3.8 \(\int \frac{x^4 (a+b \cos ^{-1}(c x))}{(d-c^2 d x^2)^2} \, dx\)

Optimal. Leaf size=180 \[ \frac{3 i b \text{PolyLog}\left (2,-e^{i \cos ^{-1}(c x)}\right )}{2 c^5 d^2}-\frac{3 i b \text{PolyLog}\left (2,e^{i \cos ^{-1}(c x)}\right )}{2 c^5 d^2}+\frac{x^3 \left (a+b \cos ^{-1}(c x)\right )}{2 c^2 d^2 \left (1-c^2 x^2\right )}+\frac{3 x \left (a+b \cos ^{-1}(c x)\right )}{2 c^4 d^2}-\frac{3 \tanh ^{-1}\left (e^{i \cos ^{-1}(c x)}\right ) \left (a+b \cos ^{-1}(c x)\right )}{c^5 d^2}-\frac{b \sqrt{1-c^2 x^2}}{c^5 d^2}+\frac{b}{2 c^5 d^2 \sqrt{1-c^2 x^2}} \]

[Out]

b/(2*c^5*d^2*Sqrt[1 - c^2*x^2]) - (b*Sqrt[1 - c^2*x^2])/(c^5*d^2) + (3*x*(a + b*ArcCos[c*x]))/(2*c^4*d^2) + (x
^3*(a + b*ArcCos[c*x]))/(2*c^2*d^2*(1 - c^2*x^2)) - (3*(a + b*ArcCos[c*x])*ArcTanh[E^(I*ArcCos[c*x])])/(c^5*d^
2) + (((3*I)/2)*b*PolyLog[2, -E^(I*ArcCos[c*x])])/(c^5*d^2) - (((3*I)/2)*b*PolyLog[2, E^(I*ArcCos[c*x])])/(c^5
*d^2)

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Rubi [A]  time = 0.230507, antiderivative size = 180, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 9, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.36, Rules used = {4704, 4716, 4658, 4183, 2279, 2391, 261, 266, 43} \[ \frac{3 i b \text{PolyLog}\left (2,-e^{i \cos ^{-1}(c x)}\right )}{2 c^5 d^2}-\frac{3 i b \text{PolyLog}\left (2,e^{i \cos ^{-1}(c x)}\right )}{2 c^5 d^2}+\frac{x^3 \left (a+b \cos ^{-1}(c x)\right )}{2 c^2 d^2 \left (1-c^2 x^2\right )}+\frac{3 x \left (a+b \cos ^{-1}(c x)\right )}{2 c^4 d^2}-\frac{3 \tanh ^{-1}\left (e^{i \cos ^{-1}(c x)}\right ) \left (a+b \cos ^{-1}(c x)\right )}{c^5 d^2}-\frac{b \sqrt{1-c^2 x^2}}{c^5 d^2}+\frac{b}{2 c^5 d^2 \sqrt{1-c^2 x^2}} \]

Antiderivative was successfully verified.

[In]

Int[(x^4*(a + b*ArcCos[c*x]))/(d - c^2*d*x^2)^2,x]

[Out]

b/(2*c^5*d^2*Sqrt[1 - c^2*x^2]) - (b*Sqrt[1 - c^2*x^2])/(c^5*d^2) + (3*x*(a + b*ArcCos[c*x]))/(2*c^4*d^2) + (x
^3*(a + b*ArcCos[c*x]))/(2*c^2*d^2*(1 - c^2*x^2)) - (3*(a + b*ArcCos[c*x])*ArcTanh[E^(I*ArcCos[c*x])])/(c^5*d^
2) + (((3*I)/2)*b*PolyLog[2, -E^(I*ArcCos[c*x])])/(c^5*d^2) - (((3*I)/2)*b*PolyLog[2, E^(I*ArcCos[c*x])])/(c^5
*d^2)

Rule 4704

Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d_) + (e_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(
f*(f*x)^(m - 1)*(d + e*x^2)^(p + 1)*(a + b*ArcCos[c*x])^n)/(2*e*(p + 1)), x] + (-Dist[(f^2*(m - 1))/(2*e*(p +
1)), Int[(f*x)^(m - 2)*(d + e*x^2)^(p + 1)*(a + b*ArcCos[c*x])^n, x], x] - Dist[(b*f*n*d^IntPart[p]*(d + e*x^2
)^FracPart[p])/(2*c*(p + 1)*(1 - c^2*x^2)^FracPart[p]), Int[(f*x)^(m - 1)*(1 - c^2*x^2)^(p + 1/2)*(a + b*ArcCo
s[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && LtQ[p, -1] && Gt
Q[m, 1]

