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

Optimal. Leaf size=124 \[ -\frac {c^2 (a x+6) \sqrt {1-a^2 x^2}}{2 a^2 x}+\frac {c^2 \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )}{2 a}-\frac {c^2 \left (1-a^2 x^2\right )^{3/2}}{3 a^4 x^3}-\frac {3 c^2 \left (1-a^2 x^2\right )^{3/2}}{2 a^3 x^2}-\frac {3 c^2 \sin ^{-1}(a x)}{a} \]

[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 = 0.27, antiderivative size = 124, 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, 6148, 1807, 813, 844, 216, 266, 63, 208} \[ -\frac {3 c^2 \left (1-a^2 x^2\right )^{3/2}}{2 a^3 x^2}-\frac {c^2 \left (1-a^2 x^2\right )^{3/2}}{3 a^4 x^3}-\frac {c^2 (a x+6) \sqrt {1-a^2 x^2}}{2 a^2 x}+\frac {c^2 \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )}{2 a}-\frac {3 c^2 \sin ^{-1}(a x)}{a} \]

Antiderivative was successfully verified.

[In]

Int[E^(3*ArcTanh[a*x])*(c - c/(a^2*x^2))^2,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 6148

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] || Gt
Q[c, 0]) && IGtQ[(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 = 0.07, size = 128, normalized size = 1.03 \[ -\frac {c^2 \left (6 a^5 x^5-16 a^4 x^4-15 a^3 x^3+14 a^2 x^2+18 a^3 x^3 \sqrt {1-a^2 x^2} \sin ^{-1}(a x)-3 a^3 x^3 \sqrt {1-a^2 x^2} \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )+9 a x+2\right )}{6 a^4 x^3 \sqrt {1-a^2 x^2}} \]

Antiderivative was successfully verified.

[In]

Integrate[E^(3*ArcTanh[a*x])*(c - c/(a^2*x^2))^2,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 = 0.58, size = 131, 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((a*x+1)^3/(-a^2*x^2+1)^(3/2)*(c-c/a^2/x^2)^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 = 0.31, size = 262, normalized size = 2.11 \[ \frac {{\left (c^{2} + \frac {9 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )} c^{2}}{a^{2} x} + \frac {33 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{2} c^{2}}{a^{4} x^{2}}\right )} a^{6} x^{3}}{24 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{3} {\left | a \right |}} - \frac {3 \, c^{2} \arcsin \left (a x\right ) \mathrm {sgn}\relax (a)}{{\left | a \right |}} + \frac {c^{2} \log \left (\frac {{\left | -2 \, \sqrt {-a^{2} x^{2} + 1} {\left | a \right |} - 2 \, a \right |}}{2 \, a^{2} {\left | x \right |}}\right )}{2 \, {\left | a \right |}} + \frac {\sqrt {-a^{2} x^{2} + 1} c^{2}}{a} - \frac {\frac {33 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )} c^{2}}{x} + \frac {9 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{2} c^{2}}{a^{2} x^{2}} + \frac {{\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{3} c^{2}}{a^{4} x^{3}}}{24 \, a^{2} {\left | a \right |}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)^3/(-a^2*x^2+1)^(3/2)*(c-c/a^2/x^2)^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 [A]  time = 0.05, size = 181, normalized size = 1.46 \[ -\frac {c^{2} a \,x^{2}}{\sqrt {-a^{2} x^{2}+1}}+\frac {5 c^{2}}{2 a \sqrt {-a^{2} x^{2}+1}}+\frac {8 c^{2} x}{3 \sqrt {-a^{2} x^{2}+1}}-\frac {3 c^{2} \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{\sqrt {a^{2}}}+\frac {c^{2} \arctanh \left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )}{2 a}-\frac {7 c^{2}}{3 a^{2} x \sqrt {-a^{2} x^{2}+1}}-\frac {c^{2}}{3 a^{4} x^{3} \sqrt {-a^{2} x^{2}+1}}-\frac {3 c^{2}}{2 a^{3} x^{2} \sqrt {-a^{2} x^{2}+1}} \]

