3.7.45 \(\int e^{3 \tanh ^{-1}(a x)} (c-\frac {c}{a^2 x^2})^4 \, dx\) [645]

Optimal. Leaf size=191 \[ -\frac {3 c^4 (16-5 a x) \sqrt {1-a^2 x^2}}{16 a^2 x}+\frac {c^4 (16+5 a x) \left (1-a^2 x^2\right )^{3/2}}{16 a^4 x^3}-\frac {c^4 (24+5 a x) \left (1-a^2 x^2\right )^{5/2}}{40 a^6 x^5}-\frac {c^4 \left (1-a^2 x^2\right )^{7/2}}{7 a^8 x^7}-\frac {c^4 \left (1-a^2 x^2\right )^{7/2}}{2 a^7 x^6}-\frac {3 c^4 \text {ArcSin}(a x)}{a}-\frac {15 c^4 \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )}{16 a} \]

[Out]

1/16*c^4*(5*a*x+16)*(-a^2*x^2+1)^(3/2)/a^4/x^3-1/40*c^4*(5*a*x+24)*(-a^2*x^2+1)^(5/2)/a^6/x^5-1/7*c^4*(-a^2*x^
2+1)^(7/2)/a^8/x^7-1/2*c^4*(-a^2*x^2+1)^(7/2)/a^7/x^6-3*c^4*arcsin(a*x)/a-15/16*c^4*arctanh((-a^2*x^2+1)^(1/2)
)/a-3/16*c^4*(-5*a*x+16)*(-a^2*x^2+1)^(1/2)/a^2/x

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Rubi [A]
time = 0.24, antiderivative size = 191, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 10, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.454, Rules used = {6292, 6283, 1821, 825, 827, 858, 222, 272, 65, 214} \begin {gather*} -\frac {3 c^4 (16-5 a x) \sqrt {1-a^2 x^2}}{16 a^2 x}-\frac {15 c^4 \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )}{16 a}-\frac {c^4 \left (1-a^2 x^2\right )^{7/2}}{7 a^8 x^7}-\frac {c^4 \left (1-a^2 x^2\right )^{7/2}}{2 a^7 x^6}-\frac {c^4 (5 a x+24) \left (1-a^2 x^2\right )^{5/2}}{40 a^6 x^5}+\frac {c^4 (5 a x+16) \left (1-a^2 x^2\right )^{3/2}}{16 a^4 x^3}-\frac {3 c^4 \text {ArcSin}(a x)}{a} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[E^(3*ArcTanh[a*x])*(c - c/(a^2*x^2))^4,x]

[Out]

(-3*c^4*(16 - 5*a*x)*Sqrt[1 - a^2*x^2])/(16*a^2*x) + (c^4*(16 + 5*a*x)*(1 - a^2*x^2)^(3/2))/(16*a^4*x^3) - (c^
4*(24 + 5*a*x)*(1 - a^2*x^2)^(5/2))/(40*a^6*x^5) - (c^4*(1 - a^2*x^2)^(7/2))/(7*a^8*x^7) - (c^4*(1 - a^2*x^2)^
(7/2))/(2*a^7*x^6) - (3*c^4*ArcSin[a*x])/a - (15*c^4*ArcTanh[Sqrt[1 - a^2*x^2]])/(16*a)

Rule 65

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 214

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x/Rt[-a/b, 2]], x] /; FreeQ[{a, b},
x] && NegQ[a/b]

Rule 222

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 272

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 825

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/(e^2*(m + 1)*(m + 2)*(c*d^2 + a*e^2)))*((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), 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 827

