3.95 \(\int x^2 \cos (a+b \sqrt [3]{c+d x}) \, dx\)

Optimal. Leaf size=537 \[ -\frac{6 c^2 \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d^3}+\frac{6 c^2 \sqrt [3]{c+d x} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^3}-\frac{168 (c+d x)^2 \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d^3}+\frac{5040 (c+d x)^{4/3} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^5 d^3}+\frac{120 c (c+d x) \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d^3}-\frac{60480 (c+d x)^{2/3} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^7 d^3}-\frac{720 c \sqrt [3]{c+d x} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^5 d^3}+\frac{120960 \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^9 d^3}+\frac{24 (c+d x)^{7/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^3}-\frac{1008 (c+d x)^{5/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^4 d^3}-\frac{30 c (c+d x)^{4/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^3}+\frac{20160 (c+d x) \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^6 d^3}+\frac{360 c (c+d x)^{2/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^4 d^3}-\frac{120960 \sqrt [3]{c+d x} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^8 d^3}-\frac{720 c \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^6 d^3}+\frac{3 c^2 (c+d x)^{2/3} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b d^3}+\frac{3 (c+d x)^{8/3} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b d^3}-\frac{6 c (c+d x)^{5/3} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b d^3} \]

[Out]

(-720*c*Cos[a + b*(c + d*x)^(1/3)])/(b^6*d^3) - (120960*(c + d*x)^(1/3)*Cos[a + b*(c + d*x)^(1/3)])/(b^8*d^3)
+ (6*c^2*(c + d*x)^(1/3)*Cos[a + b*(c + d*x)^(1/3)])/(b^2*d^3) + (360*c*(c + d*x)^(2/3)*Cos[a + b*(c + d*x)^(1
/3)])/(b^4*d^3) + (20160*(c + d*x)*Cos[a + b*(c + d*x)^(1/3)])/(b^6*d^3) - (30*c*(c + d*x)^(4/3)*Cos[a + b*(c
+ d*x)^(1/3)])/(b^2*d^3) - (1008*(c + d*x)^(5/3)*Cos[a + b*(c + d*x)^(1/3)])/(b^4*d^3) + (24*(c + d*x)^(7/3)*C
os[a + b*(c + d*x)^(1/3)])/(b^2*d^3) + (120960*Sin[a + b*(c + d*x)^(1/3)])/(b^9*d^3) - (6*c^2*Sin[a + b*(c + d
*x)^(1/3)])/(b^3*d^3) - (720*c*(c + d*x)^(1/3)*Sin[a + b*(c + d*x)^(1/3)])/(b^5*d^3) - (60480*(c + d*x)^(2/3)*
Sin[a + b*(c + d*x)^(1/3)])/(b^7*d^3) + (3*c^2*(c + d*x)^(2/3)*Sin[a + b*(c + d*x)^(1/3)])/(b*d^3) + (120*c*(c
+ d*x)*Sin[a + b*(c + d*x)^(1/3)])/(b^3*d^3) + (5040*(c + d*x)^(4/3)*Sin[a + b*(c + d*x)^(1/3)])/(b^5*d^3) -
(6*c*(c + d*x)^(5/3)*Sin[a + b*(c + d*x)^(1/3)])/(b*d^3) - (168*(c + d*x)^2*Sin[a + b*(c + d*x)^(1/3)])/(b^3*d
^3) + (3*(c + d*x)^(8/3)*Sin[a + b*(c + d*x)^(1/3)])/(b*d^3)

________________________________________________________________________________________

Rubi [A]  time = 0.51131, antiderivative size = 537, normalized size of antiderivative = 1., number of steps used = 20, number of rules used = 4, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {3432, 3296, 2637, 2638} \[ -\frac{6 c^2 \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d^3}+\frac{6 c^2 \sqrt [3]{c+d x} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^3}-\frac{168 (c+d x)^2 \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d^3}+\frac{5040 (c+d x)^{4/3} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^5 d^3}+\frac{120 c (c+d x) \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d^3}-\frac{60480 (c+d x)^{2/3} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^7 d^3}-\frac{720 c \sqrt [3]{c+d x} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^5 d^3}+\frac{120960 \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^9 d^3}+\frac{24 (c+d x)^{7/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^3}-\frac{1008 (c+d x)^{5/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^4 d^3}-\frac{30 c (c+d x)^{4/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^3}+\frac{20160 (c+d x) \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^6 d^3}+\frac{360 c (c+d x)^{2/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^4 d^3}-\frac{120960 \sqrt [3]{c+d x} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^8 d^3}-\frac{720 c \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^6 d^3}+\frac{3 c^2 (c+d x)^{2/3} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b d^3}+\frac{3 (c+d x)^{8/3} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b d^3}-\frac{6 c (c+d x)^{5/3} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b d^3} \]

Antiderivative was successfully verified.

