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

Optimal. Leaf size=157 \[ -\frac {3 c^3 (8-a x) \sqrt {1-a^2 x^2}}{8 a^2 x}-\frac {3 c^3 \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )}{8 a}+\frac {c^3 \left (1-a^2 x^2\right )^{5/2}}{5 a^6 x^5}+\frac {3 c^3 \left (1-a^2 x^2\right )^{5/2}}{4 a^5 x^4}+\frac {c^3 (a x+8) \left (1-a^2 x^2\right )^{3/2}}{8 a^4 x^3}-\frac {3 c^3 \sin ^{-1}(a x)}{a} \]

[Out]

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

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Rubi [A]  time = 0.32, antiderivative size = 157, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 10, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.454, Rules used = {6157, 6148, 1807, 811, 813, 844, 216, 266, 63, 208} \[ \frac {3 c^3 \left (1-a^2 x^2\right )^{5/2}}{4 a^5 x^4}+\frac {c^3 \left (1-a^2 x^2\right )^{5/2}}{5 a^6 x^5}+\frac {c^3 (a x+8) \left (1-a^2 x^2\right )^{3/2}}{8 a^4 x^3}-\frac {3 c^3 (8-a x) \sqrt {1-a^2 x^2}}{8 a^2 x}-\frac {3 c^3 \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )}{8 a}-\frac {3 c^3 \sin ^{-1}(a x)}{a} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-3*c^3*(8 - a*x)*Sqrt[1 - a^2*x^2])/(8*a^2*x) + (c^3*(8 + a*x)*(1 - a^2*x^2)^(3/2))/(8*a^4*x^3) + (c^3*(1 - a
^2*x^2)^(5/2))/(5*a^6*x^5) + (3*c^3*(1 - a^2*x^2)^(5/2))/(4*a^5*x^4) - (3*c^3*ArcSin[a*x])/a - (3*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 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 )^3 \, dx &=-\frac {c^3 \int \frac {e^{3 \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)^3 \left (1-a^2 x^2\right )^{3/2}}{x^6} \, dx}{a^6}\\ &=\frac {c^3 \left (1-a^2 x^2\right )^{5/2}}{5 a^6 x^5}+\frac {c^3 \int \frac {\left (1-a^2 x^2\right )^{3/2} \left (-15 a-15 a^2 x-5 a^3 x^2\right )}{x^5} \, dx}{5 a^6}\\ &=\frac {c^3 \left (1-a^2 x^2\right )^{5/2}}{5 a^6 x^5}+\frac {3 c^3 \left (1-a^2 x^2\right )^{5/2}}{4 a^5 x^4}-\frac {c^3 \int \frac {\left (60 a^2+5 a^3 x\right ) \left (1-a^2 x^2\right )^{3/2}}{x^4} \, dx}{20 a^6}\\ &=\frac {c^3 (8+a x) \left (1-a^2 x^2\right )^{3/2}}{8 a^4 x^3}+\frac {c^3 \left (1-a^2 x^2\right )^{5/2}}{5 a^6 x^5}+\frac {3 c^3 \left (1-a^2 x^2\right )^{5/2}}{4 a^5 x^4}+\frac {c^3 \int \frac {\left (240 a^4+30 a^5 x\right ) \sqrt {1-a^2 x^2}}{x^2} \, dx}{80 a^6}\\ &=-\frac {3 c^3 (8-a x) \sqrt {1-a^2 x^2}}{8 a^2 x}+\frac {c^3 (8+a x) \left (1-a^2 x^2\right )^{3/2}}{8 a^4 x^3}+\frac {c^3 \left (1-a^2 x^2\right )^{5/2}}{5 a^6 x^5}+\frac {3 c^3 \left (1-a^2 x^2\right )^{5/2}}{4 a^5 x^4}-\frac {c^3 \int \frac {-60 a^5+480 a^6 x}{x \sqrt {1-a^2 x^2}} \, dx}{160 a^6}\\ &=-\frac {3 c^3 (8-a x) \sqrt {1-a^2 x^2}}{8 a^2 x}+\frac {c^3 (8+a x) \left (1-a^2 x^2\right )^{3/2}}{8 a^4 x^3}+\frac {c^3 \left (1-a^2 x^2\right )^{5/2}}{5 a^6 x^5}+\frac {3 c^3 \left (1-a^2 x^2\right )^{5/2}}{4 a^5 x^4}-\left (3 c^3\right ) \int \frac {1}{\sqrt {1-a^2 x^2}} \, dx+\frac {\left (3 c^3\right ) \int \frac {1}{x \sqrt {1-a^2 x^2}} \, dx}{8 a}\\ &=-\frac {3 c^3 (8-a x) \sqrt {1-a^2 x^2}}{8 a^2 x}+\frac {c^3 (8+a x) \left (1-a^2 x^2\right )^{3/2}}{8 a^4 x^3}+\frac {c^3 \left (1-a^2 x^2\right )^{5/2}}{5 a^6 x^5}+\frac {3 c^3 \left (1-a^2 x^2\right )^{5/2}}{4 a^5 x^4}-\frac {3 c^3 \sin ^{-1}(a x)}{a}+\frac {\left (3 c^3\right ) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {1-a^2 x}} \, dx,x,x^2\right )}{16 a}\\ &=-\frac {3 c^3 (8-a x) \sqrt {1-a^2 x^2}}{8 a^2 x}+\frac {c^3 (8+a x) \left (1-a^2 x^2\right )^{3/2}}{8 a^4 x^3}+\frac {c^3 \left (1-a^2 x^2\right )^{5/2}}{5 a^6 x^5}+\frac {3 c^3 \left (1-a^2 x^2\right )^{5/2}}{4 a^5 x^4}-\frac {3 c^3 \sin ^{-1}(a x)}{a}-\frac {\left (3 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 {3 c^3 (8-a x) \sqrt {1-a^2 x^2}}{8 a^2 x}+\frac {c^3 (8+a x) \left (1-a^2 x^2\right )^{3/2}}{8 a^4 x^3}+\frac {c^3 \left (1-a^2 x^2\right )^{5/2}}{5 a^6 x^5}+\frac {3 c^3 \left (1-a^2 x^2\right )^{5/2}}{4 a^5 x^4}-\frac {3 c^3 \sin ^{-1}(a x)}{a}-\frac {3 c^3 \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )}{8 a}\\ \end {align*}

