\(\int x \cos (a+b \sqrt [3]{c+d x}) \, dx\) [96]

Optimal result

Integrand size = 16, antiderivative size = 261 \[ \int x \cos \left (a+b \sqrt [3]{c+d x}\right ) \, dx=\frac {360 \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^6 d^2}-\frac {6 c \sqrt [3]{c+d x} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^2}-\frac {180 (c+d x)^{2/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^4 d^2}+\frac {15 (c+d x)^{4/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^2}+\frac {6 c \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d^2}+\frac {360 \sqrt [3]{c+d x} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^5 d^2}-\frac {3 c (c+d x)^{2/3} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b d^2}-\frac {60 (c+d x) \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d^2}+\frac {3 (c+d x)^{5/3} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b d^2} \]

[Out]

360*cos(a+b*(d*x+c)^(1/3))/b^6/d^2-6*c*(d*x+c)^(1/3)*cos(a+b*(d*x+c)^(1/3))/b^2/d^2-180*(d*x+c)^(2/3)*cos(a+b*
(d*x+c)^(1/3))/b^4/d^2+15*(d*x+c)^(4/3)*cos(a+b*(d*x+c)^(1/3))/b^2/d^2+6*c*sin(a+b*(d*x+c)^(1/3))/b^3/d^2+360*
(d*x+c)^(1/3)*sin(a+b*(d*x+c)^(1/3))/b^5/d^2-3*c*(d*x+c)^(2/3)*sin(a+b*(d*x+c)^(1/3))/b/d^2-60*(d*x+c)*sin(a+b
*(d*x+c)^(1/3))/b^3/d^2+3*(d*x+c)^(5/3)*sin(a+b*(d*x+c)^(1/3))/b/d^2

Rubi [A] (verified)

Time = 0.29 (sec) , antiderivative size = 261, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3513, 3377, 2717, 2718} \[ \int x \cos \left (a+b \sqrt [3]{c+d x}\right ) \, dx=\frac {360 \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^6 d^2}+\frac {360 \sqrt [3]{c+d x} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^5 d^2}-\frac {180 (c+d x)^{2/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^4 d^2}-\frac {60 (c+d x) \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d^2}+\frac {6 c \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d^2}+\frac {15 (c+d x)^{4/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^2}-\frac {6 c \sqrt [3]{c+d x} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^2}+\frac {3 (c+d x)^{5/3} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b d^2}-\frac {3 c (c+d x)^{2/3} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b d^2} \]

[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 2717

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

Rule 2718

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

Rule 3377

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 3513

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]

Rubi steps \begin{align*} \text {integral}& = \frac {3 \text {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 \text {Subst}\left (\int x^5 \cos (a+b x) \, dx,x,\sqrt [3]{c+d x}\right )}{d^2}-\frac {(3 c) \text {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 \text {Subst}\left (\int x^4 \sin (a+b x) \, dx,x,\sqrt [3]{c+d x}\right )}{b d^2}+\frac {(6 c) \text {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 \text {Subst}\left (\int x^3 \cos (a+b x) \, dx,x,\sqrt [3]{c+d x}\right )}{b^2 d^2}+\frac {(6 c) \text {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 \text {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 \text {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 \text {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] (verified)

Time = 0.53 (sec) , antiderivative size = 117, normalized size of antiderivative = 0.45 \[ \int x \cos \left (a+b \sqrt [3]{c+d x}\right ) \, dx=\frac {3 \left (\left (120-60 b^2 (c+d x)^{2/3}+b^4 \sqrt [3]{c+d x} (3 c+5 d x)\right ) \cos \left (a+b \sqrt [3]{c+d x}\right )+b \left (120 \sqrt [3]{c+d x}+b^4 d x (c+d x)^{2/3}-2 b^2 (9 c+10 d x)\right ) \sin \left (a+b \sqrt [3]{c+d x}\right )\right )}{b^6 d^2} \]

[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)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(654\) vs. \(2(231)=462\).

Time = 2.10 (sec) , antiderivative size = 655, normalized size of antiderivative = 2.51