[In]

Int[x^2*Cos[a + b*(c + d*x)^(1/3)],x]

[Out]

(-720*c*Cos[a + b*(c + d*x)^(1/3)])/(b^6*d^3) - (120960*(c + d*x)^(1/3)*Cos[a + b*(c + d*x)^(1/3)])/(b^8*d^3)
+ (6*c^2*(c + d*x)^(1/3)*Cos[a + b*(c + d*x)^(1/3)])/(b^2*d^3) + (360*c*(c + d*x)^(2/3)*Cos[a + b*(c + d*x)^(1
/3)])/(b^4*d^3) + (20160*(c + d*x)*Cos[a + b*(c + d*x)^(1/3)])/(b^6*d^3) - (30*c*(c + d*x)^(4/3)*Cos[a + b*(c
+ d*x)^(1/3)])/(b^2*d^3) - (1008*(c + d*x)^(5/3)*Cos[a + b*(c + d*x)^(1/3)])/(b^4*d^3) + (24*(c + d*x)^(7/3)*C
os[a + b*(c + d*x)^(1/3)])/(b^2*d^3) + (120960*Sin[a + b*(c + d*x)^(1/3)])/(b^9*d^3) - (6*c^2*Sin[a + b*(c + d
*x)^(1/3)])/(b^3*d^3) - (720*c*(c + d*x)^(1/3)*Sin[a + b*(c + d*x)^(1/3)])/(b^5*d^3) - (60480*(c + d*x)^(2/3)*
Sin[a + b*(c + d*x)^(1/3)])/(b^7*d^3) + (3*c^2*(c + d*x)^(2/3)*Sin[a + b*(c + d*x)^(1/3)])/(b*d^3) + (120*c*(c
+ d*x)*Sin[a + b*(c + d*x)^(1/3)])/(b^3*d^3) + (5040*(c + d*x)^(4/3)*Sin[a + b*(c + d*x)^(1/3)])/(b^5*d^3) -
(6*c*(c + d*x)^(5/3)*Sin[a + b*(c + d*x)^(1/3)])/(b*d^3) - (168*(c + d*x)^2*Sin[a + b*(c + d*x)^(1/3)])/(b^3*d
^3) + (3*(c + d*x)^(8/3)*Sin[a + b*(c + d*x)^(1/3)])/(b*d^3)

Rule 3432

Int[((a_.) + Cos[(c_.) + (d_.)*((e_.) + (f_.)*(x_))^(n_)]*(b_.))^(p_.)*((g_.) + (h_.)*(x_))^(m_.), x_Symbol] :
> Dist[1/(n*f), Subst[Int[ExpandIntegrand[(a + b*Cos[c + d*x])^p, x^(1/n - 1)*(g - (e*h)/f + (h*x^(1/n))/f)^m,
x], x], x, (e + f*x)^n], x] /; FreeQ[{a, b, c, d, e, f, g, h, m}, x] && IGtQ[p, 0] && IntegerQ[1/n]

Rule 3296

Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> -Simp[((c + d*x)^m*Cos[e + f*x])/f, x] +
Dist[(d*m)/f, Int[(c + d*x)^(m - 1)*Cos[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]

Rule 2637

Int[sin[Pi/2 + (c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Sin[c + d*x]/d, x] /; FreeQ[{c, d}, x]

Rule 2638

Int[sin[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[Cos[c + d*x]/d, x] /; FreeQ[{c, d}, x]