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Mathematica [C]  time = 0.10, size = 186, normalized size = 1.18 \[ \frac {c^3 \left (-8 a^6 x^6+75 a^5 x^5+24 a^4 x^4-105 a^3 x^3+40 a^2 x^2 \sqrt {1-a^2 x^2} \, _2F_1\left (-\frac {3}{2},-\frac {3}{2};-\frac {1}{2};a^2 x^2\right )-24 a^2 x^2-8 a^5 x^5 \left (a^2 x^2-1\right )^3 \, _2F_1\left (2,\frac {5}{2};\frac {7}{2};1-a^2 x^2\right )+45 a^5 x^5 \sqrt {1-a^2 x^2} \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )+30 a x+8\right )}{40 a^6 x^5 \sqrt {1-a^2 x^2}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(c^3*(8 + 30*a*x - 24*a^2*x^2 - 105*a^3*x^3 + 24*a^4*x^4 + 75*a^5*x^5 - 8*a^6*x^6 + 45*a^5*x^5*Sqrt[1 - a^2*x^
2]*ArcTanh[Sqrt[1 - a^2*x^2]] + 40*a^2*x^2*Sqrt[1 - a^2*x^2]*Hypergeometric2F1[-3/2, -3/2, -1/2, a^2*x^2] - 8*
a^5*x^5*(-1 + a^2*x^2)^3*Hypergeometric2F1[2, 5/2, 7/2, 1 - a^2*x^2]))/(40*a^6*x^5*Sqrt[1 - a^2*x^2])

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fricas [A]  time = 0.66, size = 153, normalized size = 0.97 \[ \frac {240 \, a^{5} c^{3} x^{5} \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right ) + 15 \, a^{5} c^{3} x^{5} \log \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{x}\right ) + 40 \, a^{5} c^{3} x^{5} + {\left (40 \, a^{5} c^{3} x^{5} - 152 \, a^{4} c^{3} x^{4} - 55 \, a^{3} c^{3} x^{3} + 24 \, a^{2} c^{3} x^{2} + 30 \, a c^{3} x + 8 \, c^{3}\right )} \sqrt {-a^{2} x^{2} + 1}}{40 \, a^{6} x^{5}} \]

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)^3,x, algorithm="fricas")

[Out]

1/40*(240*a^5*c^3*x^5*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) + 15*a^5*c^3*x^5*log((sqrt(-a^2*x^2 + 1) - 1)/x)
+ 40*a^5*c^3*x^5 + (40*a^5*c^3*x^5 - 152*a^4*c^3*x^4 - 55*a^3*c^3*x^3 + 24*a^2*c^3*x^2 + 30*a*c^3*x + 8*c^3)*s
qrt(-a^2*x^2 + 1))/(a^6*x^5)

