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

Optimal. Leaf size=136 \[ \frac {c^3 (15 a x+8) \sqrt {1-a^2 x^2}}{8 a^2 x}-\frac {15 c^3 \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )}{8 a}+\frac {c^3 (4-5 a x) \left (1-a^2 x^2\right )^{5/2}}{20 a^6 x^5}-\frac {c^3 (8-15 a x) \left (1-a^2 x^2\right )^{3/2}}{24 a^4 x^3}+\frac {c^3 \sin ^{-1}(a x)}{a} \]

[Out]

-1/24*c^3*(-15*a*x+8)*(-a^2*x^2+1)^(3/2)/a^4/x^3+1/20*c^3*(-5*a*x+4)*(-a^2*x^2+1)^(5/2)/a^6/x^5+c^3*arcsin(a*x
)/a-15/8*c^3*arctanh((-a^2*x^2+1)^(1/2))/a+1/8*c^3*(15*a*x+8)*(-a^2*x^2+1)^(1/2)/a^2/x

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Rubi [A]  time = 0.22, antiderivative size = 136, 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, 811, 813, 844, 216, 266, 63, 208} \[ \frac {c^3 (4-5 a x) \left (1-a^2 x^2\right )^{5/2}}{20 a^6 x^5}-\frac {c^3 (8-15 a x) \left (1-a^2 x^2\right )^{3/2}}{24 a^4 x^3}+\frac {c^3 (15 a x+8) \sqrt {1-a^2 x^2}}{8 a^2 x}-\frac {15 c^3 \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )}{8 a}+\frac {c^3 \sin ^{-1}(a x)}{a} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(c^3*(8 + 15*a*x)*Sqrt[1 - a^2*x^2])/(8*a^2*x) - (c^3*(8 - 15*a*x)*(1 - a^2*x^2)^(3/2))/(24*a^4*x^3) + (c^3*(4
- 5*a*x)*(1 - a^2*x^2)^(5/2))/(20*a^6*x^5) + (c^3*ArcSin[a*x])/a - (15*c^3*ArcTanh[Sqrt[1 - a^2*x^2]])/(8*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 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 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^{-\tanh ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^3 \, dx &=-\frac {c^3 \int \frac {e^{-\tanh ^{-1}(a x)} \left (1-a^2 x^2\right )^3}{x^6} \, dx}{a^6}\\ &=-\frac {c^3 \int \frac {(1-a x) \left (1-a^2 x^2\right )^{5/2}}{x^6} \, dx}{a^6}\\ &=\frac {c^3 (4-5 a x) \left (1-a^2 x^2\right )^{5/2}}{20 a^6 x^5}+\frac {c^3 \int \frac {\left (8 a^2-10 a^3 x\right ) \left (1-a^2 x^2\right )^{3/2}}{x^4} \, dx}{8 a^6}\\ &=-\frac {c^3 (8-15 a x) \left (1-a^2 x^2\right )^{3/2}}{24 a^4 x^3}+\frac {c^3 (4-5 a x) \left (1-a^2 x^2\right )^{5/2}}{20 a^6 x^5}-\frac {c^3 \int \frac {\left (32 a^4-60 a^5 x\right ) \sqrt {1-a^2 x^2}}{x^2} \, dx}{32 a^6}\\ &=\frac {c^3 (8+15 a x) \sqrt {1-a^2 x^2}}{8 a^2 x}-\frac {c^3 (8-15 a x) \left (1-a^2 x^2\right )^{3/2}}{24 a^4 x^3}+\frac {c^3 (4-5 a x) \left (1-a^2 x^2\right )^{5/2}}{20 a^6 x^5}+\frac {c^3 \int \frac {120 a^5+64 a^6 x}{x \sqrt {1-a^2 x^2}} \, dx}{64 a^6}\\ &=\frac {c^3 (8+15 a x) \sqrt {1-a^2 x^2}}{8 a^2 x}-\frac {c^3 (8-15 a x) \left (1-a^2 x^2\right )^{3/2}}{24 a^4 x^3}+\frac {c^3 (4-5 a x) \left (1-a^2 x^2\right )^{5/2}}{20 a^6 x^5}+c^3 \int \frac {1}{\sqrt {1-a^2 x^2}} \, dx+\frac {\left (15 c^3\right ) \int \frac {1}{x \sqrt {1-a^2 x^2}} \, dx}{8 a}\\ &=\frac {c^3 (8+15 a x) \sqrt {1-a^2 x^2}}{8 a^2 x}-\frac {c^3 (8-15 a x) \left (1-a^2 x^2\right )^{3/2}}{24 a^4 x^3}+\frac {c^3 (4-5 a x) \left (1-a^2 x^2\right )^{5/2}}{20 a^6 x^5}+\frac {c^3 \sin ^{-1}(a x)}{a}+\frac {\left (15 c^3\right ) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {1-a^2 x}} \, dx,x,x^2\right )}{16 a}\\ &=\frac {c^3 (8+15 a x) \sqrt {1-a^2 x^2}}{8 a^2 x}-\frac {c^3 (8-15 a x) \left (1-a^2 x^2\right )^{3/2}}{24 a^4 x^3}+\frac {c^3 (4-5 a x) \left (1-a^2 x^2\right )^{5/2}}{20 a^6 x^5}+\frac {c^3 \sin ^{-1}(a x)}{a}-\frac {\left (15 c^3\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 )}{8 a^3}\\ &=\frac {c^3 (8+15 a x) \sqrt {1-a^2 x^2}}{8 a^2 x}-\frac {c^3 (8-15 a x) \left (1-a^2 x^2\right )^{3/2}}{24 a^4 x^3}+\frac {c^3 (4-5 a x) \left (1-a^2 x^2\right )^{5/2}}{20 a^6 x^5}+\frac {c^3 \sin ^{-1}(a x)}{a}-\frac {15 c^3 \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )}{8 a}\\ \end {align*}

