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

Optimal. Leaf size=125 \[ -\frac {c^2 (6-a x) \sqrt {1-a^2 x^2}}{2 a^2 x}-\frac {c^2 \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )}{2 a}-\frac {c^2 \left (1-a^2 x^2\right )^{3/2}}{3 a^4 x^3}+\frac {3 c^2 \left (1-a^2 x^2\right )^{3/2}}{2 a^3 x^2}-\frac {3 c^2 \sin ^{-1}(a x)}{a} \]

[Out]

-1/3*c^2*(-a^2*x^2+1)^(3/2)/a^4/x^3+3/2*c^2*(-a^2*x^2+1)^(3/2)/a^3/x^2-3*c^2*arcsin(a*x)/a-1/2*c^2*arctanh((-a
^2*x^2+1)^(1/2))/a-1/2*c^2*(-a*x+6)*(-a^2*x^2+1)^(1/2)/a^2/x

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Rubi [A]  time = 0.28, antiderivative size = 125, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 9, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.409, Rules used = {6157, 6149, 1807, 813, 844, 216, 266, 63, 208} \[ \frac {3 c^2 \left (1-a^2 x^2\right )^{3/2}}{2 a^3 x^2}-\frac {c^2 \left (1-a^2 x^2\right )^{3/2}}{3 a^4 x^3}-\frac {c^2 (6-a x) \sqrt {1-a^2 x^2}}{2 a^2 x}-\frac {c^2 \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )}{2 a}-\frac {3 c^2 \sin ^{-1}(a x)}{a} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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]

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 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 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 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 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]

Rubi steps

\begin {align*} \int e^{-3 \tanh ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^2 \, dx &=\frac {c^2 \int \frac {e^{-3 \tanh ^{-1}(a x)} \left (1-a^2 x^2\right )^2}{x^4} \, dx}{a^4}\\ &=\frac {c^2 \int \frac {(1-a x)^3 \sqrt {1-a^2 x^2}}{x^4} \, dx}{a^4}\\ &=-\frac {c^2 \left (1-a^2 x^2\right )^{3/2}}{3 a^4 x^3}-\frac {c^2 \int \frac {\sqrt {1-a^2 x^2} \left (9 a-9 a^2 x+3 a^3 x^2\right )}{x^3} \, dx}{3 a^4}\\ &=-\frac {c^2 \left (1-a^2 x^2\right )^{3/2}}{3 a^4 x^3}+\frac {3 c^2 \left (1-a^2 x^2\right )^{3/2}}{2 a^3 x^2}+\frac {c^2 \int \frac {\left (18 a^2+3 a^3 x\right ) \sqrt {1-a^2 x^2}}{x^2} \, dx}{6 a^4}\\ &=-\frac {c^2 (6-a x) \sqrt {1-a^2 x^2}}{2 a^2 x}-\frac {c^2 \left (1-a^2 x^2\right )^{3/2}}{3 a^4 x^3}+\frac {3 c^2 \left (1-a^2 x^2\right )^{3/2}}{2 a^3 x^2}-\frac {c^2 \int \frac {-6 a^3+36 a^4 x}{x \sqrt {1-a^2 x^2}} \, dx}{12 a^4}\\ &=-\frac {c^2 (6-a x) \sqrt {1-a^2 x^2}}{2 a^2 x}-\frac {c^2 \left (1-a^2 x^2\right )^{3/2}}{3 a^4 x^3}+\frac {3 c^2 \left (1-a^2 x^2\right )^{3/2}}{2 a^3 x^2}-\left (3 c^2\right ) \int \frac {1}{\sqrt {1-a^2 x^2}} \, dx+\frac {c^2 \int \frac {1}{x \sqrt {1-a^2 x^2}} \, dx}{2 a}\\ &=-\frac {c^2 (6-a x) \sqrt {1-a^2 x^2}}{2 a^2 x}-\frac {c^2 \left (1-a^2 x^2\right )^{3/2}}{3 a^4 x^3}+\frac {3 c^2 \left (1-a^2 x^2\right )^{3/2}}{2 a^3 x^2}-\frac {3 c^2 \sin ^{-1}(a x)}{a}+\frac {c^2 \operatorname {Subst}\left (\int \frac {1}{x \sqrt {1-a^2 x}} \, dx,x,x^2\right )}{4 a}\\ &=-\frac {c^2 (6-a x) \sqrt {1-a^2 x^2}}{2 a^2 x}-\frac {c^2 \left (1-a^2 x^2\right )^{3/2}}{3 a^4 x^3}+\frac {3 c^2 \left (1-a^2 x^2\right )^{3/2}}{2 a^3 x^2}-\frac {3 c^2 \sin ^{-1}(a x)}{a}-\frac {c^2 \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^2 (6-a x) \sqrt {1-a^2 x^2}}{2 a^2 x}-\frac {c^2 \left (1-a^2 x^2\right )^{3/2}}{3 a^4 x^3}+\frac {3 c^2 \left (1-a^2 x^2\right )^{3/2}}{2 a^3 x^2}-\frac {3 c^2 \sin ^{-1}(a x)}{a}-\frac {c^2 \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )}{2 a}\\ \end {align*}