Rubi steps

\begin{align*} \int x^2 \cos \left (a+b \sqrt [3]{c+d x}\right ) \, dx &=\frac{3 \operatorname{Subst}\left (\int \left (\frac{c^2 x^2 \cos (a+b x)}{d^2}-\frac{2 c x^5 \cos (a+b x)}{d^2}+\frac{x^8 \cos (a+b x)}{d^2}\right ) \, dx,x,\sqrt [3]{c+d x}\right )}{d}\\ &=\frac{3 \operatorname{Subst}\left (\int x^8 \cos (a+b x) \, dx,x,\sqrt [3]{c+d x}\right )}{d^3}-\frac{(6 c) \operatorname{Subst}\left (\int x^5 \cos (a+b x) \, dx,x,\sqrt [3]{c+d x}\right )}{d^3}+\frac{\left (3 c^2\right ) \operatorname{Subst}\left (\int x^2 \cos (a+b x) \, dx,x,\sqrt [3]{c+d x}\right )}{d^3}\\ &=\frac{3 c^2 (c+d x)^{2/3} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b d^3}-\frac{6 c (c+d x)^{5/3} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b d^3}+\frac{3 (c+d x)^{8/3} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b d^3}-\frac{24 \operatorname{Subst}\left (\int x^7 \sin (a+b x) \, dx,x,\sqrt [3]{c+d x}\right )}{b d^3}+\frac{(30 c) \operatorname{Subst}\left (\int x^4 \sin (a+b x) \, dx,x,\sqrt [3]{c+d x}\right )}{b d^3}-\frac{\left (6 c^2\right ) \operatorname{Subst}\left (\int x \sin (a+b x) \, dx,x,\sqrt [3]{c+d x}\right )}{b d^3}\\ &=\frac{6 c^2 \sqrt [3]{c+d x} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^3}-\frac{30 c (c+d x)^{4/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^3}+\frac{24 (c+d x)^{7/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^3}+\frac{3 c^2 (c+d x)^{2/3} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b d^3}-\frac{6 c (c+d x)^{5/3} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b d^3}+\frac{3 (c+d x)^{8/3} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b d^3}-\frac{168 \operatorname{Subst}\left (\int x^6 \cos (a+b x) \, dx,x,\sqrt [3]{c+d x}\right )}{b^2 d^3}+\frac{(120 c) \operatorname{Subst}\left (\int x^3 \cos (a+b x) \, dx,x,\sqrt [3]{c+d x}\right )}{b^2 d^3}-\frac{\left (6 c^2\right ) \operatorname{Subst}\left (\int \cos (a+b x) \, dx,x,\sqrt [3]{c+d x}\right )}{b^2 d^3}\\ &=\frac{6 c^2 \sqrt [3]{c+d x} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^3}-\frac{30 c (c+d x)^{4/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^3}+\frac{24 (c+d x)^{7/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^3}-\frac{6 c^2 \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d^3}+\frac{3 c^2 (c+d x)^{2/3} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b d^3}+\frac{120 c (c+d x) \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d^3}-\frac{6 c (c+d x)^{5/3} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b d^3}-\frac{168 (c+d x)^2 \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d^3}+\frac{3 (c+d x)^{8/3} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b d^3}+\frac{1008 \operatorname{Subst}\left (\int x^5 \sin (a+b x) \, dx,x,\sqrt [3]{c+d x}\right )}{b^3 d^3}-\frac{(360 c) \operatorname{Subst}\left (\int x^2 \sin (a+b x) \, dx,x,\sqrt [3]{c+d x}\right )}{b^3 d^3}\\ &=\frac{6 c^2 \sqrt [3]{c+d x} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^3}+\frac{360 c (c+d x)^{2/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^4 d^3}-\frac{30 c (c+d x)^{4/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^3}-\frac{1008 (c+d x)^{5/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^4 d^3}+\frac{24 (c+d x)^{7/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^3}-\frac{6 c^2 \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d^3}+\frac{3 c^2 (c+d x)^{2/3} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b d^3}+\frac{120 c (c+d x) \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d^3}-\frac{6 c (c+d x)^{5/3} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b d^3}-\frac{168 (c+d x)^2 \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d^3}+\frac{3 (c+d x)^{8/3} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b d^3}+\frac{5040 \operatorname{Subst}\left (\int x^4 \cos (a+b x) \, dx,x,\sqrt [3]{c+d x}\right )}{b^4 d^3}-\frac{(720 c) \operatorname{Subst}\left (\int x \cos (a+b x) \, dx,x,\sqrt [3]{c+d x}\right )}{b^4 d^3}\\ &=\frac{6 c^2 \sqrt [3]{c+d x} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^3}+\frac{360 c (c+d x)^{2/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^4 d^3}-\frac{30 c (c+d x)^{4/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^3}-\frac{1008 (c+d x)^{5/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^4 d^3}+\frac{24 (c+d x)^{7/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^3}-\frac{6 c^2 \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d^3}-\frac{720 c \sqrt [3]{c+d x} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^5 d^3}+\frac{3 c^2 (c+d x)^{2/3} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b d^3}+\frac{120 c (c+d x) \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d^3}+\frac{5040 (c+d x)^{4/3} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^5 d^3}-\frac{6 c (c+d x)^{5/3} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b d^3}-\frac{168 (c+d x)^2 \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d^3}+\frac{3 (c+d x)^{8/3} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b d^3}-\frac{20160 \operatorname{Subst}\left (\int x^3 \sin (a+b x) \, dx,x,\sqrt [3]{c+d x}\right )}{b^5 d^3}+\frac{(720 c) \operatorname{Subst}\left (\int \sin (a+b x) \, dx,x,\sqrt [3]{c+d x}\right )}{b^5 d^3}\\ &=-\frac{720 c \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^6 d^3}+\frac{6 c^2 \sqrt [3]{c+d x} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^3}+\frac{360 c (c+d x)^{2/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^4 d^3}+\frac{20160 (c+d x) \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^6 d^3}-\frac{30 c (c+d x)^{4/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^3}-\frac{1008 (c+d x)^{5/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^4 d^3}+\frac{24 (c+d x)^{7/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^3}-\frac{6 c^2 \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d^3}-\frac{720 c \sqrt [3]{c+d x} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^5 d^3}+\frac{3 c^2 (c+d x)^{2/3} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b d^3}+\frac{120 c (c+d x) \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d^3}+\frac{5040 (c+d x)^{4/3} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^5 d^3}-\frac{6 c (c+d x)^{5/3} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b d^3}-\frac{168 (c+d x)^2 \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d^3}+\frac{3 (c+d x)^{8/3} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b d^3}-\frac{60480 \operatorname{Subst}\left (\int x^2 \cos (a+b x) \, dx,x,\sqrt [3]{c+d x}\right )}{b^6 d^3}\\ &=-\frac{720 c \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^6 d^3}+\frac{6 c^2 \sqrt [3]{c+d x} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^3}+\frac{360 c (c+d x)^{2/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^4 d^3}+\frac{20160 (c+d x) \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^6 d^3}-\frac{30 c (c+d x)^{4/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^3}-\frac{1008 (c+d x)^{5/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^4 d^3}+\frac{24 (c+d x)^{7/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^3}-\frac{6 c^2 \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d^3}-\frac{720 c \sqrt [3]{c+d x} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^5 d^3}-\frac{60480 (c+d x)^{2/3} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^7 d^3}+\frac{3 c^2 (c+d x)^{2/3} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b d^3}+\frac{120 c (c+d x) \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d^3}+\frac{5040 (c+d x)^{4/3} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^5 d^3}-\frac{6 c (c+d x)^{5/3} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b d^3}-\frac{168 (c+d x)^2 \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d^3}+\frac{3 (c+d x)^{8/3} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b d^3}+\frac{120960 \operatorname{Subst}\left (\int x \sin (a+b x) \, dx,x,\sqrt [3]{c+d x}\right )}{b^7 d^3}\\ &=-\frac{720 c \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^6 d^3}-\frac{120960 \sqrt [3]{c+d x} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^8 d^3}+\frac{6 c^2 \sqrt [3]{c+d x} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^3}+\frac{360 c (c+d x)^{2/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^4 d^3}+\frac{20160 (c+d x) \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^6 d^3}-\frac{30 c (c+d x)^{4/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^3}-\frac{1008 (c+d x)^{5/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^4 d^3}+\frac{24 (c+d x)^{7/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^3}-\frac{6 c^2 \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d^3}-\frac{720 c \sqrt [3]{c+d x} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^5 d^3}-\frac{60480 (c+d x)^{2/3} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^7 d^3}+\frac{3 c^2 (c+d x)^{2/3} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b d^3}+\frac{120 c (c+d x) \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d^3}+\frac{5040 (c+d x)^{4/3} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^5 d^3}-\frac{6 c (c+d x)^{5/3} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b d^3}-\frac{168 (c+d x)^2 \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d^3}+\frac{3 (c+d x)^{8/3} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b d^3}+\frac{120960 \operatorname{Subst}\left (\int \cos (a+b x) \, dx,x,\sqrt [3]{c+d x}\right )}{b^8 d^3}\\ &=-\frac{720 c \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^6 d^3}-\frac{120960 \sqrt [3]{c+d x} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^8 d^3}+\frac{6 c^2 \sqrt [3]{c+d x} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^3}+\frac{360 c (c+d x)^{2/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^4 d^3}+\frac{20160 (c+d x) \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^6 d^3}-\frac{30 c (c+d x)^{4/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^3}-\frac{1008 (c+d x)^{5/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^4 d^3}+\frac{24 (c+d x)^{7/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^3}+\frac{120960 \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^9 d^3}-\frac{6 c^2 \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d^3}-\frac{720 c \sqrt [3]{c+d x} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^5 d^3}-\frac{60480 (c+d x)^{2/3} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^7 d^3}+\frac{3 c^2 (c+d x)^{2/3} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b d^3}+\frac{120 c (c+d x) \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d^3}+\frac{5040 (c+d x)^{4/3} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^5 d^3}-\frac{6 c (c+d x)^{5/3} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b d^3}-\frac{168 (c+d x)^2 \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d^3}+\frac{3 (c+d x)^{8/3} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b d^3}\\ \end{align*}

