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

Optimal. Leaf size=261 \[ -\frac{60 (c+d x) \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d^2}+\frac{360 \sqrt [3]{c+d x} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^5 d^2}+\frac{6 c \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d^2}+\frac{15 (c+d x)^{4/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^2}-\frac{180 (c+d x)^{2/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^4 d^2}-\frac{6 c \sqrt [3]{c+d x} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^2}+\frac{360 \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^6 d^2}+\frac{3 (c+d x)^{5/3} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b d^2}-\frac{3 c (c+d x)^{2/3} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b d^2} \]

[Out]

(360*Cos[a + b*(c + d*x)^(1/3)])/(b^6*d^2) - (6*c*(c + d*x)^(1/3)*Cos[a + b*(c + d*x)^(1/3)])/(b^2*d^2) - (180
*(c + d*x)^(2/3)*Cos[a + b*(c + d*x)^(1/3)])/(b^4*d^2) + (15*(c + d*x)^(4/3)*Cos[a + b*(c + d*x)^(1/3)])/(b^2*
d^2) + (6*c*Sin[a + b*(c + d*x)^(1/3)])/(b^3*d^2) + (360*(c + d*x)^(1/3)*Sin[a + b*(c + d*x)^(1/3)])/(b^5*d^2)
- (3*c*(c + d*x)^(2/3)*Sin[a + b*(c + d*x)^(1/3)])/(b*d^2) - (60*(c + d*x)*Sin[a + b*(c + d*x)^(1/3)])/(b^3*d
^2) + (3*(c + d*x)^(5/3)*Sin[a + b*(c + d*x)^(1/3)])/(b*d^2)

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Rubi [A]  time = 0.231053, antiderivative size = 261, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 4, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {3432, 3296, 2637, 2638} \[ -\frac{60 (c+d x) \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d^2}+\frac{360 \sqrt [3]{c+d x} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^5 d^2}+\frac{6 c \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d^2}+\frac{15 (c+d x)^{4/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^2}-\frac{180 (c+d x)^{2/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^4 d^2}-\frac{6 c \sqrt [3]{c+d x} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^2}+\frac{360 \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^6 d^2}+\frac{3 (c+d x)^{5/3} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b d^2}-\frac{3 c (c+d x)^{2/3} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b d^2} \]

Antiderivative was successfully verified.

[In]

Int[x*Cos[a + b*(c + d*x)^(1/3)],x]

[Out]

(360*Cos[a + b*(c + d*x)^(1/3)])/(b^6*d^2) - (6*c*(c + d*x)^(1/3)*Cos[a + b*(c + d*x)^(1/3)])/(b^2*d^2) - (180
*(c + d*x)^(2/3)*Cos[a + b*(c + d*x)^(1/3)])/(b^4*d^2) + (15*(c + d*x)^(4/3)*Cos[a + b*(c + d*x)^(1/3)])/(b^2*
d^2) + (6*c*Sin[a + b*(c + d*x)^(1/3)])/(b^3*d^2) + (360*(c + d*x)^(1/3)*Sin[a + b*(c + d*x)^(1/3)])/(b^5*d^2)
- (3*c*(c + d*x)^(2/3)*Sin[a + b*(c + d*x)^(1/3)])/(b*d^2) - (60*(c + d*x)*Sin[a + b*(c + d*x)^(1/3)])/(b^3*d
^2) + (3*(c + d*x)^(5/3)*Sin[a + b*(c + d*x)^(1/3)])/(b*d^2)