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Mathematica [A]  time = 0.08, size = 128, normalized size = 1.02 \[ -\frac {c^2 \left (-6 a^5 x^5-16 a^4 x^4+15 a^3 x^3+14 a^2 x^2+18 a^3 x^3 \sqrt {1-a^2 x^2} \sin ^{-1}(a x)+3 a^3 x^3 \sqrt {1-a^2 x^2} \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )-9 a x+2\right )}{6 a^4 x^3 \sqrt {1-a^2 x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-1/6*(c^2*(2 - 9*a*x + 14*a^2*x^2 + 15*a^3*x^3 - 16*a^4*x^4 - 6*a^5*x^5 + 18*a^3*x^3*Sqrt[1 - a^2*x^2]*ArcSin[
a*x] + 3*a^3*x^3*Sqrt[1 - a^2*x^2]*ArcTanh[Sqrt[1 - a^2*x^2]]))/(a^4*x^3*Sqrt[1 - a^2*x^2])

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fricas [A]  time = 0.51, size = 132, normalized size = 1.06 \[ \frac {36 \, a^{3} c^{2} x^{3} \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right ) + 3 \, a^{3} c^{2} x^{3} \log \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{x}\right ) - 6 \, a^{3} c^{2} x^{3} - {\left (6 \, a^{3} c^{2} x^{3} + 16 \, a^{2} c^{2} x^{2} - 9 \, a c^{2} x + 2 \, c^{2}\right )} \sqrt {-a^{2} x^{2} + 1}}{6 \, a^{4} x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*(36*a^3*c^2*x^3*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) + 3*a^3*c^2*x^3*log((sqrt(-a^2*x^2 + 1) - 1)/x) - 6
*a^3*c^2*x^3 - (6*a^3*c^2*x^3 + 16*a^2*c^2*x^2 - 9*a*c^2*x + 2*c^2)*sqrt(-a^2*x^2 + 1))/(a^4*x^3)