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giac [B]  time = 0.51, size = 385, normalized size = 2.45 \[ -\frac {{\left (2 \, c^{3} + \frac {15 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )} c^{3}}{a^{2} x} + \frac {30 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{2} c^{3}}{a^{4} x^{2}} - \frac {80 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{3} c^{3}}{a^{6} x^{3}} - \frac {580 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{4} c^{3}}{a^{8} x^{4}}\right )} a^{10} x^{5}}{320 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{5} {\left | a \right |}} - \frac {3 \, c^{3} \arcsin \left (a x\right ) \mathrm {sgn}\relax (a)}{{\left | a \right |}} - \frac {3 \, c^{3} \log \left (\frac {{\left | -2 \, \sqrt {-a^{2} x^{2} + 1} {\left | a \right |} - 2 \, a \right |}}{2 \, a^{2} {\left | x \right |}}\right )}{8 \, {\left | a \right |}} + \frac {\sqrt {-a^{2} x^{2} + 1} c^{3}}{a} - \frac {\frac {580 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )} a^{2} c^{3}}{x} + \frac {80 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{2} c^{3}}{x^{2}} - \frac {30 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{3} c^{3}}{a^{2} x^{3}} - \frac {15 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{4} c^{3}}{a^{4} x^{4}} - \frac {2 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{5} c^{3}}{a^{6} x^{5}}}{320 \, a^{4} {\left | a \right |}} \]

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)^3,x, algorithm="giac")

[Out]

-1/320*(2*c^3 + 15*(sqrt(-a^2*x^2 + 1)*abs(a) + a)*c^3/(a^2*x) + 30*(sqrt(-a^2*x^2 + 1)*abs(a) + a)^2*c^3/(a^4
*x^2) - 80*(sqrt(-a^2*x^2 + 1)*abs(a) + a)^3*c^3/(a^6*x^3) - 580*(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)) - 3*c^3*arcsin(a*x)*sgn(a)/abs(a) - 3/8*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 - 1/320*(580*(sqrt(-a^2*
x^2 + 1)*abs(a) + a)*a^2*c^3/x + 80*(sqrt(-a^2*x^2 + 1)*abs(a) + a)^2*c^3/x^2 - 30*(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) - 2*(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 = 0.06, size = 227, normalized size = 1.45 \[ -\frac {c^{3} a \,x^{2}}{\sqrt {-a^{2} x^{2}+1}}+\frac {19 c^{3}}{8 a \sqrt {-a^{2} x^{2}+1}}+\frac {19 c^{3} x}{5 \sqrt {-a^{2} x^{2}+1}}-\frac {3 c^{3} \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{\sqrt {a^{2}}}-\frac {3 c^{3} \arctanh \left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )}{8 a}-\frac {22 c^{3}}{5 a^{2} x \sqrt {-a^{2} x^{2}+1}}-\frac {17 c^{3}}{8 a^{3} x^{2} \sqrt {-a^{2} x^{2}+1}}+\frac {3 c^{3}}{4 a^{5} x^{4} \sqrt {-a^{2} x^{2}+1}}+\frac {c^{3}}{5 a^{6} x^{5} \sqrt {-a^{2} x^{2}+1}}+\frac {2 c^{3}}{5 a^{4} x^{3} \sqrt {-a^{2} x^{2}+1}} \]

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)^3,x)

[Out]

-c^3*a*x^2/(-a^2*x^2+1)^(1/2)+19/8*c^3/a/(-a^2*x^2+1)^(1/2)+19/5*c^3*x/(-a^2*x^2+1)^(1/2)-3*c^3/(a^2)^(1/2)*ar
ctan((a^2)^(1/2)*x/(-a^2*x^2+1)^(1/2))-3/8*c^3/a*arctanh(1/(-a^2*x^2+1)^(1/2))-22/5*c^3/a^2/x/(-a^2*x^2+1)^(1/
2)-17/8*c^3/a^3/x^2/(-a^2*x^2+1)^(1/2)+3/4*c^3/a^5/x^4/(-a^2*x^2+1)^(1/2)+1/5*c^3/a^6/x^5/(-a^2*x^2+1)^(1/2)+2
/5*c^3/a^4/x^3/(-a^2*x^2+1)^(1/2)