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 \cos \left (a+b \sqrt [3]{c+d x}\right ) \, dx &=\frac{3 \operatorname{Subst}\left (\int \left (-\frac{c x^2 \cos (a+b x)}{d}+\frac{x^5 \cos (a+b x)}{d}\right ) \, dx,x,\sqrt [3]{c+d x}\right )}{d}\\ &=\frac{3 \operatorname{Subst}\left (\int x^5 \cos (a+b x) \, dx,x,\sqrt [3]{c+d x}\right )}{d^2}-\frac{(3 c) \operatorname{Subst}\left (\int x^2 \cos (a+b x) \, dx,x,\sqrt [3]{c+d x}\right )}{d^2}\\ &=-\frac{3 c (c+d x)^{2/3} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b d^2}+\frac{3 (c+d x)^{5/3} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b d^2}-\frac{15 \operatorname{Subst}\left (\int x^4 \sin (a+b x) \, dx,x,\sqrt [3]{c+d x}\right )}{b d^2}+\frac{(6 c) \operatorname{Subst}\left (\int x \sin (a+b x) \, dx,x,\sqrt [3]{c+d x}\right )}{b d^2}\\ &=-\frac{6 c \sqrt [3]{c+d x} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^2}+\frac{15 (c+d x)^{4/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^2}-\frac{3 c (c+d x)^{2/3} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b d^2}+\frac{3 (c+d x)^{5/3} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b d^2}-\frac{60 \operatorname{Subst}\left (\int x^3 \cos (a+b x) \, dx,x,\sqrt [3]{c+d x}\right )}{b^2 d^2}+\frac{(6 c) \operatorname{Subst}\left (\int \cos (a+b x) \, dx,x,\sqrt [3]{c+d x}\right )}{b^2 d^2}\\ &=-\frac{6 c \sqrt [3]{c+d x} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^2}+\frac{15 (c+d x)^{4/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^2}+\frac{6 c \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d^2}-\frac{3 c (c+d x)^{2/3} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b d^2}-\frac{60 (c+d x) \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d^2}+\frac{3 (c+d x)^{5/3} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b d^2}+\frac{180 \operatorname{Subst}\left (\int x^2 \sin (a+b x) \, dx,x,\sqrt [3]{c+d x}\right )}{b^3 d^2}\\ &=-\frac{6 c \sqrt [3]{c+d x} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^2}-\frac{180 (c+d x)^{2/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^4 d^2}+\frac{15 (c+d x)^{4/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^2}+\frac{6 c \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d^2}-\frac{3 c (c+d x)^{2/3} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b d^2}-\frac{60 (c+d x) \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d^2}+\frac{3 (c+d x)^{5/3} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b d^2}+\frac{360 \operatorname{Subst}\left (\int x \cos (a+b x) \, dx,x,\sqrt [3]{c+d x}\right )}{b^4 d^2}\\ &=-\frac{6 c \sqrt [3]{c+d x} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^2}-\frac{180 (c+d x)^{2/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^4 d^2}+\frac{15 (c+d x)^{4/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^2}+\frac{6 c \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d^2}+\frac{360 \sqrt [3]{c+d x} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^5 d^2}-\frac{3 c (c+d x)^{2/3} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b d^2}-\frac{60 (c+d x) \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d^2}+\frac{3 (c+d x)^{5/3} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b d^2}-\frac{360 \operatorname{Subst}\left (\int \sin (a+b x) \, dx,x,\sqrt [3]{c+d x}\right )}{b^5 d^2}\\ &=\frac{360 \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^6 d^2}-\frac{6 c \sqrt [3]{c+d x} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^2}-\frac{180 (c+d x)^{2/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^4 d^2}+\frac{15 (c+d x)^{4/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^2}+\frac{6 c \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d^2}+\frac{360 \sqrt [3]{c+d x} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^5 d^2}-\frac{3 c (c+d x)^{2/3} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b d^2}-\frac{60 (c+d x) \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d^2}+\frac{3 (c+d x)^{5/3} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b d^2}\\ \end{align*}

Mathematica [A]  time = 0.417346, size = 117, normalized size = 0.45 \[ \frac{3 \left (b \left (b^4 d x (c+d x)^{2/3}-2 b^2 (9 c+10 d x)+120 \sqrt [3]{c+d x}\right ) \sin \left (a+b \sqrt [3]{c+d x}\right )+\left (b^4 \sqrt [3]{c+d x} (3 c+5 d x)-60 b^2 (c+d x)^{2/3}+120\right ) \cos \left (a+b \sqrt [3]{c+d x}\right )\right )}{b^6 d^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(3*((120 - 60*b^2*(c + d*x)^(2/3) + b^4*(c + d*x)^(1/3)*(3*c + 5*d*x))*Cos[a + b*(c + d*x)^(1/3)] + b*(120*(c
+ d*x)^(1/3) + b^4*d*x*(c + d*x)^(2/3) - 2*b^2*(9*c + 10*d*x))*Sin[a + b*(c + d*x)^(1/3)]))/(b^6*d^2)

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Maple [B]  time = 0.035, size = 655, normalized size = 2.5 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/d^2/b^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)))+2*c*a*(cos(a+b*(d*x+c)^(1/3))+(a+b*(d*x+c)^(1/3))*sin(a+b*(d*x+c)^(1/3)))-a^2*c*sin(a+b*(
d*x+c)^(1/3))+1/b^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)))-5/b^3*a*((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)))+10/b^3*a^2*((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)))-10/b^3*a^3*((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)))+5/b^3*a^4*(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^3*a^5*sin(a+b*(d*x+c)^(1/3)))

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Maxima [B]  time = 1.25364, size = 706, normalized size = 2.7 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3*(a^2*c*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 + a^5*sin((d*x + c)^(1/3)*b + a)/b^3 - 5*(((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/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 + 10*(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/b^3 - 10*(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/b^3 + 5*(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/b^3 - (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))/b^3)/(b^3*d^2)