Rule 4716

Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d_) + (e_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(
f*(f*x)^(m - 1)*(d + e*x^2)^(p + 1)*(a + b*ArcCos[c*x])^n)/(e*(m + 2*p + 1)), x] + (Dist[(f^2*(m - 1))/(c^2*(m
+ 2*p + 1)), Int[(f*x)^(m - 2)*(d + e*x^2)^p*(a + b*ArcCos[c*x])^n, x], x] - Dist[(b*f*n*d^IntPart[p]*(d + e*
x^2)^FracPart[p])/(c*(m + 2*p + 1)*(1 - c^2*x^2)^FracPart[p]), Int[(f*x)^(m - 1)*(1 - c^2*x^2)^(p + 1/2)*(a +
b*ArcCos[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e, f, p}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && GtQ[m,
1] && NeQ[m + 2*p + 1, 0] && IntegerQ[m]

Rule 4658

Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)/((d_) + (e_.)*(x_)^2), x_Symbol] :> -Dist[(c*d)^(-1), Subst[Int[(
a + b*x)^n*Csc[x], x], x, ArcCos[c*x]], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[n, 0]

Rule 4183

Int[csc[(e_.) + (f_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[(-2*(c + d*x)^m*ArcTanh[E^(I*(e + f*
x))])/f, x] + (-Dist[(d*m)/f, Int[(c + d*x)^(m - 1)*Log[1 - E^(I*(e + f*x))], x], x] + Dist[(d*m)/f, Int[(c +
d*x)^(m - 1)*Log[1 + E^(I*(e + f*x))], x], x]) /; FreeQ[{c, d, e, f}, x] && IGtQ[m, 0]

Rule 2279

Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Dist[1/(d*e*n*Log[F]), Subst[Int
[Log[a + b*x]/x, x], x, (F^(e*(c + d*x)))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]

Rule 2391

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> -Simp[PolyLog[2, -(c*e*x^n)]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]

Rule 261

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a + b*x^n)^(p + 1)/(b*n*(p + 1)), x] /; FreeQ
[{a, b, m, n, p}, x] && EqQ[m, n - 1] && NeQ[p, -1]