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 858

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 1821

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 6283

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 6292

Int[E^(ArcTanh[(a_.)*(x_)]*(n_.))*(u_.)*((c_) + (d_.)/(x_)^2)^(p_.), x_Symbol] :> Dist[d^p, Int[(u/x^(2*p))*(1
- a^2*x^2)^p*E^(n*ArcTanh[a*x]), 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 )^4 \, dx &=\frac {c^4 \int \frac {e^{3 \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)^3 \left (1-a^2 x^2\right )^{5/2}}{x^8} \, dx}{a^8}\\ &=-\frac {c^4 \left (1-a^2 x^2\right )^{7/2}}{7 a^8 x^7}-\frac {c^4 \int \frac {\left (1-a^2 x^2\right )^{5/2} \left (-21 a-21 a^2 x-7 a^3 x^2\right )}{x^7} \, dx}{7 a^8}\\ &=-\frac {c^4 \left (1-a^2 x^2\right )^{7/2}}{7 a^8 x^7}-\frac {c^4 \left (1-a^2 x^2\right )^{7/2}}{2 a^7 x^6}+\frac {c^4 \int \frac {\left (126 a^2+21 a^3 x\right ) \left (1-a^2 x^2\right )^{5/2}}{x^6} \, dx}{42 a^8}\\ &=-\frac {c^4 (24+5 a x) \left (1-a^2 x^2\right )^{5/2}}{40 a^6 x^5}-\frac {c^4 \left (1-a^2 x^2\right )^{7/2}}{7 a^8 x^7}-\frac {c^4 \left (1-a^2 x^2\right )^{7/2}}{2 a^7 x^6}-\frac {c^4 \int \frac {\left (1008 a^4+210 a^5 x\right ) \left (1-a^2 x^2\right )^{3/2}}{x^4} \, dx}{336 a^8}\\ &=\frac {c^4 (16+5 a x) \left (1-a^2 x^2\right )^{3/2}}{16 a^4 x^3}-\frac {c^4 (24+5 a x) \left (1-a^2 x^2\right )^{5/2}}{40 a^6 x^5}-\frac {c^4 \left (1-a^2 x^2\right )^{7/2}}{7 a^8 x^7}-\frac {c^4 \left (1-a^2 x^2\right )^{7/2}}{2 a^7 x^6}+\frac {c^4 \int \frac {\left (4032 a^6+1260 a^7 x\right ) \sqrt {1-a^2 x^2}}{x^2} \, dx}{1344 a^8}\\ &=-\frac {3 c^4 (16-5 a x) \sqrt {1-a^2 x^2}}{16 a^2 x}+\frac {c^4 (16+5 a x) \left (1-a^2 x^2\right )^{3/2}}{16 a^4 x^3}-\frac {c^4 (24+5 a x) \left (1-a^2 x^2\right )^{5/2}}{40 a^6 x^5}-\frac {c^4 \left (1-a^2 x^2\right )^{7/2}}{7 a^8 x^7}-\frac {c^4 \left (1-a^2 x^2\right )^{7/2}}{2 a^7 x^6}-\frac {c^4 \int \frac {-2520 a^7+8064 a^8 x}{x \sqrt {1-a^2 x^2}} \, dx}{2688 a^8}\\ &=-\frac {3 c^4 (16-5 a x) \sqrt {1-a^2 x^2}}{16 a^2 x}+\frac {c^4 (16+5 a x) \left (1-a^2 x^2\right )^{3/2}}{16 a^4 x^3}-\frac {c^4 (24+5 a x) \left (1-a^2 x^2\right )^{5/2}}{40 a^6 x^5}-\frac {c^4 \left (1-a^2 x^2\right )^{7/2}}{7 a^8 x^7}-\frac {c^4 \left (1-a^2 x^2\right )^{7/2}}{2 a^7 x^6}-\left (3 c^4\right ) \int \frac {1}{\sqrt {1-a^2 x^2}} \, dx+\frac {\left (15 c^4\right ) \int \frac {1}{x \sqrt {1-a^2 x^2}} \, dx}{16 a}\\ &=-\frac {3 c^4 (16-5 a x) \sqrt {1-a^2 x^2}}{16 a^2 x}+\frac {c^4 (16+5 a x) \left (1-a^2 x^2\right )^{3/2}}{16 a^4 x^3}-\frac {c^4 (24+5 a x) \left (1-a^2 x^2\right )^{5/2}}{40 a^6 x^5}-\frac {c^4 \left (1-a^2 x^2\right )^{7/2}}{7 a^8 x^7}-\frac {c^4 \left (1-a^2 x^2\right )^{7/2}}{2 a^7 x^6}-\frac {3 c^4 \sin ^{-1}(a x)}{a}+\frac {\left (15 c^4\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {1-a^2 x}} \, dx,x,x^2\right )}{32 a}\\ &=-\frac {3 c^4 (16-5 a x) \sqrt {1-a^2 x^2}}{16 a^2 x}+\frac {c^4 (16+5 a x) \left (1-a^2 x^2\right )^{3/2}}{16 a^4 x^3}-\frac {c^4 (24+5 a x) \left (1-a^2 x^2\right )^{5/2}}{40 a^6 x^5}-\frac {c^4 \left (1-a^2 x^2\right )^{7/2}}{7 a^8 x^7}-\frac {c^4 \left (1-a^2 x^2\right )^{7/2}}{2 a^7 x^6}-\frac {3 c^4 \sin ^{-1}(a x)}{a}-\frac {\left (15 c^4\right ) \text {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 {3 c^4 (16-5 a x) \sqrt {1-a^2 x^2}}{16 a^2 x}+\frac {c^4 (16+5 a x) \left (1-a^2 x^2\right )^{3/2}}{16 a^4 x^3}-\frac {c^4 (24+5 a x) \left (1-a^2 x^2\right )^{5/2}}{40 a^6 x^5}-\frac {c^4 \left (1-a^2 x^2\right )^{7/2}}{7 a^8 x^7}-\frac {c^4 \left (1-a^2 x^2\right )^{7/2}}{2 a^7 x^6}-\frac {3 c^4 \sin ^{-1}(a x)}{a}-\frac {15 c^4 \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )}{16 a}\\ \end {align*}