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giac [B]  time = 0.18, size = 263, normalized size = 2.10 \[ \frac {{\left (c^{2} - \frac {9 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )} c^{2}}{a^{2} x} + \frac {33 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{2} c^{2}}{a^{4} x^{2}}\right )} a^{6} x^{3}}{24 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{3} {\left | a \right |}} - \frac {3 \, c^{2} \arcsin \left (a x\right ) \mathrm {sgn}\relax (a)}{{\left | a \right |}} - \frac {c^{2} \log \left (\frac {{\left | -2 \, \sqrt {-a^{2} x^{2} + 1} {\left | a \right |} - 2 \, a \right |}}{2 \, a^{2} {\left | x \right |}}\right )}{2 \, {\left | a \right |}} - \frac {\sqrt {-a^{2} x^{2} + 1} c^{2}}{a} - \frac {\frac {33 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )} c^{2}}{x} - \frac {9 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{2} c^{2}}{a^{2} x^{2}} + \frac {{\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{3} c^{2}}{a^{4} x^{3}}}{24 \, a^{2} {\left | a \right |}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/24*(c^2 - 9*(sqrt(-a^2*x^2 + 1)*abs(a) + a)*c^2/(a^2*x) + 33*(sqrt(-a^2*x^2 + 1)*abs(a) + a)^2*c^2/(a^4*x^2)
)*a^6*x^3/((sqrt(-a^2*x^2 + 1)*abs(a) + a)^3*abs(a)) - 3*c^2*arcsin(a*x)*sgn(a)/abs(a) - 1/2*c^2*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^2/a - 1/24*(33*(sqrt(-a^2*x^2 +
1)*abs(a) + a)*c^2/x - 9*(sqrt(-a^2*x^2 + 1)*abs(a) + a)^2*c^2/(a^2*x^2) + (sqrt(-a^2*x^2 + 1)*abs(a) + a)^3*
c^2/(a^4*x^3))/(a^2*abs(a))

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maple [B]  time = 0.05, size = 299, normalized size = 2.39 \[ \frac {c^{2} \left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}}{6 a}+\frac {c^{2} \sqrt {-a^{2} x^{2}+1}}{2 a}-\frac {c^{2} \arctanh \left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )}{2 a}-\frac {10 c^{2} \left (-a^{2} x^{2}+1\right )^{\frac {5}{2}}}{3 a^{2} x}-\frac {10 c^{2} x \left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}}{3}-5 c^{2} x \sqrt {-a^{2} x^{2}+1}-\frac {5 c^{2} \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{\sqrt {a^{2}}}-\frac {c^{2} \left (-a^{2} x^{2}+1\right )^{\frac {5}{2}}}{3 a^{4} x^{3}}+\frac {3 c^{2} \left (-a^{2} x^{2}+1\right )^{\frac {5}{2}}}{2 a^{3} x^{2}}+\frac {4 c^{2} \left (-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )\right )^{\frac {3}{2}}}{3 a}+2 c^{2} \sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}\, x +\frac {2 c^{2} \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}\right )}{\sqrt {a^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*c^2*(-a^2*x^2+1)^(3/2)/a+1/2*c^2*(-a^2*x^2+1)^(1/2)/a-1/2*c^2/a*arctanh(1/(-a^2*x^2+1)^(1/2))-10/3*c^2/a^2
/x*(-a^2*x^2+1)^(5/2)-10/3*c^2*x*(-a^2*x^2+1)^(3/2)-5*c^2*x*(-a^2*x^2+1)^(1/2)-5*c^2/(a^2)^(1/2)*arctan((a^2)^
(1/2)*x/(-a^2*x^2+1)^(1/2))-1/3*c^2/a^4/x^3*(-a^2*x^2+1)^(5/2)+3/2*c^2/a^3/x^2*(-a^2*x^2+1)^(5/2)+4/3*c^2/a*(-
a^2*(x+1/a)^2+2*a*(x+1/a))^(3/2)+2*c^2*(-a^2*(x+1/a)^2+2*a*(x+1/a))^(1/2)*x+2*c^2/(a^2)^(1/2)*arctan((a^2)^(1/
2)*x/(-a^2*(x+1/a)^2+2*a*(x+1/a))^(1/2))