Mathematica [C]  time = 1.10729, size = 382, normalized size = 0.71 \[ \frac{3 e^{-i \left (a+b \sqrt [3]{c+d x}\right )} \left (2 i b^6 \left (9 c^2+36 c d x+28 d^2 x^2\right ) \left (-1+e^{2 i \left (a+b \sqrt [3]{c+d x}\right )}\right )-i b^8 d^2 x^2 (c+d x)^{2/3} \left (-1+e^{2 i \left (a+b \sqrt [3]{c+d x}\right )}\right )+2 b^7 d x \sqrt [3]{c+d x} (3 c+4 d x) \left (1+e^{2 i \left (a+b \sqrt [3]{c+d x}\right )}\right )-24 b^5 (c+d x)^{2/3} (9 c+14 d x) \left (1+e^{2 i \left (a+b \sqrt [3]{c+d x}\right )}\right )-240 i b^4 \sqrt [3]{c+d x} (6 c+7 d x) \left (-1+e^{2 i \left (a+b \sqrt [3]{c+d x}\right )}\right )+240 b^3 (27 c+28 d x) \left (1+e^{2 i \left (a+b \sqrt [3]{c+d x}\right )}\right )+20160 i b^2 (c+d x)^{2/3} \left (-1+e^{2 i \left (a+b \sqrt [3]{c+d x}\right )}\right )-40320 b \sqrt [3]{c+d x} \left (1+e^{2 i \left (a+b \sqrt [3]{c+d x}\right )}\right )-40320 i \left (-1+e^{2 i \left (a+b \sqrt [3]{c+d x}\right )}\right )\right )}{2 b^9 d^3} \]

Antiderivative was successfully verified.