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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
time = 0.27, size = 191, normalized size = 1.00 \begin {gather*} \frac {c^4 \left (-336 a^2 x^2 \, _2F_1\left (-\frac {5}{2},-\frac {5}{2};-\frac {3}{2};a^2 x^2\right )-\frac {5 \left (16+56 a x-64 a^2 x^2-238 a^3 x^3+96 a^4 x^4+413 a^5 x^5-64 a^6 x^6-231 a^7 x^7+16 a^8 x^8-105 a^7 x^7 \sqrt {1-a^2 x^2} \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )+16 a^7 x^7 \left (-1+a^2 x^2\right )^4 \, _2F_1\left (3,\frac {7}{2};\frac {9}{2};1-a^2 x^2\right )\right )}{\sqrt {1-a^2 x^2}}\right )}{560 a^8 x^7} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Integrate[E^(3*ArcTanh[a*x])*(c - c/(a^2*x^2))^4,x]

[Out]

(c^4*(-336*a^2*x^2*Hypergeometric2F1[-5/2, -5/2, -3/2, a^2*x^2] - (5*(16 + 56*a*x - 64*a^2*x^2 - 238*a^3*x^3 +
96*a^4*x^4 + 413*a^5*x^5 - 64*a^6*x^6 - 231*a^7*x^7 + 16*a^8*x^8 - 105*a^7*x^7*Sqrt[1 - a^2*x^2]*ArcTanh[Sqrt
[1 - a^2*x^2]] + 16*a^7*x^7*(-1 + a^2*x^2)^4*Hypergeometric2F1[3, 7/2, 9/2, 1 - a^2*x^2]))/Sqrt[1 - a^2*x^2]))
/(560*a^8*x^7)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(615\) vs. \(2(167)=334\).
time = 1.48, size = 616, normalized size = 3.23 Too large to display

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)^4,x,method=_RETURNVERBOSE)

[Out]

