3.1.50 \(\int \frac {a+b \text {ArcSin}(c x)}{(d-c^2 d x^2)^3} \, dx\) [50]

Optimal. Leaf size=196 \[ -\frac {b}{12 c d^3 \left (1-c^2 x^2\right )^{3/2}}-\frac {3 b}{8 c d^3 \sqrt {1-c^2 x^2}}+\frac {x (a+b \text {ArcSin}(c x))}{4 d^3 \left (1-c^2 x^2\right )^2}+\frac {3 x (a+b \text {ArcSin}(c x))}{8 d^3 \left (1-c^2 x^2\right )}-\frac {3 i (a+b \text {ArcSin}(c x)) \text {ArcTan}\left (e^{i \text {ArcSin}(c x)}\right )}{4 c d^3}+\frac {3 i b \text {PolyLog}\left (2,-i e^{i \text {ArcSin}(c x)}\right )}{8 c d^3}-\frac {3 i b \text {PolyLog}\left (2,i e^{i \text {ArcSin}(c x)}\right )}{8 c d^3} \]

[Out]

-1/12*b/c/d^3/(-c^2*x^2+1)^(3/2)+1/4*x*(a+b*arcsin(c*x))/d^3/(-c^2*x^2+1)^2+3/8*x*(a+b*arcsin(c*x))/d^3/(-c^2*
x^2+1)-3/4*I*(a+b*arcsin(c*x))*arctan(I*c*x+(-c^2*x^2+1)^(1/2))/c/d^3+3/8*I*b*polylog(2,-I*(I*c*x+(-c^2*x^2+1)
^(1/2)))/c/d^3-3/8*I*b*polylog(2,I*(I*c*x+(-c^2*x^2+1)^(1/2)))/c/d^3-3/8*b/c/d^3/(-c^2*x^2+1)^(1/2)

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Rubi [A]
time = 0.10, antiderivative size = 196, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 6, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {4747, 4749, 4266, 2317, 2438, 267} \begin {gather*} -\frac {3 i \text {ArcTan}\left (e^{i \text {ArcSin}(c x)}\right ) (a+b \text {ArcSin}(c x))}{4 c d^3}+\frac {3 x (a+b \text {ArcSin}(c x))}{8 d^3 \left (1-c^2 x^2\right )}+\frac {x (a+b \text {ArcSin}(c x))}{4 d^3 \left (1-c^2 x^2\right )^2}+\frac {3 i b \text {Li}_2\left (-i e^{i \text {ArcSin}(c x)}\right )}{8 c d^3}-\frac {3 i b \text {Li}_2\left (i e^{i \text {ArcSin}(c x)}\right )}{8 c d^3}-\frac {3 b}{8 c d^3 \sqrt {1-c^2 x^2}}-\frac {b}{12 c d^3 \left (1-c^2 x^2\right )^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*ArcSin[c*x])/(d - c^2*d*x^2)^3,x]

[Out]

-1/12*b/(c*d^3*(1 - c^2*x^2)^(3/2)) - (3*b)/(8*c*d^3*Sqrt[1 - c^2*x^2]) + (x*(a + b*ArcSin[c*x]))/(4*d^3*(1 -
c^2*x^2)^2) + (3*x*(a + b*ArcSin[c*x]))/(8*d^3*(1 - c^2*x^2)) - (((3*I)/4)*(a + b*ArcSin[c*x])*ArcTan[E^(I*Arc
Sin[c*x])])/(c*d^3) + (((3*I)/8)*b*PolyLog[2, (-I)*E^(I*ArcSin[c*x])])/(c*d^3) - (((3*I)/8)*b*PolyLog[2, I*E^(
I*ArcSin[c*x])])/(c*d^3)