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^2/x^2)^2,x)

[Out]

-c^2*a*x^2/(-a^2*x^2+1)^(1/2)+5/2*c^2/a/(-a^2*x^2+1)^(1/2)+8/3*c^2*x/(-a^2*x^2+1)^(1/2)-3*c^2/(a^2)^(1/2)*arct
an((a^2)^(1/2)*x/(-a^2*x^2+1)^(1/2))+1/2*c^2/a*arctanh(1/(-a^2*x^2+1)^(1/2))-7/3*c^2/a^2/x/(-a^2*x^2+1)^(1/2)-
1/3*c^2/a^4/x^3/(-a^2*x^2+1)^(1/2)-3/2*c^2/a^3/x^2/(-a^2*x^2+1)^(1/2)

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maxima [B]  time = 0.42, size = 347, normalized size = 2.80 \[ -a^{3} c^{2} {\left (\frac {x^{2}}{\sqrt {-a^{2} x^{2} + 1} a^{2}} - \frac {2}{\sqrt {-a^{2} x^{2} + 1} a^{4}}\right )} + 3 \, a^{2} c^{2} {\left (\frac {x}{\sqrt {-a^{2} x^{2} + 1} a^{2}} - \frac {\arcsin \left (a x\right )}{a^{3}}\right )} - \frac {5 \, c^{2} x}{\sqrt {-a^{2} x^{2} + 1}} - \frac {5 \, c^{2} {\left (\frac {1}{\sqrt {-a^{2} x^{2} + 1}} - \log \left (\frac {2 \, \sqrt {-a^{2} x^{2} + 1}}{{\left | x \right |}} + \frac {2}{{\left | x \right |}}\right )\right )}}{a} + \frac {{\left (\frac {2 \, a^{2} x}{\sqrt {-a^{2} x^{2} + 1}} - \frac {1}{\sqrt {-a^{2} x^{2} + 1} x}\right )} c^{2}}{a^{2}} + \frac {c^{2}}{\sqrt {-a^{2} x^{2} + 1} a} - \frac {3 \, {\left (3 \, a^{2} \log \left (\frac {2 \, \sqrt {-a^{2} x^{2} + 1}}{{\left | x \right |}} + \frac {2}{{\left | x \right |}}\right ) - \frac {3 \, a^{2}}{\sqrt {-a^{2} x^{2} + 1}} + \frac {1}{\sqrt {-a^{2} x^{2} + 1} x^{2}}\right )} c^{2}}{2 \, a^{3}} + \frac {{\left (\frac {8 \, a^{4} x}{\sqrt {-a^{2} x^{2} + 1}} - \frac {4 \, a^{2}}{\sqrt {-a^{2} x^{2} + 1} x} - \frac {1}{\sqrt {-a^{2} x^{2} + 1} x^{3}}\right )} c^{2}}{3 \, a^{4}} \]

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^2/x^2)^2,x, algorithm="maxima")

[Out]

-a^3*c^2*(x^2/(sqrt(-a^2*x^2 + 1)*a^2) - 2/(sqrt(-a^2*x^2 + 1)*a^4)) + 3*a^2*c^2*(x/(sqrt(-a^2*x^2 + 1)*a^2) -
arcsin(a*x)/a^3) - 5*c^2*x/sqrt(-a^2*x^2 + 1) - 5*c^2*(1/sqrt(-a^2*x^2 + 1) - log(2*sqrt(-a^2*x^2 + 1)/abs(x)
+ 2/abs(x)))/a + (2*a^2*x/sqrt(-a^2*x^2 + 1) - 1/(sqrt(-a^2*x^2 + 1)*x))*c^2/a^2 + c^2/(sqrt(-a^2*x^2 + 1)*a)
- 3/2*(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^2/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/(sqrt(-a^2*x^2 + 1)*x^3))*c^
2/a^4