c^4/a^8*(a^11*(-x^2/a^2/(-a^2*x^2+1)^(1/2)+2/a^4/(-a^2*x^2+1)^(1/2))+3*a^10*(x/a^2/(-a^2*x^2+1)^(1/2)-1/a^2/(a
^2)^(1/2)*arctan((a^2)^(1/2)*x/(-a^2*x^2+1)^(1/2)))-a^7/(-a^2*x^2+1)^(1/2)-11*a^8*x/(-a^2*x^2+1)^(1/2)-6*a^4*(
-1/3/x^3/(-a^2*x^2+1)^(1/2)+4/3*a^2*(-1/x/(-a^2*x^2+1)^(1/2)+2*a^2*x/(-a^2*x^2+1)^(1/2)))-11*a^3*(-1/4/x^4/(-a
^2*x^2+1)^(1/2)+5/4*a^2*(-1/2/x^2/(-a^2*x^2+1)^(1/2)+3/2*a^2*(1/(-a^2*x^2+1)^(1/2)-arctanh(1/(-a^2*x^2+1)^(1/2
)))))+14*a^5*(-1/2/x^2/(-a^2*x^2+1)^(1/2)+3/2*a^2*(1/(-a^2*x^2+1)^(1/2)-arctanh(1/(-a^2*x^2+1)^(1/2))))+14*a^6
*(-1/x/(-a^2*x^2+1)^(1/2)+2*a^2*x/(-a^2*x^2+1)^(1/2))+1/7*a^2*(-1/5/x^5/(-a^2*x^2+1)^(1/2)+6/5*a^2*(-1/3/x^3/(
-a^2*x^2+1)^(1/2)+4/3*a^2*(-1/x/(-a^2*x^2+1)^(1/2)+2*a^2*x/(-a^2*x^2+1)^(1/2))))-1/7/x^7/(-a^2*x^2+1)^(1/2)+3*
a*(-1/6/x^6/(-a^2*x^2+1)^(1/2)+7/6*a^2*(-1/4/x^4/(-a^2*x^2+1)^(1/2)+5/4*a^2*(-1/2/x^2/(-a^2*x^2+1)^(1/2)+3/2*a
^2*(1/(-a^2*x^2+1)^(1/2)-arctanh(1/(-a^2*x^2+1)^(1/2))))))-6*a^7*(1/(-a^2*x^2+1)^(1/2)-arctanh(1/(-a^2*x^2+1)^
(1/2))))

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 745 vs. \(2 (167) = 334\).
time = 0.48, size = 745, normalized size = 3.90 \begin {gather*} -a^{3} c^{4} {\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^{4} {\left (\frac {x}{\sqrt {-a^{2} x^{2} + 1} a^{2}} - \frac {\arcsin \left (a x\right )}{a^{3}}\right )} - \frac {11 \, c^{4} x}{\sqrt {-a^{2} x^{2} + 1}} - \frac {6 \, c^{4} {\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 {14 \, {\left (\frac {2 \, a^{2} x}{\sqrt {-a^{2} x^{2} + 1}} - \frac {1}{\sqrt {-a^{2} x^{2} + 1} x}\right )} c^{4}}{a^{2}} - \frac {c^{4}}{\sqrt {-a^{2} x^{2} + 1} a} - \frac {7 \, {\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^{4}}{a^{3}} - \frac {2 \, {\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^{4}}{a^{4}} + \frac {11 \, {\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^{4}}{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^{4}}{5 \, a^{6}} - \frac {{\left (105 \, a^{6} \log \left (\frac {2 \, \sqrt {-a^{2} x^{2} + 1}}{{\left | x \right |}} + \frac {2}{{\left | x \right |}}\right ) - \frac {105 \, a^{6}}{\sqrt {-a^{2} x^{2} + 1}} + \frac {35 \, a^{4}}{\sqrt {-a^{2} x^{2} + 1} x^{2}} + \frac {14 \, a^{2}}{\sqrt {-a^{2} x^{2} + 1} x^{4}} + \frac {8}{\sqrt {-a^{2} x^{2} + 1} x^{6}}\right )} c^{4}}{16 \, a^{7}} + \frac {{\left (\frac {128 \, a^{8} x}{\sqrt {-a^{2} x^{2} + 1}} - \frac {64 \, a^{6}}{\sqrt {-a^{2} x^{2} + 1} x} - \frac {16 \, a^{4}}{\sqrt {-a^{2} x^{2} + 1} x^{3}} - \frac {8 \, a^{2}}{\sqrt {-a^{2} x^{2} + 1} x^{5}} - \frac {5}{\sqrt {-a^{2} x^{2} + 1} x^{7}}\right )} c^{4}}{35 \, a^{8}} \end {gather*}

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)^4,x, algorithm="maxima")

[Out]