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Mathematica [C]  time = 0.03, size = 70, normalized size = 0.51 \[ \frac {c^3 \left (\frac {7 \, _2F_1\left (-\frac {5}{2},-\frac {5}{2};-\frac {3}{2};a^2 x^2\right )}{x^5}-5 a^5 \left (1-a^2 x^2\right )^{7/2} \, _2F_1\left (3,\frac {7}{2};\frac {9}{2};1-a^2 x^2\right )\right )}{35 a^6} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(c^3*((7*Hypergeometric2F1[-5/2, -5/2, -3/2, a^2*x^2])/x^5 - 5*a^5*(1 - a^2*x^2)^(7/2)*Hypergeometric2F1[3, 7/
2, 9/2, 1 - a^2*x^2]))/(35*a^6)

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fricas [A]  time = 1.15, size = 154, normalized size = 1.13 \[ -\frac {240 \, a^{5} c^{3} x^{5} \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right ) - 225 \, a^{5} c^{3} x^{5} \log \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{x}\right ) - 120 \, a^{5} c^{3} x^{5} - {\left (120 \, a^{5} c^{3} x^{5} + 184 \, a^{4} c^{3} x^{4} + 135 \, a^{3} c^{3} x^{3} - 88 \, a^{2} c^{3} x^{2} - 30 \, a c^{3} x + 24 \, c^{3}\right )} \sqrt {-a^{2} x^{2} + 1}}{120 \, a^{6} x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/120*(240*a^5*c^3*x^5*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) - 225*a^5*c^3*x^5*log((sqrt(-a^2*x^2 + 1) - 1)/
x) - 120*a^5*c^3*x^5 - (120*a^5*c^3*x^5 + 184*a^4*c^3*x^4 + 135*a^3*c^3*x^3 - 88*a^2*c^3*x^2 - 30*a*c^3*x + 24
*c^3)*sqrt(-a^2*x^2 + 1))/(a^6*x^5)