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 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int \frac{x^4 \left (a+b \cos ^{-1}(c x)\right )}{\left (d-c^2 d x^2\right )^2} \, dx &=\frac{x^3 \left (a+b \cos ^{-1}(c x)\right )}{2 c^2 d^2 \left (1-c^2 x^2\right )}+\frac{b \int \frac{x^3}{\left (1-c^2 x^2\right )^{3/2}} \, dx}{2 c d^2}-\frac{3 \int \frac{x^2 \left (a+b \cos ^{-1}(c x)\right )}{d-c^2 d x^2} \, dx}{2 c^2 d}\\ &=\frac{3 x \left (a+b \cos ^{-1}(c x)\right )}{2 c^4 d^2}+\frac{x^3 \left (a+b \cos ^{-1}(c x)\right )}{2 c^2 d^2 \left (1-c^2 x^2\right )}+\frac{(3 b) \int \frac{x}{\sqrt{1-c^2 x^2}} \, dx}{2 c^3 d^2}+\frac{b \operatorname{Subst}\left (\int \frac{x}{\left (1-c^2 x\right )^{3/2}} \, dx,x,x^2\right )}{4 c d^2}-\frac{3 \int \frac{a+b \cos ^{-1}(c x)}{d-c^2 d x^2} \, dx}{2 c^4 d}\\ &=-\frac{3 b \sqrt{1-c^2 x^2}}{2 c^5 d^2}+\frac{3 x \left (a+b \cos ^{-1}(c x)\right )}{2 c^4 d^2}+\frac{x^3 \left (a+b \cos ^{-1}(c x)\right )}{2 c^2 d^2 \left (1-c^2 x^2\right )}+\frac{3 \operatorname{Subst}\left (\int (a+b x) \csc (x) \, dx,x,\cos ^{-1}(c x)\right )}{2 c^5 d^2}+\frac{b \operatorname{Subst}\left (\int \left (\frac{1}{c^2 \left (1-c^2 x\right )^{3/2}}-\frac{1}{c^2 \sqrt{1-c^2 x}}\right ) \, dx,x,x^2\right )}{4 c d^2}\\ &=\frac{b}{2 c^5 d^2 \sqrt{1-c^2 x^2}}-\frac{b \sqrt{1-c^2 x^2}}{c^5 d^2}+\frac{3 x \left (a+b \cos ^{-1}(c x)\right )}{2 c^4 d^2}+\frac{x^3 \left (a+b \cos ^{-1}(c x)\right )}{2 c^2 d^2 \left (1-c^2 x^2\right )}-\frac{3 \left (a+b \cos ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{i \cos ^{-1}(c x)}\right )}{c^5 d^2}-\frac{(3 b) \operatorname{Subst}\left (\int \log \left (1-e^{i x}\right ) \, dx,x,\cos ^{-1}(c x)\right )}{2 c^5 d^2}+\frac{(3 b) \operatorname{Subst}\left (\int \log \left (1+e^{i x}\right ) \, dx,x,\cos ^{-1}(c x)\right )}{2 c^5 d^2}\\ &=\frac{b}{2 c^5 d^2 \sqrt{1-c^2 x^2}}-\frac{b \sqrt{1-c^2 x^2}}{c^5 d^2}+\frac{3 x \left (a+b \cos ^{-1}(c x)\right )}{2 c^4 d^2}+\frac{x^3 \left (a+b \cos ^{-1}(c x)\right )}{2 c^2 d^2 \left (1-c^2 x^2\right )}-\frac{3 \left (a+b \cos ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{i \cos ^{-1}(c x)}\right )}{c^5 d^2}+\frac{(3 i b) \operatorname{Subst}\left (\int \frac{\log (1-x)}{x} \, dx,x,e^{i \cos ^{-1}(c x)}\right )}{2 c^5 d^2}-\frac{(3 i b) \operatorname{Subst}\left (\int \frac{\log (1+x)}{x} \, dx,x,e^{i \cos ^{-1}(c x)}\right )}{2 c^5 d^2}\\ &=\frac{b}{2 c^5 d^2 \sqrt{1-c^2 x^2}}-\frac{b \sqrt{1-c^2 x^2}}{c^5 d^2}+\frac{3 x \left (a+b \cos ^{-1}(c x)\right )}{2 c^4 d^2}+\frac{x^3 \left (a+b \cos ^{-1}(c x)\right )}{2 c^2 d^2 \left (1-c^2 x^2\right )}-\frac{3 \left (a+b \cos ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{i \cos ^{-1}(c x)}\right )}{c^5 d^2}+\frac{3 i b \text{Li}_2\left (-e^{i \cos ^{-1}(c x)}\right )}{2 c^5 d^2}-\frac{3 i b \text{Li}_2\left (e^{i \cos ^{-1}(c x)}\right )}{2 c^5 d^2}\\ \end{align*}

Mathematica [A]  time = 0.335002, size = 294, normalized size = 1.63 \[ \frac{b \left (-\frac{3 \left (-\frac{2 i \text{PolyLog}\left (2,-e^{i \cos ^{-1}(c x)}\right )}{c}-\frac{i \cos ^{-1}(c x)^2}{2 c}+\frac{2 \cos ^{-1}(c x) \log \left (1+e^{i \cos ^{-1}(c x)}\right )}{c}\right )}{4 c^4}-\frac{3 i \left (4 \text{PolyLog}\left (2,e^{i \cos ^{-1}(c x)}\right )+\cos ^{-1}(c x) \left (\cos ^{-1}(c x)+4 i \log \left (1-e^{i \cos ^{-1}(c x)}\right )\right )\right )}{8 c^5}+\frac{\sqrt{1-c^2 x^2}-\cos ^{-1}(c x)}{4 c^4 \left (c^2 x+c\right )}+\frac{\sqrt{1-c^2 x^2}+\cos ^{-1}(c x)}{4 c^4 \left (c-c^2 x\right )}+\frac{c x \cos ^{-1}(c x)-\sqrt{1-c^2 x^2}}{c^5}\right )}{d^2}-\frac{a x}{2 c^4 d^2 \left (c^2 x^2-1\right )}+\frac{a x}{c^4 d^2}+\frac{3 a \log (1-c x)}{4 c^5 d^2}-\frac{3 a \log (c x+1)}{4 c^5 d^2} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[(x^4*(a + b*ArcCos[c*x]))/(d - c^2*d*x^2)^2,x]