-a^3*c^4*(x^2/(sqrt(-a^2*x^2 + 1)*a^2) - 2/(sqrt(-a^2*x^2 + 1)*a^4)) + 3*a^2*c^4*(x/(sqrt(-a^2*x^2 + 1)*a^2) -
arcsin(a*x)/a^3) - 11*c^4*x/sqrt(-a^2*x^2 + 1) - 6*c^4*(1/sqrt(-a^2*x^2 + 1) - log(2*sqrt(-a^2*x^2 + 1)/abs(x
) + 2/abs(x)))/a + 14*(2*a^2*x/sqrt(-a^2*x^2 + 1) - 1/(sqrt(-a^2*x^2 + 1)*x))*c^4/a^2 - c^4/(sqrt(-a^2*x^2 + 1
)*a) - 7*(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^4/a^3 - 2*(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^
4/a^4 + 11/8*(15*a^4*log(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/(sqrt(-a^2*x^2 + 1)*x^4))*c^4/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^4/a^6 - 1/16*(105*a^6*log(2*sqrt(-a
^2*x^2 + 1)/abs(x) + 2/abs(x)) - 105*a^6/sqrt(-a^2*x^2 + 1) + 35*a^4/(sqrt(-a^2*x^2 + 1)*x^2) + 14*a^2/(sqrt(-
a^2*x^2 + 1)*x^4) + 8/(sqrt(-a^2*x^2 + 1)*x^6))*c^4/a^7 + 1/35*(128*a^8*x/sqrt(-a^2*x^2 + 1) - 64*a^6/(sqrt(-a
^2*x^2 + 1)*x) - 16*a^4/(sqrt(-a^2*x^2 + 1)*x^3) - 8*a^2/(sqrt(-a^2*x^2 + 1)*x^5) - 5/(sqrt(-a^2*x^2 + 1)*x^7)
)*c^4/a^8

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Fricas [A]
time = 0.38, size = 175, normalized size = 0.92 \begin {gather*} \frac {3360 \, a^{7} c^{4} x^{7} \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right ) + 525 \, a^{7} c^{4} x^{7} \log \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{x}\right ) + 560 \, a^{7} c^{4} x^{7} + {\left (560 \, a^{7} c^{4} x^{7} - 2496 \, a^{6} c^{4} x^{6} - 525 \, a^{5} c^{4} x^{5} + 992 \, a^{4} c^{4} x^{4} + 770 \, a^{3} c^{4} x^{3} - 96 \, a^{2} c^{4} x^{2} - 280 \, a c^{4} x - 80 \, c^{4}\right )} \sqrt {-a^{2} x^{2} + 1}}{560 \, a^{8} x^{7}} \end {gather*}

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)^4,x, algorithm="fricas")

[Out]

1/560*(3360*a^7*c^4*x^7*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) + 525*a^7*c^4*x^7*log((sqrt(-a^2*x^2 + 1) - 1)/
x) + 560*a^7*c^4*x^7 + (560*a^7*c^4*x^7 - 2496*a^6*c^4*x^6 - 525*a^5*c^4*x^5 + 992*a^4*c^4*x^4 + 770*a^3*c^4*x
^3 - 96*a^2*c^4*x^2 - 280*a*c^4*x - 80*c^4)*sqrt(-a^2*x^2 + 1))/(a^8*x^7)

