sqrt(-1 + 1/(a**2*x**2))/(3*x**2), 1/Abs(a**2*x**2) > 1), (I*a**3*sqrt(1 - 1/(a**2*x**2))/3 - I*a*sqrt(1 - 1
/(a**2*x**2))/(3*x**2), True))/a**4 + 3*c**4*Piecewise((a**4*acosh(1/(a*x))/8 - a**3/(8*x*sqrt(-1 + 1/(a**2*x*
*2))) + 3*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),
(-I*a**4*asin(1/(a*x))/8 + I*a**3/(8*x*sqrt(1 - 1/(a**2*x**2))) - 3*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 - 3*c**4*Piecewise((2*I*a**4*sqrt(a**2*x**2 - 1)/(15*x) + I*a**
2*sqrt(a**2*x**2 - 1)/(15*x**3) - I*sqrt(a**2*x**2 - 1)/(5*x**5), Abs(a**2*x**2) > 1), (2*a**4*sqrt(-a**2*x**2
+ 1)/(15*x) + a**2*sqrt(-a**2*x**2 + 1)/(15*x**3) - sqrt(-a**2*x**2 + 1)/(5*x**5), True))/a**6 - c**4*Piecewi
se((a**6*acosh(1/(a*x))/16 - a**5/(16*x*sqrt(-1 + 1/(a**2*x**2))) + a**3/(48*x**3*sqrt(-1 + 1/(a**2*x**2))) +
5*a/(24*x**5*sqrt(-1 + 1/(a**2*x**2))) - 1/(6*a*x**7*sqrt(-1 + 1/(a**2*x**2))), 1/Abs(a**2*x**2) > 1), (-I*a**
6*asin(1/(a*x))/16 + I*a**5/(16*x*sqrt(1 - 1/(a**2*x**2))) - I*a**3/(48*x**3*sqrt(1 - 1/(a**2*x**2))) - 5*I*a/
(24*x**5*sqrt(1 - 1/(a**2*x**2))) + I/(6*a*x**7*sqrt(1 - 1/(a**2*x**2))), True))/a**7 + c**4*Piecewise((8*a**7
*sqrt(-1 + 1/(a**2*x**2))/105 + 4*a**5*sqrt(-1 + 1/(a**2*x**2))/(105*x**2) + a**3*sqrt(-1 + 1/(a**2*x**2))/(35
*x**4) - a*sqrt(-1 + 1/(a**2*x**2))/(7*x**6), 1/Abs(a**2*x**2) > 1), (8*I*a**7*sqrt(1 - 1/(a**2*x**2))/105 + 4
*I*a**5*sqrt(1 - 1/(a**2*x**2))/(105*x**2) + I*a**3*sqrt(1 - 1/(a**2*x**2))/(35*x**4) - I*a*sqrt(1 - 1/(a**2*x
**2))/(7*x**6), True))/a**8

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Giac [B]  time = 1.28195, size = 680, normalized size = 4.02 \begin{align*} \frac{{\left (15 \, c^{4} - \frac{35 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )} c^{4}}{a^{2} x} - \frac{189 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{2} c^{4}}{a^{4} x^{2}} + \frac{525 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{3} c^{4}}{a^{6} x^{3}} + \frac{1295 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{4} c^{4}}{a^{8} x^{4}} - \frac{4935 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{5} c^{4}}{a^{10} x^{5}} - \frac{9765 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{6} c^{4}}{a^{12} x^{6}}\right )} a^{14} x^{7}}{13440 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{7}{\left | a \right |}} + \frac{c^{4} \arcsin \left (a x\right ) \mathrm{sgn}\left (a\right )}{{\left | a \right |}} - \frac{35 \, 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 )}{16 \,{\left | a \right |}} + \frac{\sqrt{-a^{2} x^{2} + 1} c^{4}}{a} + \frac{\frac{9765 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )} a^{4} c^{4}}{x} + \frac{4935 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{2} a^{2} c^{4}}{x^{2}} - \frac{1295 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{3} c^{4}}{x^{3}} - \frac{525 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{4} c^{4}}{a^{2} x^{4}} + \frac{189 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{5} c^{4}}{a^{4} x^{5}} + \frac{35 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{6} c^{4}}{a^{6} x^{6}} - \frac{15 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{7} c^{4}}{a^{8} x^{7}}}{13440 \, a^{6}{\left | a \right |}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/13440*(15*c^4 - 35*(sqrt(-a^2*x^2 + 1)*abs(a) + a)*c^4/(a^2*x) - 189*(sqrt(-a^2*x^2 + 1)*abs(a) + a)^2*c^4/(
a^4*x^2) + 525*(sqrt(-a^2*x^2 + 1)*abs(a) + a)^3*c^4/(a^6*x^3) + 1295*(sqrt(-a^2*x^2 + 1)*abs(a) + a)^4*c^4/(a
^8*x^4) - 4935*(sqrt(-a^2*x^2 + 1)*abs(a) + a)^5*c^4/(a^10*x^5) - 9765*(sqrt(-a^2*x^2 + 1)*abs(a) + a)^6*c^4/(
a^12*x^6))*a^14*x^7/((sqrt(-a^2*x^2 + 1)*abs(a) + a)^7*abs(a)) + c^4*arcsin(a*x)*sgn(a)/abs(a) - 35/16*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 + 1/13440*(9765*(
sqrt(-a^2*x^2 + 1)*abs(a) + a)*a^4*c^4/x + 4935*(sqrt(-a^2*x^2 + 1)*abs(a) + a)^2*a^2*c^4/x^2 - 1295*(sqrt(-a^
2*x^2 + 1)*abs(a) + a)^3*c^4/x^3 - 525*(sqrt(-a^2*x^2 + 1)*abs(a) + a)^4*c^4/(a^2*x^4) + 189*(sqrt(-a^2*x^2 +
1)*abs(a) + a)^5*c^4/(a^4*x^5) + 35*(sqrt(-a^2*x^2 + 1)*abs(a) + a)^6*c^4/(a^6*x^6) - 15*(sqrt(-a^2*x^2 + 1)*a
bs(a) + a)^7*c^4/(a^8*x^7))/(a^6*abs(a))