Rule 267

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 2317

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 2438

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 4266

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

Rule 4747

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-x)*(d + e*x^2)^(p
+ 1)*((a + b*ArcSin[c*x])^n/(2*d*(p + 1))), x] + (Dist[(2*p + 3)/(2*d*(p + 1)), Int[(d + e*x^2)^(p + 1)*(a +
b*ArcSin[c*x])^n, x], x] + Dist[b*c*(n/(2*(p + 1)))*Simp[(d + e*x^2)^p/(1 - c^2*x^2)^p], Int[x*(1 - c^2*x^2)^(
p + 1/2)*(a + b*ArcSin[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] &
& LtQ[p, -1] && NeQ[p, -3/2]

Rule 4749

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

Rubi steps

\begin {align*} \int \frac {a+b \sin ^{-1}(c x)}{\left (d-c^2 d x^2\right )^3} \, dx &=\frac {x \left (a+b \sin ^{-1}(c x)\right )}{4 d^3 \left (1-c^2 x^2\right )^2}-\frac {(b c) \int \frac {x}{\left (1-c^2 x^2\right )^{5/2}} \, dx}{4 d^3}+\frac {3 \int \frac {a+b \sin ^{-1}(c x)}{\left (d-c^2 d x^2\right )^2} \, dx}{4 d}\\ &=-\frac {b}{12 c d^3 \left (1-c^2 x^2\right )^{3/2}}+\frac {x \left (a+b \sin ^{-1}(c x)\right )}{4 d^3 \left (1-c^2 x^2\right )^2}+\frac {3 x \left (a+b \sin ^{-1}(c x)\right )}{8 d^3 \left (1-c^2 x^2\right )}-\frac {(3 b c) \int \frac {x}{\left (1-c^2 x^2\right )^{3/2}} \, dx}{8 d^3}+\frac {3 \int \frac {a+b \sin ^{-1}(c x)}{d-c^2 d x^2} \, dx}{8 d^2}\\ &=-\frac {b}{12 c d^3 \left (1-c^2 x^2\right )^{3/2}}-\frac {3 b}{8 c d^3 \sqrt {1-c^2 x^2}}+\frac {x \left (a+b \sin ^{-1}(c x)\right )}{4 d^3 \left (1-c^2 x^2\right )^2}+\frac {3 x \left (a+b \sin ^{-1}(c x)\right )}{8 d^3 \left (1-c^2 x^2\right )}+\frac {3 \text {Subst}\left (\int (a+b x) \sec (x) \, dx,x,\sin ^{-1}(c x)\right )}{8 c d^3}\\ &=-\frac {b}{12 c d^3 \left (1-c^2 x^2\right )^{3/2}}-\frac {3 b}{8 c d^3 \sqrt {1-c^2 x^2}}+\frac {x \left (a+b \sin ^{-1}(c x)\right )}{4 d^3 \left (1-c^2 x^2\right )^2}+\frac {3 x \left (a+b \sin ^{-1}(c x)\right )}{8 d^3 \left (1-c^2 x^2\right )}-\frac {3 i \left (a+b \sin ^{-1}(c x)\right ) \tan ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )}{4 c d^3}-\frac {(3 b) \text {Subst}\left (\int \log \left (1-i e^{i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{8 c d^3}+\frac {(3 b) \text {Subst}\left (\int \log \left (1+i e^{i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{8 c d^3}\\ &=-\frac {b}{12 c d^3 \left (1-c^2 x^2\right )^{3/2}}-\frac {3 b}{8 c d^3 \sqrt {1-c^2 x^2}}+\frac {x \left (a+b \sin ^{-1}(c x)\right )}{4 d^3 \left (1-c^2 x^2\right )^2}+\frac {3 x \left (a+b \sin ^{-1}(c x)\right )}{8 d^3 \left (1-c^2 x^2\right )}-\frac {3 i \left (a+b \sin ^{-1}(c x)\right ) \tan ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )}{4 c d^3}+\frac {(3 i b) \text {Subst}\left (\int \frac {\log (1-i x)}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )}{8 c d^3}-\frac {(3 i b) \text {Subst}\left (\int \frac {\log (1+i x)}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )}{8 c d^3}\\ &=-\frac {b}{12 c d^3 \left (1-c^2 x^2\right )^{3/2}}-\frac {3 b}{8 c d^3 \sqrt {1-c^2 x^2}}+\frac {x \left (a+b \sin ^{-1}(c x)\right )}{4 d^3 \left (1-c^2 x^2\right )^2}+\frac {3 x \left (a+b \sin ^{-1}(c x)\right )}{8 d^3 \left (1-c^2 x^2\right )}-\frac {3 i \left (a+b \sin ^{-1}(c x)\right ) \tan ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )}{4 c d^3}+\frac {3 i b \text {Li}_2\left (-i e^{i \sin ^{-1}(c x)}\right )}{8 c d^3}-\frac {3 i b \text {Li}_2\left (i e^{i \sin ^{-1}(c x)}\right )}{8 c d^3}\\ \end {align*}