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Sympy [A]
time = 24.09, size = 933, normalized size = 4.88 \begin {gather*} - a c^{4} \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^{4} \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 {8 c^{4} \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 {6 c^{4} \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 {6 c^{4} \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 {8 c^{4} \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 {3 c^{4} \left (\begin {cases} - \frac {5 a^{6} \operatorname {acosh}{\left (\frac {1}{a x} \right )}}{16} + \frac {5 a^{5}}{16 x \sqrt {-1 + \frac {1}{a^{2} x^{2}}}} - \frac {5 a^{3}}{48 x^{3} \sqrt {-1 + \frac {1}{a^{2} x^{2}}}} - \frac {a}{24 x^{5} \sqrt {-1 + \frac {1}{a^{2} x^{2}}}} - \frac {1}{6 a x^{7} \sqrt {-1 + \frac {1}{a^{2} x^{2}}}} & \text {for}\: \frac {1}{\left |{a^{2} x^{2}}\right |} > 1 \\\frac {5 i a^{6} \operatorname {asin}{\left (\frac {1}{a x} \right )}}{16} - \frac {5 i a^{5}}{16 x \sqrt {1 - \frac {1}{a^{2} x^{2}}}} + \frac {5 i a^{3}}{48 x^{3} \sqrt {1 - \frac {1}{a^{2} x^{2}}}} + \frac {i a}{24 x^{5} \sqrt {1 - \frac {1}{a^{2} x^{2}}}} + \frac {i}{6 a x^{7} \sqrt {1 - \frac {1}{a^{2} x^{2}}}} & \text {otherwise} \end {cases}\right )}{a^{7}} + \frac {c^{4} \left (\begin {cases} - \frac {16 a^{7} \sqrt {-1 + \frac {1}{a^{2} x^{2}}}}{35} - \frac {8 a^{5} \sqrt {-1 + \frac {1}{a^{2} x^{2}}}}{35 x^{2}} - \frac {6 a^{3} \sqrt {-1 + \frac {1}{a^{2} x^{2}}}}{35 x^{4}} - \frac {a \sqrt {-1 + \frac {1}{a^{2} x^{2}}}}{7 x^{6}} & \text {for}\: \frac {1}{\left |{a^{2} x^{2}}\right |} > 1 \\- \frac {16 i a^{7} \sqrt {1 - \frac {1}{a^{2} x^{2}}}}{35} - \frac {8 i a^{5} \sqrt {1 - \frac {1}{a^{2} x^{2}}}}{35 x^{2}} - \frac {6 i a^{3} \sqrt {1 - \frac {1}{a^{2} x^{2}}}}{35 x^{4}} - \frac {i a \sqrt {1 - \frac {1}{a^{2} x^{2}}}}{7 x^{6}} & \text {otherwise} \end {cases}\right )}{a^{8}} \end {gather*}

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)**4,x)

[Out]

-a*c**4*Piecewise((x**2/2, Eq(a**2, 0)), (-sqrt(-a**2*x**2 + 1)/a**2, True)) - 3*c**4*Piecewise((sqrt(a**(-2))
*asin(x*sqrt(a**2)), a**2 > 0), (sqrt(-1/a**2)*asinh(x*sqrt(-a**2)), a**2 < 0)) + 8*c**4*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 + 6*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 - 6*c**4*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) - s
qrt(-a**2*x**2 + 1)/(3*x**3), True))/a**4 - 8*c**4*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 + 3*c**4*Piecewise((-5*a**6*acosh(1/(a*x))/16 + 5*a**
5/(16*x*sqrt(-1 + 1/(a**2*x**2))) - 5*a**3/(48*x**3*sqrt(-1 + 1/(a**2*x**2))) - 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), (5*I*a**6*asin(1/(a*x))/16 - 5*I*a**5/(
16*x*sqrt(1 - 1/(a**2*x**2))) + 5*I*a**3/(48*x**3*sqrt(1 - 1/(a**2*x**2))) + 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((-16*a**7*sqrt(-1 + 1/(a**2*x**2))/3
5 - 8*a**5*sqrt(-1 + 1/(a**2*x**2))/(35*x**2) - 6*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), (-16*I*a**7*sqrt(1 - 1/(a**2*x**2))/35 - 8*I*a**5*sqrt(1 - 1/(a**2*x
**2))/(35*x**2) - 6*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] Leaf count of result is larger than twice the leaf count of optimal. 505 vs. \(2 (167) = 334\).
time = 0.44, size = 505, normalized size = 2.64 \begin {gather*} \frac {{\left (5 \, c^{4} + \frac {35 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )} c^{4}}{a^{2} x} + \frac {49 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{2} c^{4}}{a^{4} x^{2}} - \frac {245 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{3} c^{4}}{a^{6} x^{3}} - \frac {875 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{4} c^{4}}{a^{8} x^{4}} + \frac {455 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{5} c^{4}}{a^{10} x^{5}} + \frac {9065 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{6} c^{4}}{a^{12} x^{6}}\right )} a^{14} x^{7}}{4480 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{7} {\left | a \right |}} - \frac {3 \, c^{4} \arcsin \left (a x\right ) \mathrm {sgn}\left (a\right )}{{\left | a \right |}} - \frac {15 \, 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 {9065 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )} a^{4} c^{4}}{x} + \frac {455 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{2} a^{2} c^{4}}{x^{2}} - \frac {875 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{3} c^{4}}{x^{3}} - \frac {245 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{4} c^{4}}{a^{2} x^{4}} + \frac {49 \, {\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 {5 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{7} c^{4}}{a^{8} x^{7}}}{4480 \, a^{6} {\left | a \right |}} \end {gather*}