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Fricas [A]  time = 1.63794, size = 293, normalized size = 1.12 \begin{align*} -\frac{3 \,{\left ({\left (60 \,{\left (d x + c\right )}^{\frac{2}{3}} b^{2} -{\left (5 \, b^{4} d x + 3 \, b^{4} c\right )}{\left (d x + c\right )}^{\frac{1}{3}} - 120\right )} \cos \left ({\left (d x + c\right )}^{\frac{1}{3}} b + a\right ) -{\left ({\left (d x + c\right )}^{\frac{2}{3}} b^{5} d x - 20 \, b^{3} d x - 18 \, b^{3} c + 120 \,{\left (d x + c\right )}^{\frac{1}{3}} b\right )} \sin \left ({\left (d x + c\right )}^{\frac{1}{3}} b + a\right )\right )}}{b^{6} d^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3*((60*(d*x + c)^(2/3)*b^2 - (5*b^4*d*x + 3*b^4*c)*(d*x + c)^(1/3) - 120)*cos((d*x + c)^(1/3)*b + a) - ((d*x
+ c)^(2/3)*b^5*d*x - 20*b^3*d*x - 18*b^3*c + 120*(d*x + c)^(1/3)*b)*sin((d*x + c)^(1/3)*b + a))/(b^6*d^2)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x \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*cos(a+b*(d*x+c)**(1/3)),x)

[Out]

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

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Giac [A]  time = 1.23773, size = 500, normalized size = 1.92 \begin{align*} -\frac{3 \,{\left (\frac{{\left (2 \,{\left ({\left (d x + c\right )}^{\frac{1}{3}} b + a\right )} b^{3} c - 2 \, a b^{3} c - 5 \,{\left ({\left (d x + c\right )}^{\frac{1}{3}} b + a\right )}^{4} + 20 \,{\left ({\left (d x + c\right )}^{\frac{1}{3}} b + a\right )}^{3} a - 30 \,{\left ({\left (d x + c\right )}^{\frac{1}{3}} b + a\right )}^{2} a^{2} + 20 \,{\left ({\left (d x + c\right )}^{\frac{1}{3}} b + a\right )} a^{3} - 5 \, a^{4} + 60 \,{\left ({\left (d x + c\right )}^{\frac{1}{3}} b + a\right )}^{2} - 120 \,{\left ({\left (d x + c\right )}^{\frac{1}{3}} b + a\right )} a + 60 \, a^{2} - 120\right )} \cos \left ({\left (d x + c\right )}^{\frac{1}{3}} b + a\right )}{b^{5}} + \frac{{\left ({\left ({\left (d x + c\right )}^{\frac{1}{3}} b + a\right )}^{2} b^{3} c - 2 \,{\left ({\left (d x + c\right )}^{\frac{1}{3}} b + a\right )} a b^{3} c + a^{2} b^{3} c -{\left ({\left (d x + c\right )}^{\frac{1}{3}} b + a\right )}^{5} + 5 \,{\left ({\left (d x + c\right )}^{\frac{1}{3}} b + a\right )}^{4} a - 10 \,{\left ({\left (d x + c\right )}^{\frac{1}{3}} b + a\right )}^{3} a^{2} + 10 \,{\left ({\left (d x + c\right )}^{\frac{1}{3}} b + a\right )}^{2} a^{3} - 5 \,{\left ({\left (d x + c\right )}^{\frac{1}{3}} b + a\right )} a^{4} + a^{5} - 2 \, b^{3} c + 20 \,{\left ({\left (d x + c\right )}^{\frac{1}{3}} b + a\right )}^{3} - 60 \,{\left ({\left (d x + c\right )}^{\frac{1}{3}} b + a\right )}^{2} a + 60 \,{\left ({\left (d x + c\right )}^{\frac{1}{3}} b + a\right )} a^{2} - 20 \, a^{3} - 120 \,{\left (d x + c\right )}^{\frac{1}{3}} b\right )} \sin \left ({\left (d x + c\right )}^{\frac{1}{3}} b + a\right )}{b^{5}}\right )}}{b d^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3*((2*((d*x + c)^(1/3)*b + a)*b^3*c - 2*a*b^3*c - 5*((d*x + c)^(1/3)*b + a)^4 + 20*((d*x + c)^(1/3)*b + a)^3*
a - 30*((d*x + c)^(1/3)*b + a)^2*a^2 + 20*((d*x + c)^(1/3)*b + a)*a^3 - 5*a^4 + 60*((d*x + c)^(1/3)*b + a)^2 -
120*((d*x + c)^(1/3)*b + a)*a + 60*a^2 - 120)*cos((d*x + c)^(1/3)*b + a)/b^5 + (((d*x + c)^(1/3)*b + a)^2*b^3
*c - 2*((d*x + c)^(1/3)*b + a)*a*b^3*c + a^2*b^3*c - ((d*x + c)^(1/3)*b + a)^5 + 5*((d*x + c)^(1/3)*b + a)^4*a
- 10*((d*x + c)^(1/3)*b + a)^3*a^2 + 10*((d*x + c)^(1/3)*b + a)^2*a^3 - 5*((d*x + c)^(1/3)*b + a)*a^4 + a^5 -
2*b^3*c + 20*((d*x + c)^(1/3)*b + a)^3 - 60*((d*x + c)^(1/3)*b + a)^2*a + 60*((d*x + c)^(1/3)*b + a)*a^2 - 20
*a^3 - 120*(d*x + c)^(1/3)*b)*sin((d*x + c)^(1/3)*b + a)/b^5)/(b*d^2)