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maxima [B]  time = 0.42, size = 443, normalized size = 2.82 \[ -a^{3} c^{3} {\left (\frac {x^{2}}{\sqrt {-a^{2} x^{2} + 1} a^{2}} - \frac {2}{\sqrt {-a^{2} x^{2} + 1} a^{4}}\right )} + 3 \, a^{2} c^{3} {\left (\frac {x}{\sqrt {-a^{2} x^{2} + 1} a^{2}} - \frac {\arcsin \left (a x\right )}{a^{3}}\right )} - \frac {8 \, c^{3} x}{\sqrt {-a^{2} x^{2} + 1}} - \frac {6 \, c^{3} {\left (\frac {1}{\sqrt {-a^{2} x^{2} + 1}} - \log \left (\frac {2 \, \sqrt {-a^{2} x^{2} + 1}}{{\left | x \right |}} + \frac {2}{{\left | x \right |}}\right )\right )}}{a} + \frac {6 \, {\left (\frac {2 \, a^{2} x}{\sqrt {-a^{2} x^{2} + 1}} - \frac {1}{\sqrt {-a^{2} x^{2} + 1} x}\right )} c^{3}}{a^{2}} - \frac {4 \, {\left (3 \, a^{2} \log \left (\frac {2 \, \sqrt {-a^{2} x^{2} + 1}}{{\left | x \right |}} + \frac {2}{{\left | x \right |}}\right ) - \frac {3 \, a^{2}}{\sqrt {-a^{2} x^{2} + 1}} + \frac {1}{\sqrt {-a^{2} x^{2} + 1} x^{2}}\right )} c^{3}}{a^{3}} + \frac {3 \, {\left (15 \, a^{4} \log \left (\frac {2 \, \sqrt {-a^{2} x^{2} + 1}}{{\left | x \right |}} + \frac {2}{{\left | x \right |}}\right ) - \frac {15 \, a^{4}}{\sqrt {-a^{2} x^{2} + 1}} + \frac {5 \, a^{2}}{\sqrt {-a^{2} x^{2} + 1} x^{2}} + \frac {2}{\sqrt {-a^{2} x^{2} + 1} x^{4}}\right )} c^{3}}{8 \, a^{5}} - \frac {{\left (\frac {16 \, a^{6} x}{\sqrt {-a^{2} x^{2} + 1}} - \frac {8 \, a^{4}}{\sqrt {-a^{2} x^{2} + 1} x} - \frac {2 \, a^{2}}{\sqrt {-a^{2} x^{2} + 1} x^{3}} - \frac {1}{\sqrt {-a^{2} x^{2} + 1} x^{5}}\right )} c^{3}}{5 \, a^{6}} \]

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)^3,x, algorithm="maxima")

[Out]

-a^3*c^3*(x^2/(sqrt(-a^2*x^2 + 1)*a^2) - 2/(sqrt(-a^2*x^2 + 1)*a^4)) + 3*a^2*c^3*(x/(sqrt(-a^2*x^2 + 1)*a^2) -
arcsin(a*x)/a^3) - 8*c^3*x/sqrt(-a^2*x^2 + 1) - 6*c^3*(1/sqrt(-a^2*x^2 + 1) - log(2*sqrt(-a^2*x^2 + 1)/abs(x)
+ 2/abs(x)))/a + 6*(2*a^2*x/sqrt(-a^2*x^2 + 1) - 1/(sqrt(-a^2*x^2 + 1)*x))*c^3/a^2 - 4*(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^3/a^3 + 3/8*(15*a^4*lo
g(2*sqrt(-a^2*x^2 + 1)/abs(x) + 2/abs(x)) - 15*a^4/sqrt(-a^2*x^2 + 1) + 5*a^2/(sqrt(-a^2*x^2 + 1)*x^2) + 2/(sq
rt(-a^2*x^2 + 1)*x^4))*c^3/a^5 - 1/5*(16*a^6*x/sqrt(-a^2*x^2 + 1) - 8*a^4/(sqrt(-a^2*x^2 + 1)*x) - 2*a^2/(sqrt
(-a^2*x^2 + 1)*x^3) - 1/(sqrt(-a^2*x^2 + 1)*x^5))*c^3/a^6