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)^4,x, algorithm="giac")

[Out]

1/4480*(5*c^4 + 35*(sqrt(-a^2*x^2 + 1)*abs(a) + a)*c^4/(a^2*x) + 49*(sqrt(-a^2*x^2 + 1)*abs(a) + a)^2*c^4/(a^4
*x^2) - 245*(sqrt(-a^2*x^2 + 1)*abs(a) + a)^3*c^4/(a^6*x^3) - 875*(sqrt(-a^2*x^2 + 1)*abs(a) + a)^4*c^4/(a^8*x
^4) + 455*(sqrt(-a^2*x^2 + 1)*abs(a) + a)^5*c^4/(a^10*x^5) + 9065*(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)) - 3*c^4*arcsin(a*x)*sgn(a)/abs(a) - 15/16*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 - 1/4480*(9065*(sqrt
(-a^2*x^2 + 1)*abs(a) + a)*a^4*c^4/x + 455*(sqrt(-a^2*x^2 + 1)*abs(a) + a)^2*a^2*c^4/x^2 - 875*(sqrt(-a^2*x^2
+ 1)*abs(a) + a)^3*c^4/x^3 - 245*(sqrt(-a^2*x^2 + 1)*abs(a) + a)^4*c^4/(a^2*x^4) + 49*(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) + 5*(sqrt(-a^2*x^2 + 1)*abs(a) +
a)^7*c^4/(a^8*x^7))/(a^6*abs(a))

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Mupad [B]
time = 0.89, size = 228, normalized size = 1.19 \begin {gather*} \frac {c^4\,\sqrt {1-a^2\,x^2}}{a}-\frac {3\,c^4\,\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )}{\sqrt {-a^2}}-\frac {156\,c^4\,\sqrt {1-a^2\,x^2}}{35\,a^2\,x}-\frac {15\,c^4\,\sqrt {1-a^2\,x^2}}{16\,a^3\,x^2}+\frac {62\,c^4\,\sqrt {1-a^2\,x^2}}{35\,a^4\,x^3}+\frac {11\,c^4\,\sqrt {1-a^2\,x^2}}{8\,a^5\,x^4}-\frac {6\,c^4\,\sqrt {1-a^2\,x^2}}{35\,a^6\,x^5}-\frac {c^4\,\sqrt {1-a^2\,x^2}}{2\,a^7\,x^6}-\frac {c^4\,\sqrt {1-a^2\,x^2}}{7\,a^8\,x^7}+\frac {c^4\,\mathrm {atan}\left (\sqrt {1-a^2\,x^2}\,1{}\mathrm {i}\right )\,15{}\mathrm {i}}{16\,a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((c - c/(a^2*x^2))^4*(a*x + 1)^3)/(1 - a^2*x^2)^(3/2),x)

[Out]

(c^4*atan((1 - a^2*x^2)^(1/2)*1i)*15i)/(16*a) - (3*c^4*asinh(x*(-a^2)^(1/2)))/(-a^2)^(1/2) + (c^4*(1 - a^2*x^2
)^(1/2))/a - (156*c^4*(1 - a^2*x^2)^(1/2))/(35*a^2*x) - (15*c^4*(1 - a^2*x^2)^(1/2))/(16*a^3*x^2) + (62*c^4*(1
- a^2*x^2)^(1/2))/(35*a^4*x^3) + (11*c^4*(1 - a^2*x^2)^(1/2))/(8*a^5*x^4) - (6*c^4*(1 - a^2*x^2)^(1/2))/(35*a
^6*x^5) - (c^4*(1 - a^2*x^2)^(1/2))/(2*a^7*x^6) - (c^4*(1 - a^2*x^2)^(1/2))/(7*a^8*x^7)

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