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Mathematica [B] Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(501\) vs. \(2(196)=392\).
time = 0.80, size = 501, normalized size = 2.56 \begin {gather*} -\frac {\frac {2 b \sqrt {1-c^2 x^2}}{3 c (-1+c x)^2}-\frac {b x \sqrt {1-c^2 x^2}}{3 (-1+c x)^2}+\frac {2 b \sqrt {1-c^2 x^2}}{3 c (1+c x)^2}+\frac {b x \sqrt {1-c^2 x^2}}{3 (1+c x)^2}+\frac {3 b \sqrt {1-c^2 x^2}}{c-c^2 x}+\frac {3 b \sqrt {1-c^2 x^2}}{c+c^2 x}-\frac {4 a x}{\left (-1+c^2 x^2\right )^2}+\frac {6 a x}{-1+c^2 x^2}+\frac {3 i b \pi \text {ArcSin}(c x)}{c}-\frac {b \text {ArcSin}(c x)}{c (-1+c x)^2}+\frac {b \text {ArcSin}(c x)}{c (1+c x)^2}-\frac {3 b \text {ArcSin}(c x)}{c-c^2 x}+\frac {3 b \text {ArcSin}(c x)}{c+c^2 x}-\frac {3 b \pi \log \left (1-i e^{i \text {ArcSin}(c x)}\right )}{c}-\frac {6 b \text {ArcSin}(c x) \log \left (1-i e^{i \text {ArcSin}(c x)}\right )}{c}-\frac {3 b \pi \log \left (1+i e^{i \text {ArcSin}(c x)}\right )}{c}+\frac {6 b \text {ArcSin}(c x) \log \left (1+i e^{i \text {ArcSin}(c x)}\right )}{c}+\frac {3 a \log (1-c x)}{c}-\frac {3 a \log (1+c x)}{c}+\frac {3 b \pi \log \left (-\cos \left (\frac {1}{4} (\pi +2 \text {ArcSin}(c x))\right )\right )}{c}+\frac {3 b \pi \log \left (\sin \left (\frac {1}{4} (\pi +2 \text {ArcSin}(c x))\right )\right )}{c}-\frac {6 i b \text {PolyLog}\left (2,-i e^{i \text {ArcSin}(c x)}\right )}{c}+\frac {6 i b \text {PolyLog}\left (2,i e^{i \text {ArcSin}(c x)}\right )}{c}}{16 d^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*ArcSin[c*x])/(d - c^2*d*x^2)^3,x]

[Out]