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giac [B]  time = 0.22, size = 384, normalized size = 2.82 \[ -\frac {{\left (6 \, c^{3} - \frac {15 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )} c^{3}}{a^{2} x} - \frac {70 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{2} c^{3}}{a^{4} x^{2}} + \frac {240 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{3} c^{3}}{a^{6} x^{3}} + \frac {660 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{4} c^{3}}{a^{8} x^{4}}\right )} a^{10} x^{5}}{960 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{5} {\left | a \right |}} + \frac {c^{3} \arcsin \left (a x\right ) \mathrm {sgn}\relax (a)}{{\left | a \right |}} - \frac {15 \, c^{3} \log \left (\frac {{\left | -2 \, \sqrt {-a^{2} x^{2} + 1} {\left | a \right |} - 2 \, a \right |}}{2 \, a^{2} {\left | x \right |}}\right )}{8 \, {\left | a \right |}} + \frac {\sqrt {-a^{2} x^{2} + 1} c^{3}}{a} + \frac {\frac {660 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )} a^{2} c^{3}}{x} + \frac {240 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{2} c^{3}}{x^{2}} - \frac {70 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{3} c^{3}}{a^{2} x^{3}} - \frac {15 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{4} c^{3}}{a^{4} x^{4}} + \frac {6 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{5} c^{3}}{a^{6} x^{5}}}{960 \, a^{4} {\left | a \right |}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/960*(6*c^3 - 15*(sqrt(-a^2*x^2 + 1)*abs(a) + a)*c^3/(a^2*x) - 70*(sqrt(-a^2*x^2 + 1)*abs(a) + a)^2*c^3/(a^4
*x^2) + 240*(sqrt(-a^2*x^2 + 1)*abs(a) + a)^3*c^3/(a^6*x^3) + 660*(sqrt(-a^2*x^2 + 1)*abs(a) + a)^4*c^3/(a^8*x
^4))*a^10*x^5/((sqrt(-a^2*x^2 + 1)*abs(a) + a)^5*abs(a)) + c^3*arcsin(a*x)*sgn(a)/abs(a) - 15/8*c^3*log(1/2*ab
s(-2*sqrt(-a^2*x^2 + 1)*abs(a) - 2*a)/(a^2*abs(x)))/abs(a) + sqrt(-a^2*x^2 + 1)*c^3/a + 1/960*(660*(sqrt(-a^2*
x^2 + 1)*abs(a) + a)*a^2*c^3/x + 240*(sqrt(-a^2*x^2 + 1)*abs(a) + a)^2*c^3/x^2 - 70*(sqrt(-a^2*x^2 + 1)*abs(a)
+ a)^3*c^3/(a^2*x^3) - 15*(sqrt(-a^2*x^2 + 1)*abs(a) + a)^4*c^3/(a^4*x^4) + 6*(sqrt(-a^2*x^2 + 1)*abs(a) + a)
^5*c^3/(a^6*x^5))/(a^4*abs(a))

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maple [A]  time = 0.05, size = 203, normalized size = 1.49 \[ \frac {15 c^{3} \sqrt {-a^{2} x^{2}+1}}{8 a}-\frac {15 c^{3} \arctanh \left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )}{8 a}+\frac {c^{3} \left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}}{a^{2} x}+c^{3} x \sqrt {-a^{2} x^{2}+1}+\frac {c^{3} \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{\sqrt {a^{2}}}-\frac {8 c^{3} \left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}}{15 a^{4} x^{3}}+\frac {7 c^{3} \left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}}{8 x^{2} a^{3}}-\frac {c^{3} \left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}}{4 a^{5} x^{4}}+\frac {c^{3} \left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}}{5 a^{6} x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

15/8*c^3*(-a^2*x^2+1)^(1/2)/a-15/8*c^3/a*arctanh(1/(-a^2*x^2+1)^(1/2))+c^3/a^2/x*(-a^2*x^2+1)^(3/2)+c^3*x*(-a^
2*x^2+1)^(1/2)+c^3/(a^2)^(1/2)*arctan((a^2)^(1/2)*x/(-a^2*x^2+1)^(1/2))-8/15*c^3/a^4/x^3*(-a^2*x^2+1)^(3/2)+7/
8*c^3*(-a^2*x^2+1)^(3/2)/x^2/a^3-1/4*c^3/a^5/x^4*(-a^2*x^2+1)^(3/2)+1/5*c^3/a^6/x^5*(-a^2*x^2+1)^(3/2)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ c^{3} {\left (\frac {\arcsin \left (a x\right )}{a} + \frac {\sqrt {-a^{2} x^{2} + 1}}{a}\right )} - \int \frac {{\left (3 \, a^{4} c^{3} x^{4} - 3 \, a^{2} c^{3} x^{2} + c^{3}\right )} \sqrt {a x + 1} \sqrt {-a x + 1}}{a^{7} x^{7} + a^{6} x^{6}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

c^3*(arcsin(a*x)/a + sqrt(-a^2*x^2 + 1)/a) - integrate((3*a^4*c^3*x^4 - 3*a^2*c^3*x^2 + c^3)*sqrt(a*x + 1)*sqr
t(-a*x + 1)/(a^7*x^7 + a^6*x^6), x)