[In]

Integrate[x^2*Cos[a + b*(c + d*x)^(1/3)],x]

[Out]

(3*((-40320*I)*(-1 + E^((2*I)*(a + b*(c + d*x)^(1/3)))) - 40320*b*(1 + E^((2*I)*(a + b*(c + d*x)^(1/3))))*(c +
d*x)^(1/3) + (20160*I)*b^2*(-1 + E^((2*I)*(a + b*(c + d*x)^(1/3))))*(c + d*x)^(2/3) - I*b^8*d^2*(-1 + E^((2*I
)*(a + b*(c + d*x)^(1/3))))*x^2*(c + d*x)^(2/3) + 2*b^7*d*(1 + E^((2*I)*(a + b*(c + d*x)^(1/3))))*x*(c + d*x)^
(1/3)*(3*c + 4*d*x) - (240*I)*b^4*(-1 + E^((2*I)*(a + b*(c + d*x)^(1/3))))*(c + d*x)^(1/3)*(6*c + 7*d*x) - 24*
b^5*(1 + E^((2*I)*(a + b*(c + d*x)^(1/3))))*(c + d*x)^(2/3)*(9*c + 14*d*x) + 240*b^3*(1 + E^((2*I)*(a + b*(c +
d*x)^(1/3))))*(27*c + 28*d*x) + (2*I)*b^6*(-1 + E^((2*I)*(a + b*(c + d*x)^(1/3))))*(9*c^2 + 36*c*d*x + 28*d^2
*x^2)))/(2*b^9*d^3*E^(I*(a + b*(c + d*x)^(1/3))))

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Maple [B]  time = 0.102, size = 1809, normalized size = 3.4 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*cos(a+b*(d*x+c)^(1/3)),x)

[Out]

3/d^3/b^3*(c^2*((a+b*(d*x+c)^(1/3))^2*sin(a+b*(d*x+c)^(1/3))-2*sin(a+b*(d*x+c)^(1/3))+2*(a+b*(d*x+c)^(1/3))*co
s(a+b*(d*x+c)^(1/3)))-2*a*c^2*(cos(a+b*(d*x+c)^(1/3))+(a+b*(d*x+c)^(1/3))*sin(a+b*(d*x+c)^(1/3)))+c^2*a^2*sin(
a+b*(d*x+c)^(1/3))-2/b^3*c*((a+b*(d*x+c)^(1/3))^5*sin(a+b*(d*x+c)^(1/3))+5*(a+b*(d*x+c)^(1/3))^4*cos(a+b*(d*x+
c)^(1/3))-20*(a+b*(d*x+c)^(1/3))^3*sin(a+b*(d*x+c)^(1/3))-60*(a+b*(d*x+c)^(1/3))^2*cos(a+b*(d*x+c)^(1/3))+120*
cos(a+b*(d*x+c)^(1/3))+120*(a+b*(d*x+c)^(1/3))*sin(a+b*(d*x+c)^(1/3)))+10/b^3*a*c*((a+b*(d*x+c)^(1/3))^4*sin(a
+b*(d*x+c)^(1/3))+4*(a+b*(d*x+c)^(1/3))^3*cos(a+b*(d*x+c)^(1/3))-12*(a+b*(d*x+c)^(1/3))^2*sin(a+b*(d*x+c)^(1/3
))+24*sin(a+b*(d*x+c)^(1/3))-24*(a+b*(d*x+c)^(1/3))*cos(a+b*(d*x+c)^(1/3)))-20/b^3*a^2*c*((a+b*(d*x+c)^(1/3))^
3*sin(a+b*(d*x+c)^(1/3))+3*(a+b*(d*x+c)^(1/3))^2*cos(a+b*(d*x+c)^(1/3))-6*cos(a+b*(d*x+c)^(1/3))-6*(a+b*(d*x+c
)^(1/3))*sin(a+b*(d*x+c)^(1/3)))+20/b^3*a^3*c*((a+b*(d*x+c)^(1/3))^2*sin(a+b*(d*x+c)^(1/3))-2*sin(a+b*(d*x+c)^
(1/3))+2*(a+b*(d*x+c)^(1/3))*cos(a+b*(d*x+c)^(1/3)))-10/b^3*a^4*c*(cos(a+b*(d*x+c)^(1/3))+(a+b*(d*x+c)^(1/3))*
sin(a+b*(d*x+c)^(1/3)))+2/b^3*a^5*c*sin(a+b*(d*x+c)^(1/3))+1/b^6*((a+b*(d*x+c)^(1/3))^8*sin(a+b*(d*x+c)^(1/3))
+8*(a+b*(d*x+c)^(1/3))^7*cos(a+b*(d*x+c)^(1/3))-56*(a+b*(d*x+c)^(1/3))^6*sin(a+b*(d*x+c)^(1/3))-336*(a+b*(d*x+
c)^(1/3))^5*cos(a+b*(d*x+c)^(1/3))+1680*(a+b*(d*x+c)^(1/3))^4*sin(a+b*(d*x+c)^(1/3))+6720*(a+b*(d*x+c)^(1/3))^
3*cos(a+b*(d*x+c)^(1/3))-20160*(a+b*(d*x+c)^(1/3))^2*sin(a+b*(d*x+c)^(1/3))+40320*sin(a+b*(d*x+c)^(1/3))-40320
*(a+b*(d*x+c)^(1/3))*cos(a+b*(d*x+c)^(1/3)))-8/b^6*a*((a+b*(d*x+c)^(1/3))^7*sin(a+b*(d*x+c)^(1/3))+7*(a+b*(d*x
+c)^(1/3))^6*cos(a+b*(d*x+c)^(1/3))-42*(a+b*(d*x+c)^(1/3))^5*sin(a+b*(d*x+c)^(1/3))-210*(a+b*(d*x+c)^(1/3))^4*
cos(a+b*(d*x+c)^(1/3))+840*(a+b*(d*x+c)^(1/3))^3*sin(a+b*(d*x+c)^(1/3))+2520*(a+b*(d*x+c)^(1/3))^2*cos(a+b*(d*
x+c)^(1/3))-5040*cos(a+b*(d*x+c)^(1/3))-5040*(a+b*(d*x+c)^(1/3))*sin(a+b*(d*x+c)^(1/3)))+28/b^6*a^2*((a+b*(d*x
+c)^(1/3))^6*sin(a+b*(d*x+c)^(1/3))+6*(a+b*(d*x+c)^(1/3))^5*cos(a+b*(d*x+c)^(1/3))-30*(a+b*(d*x+c)^(1/3))^4*si
n(a+b*(d*x+c)^(1/3))-120*(a+b*(d*x+c)^(1/3))^3*cos(a+b*(d*x+c)^(1/3))+360*(a+b*(d*x+c)^(1/3))^2*sin(a+b*(d*x+c
)^(1/3))-720*sin(a+b*(d*x+c)^(1/3))+720*(a+b*(d*x+c)^(1/3))*cos(a+b*(d*x+c)^(1/3)))-56/b^6*a^3*((a+b*(d*x+c)^(
1/3))^5*sin(a+b*(d*x+c)^(1/3))+5*(a+b*(d*x+c)^(1/3))^4*cos(a+b*(d*x+c)^(1/3))-20*(a+b*(d*x+c)^(1/3))^3*sin(a+b
*(d*x+c)^(1/3))-60*(a+b*(d*x+c)^(1/3))^2*cos(a+b*(d*x+c)^(1/3))+120*cos(a+b*(d*x+c)^(1/3))+120*(a+b*(d*x+c)^(1
/3))*sin(a+b*(d*x+c)^(1/3)))+70/b^6*a^4*((a+b*(d*x+c)^(1/3))^4*sin(a+b*(d*x+c)^(1/3))+4*(a+b*(d*x+c)^(1/3))^3*
cos(a+b*(d*x+c)^(1/3))-12*(a+b*(d*x+c)^(1/3))^2*sin(a+b*(d*x+c)^(1/3))+24*sin(a+b*(d*x+c)^(1/3))-24*(a+b*(d*x+
c)^(1/3))*cos(a+b*(d*x+c)^(1/3)))-56/b^6*a^5*((a+b*(d*x+c)^(1/3))^3*sin(a+b*(d*x+c)^(1/3))+3*(a+b*(d*x+c)^(1/3
))^2*cos(a+b*(d*x+c)^(1/3))-6*cos(a+b*(d*x+c)^(1/3))-6*(a+b*(d*x+c)^(1/3))*sin(a+b*(d*x+c)^(1/3)))+28/b^6*a^6*
((a+b*(d*x+c)^(1/3))^2*sin(a+b*(d*x+c)^(1/3))-2*sin(a+b*(d*x+c)^(1/3))+2*(a+b*(d*x+c)^(1/3))*cos(a+b*(d*x+c)^(
1/3)))-8/b^6*a^7*(cos(a+b*(d*x+c)^(1/3))+(a+b*(d*x+c)^(1/3))*sin(a+b*(d*x+c)^(1/3)))+1/b^6*a^8*sin(a+b*(d*x+c)
^(1/3)))