-1/16*((2*b*Sqrt[1 - c^2*x^2])/(3*c*(-1 + c*x)^2) - (b*x*Sqrt[1 - c^2*x^2])/(3*(-1 + c*x)^2) + (2*b*Sqrt[1 - c
^2*x^2])/(3*c*(1 + c*x)^2) + (b*x*Sqrt[1 - c^2*x^2])/(3*(1 + c*x)^2) + (3*b*Sqrt[1 - c^2*x^2])/(c - c^2*x) + (
3*b*Sqrt[1 - c^2*x^2])/(c + c^2*x) - (4*a*x)/(-1 + c^2*x^2)^2 + (6*a*x)/(-1 + c^2*x^2) + ((3*I)*b*Pi*ArcSin[c*
x])/c - (b*ArcSin[c*x])/(c*(-1 + c*x)^2) + (b*ArcSin[c*x])/(c*(1 + c*x)^2) - (3*b*ArcSin[c*x])/(c - c^2*x) + (
3*b*ArcSin[c*x])/(c + c^2*x) - (3*b*Pi*Log[1 - I*E^(I*ArcSin[c*x])])/c - (6*b*ArcSin[c*x]*Log[1 - I*E^(I*ArcSi
n[c*x])])/c - (3*b*Pi*Log[1 + I*E^(I*ArcSin[c*x])])/c + (6*b*ArcSin[c*x]*Log[1 + I*E^(I*ArcSin[c*x])])/c + (3*
a*Log[1 - c*x])/c - (3*a*Log[1 + c*x])/c + (3*b*Pi*Log[-Cos[(Pi + 2*ArcSin[c*x])/4]])/c + (3*b*Pi*Log[Sin[(Pi
+ 2*ArcSin[c*x])/4]])/c - ((6*I)*b*PolyLog[2, (-I)*E^(I*ArcSin[c*x])])/c + ((6*I)*b*PolyLog[2, I*E^(I*ArcSin[c
*x])])/c)/d^3

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Maple [A]
time = 0.15, size = 358, normalized size = 1.83

method	result	size
derivativedivides	\(\frac {-\frac {a}{16 d^{3} \left (c x +1\right )^{2}}-\frac {3 a}{16 d^{3} \left (c x +1\right )}+\frac {3 a \ln \left (c x +1\right )}{16 d^{3}}+\frac {a}{16 d^{3} \left (c x -1\right )^{2}}-\frac {3 a}{16 d^{3} \left (c x -1\right )}-\frac {3 a \ln \left (c x -1\right )}{16 d^{3}}-\frac {3 b \arcsin \left (c x \right ) c^{3} x^{3}}{8 d^{3} \left (c^{4} x^{4}-2 c^{2} x^{2}+1\right )}+\frac {3 b \,c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{8 d^{3} \left (c^{4} x^{4}-2 c^{2} x^{2}+1\right )}+\frac {5 b \arcsin \left (c x \right ) c x}{8 d^{3} \left (c^{4} x^{4}-2 c^{2} x^{2}+1\right )}-\frac {11 b \sqrt {-c^{2} x^{2}+1}}{24 d^{3} \left (c^{4} x^{4}-2 c^{2} x^{2}+1\right )}-\frac {3 b \arcsin \left (c x \right ) \ln \left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{8 d^{3}}+\frac {3 b \arcsin \left (c x \right ) \ln \left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{8 d^{3}}+\frac {3 i b \dilog \left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{8 d^{3}}-\frac {3 i b \dilog \left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{8 d^{3}}}{c}\)	\(358\)
default	\(\frac {-\frac {a}{16 d^{3} \left (c x +1\right )^{2}}-\frac {3 a}{16 d^{3} \left (c x +1\right )}+\frac {3 a \ln \left (c x +1\right )}{16 d^{3}}+\frac {a}{16 d^{3} \left (c x -1\right )^{2}}-\frac {3 a}{16 d^{3} \left (c x -1\right )}-\frac {3 a \ln \left (c x -1\right )}{16 d^{3}}-\frac {3 b \arcsin \left (c x \right ) c^{3} x^{3}}{8 d^{3} \left (c^{4} x^{4}-2 c^{2} x^{2}+1\right )}+\frac {3 b \,c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{8 d^{3} \left (c^{4} x^{4}-2 c^{2} x^{2}+1\right )}+\frac {5 b \arcsin \left (c x \right ) c x}{8 d^{3} \left (c^{4} x^{4}-2 c^{2} x^{2}+1\right )}-\frac {11 b \sqrt {-c^{2} x^{2}+1}}{24 d^{3} \left (c^{4} x^{4}-2 c^{2} x^{2}+1\right )}-\frac {3 b \arcsin \left (c x \right ) \ln \left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{8 d^{3}}+\frac {3 b \arcsin \left (c x \right ) \ln \left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{8 d^{3}}+\frac {3 i b \dilog \left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{8 d^{3}}-\frac {3 i b \dilog \left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{8 d^{3}}}{c}\)	\(358\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*arcsin(c*x))/(-c^2*d*x^2+d)^3,x,method=_RETURNVERBOSE)