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mupad [B]  time = 0.87, size = 181, normalized size = 1.33 \[ \frac {c^3\,\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )}{\sqrt {-a^2}}+\frac {c^3\,\sqrt {1-a^2\,x^2}}{a}+\frac {23\,c^3\,\sqrt {1-a^2\,x^2}}{15\,a^2\,x}+\frac {9\,c^3\,\sqrt {1-a^2\,x^2}}{8\,a^3\,x^2}-\frac {11\,c^3\,\sqrt {1-a^2\,x^2}}{15\,a^4\,x^3}-\frac {c^3\,\sqrt {1-a^2\,x^2}}{4\,a^5\,x^4}+\frac {c^3\,\sqrt {1-a^2\,x^2}}{5\,a^6\,x^5}+\frac {c^3\,\mathrm {atan}\left (\sqrt {1-a^2\,x^2}\,1{}\mathrm {i}\right )\,15{}\mathrm {i}}{8\,a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(c^3*asinh(x*(-a^2)^(1/2)))/(-a^2)^(1/2) + (c^3*atan((1 - a^2*x^2)^(1/2)*1i)*15i)/(8*a) + (c^3*(1 - a^2*x^2)^(
1/2))/a + (23*c^3*(1 - a^2*x^2)^(1/2))/(15*a^2*x) + (9*c^3*(1 - a^2*x^2)^(1/2))/(8*a^3*x^2) - (11*c^3*(1 - a^2
*x^2)^(1/2))/(15*a^4*x^3) - (c^3*(1 - a^2*x^2)^(1/2))/(4*a^5*x^4) + (c^3*(1 - a^2*x^2)^(1/2))/(5*a^6*x^5)

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sympy [C]  time = 10.02, size = 692, normalized size = 5.09 \[ \frac {c^{3} \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 {c^{3} \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 {2 c^{3} \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 {2 c^{3} \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}} + \frac {c^{3} \left (\begin {cases} \frac {a^{4} \operatorname {acosh}{\left (\frac {1}{a x} \right )}}{8} - \frac {a^{3}}{8 x \sqrt {-1 + \frac {1}{a^{2} x^{2}}}} + \frac {3 a}{8 x^{3} \sqrt {-1 + \frac {1}{a^{2} x^{2}}}} - \frac {1}{4 a x^{5} \sqrt {-1 + \frac {1}{a^{2} x^{2}}}} & \text {for}\: \frac {1}{\left |{a^{2} x^{2}}\right |} > 1 \\- \frac {i a^{4} \operatorname {asin}{\left (\frac {1}{a x} \right )}}{8} + \frac {i a^{3}}{8 x \sqrt {1 - \frac {1}{a^{2} x^{2}}}} - \frac {3 i a}{8 x^{3} \sqrt {1 - \frac {1}{a^{2} x^{2}}}} + \frac {i}{4 a x^{5} \sqrt {1 - \frac {1}{a^{2} x^{2}}}} & \text {otherwise} \end {cases}\right )}{a^{5}} - \frac {c^{3} \left (\begin {cases} \frac {2 i a^{4} \sqrt {a^{2} x^{2} - 1}}{15 x} + \frac {i a^{2} \sqrt {a^{2} x^{2} - 1}}{15 x^{3}} - \frac {i \sqrt {a^{2} x^{2} - 1}}{5 x^{5}} & \text {for}\: \left |{a^{2} x^{2}}\right | > 1 \\\frac {2 a^{4} \sqrt {- a^{2} x^{2} + 1}}{15 x} + \frac {a^{2} \sqrt {- a^{2} x^{2} + 1}}{15 x^{3}} - \frac {\sqrt {- a^{2} x^{2} + 1}}{5 x^{5}} & \text {otherwise} \end {cases}\right )}{a^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

c**3*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**3*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 - 2*c**3*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 + 2*c**3*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 + c**3*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 - c**3*Piecewise((2*I*a**4*sqrt(a**2*x**2 - 1)/(15*x) + I*a**2*sq
rt(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

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