[Out]

(a*x)/(c^4*d^2) - (a*x)/(2*c^4*d^2*(-1 + c^2*x^2)) + (3*a*Log[1 - c*x])/(4*c^5*d^2) - (3*a*Log[1 + c*x])/(4*c^
5*d^2) + (b*((Sqrt[1 - c^2*x^2] - ArcCos[c*x])/(4*c^4*(c + c^2*x)) + (Sqrt[1 - c^2*x^2] + ArcCos[c*x])/(4*c^4*
(c - c^2*x)) + (-Sqrt[1 - c^2*x^2] + c*x*ArcCos[c*x])/c^5 - (3*(((-I/2)*ArcCos[c*x]^2)/c + (2*ArcCos[c*x]*Log[
1 + E^(I*ArcCos[c*x])])/c - ((2*I)*PolyLog[2, -E^(I*ArcCos[c*x])])/c))/(4*c^4) - (((3*I)/8)*(ArcCos[c*x]*(ArcC
os[c*x] + (4*I)*Log[1 - E^(I*ArcCos[c*x])]) + 4*PolyLog[2, E^(I*ArcCos[c*x])]))/c^5))/d^2

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Maple [A]  time = 0.306, size = 296, normalized size = 1.6 \begin{align*}{\frac{ax}{{d}^{2}{c}^{4}}}-{\frac{a}{4\,{d}^{2}{c}^{5} \left ( cx-1 \right ) }}+{\frac{3\,a\ln \left ( cx-1 \right ) }{4\,{d}^{2}{c}^{5}}}-{\frac{a}{4\,{d}^{2}{c}^{5} \left ( cx+1 \right ) }}-{\frac{3\,a\ln \left ( cx+1 \right ) }{4\,{d}^{2}{c}^{5}}}-{\frac{b}{{d}^{2}{c}^{5}}\sqrt{-{c}^{2}{x}^{2}+1}}+{\frac{b\arccos \left ( cx \right ) x}{{d}^{2}{c}^{4}}}-{\frac{b\arccos \left ( cx \right ) x}{2\,{d}^{2}{c}^{4} \left ({c}^{2}{x}^{2}-1 \right ) }}-{\frac{b}{2\,{d}^{2}{c}^{5} \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{-{c}^{2}{x}^{2}+1}}-{\frac{3\,b\arccos \left ( cx \right ) }{2\,{d}^{2}{c}^{5}}\ln \left ( 1+cx+i\sqrt{-{c}^{2}{x}^{2}+1} \right ) }+{\frac{{\frac{3\,i}{2}}b}{{d}^{2}{c}^{5}}{\it polylog} \left ( 2,-cx-i\sqrt{-{c}^{2}{x}^{2}+1} \right ) }+{\frac{3\,b\arccos \left ( cx \right ) }{2\,{d}^{2}{c}^{5}}\ln \left ( 1-cx-i\sqrt{-{c}^{2}{x}^{2}+1} \right ) }-{\frac{{\frac{3\,i}{2}}b}{{d}^{2}{c}^{5}}{\it polylog} \left ( 2,cx+i\sqrt{-{c}^{2}{x}^{2}+1} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4*(a+b*arccos(c*x))/(-c^2*d*x^2+d)^2,x)

[Out]

1/c^4*a/d^2*x-1/4/c^5*a/d^2/(c*x-1)+3/4/c^5*a/d^2*ln(c*x-1)-1/4/c^5*a/d^2/(c*x+1)-3/4/c^5*a/d^2*ln(c*x+1)-b*(-
c^2*x^2+1)^(1/2)/d^2/c^5+1/c^4*b/d^2*arccos(c*x)*x-1/2/c^4*b/d^2/(c^2*x^2-1)*arccos(c*x)*x-1/2/c^5*b/d^2/(c^2*
x^2-1)*(-c^2*x^2+1)^(1/2)-3/2/c^5*b/d^2*arccos(c*x)*ln(1+c*x+I*(-c^2*x^2+1)^(1/2))+3/2*I*b*polylog(2,-c*x-I*(-
c^2*x^2+1)^(1/2))/d^2/c^5+3/2/c^5*b/d^2*arccos(c*x)*ln(1-c*x-I*(-c^2*x^2+1)^(1/2))-3/2*I*b*polylog(2,c*x+I*(-c
^2*x^2+1)^(1/2))/d^2/c^5