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} {\left (c - \frac {c}{a^{2} x^{2}}\right )}^{2}}{{\left (a x + 1\right )}^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.86, size = 137, normalized size = 1.10 \[ \frac {3\,c^2\,\sqrt {1-a^2\,x^2}}{2\,a^3\,x^2}-\frac {c^2\,\sqrt {1-a^2\,x^2}}{a}-\frac {8\,c^2\,\sqrt {1-a^2\,x^2}}{3\,a^2\,x}-\frac {3\,c^2\,\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )}{\sqrt {-a^2}}-\frac {c^2\,\sqrt {1-a^2\,x^2}}{3\,a^4\,x^3}+\frac {c^2\,\mathrm {atan}\left (\sqrt {1-a^2\,x^2}\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{2\,a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(c^2*atan((1 - a^2*x^2)^(1/2)*1i)*1i)/(2*a) - (3*c^2*asinh(x*(-a^2)^(1/2)))/(-a^2)^(1/2) - (c^2*(1 - a^2*x^2)^
(1/2))/a - (8*c^2*(1 - a^2*x^2)^(1/2))/(3*a^2*x) + (3*c^2*(1 - a^2*x^2)^(1/2))/(2*a^3*x^2) - (c^2*(1 - a^2*x^2
)^(1/2))/(3*a^4*x^3)

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sympy [C]  time = 7.63, size = 384, normalized size = 3.07 \[ - \frac {c^{2} \left (\begin {cases} i \sqrt {a^{2} x^{2} - 1} - \log {\left (a x \right )} + \frac {\log {\left (a^{2} x^{2} \right )}}{2} + i \operatorname {asin}{\left (\frac {1}{a x} \right )} & \text {for}\: \left |{a^{2} x^{2}}\right | > 1 \\\sqrt {- a^{2} x^{2} + 1} + \frac {\log {\left (a^{2} x^{2} \right )}}{2} - \log {\left (\sqrt {- a^{2} x^{2} + 1} + 1 \right )} & \text {otherwise} \end {cases}\right )}{a} + \frac {3 c^{2} \left (\begin {cases} - \frac {i a^{2} x}{\sqrt {a^{2} x^{2} - 1}} + i a \operatorname {acosh}{\left (a x \right )} + \frac {i}{x \sqrt {a^{2} x^{2} - 1}} & \text {for}\: \left |{a^{2} x^{2}}\right | > 1 \\\frac {a^{2} x}{\sqrt {- a^{2} x^{2} + 1}} - a \operatorname {asin}{\left (a x \right )} - \frac {1}{x \sqrt {- a^{2} x^{2} + 1}} & \text {otherwise} \end {cases}\right )}{a^{2}} - \frac {3 c^{2} \left (\begin {cases} \frac {a^{2} \operatorname {acosh}{\left (\frac {1}{a x} \right )}}{2} + \frac {a}{2 x \sqrt {-1 + \frac {1}{a^{2} x^{2}}}} - \frac {1}{2 a x^{3} \sqrt {-1 + \frac {1}{a^{2} x^{2}}}} & \text {for}\: \frac {1}{\left |{a^{2} x^{2}}\right |} > 1 \\- \frac {i a^{2} \operatorname {asin}{\left (\frac {1}{a x} \right )}}{2} - \frac {i a \sqrt {1 - \frac {1}{a^{2} x^{2}}}}{2 x} & \text {otherwise} \end {cases}\right )}{a^{3}} + \frac {c^{2} \left (\begin {cases} \frac {a^{3} \sqrt {-1 + \frac {1}{a^{2} x^{2}}}}{3} - \frac {a \sqrt {-1 + \frac {1}{a^{2} x^{2}}}}{3 x^{2}} & \text {for}\: \frac {1}{\left |{a^{2} x^{2}}\right |} > 1 \\\frac {i a^{3} \sqrt {1 - \frac {1}{a^{2} x^{2}}}}{3} - \frac {i a \sqrt {1 - \frac {1}{a^{2} x^{2}}}}{3 x^{2}} & \text {otherwise} \end {cases}\right )}{a^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-c**2*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), (
sqrt(-a**2*x**2 + 1) + log(a**2*x**2)/2 - log(sqrt(-a**2*x**2 + 1) + 1), True))/a + 3*c**2*Piecewise((-I*a**2*
x/sqrt(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**2*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 + c**2*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

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