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Maxima [B]  time = 1.47874, size = 1821, normalized size = 3.39 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3*(a^2*c^2*sin((d*x + c)^(1/3)*b + a) - 2*(((d*x + c)^(1/3)*b + a)*sin((d*x + c)^(1/3)*b + a) + cos((d*x + c)^
(1/3)*b + a))*a*c^2 + 2*a^5*c*sin((d*x + c)^(1/3)*b + a)/b^3 - 10*(((d*x + c)^(1/3)*b + a)*sin((d*x + c)^(1/3)
*b + a) + cos((d*x + c)^(1/3)*b + a))*a^4*c/b^3 + (2*((d*x + c)^(1/3)*b + a)*cos((d*x + c)^(1/3)*b + a) + (((d
*x + c)^(1/3)*b + a)^2 - 2)*sin((d*x + c)^(1/3)*b + a))*c^2 + a^8*sin((d*x + c)^(1/3)*b + a)/b^6 - 8*(((d*x +
c)^(1/3)*b + a)*sin((d*x + c)^(1/3)*b + a) + cos((d*x + c)^(1/3)*b + a))*a^7/b^6 + 20*(2*((d*x + c)^(1/3)*b +
a)*cos((d*x + c)^(1/3)*b + a) + (((d*x + c)^(1/3)*b + a)^2 - 2)*sin((d*x + c)^(1/3)*b + a))*a^3*c/b^3 + 28*(2*
((d*x + c)^(1/3)*b + a)*cos((d*x + c)^(1/3)*b + a) + (((d*x + c)^(1/3)*b + a)^2 - 2)*sin((d*x + c)^(1/3)*b + a
))*a^6/b^6 - 20*(3*(((d*x + c)^(1/3)*b + a)^2 - 2)*cos((d*x + c)^(1/3)*b + a) + (((d*x + c)^(1/3)*b + a)^3 - 6
*(d*x + c)^(1/3)*b - 6*a)*sin((d*x + c)^(1/3)*b + a))*a^2*c/b^3 - 56*(3*(((d*x + c)^(1/3)*b + a)^2 - 2)*cos((d
*x + c)^(1/3)*b + a) + (((d*x + c)^(1/3)*b + a)^3 - 6*(d*x + c)^(1/3)*b - 6*a)*sin((d*x + c)^(1/3)*b + a))*a^5
/b^6 + 10*(4*(((d*x + c)^(1/3)*b + a)^3 - 6*(d*x + c)^(1/3)*b - 6*a)*cos((d*x + c)^(1/3)*b + a) + (((d*x + c)^
(1/3)*b + a)^4 - 12*((d*x + c)^(1/3)*b + a)^2 + 24)*sin((d*x + c)^(1/3)*b + a))*a*c/b^3 + 70*(4*(((d*x + c)^(1
/3)*b + a)^3 - 6*(d*x + c)^(1/3)*b - 6*a)*cos((d*x + c)^(1/3)*b + a) + (((d*x + c)^(1/3)*b + a)^4 - 12*((d*x +
c)^(1/3)*b + a)^2 + 24)*sin((d*x + c)^(1/3)*b + a))*a^4/b^6 - 2*(5*(((d*x + c)^(1/3)*b + a)^4 - 12*((d*x + c)
^(1/3)*b + a)^2 + 24)*cos((d*x + c)^(1/3)*b + a) + (((d*x + c)^(1/3)*b + a)^5 - 20*((d*x + c)^(1/3)*b + a)^3 +
120*(d*x + c)^(1/3)*b + 120*a)*sin((d*x + c)^(1/3)*b + a))*c/b^3 - 56*(5*(((d*x + c)^(1/3)*b + a)^4 - 12*((d*
x + c)^(1/3)*b + a)^2 + 24)*cos((d*x + c)^(1/3)*b + a) + (((d*x + c)^(1/3)*b + a)^5 - 20*((d*x + c)^(1/3)*b +
a)^3 + 120*(d*x + c)^(1/3)*b + 120*a)*sin((d*x + c)^(1/3)*b + a))*a^3/b^6 + 28*(6*(((d*x + c)^(1/3)*b + a)^5 -
20*((d*x + c)^(1/3)*b + a)^3 + 120*(d*x + c)^(1/3)*b + 120*a)*cos((d*x + c)^(1/3)*b + a) + (((d*x + c)^(1/3)*
b + a)^6 - 30*((d*x + c)^(1/3)*b + a)^4 + 360*((d*x + c)^(1/3)*b + a)^2 - 720)*sin((d*x + c)^(1/3)*b + a))*a^2
/b^6 - 8*(7*(((d*x + c)^(1/3)*b + a)^6 - 30*((d*x + c)^(1/3)*b + a)^4 + 360*((d*x + c)^(1/3)*b + a)^2 - 720)*c
os((d*x + c)^(1/3)*b + a) + (((d*x + c)^(1/3)*b + a)^7 - 42*((d*x + c)^(1/3)*b + a)^5 + 840*((d*x + c)^(1/3)*b
+ a)^3 - 5040*(d*x + c)^(1/3)*b - 5040*a)*sin((d*x + c)^(1/3)*b + a))*a/b^6 + (8*(((d*x + c)^(1/3)*b + a)^7 -
42*((d*x + c)^(1/3)*b + a)^5 + 840*((d*x + c)^(1/3)*b + a)^3 - 5040*(d*x + c)^(1/3)*b - 5040*a)*cos((d*x + c)
^(1/3)*b + a) + (((d*x + c)^(1/3)*b + a)^8 - 56*((d*x + c)^(1/3)*b + a)^6 + 1680*((d*x + c)^(1/3)*b + a)^4 - 2
0160*((d*x + c)^(1/3)*b + a)^2 + 40320)*sin((d*x + c)^(1/3)*b + a))/b^6)/(b^3*d^3)