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mupad [B]  time = 0.07, size = 182, normalized size = 1.16 \[ \frac {c^3\,\sqrt {1-a^2\,x^2}}{a}-\frac {3\,c^3\,\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )}{\sqrt {-a^2}}-\frac {19\,c^3\,\sqrt {1-a^2\,x^2}}{5\,a^2\,x}-\frac {11\,c^3\,\sqrt {1-a^2\,x^2}}{8\,a^3\,x^2}+\frac {3\,c^3\,\sqrt {1-a^2\,x^2}}{5\,a^4\,x^3}+\frac {3\,c^3\,\sqrt {1-a^2\,x^2}}{4\,a^5\,x^4}+\frac {c^3\,\sqrt {1-a^2\,x^2}}{5\,a^6\,x^5}+\frac {c^3\,\mathrm {atan}\left (\sqrt {1-a^2\,x^2}\,1{}\mathrm {i}\right )\,3{}\mathrm {i}}{8\,a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(c^3*atan((1 - a^2*x^2)^(1/2)*1i)*3i)/(8*a) - (3*c^3*asinh(x*(-a^2)^(1/2)))/(-a^2)^(1/2) + (c^3*(1 - a^2*x^2)^
(1/2))/a - (19*c^3*(1 - a^2*x^2)^(1/2))/(5*a^2*x) - (11*c^3*(1 - a^2*x^2)^(1/2))/(8*a^3*x^2) + (3*c^3*(1 - a^2
*x^2)^(1/2))/(5*a^4*x^3) + (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 [A]  time = 19.49, size = 687, normalized size = 4.38 \[ - a c^{3} \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^{3} \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 {c^{3} \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 {5 c^{3} \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 {5 c^{3} \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^{3} \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}} - \frac {3 c^{3} \left (\begin {cases} - \frac {3 a^{4} \operatorname {acosh}{\left (\frac {1}{a x} \right )}}{8} + \frac {3 a^{3}}{8 x \sqrt {-1 + \frac {1}{a^{2} x^{2}}}} - \frac {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 {3 i a^{4} \operatorname {asin}{\left (\frac {1}{a x} \right )}}{8} - \frac {3 i a^{3}}{8 x \sqrt {1 - \frac {1}{a^{2} x^{2}}}} + \frac {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 {8 a^{5} \sqrt {-1 + \frac {1}{a^{2} x^{2}}}}{15} - \frac {4 a^{3} \sqrt {-1 + \frac {1}{a^{2} x^{2}}}}{15 x^{2}} - \frac {a \sqrt {-1 + \frac {1}{a^{2} x^{2}}}}{5 x^{4}} & \text {for}\: \frac {1}{\left |{a^{2} x^{2}}\right |} > 1 \\- \frac {8 i a^{5} \sqrt {1 - \frac {1}{a^{2} x^{2}}}}{15} - \frac {4 i a^{3} \sqrt {1 - \frac {1}{a^{2} x^{2}}}}{15 x^{2}} - \frac {i a \sqrt {1 - \frac {1}{a^{2} x^{2}}}}{5 x^{4}} & \text {otherwise} \end {cases}\right )}{a^{6}} \]

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)**3,x)

[Out]

-a*c**3*Piecewise((x**2/2, Eq(a**2, 0)), (-sqrt(-a**2*x**2 + 1)/a**2, True)) - 3*c**3*Piecewise((sqrt(a**(-2))
*asin(x*sqrt(a**2)), a**2 > 0), (sqrt(-1/a**2)*asinh(x*sqrt(-a**2)), a**2 < 0)) - c**3*Piecewise((-acosh(1/(a*
x)), 1/Abs(a**2*x**2) > 1), (I*asin(1/(a*x)), True))/a + 5*c**3*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 + 5*c**3*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**3*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 - 3*c**3*Piecewise((-3*a**4*acosh(1/(a*x))/8 + 3*a**3/(8*x*sqrt(-1 + 1/(a**2*x**2))) - 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), (3*I*a**4*asin(1/(a
*x))/8 - 3*I*a**3/(8*x*sqrt(1 - 1/(a**2*x**2))) + 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((-8*a**5*sqrt(-1 + 1/(a**2*x**2))/15 - 4*a**3*sqrt(-1 + 1/(a**2*
x**2))/(15*x**2) - a*sqrt(-1 + 1/(a**2*x**2))/(5*x**4), 1/Abs(a**2*x**2) > 1), (-8*I*a**5*sqrt(1 - 1/(a**2*x**
2))/15 - 4*I*a**3*sqrt(1 - 1/(a**2*x**2))/(15*x**2) - I*a*sqrt(1 - 1/(a**2*x**2))/(5*x**4), True))/a**6

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