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mupad [B]  time = 0.82, size = 136, normalized size = 1.10 \[ \frac {c^2\,\sqrt {1-a^2\,x^2}}{a}-\frac {3\,c^2\,\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )}{\sqrt {-a^2}}-\frac {8\,c^2\,\sqrt {1-a^2\,x^2}}{3\,a^2\,x}-\frac {3\,c^2\,\sqrt {1-a^2\,x^2}}{2\,a^3\,x^2}-\frac {c^2\,\sqrt {1-a^2\,x^2}}{3\,a^4\,x^3}-\frac {c^2\,\mathrm {atan}\left (\sqrt {1-a^2\,x^2}\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{2\,a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(c^2*(1 - a^2*x^2)^(1/2))/a - (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) - (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 [A]  time = 16.83, size = 357, normalized size = 2.88 \[ - a c^{2} \left (\begin {cases} \frac {x^{2}}{2} & \text {for}\: a^{2} = 0 \\- \frac {\sqrt {- a^{2} x^{2} + 1}}{a^{2}} & \text {otherwise} \end {cases}\right ) - 3 c^{2} \left (\begin {cases} \sqrt {\frac {1}{a^{2}}} \operatorname {asin}{\left (x \sqrt {a^{2}} \right )} & \text {for}\: a^{2} > 0 \\\sqrt {- \frac {1}{a^{2}}} \operatorname {asinh}{\left (x \sqrt {- a^{2}} \right )} & \text {for}\: a^{2} < 0 \end {cases}\right ) - \frac {2 c^{2} \left (\begin {cases} - \operatorname {acosh}{\left (\frac {1}{a x} \right )} & \text {for}\: \frac {1}{\left |{a^{2} x^{2}}\right |} > 1 \\i \operatorname {asin}{\left (\frac {1}{a x} \right )} & \text {otherwise} \end {cases}\right )}{a} + \frac {2 c^{2} \left (\begin {cases} - \frac {i \sqrt {a^{2} x^{2} - 1}}{x} & \text {for}\: \left |{a^{2} x^{2}}\right | > 1 \\- \frac {\sqrt {- a^{2} x^{2} + 1}}{x} & \text {otherwise} \end {cases}\right )}{a^{2}} + \frac {3 c^{2} \left (\begin {cases} - \frac {a^{2} \operatorname {acosh}{\left (\frac {1}{a x} \right )}}{2} - \frac {a \sqrt {-1 + \frac {1}{a^{2} x^{2}}}}{2 x} & \text {for}\: \frac {1}{\left |{a^{2} x^{2}}\right |} > 1 \\\frac {i a^{2} \operatorname {asin}{\left (\frac {1}{a x} \right )}}{2} - \frac {i a}{2 x \sqrt {1 - \frac {1}{a^{2} x^{2}}}} + \frac {i}{2 a x^{3} \sqrt {1 - \frac {1}{a^{2} x^{2}}}} & \text {otherwise} \end {cases}\right )}{a^{3}} + \frac {c^{2} \left (\begin {cases} - \frac {2 i a^{2} \sqrt {a^{2} x^{2} - 1}}{3 x} - \frac {i \sqrt {a^{2} x^{2} - 1}}{3 x^{3}} & \text {for}\: \left |{a^{2} x^{2}}\right | > 1 \\- \frac {2 a^{2} \sqrt {- a^{2} x^{2} + 1}}{3 x} - \frac {\sqrt {- a^{2} x^{2} + 1}}{3 x^{3}} & \text {otherwise} \end {cases}\right )}{a^{4}} \]

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**2/x**2)**2,x)

[Out]

-a*c**2*Piecewise((x**2/2, Eq(a**2, 0)), (-sqrt(-a**2*x**2 + 1)/a**2, True)) - 3*c**2*Piecewise((sqrt(a**(-2))
*asin(x*sqrt(a**2)), a**2 > 0), (sqrt(-1/a**2)*asinh(x*sqrt(-a**2)), a**2 < 0)) - 2*c**2*Piecewise((-acosh(1/(
a*x)), 1/Abs(a**2*x**2) > 1), (I*asin(1/(a*x)), True))/a + 2*c**2*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 + 3*c**2*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**2*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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