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Fricas [A]  time = 1.70366, size = 464, normalized size = 0.86 \begin{align*} \frac{3 \,{\left (2 \,{\left (3360 \, b^{3} d x + 3240 \, b^{3} c - 12 \,{\left (14 \, b^{5} d x + 9 \, b^{5} c\right )}{\left (d x + c\right )}^{\frac{2}{3}} +{\left (4 \, b^{7} d^{2} x^{2} + 3 \, b^{7} c d x - 20160 \, b\right )}{\left (d x + c\right )}^{\frac{1}{3}}\right )} \cos \left ({\left (d x + c\right )}^{\frac{1}{3}} b + a\right ) -{\left (56 \, b^{6} d^{2} x^{2} + 72 \, b^{6} c d x + 18 \, b^{6} c^{2} -{\left (b^{8} d^{2} x^{2} - 20160 \, b^{2}\right )}{\left (d x + c\right )}^{\frac{2}{3}} - 240 \,{\left (7 \, b^{4} d x + 6 \, b^{4} c\right )}{\left (d x + c\right )}^{\frac{1}{3}} - 40320\right )} \sin \left ({\left (d x + c\right )}^{\frac{1}{3}} b + a\right )\right )}}{b^{9} d^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3*(2*(3360*b^3*d*x + 3240*b^3*c - 12*(14*b^5*d*x + 9*b^5*c)*(d*x + c)^(2/3) + (4*b^7*d^2*x^2 + 3*b^7*c*d*x - 2
0160*b)*(d*x + c)^(1/3))*cos((d*x + c)^(1/3)*b + a) - (56*b^6*d^2*x^2 + 72*b^6*c*d*x + 18*b^6*c^2 - (b^8*d^2*x
^2 - 20160*b^2)*(d*x + c)^(2/3) - 240*(7*b^4*d*x + 6*b^4*c)*(d*x + c)^(1/3) - 40320)*sin((d*x + c)^(1/3)*b + a
))/(b^9*d^3)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{2} \cos{\left (a + b \sqrt [3]{c + d x} \right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**2*cos(a+b*(d*x+c)**(1/3)),x)