[Out]

1/c*(-1/16*a/d^3/(c*x+1)^2-3/16*a/d^3/(c*x+1)+3/16*a/d^3*ln(c*x+1)+1/16*a/d^3/(c*x-1)^2-3/16*a/d^3/(c*x-1)-3/1
6*a/d^3*ln(c*x-1)-3/8*b/d^3/(c^4*x^4-2*c^2*x^2+1)*arcsin(c*x)*c^3*x^3+3/8*b/d^3/(c^4*x^4-2*c^2*x^2+1)*c^2*x^2*
(-c^2*x^2+1)^(1/2)+5/8*b/d^3/(c^4*x^4-2*c^2*x^2+1)*arcsin(c*x)*c*x-11/24*b/d^3/(c^4*x^4-2*c^2*x^2+1)*(-c^2*x^2
+1)^(1/2)-3/8*b/d^3*arcsin(c*x)*ln(1+I*(I*c*x+(-c^2*x^2+1)^(1/2)))+3/8*b/d^3*arcsin(c*x)*ln(1-I*(I*c*x+(-c^2*x
^2+1)^(1/2)))+3/8*I*b/d^3*dilog(1+I*(I*c*x+(-c^2*x^2+1)^(1/2)))-3/8*I*b/d^3*dilog(1-I*(I*c*x+(-c^2*x^2+1)^(1/2
))))

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/16*a*(2*(3*c^2*x^3 - 5*x)/(c^4*d^3*x^4 - 2*c^2*d^3*x^2 + d^3) - 3*log(c*x + 1)/(c*d^3) + 3*log(c*x - 1)/(c*
d^3)) + 1/16*(3*(c^4*x^4 - 2*c^2*x^2 + 1)*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1))*log(c*x + 1) - 3*(c^4*x^4
- 2*c^2*x^2 + 1)*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1))*log(-c*x + 1) - 2*(3*c^3*x^3 - 5*c*x)*arctan2(c*x
, sqrt(c*x + 1)*sqrt(-c*x + 1)) + 16*(c^5*d^3*x^4 - 2*c^3*d^3*x^2 + c*d^3)*integrate(-1/16*(6*c^3*x^3 - 10*c*x
- 3*(c^4*x^4 - 2*c^2*x^2 + 1)*log(c*x + 1) + 3*(c^4*x^4 - 2*c^2*x^2 + 1)*log(-c*x + 1))*sqrt(c*x + 1)*sqrt(-c
*x + 1)/(c^6*d^3*x^6 - 3*c^4*d^3*x^4 + 3*c^2*d^3*x^2 - d^3), x))*b/(c^5*d^3*x^4 - 2*c^3*d^3*x^2 + c*d^3)

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(-(b*arcsin(c*x) + a)/(c^6*d^3*x^6 - 3*c^4*d^3*x^4 + 3*c^2*d^3*x^2 - d^3), x)

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*asin(c*x))/(-c**2*d*x**2+d)**3,x)

[Out]

Timed out

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(-(b*arcsin(c*x) + a)/(c^2*d*x^2 - d)^3, x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {a+b\,\mathrm {asin}\left (c\,x\right )}{{\left (d-c^2\,d\,x^2\right )}^3} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*asin(c*x))/(d - c^2*d*x^2)^3,x)

[Out]

int((a + b*asin(c*x))/(d - c^2*d*x^2)^3, x)

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