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{1}{4} \, a{\left (\frac{2 \, x}{c^{6} d^{2} x^{2} - c^{4} d^{2}} - \frac{4 \, x}{c^{4} d^{2}} + \frac{3 \, \log \left (c x + 1\right )}{c^{5} d^{2}} - \frac{3 \, \log \left (c x - 1\right )}{c^{5} d^{2}}\right )} + \frac{{\left ({\left (4 \, c^{3} x^{3} - 6 \, c x - 3 \,{\left (c^{2} x^{2} - 1\right )} \log \left (c x + 1\right ) + 3 \,{\left (c^{2} x^{2} - 1\right )} \log \left (-c x + 1\right )\right )} \arctan \left (\sqrt{c x + 1} \sqrt{-c x + 1}, c x\right ) -{\left (c^{7} d^{2} x^{2} - c^{5} d^{2}\right )} \int \frac{{\left (4 \, c^{3} x^{3} - 6 \, c x - 3 \,{\left (c^{2} x^{2} - 1\right )} \log \left (c x + 1\right ) + 3 \,{\left (c^{2} x^{2} - 1\right )} \log \left (-c x + 1\right )\right )} \sqrt{c x + 1} \sqrt{-c x + 1}}{c^{8} d^{2} x^{4} - 2 \, c^{6} d^{2} x^{2} + c^{4} d^{2}}\,{d x}\right )} b}{4 \,{\left (c^{7} d^{2} x^{2} - c^{5} d^{2}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(a+b*arccos(c*x))/(-c^2*d*x^2+d)^2,x, algorithm="maxima")

[Out]

-1/4*a*(2*x/(c^6*d^2*x^2 - c^4*d^2) - 4*x/(c^4*d^2) + 3*log(c*x + 1)/(c^5*d^2) - 3*log(c*x - 1)/(c^5*d^2)) + 1
/4*((4*c^3*x^3 - 6*c*x - 3*(c^2*x^2 - 1)*log(c*x + 1) + 3*(c^2*x^2 - 1)*log(-c*x + 1))*arctan2(sqrt(c*x + 1)*s
qrt(-c*x + 1), c*x) + 4*(c^7*d^2*x^2 - c^5*d^2)*integrate(-1/4*(4*c^3*x^3 - 6*c*x - 3*(c^2*x^2 - 1)*log(c*x +
1) + 3*(c^2*x^2 - 1)*log(-c*x + 1))*sqrt(c*x + 1)*sqrt(-c*x + 1)/(c^8*d^2*x^4 - 2*c^6*d^2*x^2 + c^4*d^2), x))*
b/(c^7*d^2*x^2 - c^5*d^2)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{b x^{4} \arccos \left (c x\right ) + a x^{4}}{c^{4} d^{2} x^{4} - 2 \, c^{2} d^{2} x^{2} + d^{2}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(a+b*arccos(c*x))/(-c^2*d*x^2+d)^2,x, algorithm="fricas")

[Out]

integral((b*x^4*arccos(c*x) + a*x^4)/(c^4*d^2*x^4 - 2*c^2*d^2*x^2 + d^2), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\int \frac{a x^{4}}{c^{4} x^{4} - 2 c^{2} x^{2} + 1}\, dx + \int \frac{b x^{4} \operatorname{acos}{\left (c x \right )}}{c^{4} x^{4} - 2 c^{2} x^{2} + 1}\, dx}{d^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**4*(a+b*acos(c*x))/(-c**2*d*x**2+d)**2,x)

[Out]

(Integral(a*x**4/(c**4*x**4 - 2*c**2*x**2 + 1), x) + Integral(b*x**4*acos(c*x)/(c**4*x**4 - 2*c**2*x**2 + 1),
x))/d**2

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b \arccos \left (c x\right ) + a\right )} x^{4}}{{\left (c^{2} d x^{2} - d\right )}^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(a+b*arccos(c*x))/(-c^2*d*x^2+d)^2,x, algorithm="giac")

[Out]

integrate((b*arccos(c*x) + a)*x^4/(c^2*d*x^2 - d)^2, x)