[Out]

Integral(x**2*cos(a + b*(c + d*x)**(1/3)), x)

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Giac [B]  time = 1.61316, size = 1490, normalized size = 2.77 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3*(2*(((d*x + c)^(1/3)*b + a)*b^6*c^2 - a*b^6*c^2 - 5*((d*x + c)^(1/3)*b + a)^4*b^3*c + 20*((d*x + c)^(1/3)*b
+ a)^3*a*b^3*c - 30*((d*x + c)^(1/3)*b + a)^2*a^2*b^3*c + 20*((d*x + c)^(1/3)*b + a)*a^3*b^3*c - 5*a^4*b^3*c +
4*((d*x + c)^(1/3)*b + a)^7 - 28*((d*x + c)^(1/3)*b + a)^6*a + 84*((d*x + c)^(1/3)*b + a)^5*a^2 - 140*((d*x +
c)^(1/3)*b + a)^4*a^3 + 140*((d*x + c)^(1/3)*b + a)^3*a^4 - 84*((d*x + c)^(1/3)*b + a)^2*a^5 + 28*((d*x + c)^
(1/3)*b + a)*a^6 - 4*a^7 + 60*((d*x + c)^(1/3)*b + a)^2*b^3*c - 120*((d*x + c)^(1/3)*b + a)*a*b^3*c + 60*a^2*b
^3*c - 168*((d*x + c)^(1/3)*b + a)^5 + 840*((d*x + c)^(1/3)*b + a)^4*a - 1680*((d*x + c)^(1/3)*b + a)^3*a^2 +
1680*((d*x + c)^(1/3)*b + a)^2*a^3 - 840*((d*x + c)^(1/3)*b + a)*a^4 + 168*a^5 - 120*b^3*c + 3360*((d*x + c)^(
1/3)*b + a)^3 - 10080*((d*x + c)^(1/3)*b + a)^2*a + 10080*((d*x + c)^(1/3)*b + a)*a^2 - 3360*a^3 - 20160*(d*x
+ c)^(1/3)*b)*cos((d*x + c)^(1/3)*b + a)/(b^8*d^2) + (((d*x + c)^(1/3)*b + a)^2*b^6*c^2 - 2*((d*x + c)^(1/3)*b
+ a)*a*b^6*c^2 + a^2*b^6*c^2 - 2*((d*x + c)^(1/3)*b + a)^5*b^3*c + 10*((d*x + c)^(1/3)*b + a)^4*a*b^3*c - 20*
((d*x + c)^(1/3)*b + a)^3*a^2*b^3*c + 20*((d*x + c)^(1/3)*b + a)^2*a^3*b^3*c - 10*((d*x + c)^(1/3)*b + a)*a^4*
b^3*c + 2*a^5*b^3*c + ((d*x + c)^(1/3)*b + a)^8 - 8*((d*x + c)^(1/3)*b + a)^7*a + 28*((d*x + c)^(1/3)*b + a)^6
*a^2 - 56*((d*x + c)^(1/3)*b + a)^5*a^3 + 70*((d*x + c)^(1/3)*b + a)^4*a^4 - 56*((d*x + c)^(1/3)*b + a)^3*a^5
+ 28*((d*x + c)^(1/3)*b + a)^2*a^6 - 8*((d*x + c)^(1/3)*b + a)*a^7 + a^8 - 2*b^6*c^2 + 40*((d*x + c)^(1/3)*b +
a)^3*b^3*c - 120*((d*x + c)^(1/3)*b + a)^2*a*b^3*c + 120*((d*x + c)^(1/3)*b + a)*a^2*b^3*c - 40*a^3*b^3*c - 5
6*((d*x + c)^(1/3)*b + a)^6 + 336*((d*x + c)^(1/3)*b + a)^5*a - 840*((d*x + c)^(1/3)*b + a)^4*a^2 + 1120*((d*x
+ c)^(1/3)*b + a)^3*a^3 - 840*((d*x + c)^(1/3)*b + a)^2*a^4 + 336*((d*x + c)^(1/3)*b + a)*a^5 - 56*a^6 - 240*
((d*x + c)^(1/3)*b + a)*b^3*c + 240*a*b^3*c + 1680*((d*x + c)^(1/3)*b + a)^4 - 6720*((d*x + c)^(1/3)*b + a)^3*
a + 10080*((d*x + c)^(1/3)*b + a)^2*a^2 - 6720*((d*x + c)^(1/3)*b + a)*a^3 + 1680*a^4 - 20160*((d*x + c)^(1/3)
*b + a)^2 + 40320*((d*x + c)^(1/3)*b + a)*a - 20160*a^2 + 40320)*sin((d*x + c)^(1/3)*b + a)/(b^8*d^2))/(b*d)