method	result	size
derivativedivides	\(\frac {-3 a^{2} c \sin \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )+6 a c \left (\cos \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )+\left (a +b \left (d x +c \right )^{\frac {1}{3}}\right ) \sin \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )\right )-3 c \left (\left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )^{2} \sin \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )-2 \sin \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )+2 \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right ) \cos \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )\right )-\frac {3 a^{5} \sin \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )}{b^{3}}+\frac {15 a^{4} \left (\cos \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )+\left (a +b \left (d x +c \right )^{\frac {1}{3}}\right ) \sin \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )\right )}{b^{3}}-\frac {30 a^{3} \left (\left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )^{2} \sin \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )-2 \sin \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )+2 \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right ) \cos \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )\right )}{b^{3}}+\frac {30 a^{2} \left (\left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )^{3} \sin \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )+3 \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )^{2} \cos \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )-6 \cos \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )-6 \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right ) \sin \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )\right )}{b^{3}}-\frac {15 a \left (\left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )^{4} \sin \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )+4 \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )^{3} \cos \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )-12 \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )^{2} \sin \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )+24 \sin \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )-24 \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right ) \cos \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )\right )}{b^{3}}+\frac {3 \left (\left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )^{5} \sin \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )+5 \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )^{4} \cos \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )-20 \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )^{3} \sin \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )-60 \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )^{2} \cos \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )+120 \cos \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )+120 \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right ) \sin \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )\right )}{b^{3}}}{d^{2} b^{3}}\)	\(655\)
default	\(\frac {-3 a^{2} c \sin \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )+6 a c \left (\cos \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )+\left (a +b \left (d x +c \right )^{\frac {1}{3}}\right ) \sin \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )\right )-3 c \left (\left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )^{2} \sin \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )-2 \sin \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )+2 \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right ) \cos \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )\right )-\frac {3 a^{5} \sin \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )}{b^{3}}+\frac {15 a^{4} \left (\cos \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )+\left (a +b \left (d x +c \right )^{\frac {1}{3}}\right ) \sin \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )\right )}{b^{3}}-\frac {30 a^{3} \left (\left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )^{2} \sin \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )-2 \sin \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )+2 \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right ) \cos \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )\right )}{b^{3}}+\frac {30 a^{2} \left (\left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )^{3} \sin \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )+3 \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )^{2} \cos \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )-6 \cos \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )-6 \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right ) \sin \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )\right )}{b^{3}}-\frac {15 a \left (\left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )^{4} \sin \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )+4 \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )^{3} \cos \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )-12 \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )^{2} \sin \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )+24 \sin \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )-24 \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right ) \cos \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )\right )}{b^{3}}+\frac {3 \left (\left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )^{5} \sin \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )+5 \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )^{4} \cos \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )-20 \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )^{3} \sin \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )-60 \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )^{2} \cos \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )+120 \cos \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )+120 \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right ) \sin \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )\right )}{b^{3}}}{d^{2} b^{3}}\)	\(655\)
parts	\(\text {Expression too large to display}\)	\(1213\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.42 \[ \int x \cos \left (a+b \sqrt [3]{c+d x}\right ) \, dx=-\frac {3 \, {\left ({\left (60 \, {\left (d x + c\right )}^{\frac {2}{3}} b^{2} - {\left (5 \, b^{4} d x + 3 \, b^{4} c\right )} {\left (d x + c\right )}^{\frac {1}{3}} - 120\right )} \cos \left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right ) - {\left ({\left (d x + c\right )}^{\frac {2}{3}} b^{5} d x - 20 \, b^{3} d x - 18 \, b^{3} c + 120 \, {\left (d x + c\right )}^{\frac {1}{3}} b\right )} \sin \left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )\right )}}{b^{6} d^{2}} \]

[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)

Sympy [F]

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

[In]

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

[Out]

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

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 523 vs. \(2 (231) = 462\).

Time = 0.24 (sec) , antiderivative size = 523, normalized size of antiderivative = 2.00 \[ \int x \cos \left (a+b \sqrt [3]{c+d x}\right ) \, dx=-\frac {3 \, {\left (a^{2} c \sin \left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right ) - 2 \, {\left ({\left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )} \sin \left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right ) + \cos \left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )\right )} a c + \frac {a^{5} \sin \left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )}{b^{3}} - \frac {5 \, {\left ({\left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )} \sin \left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right ) + \cos \left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )\right )} a^{4}}{b^{3}} + {\left (2 \, {\left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )} \cos \left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right ) + {\left ({\left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )}^{2} - 2\right )} \sin \left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )\right )} c + \frac {10 \, {\left (2 \, {\left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )} \cos \left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right ) + {\left ({\left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )}^{2} - 2\right )} \sin \left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )\right )} a^{3}}{b^{3}} - \frac {10 \, {\left (3 \, {\left ({\left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )}^{2} - 2\right )} \cos \left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right ) + {\left ({\left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )}^{3} - 6 \, {\left (d x + c\right )}^{\frac {1}{3}} b - 6 \, a\right )} \sin \left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )\right )} a^{2}}{b^{3}} + \frac {5 \, {\left (4 \, {\left ({\left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )}^{3} - 6 \, {\left (d x + c\right )}^{\frac {1}{3}} b - 6 \, a\right )} \cos \left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right ) + {\left ({\left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )}^{4} - 12 \, {\left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )}^{2} + 24\right )} \sin \left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )\right )} a}{b^{3}} - \frac {5 \, {\left ({\left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )}^{4} - 12 \, {\left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )}^{2} + 24\right )} \cos \left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right ) + {\left ({\left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )}^{5} - 20 \, {\left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )}^{3} + 120 \, {\left (d x + c\right )}^{\frac {1}{3}} b + 120 \, a\right )} \sin \left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )}{b^{3}}\right )}}{b^{3} d^{2}} \]

[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)

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 370, normalized size of antiderivative = 1.42 \[ \int x \cos \left (a+b \sqrt [3]{c+d x}\right ) \, dx=-\frac {3 \, {\left (\frac {{\left (2 \, {\left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )} b^{3} c - 2 \, a b^{3} c - 5 \, {\left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )}^{4} + 20 \, {\left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )}^{3} a - 30 \, {\left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )}^{2} a^{2} + 20 \, {\left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )} a^{3} - 5 \, a^{4} + 60 \, {\left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )}^{2} - 120 \, {\left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )} a + 60 \, a^{2} - 120\right )} \cos \left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )}{b^{5}} + \frac {{\left ({\left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )}^{2} b^{3} c - 2 \, {\left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )} a b^{3} c + a^{2} b^{3} c - {\left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )}^{5} + 5 \, {\left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )}^{4} a - 10 \, {\left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )}^{3} a^{2} + 10 \, {\left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )}^{2} a^{3} - 5 \, {\left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )} a^{4} + a^{5} - 2 \, b^{3} c + 20 \, {\left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )}^{3} - 60 \, {\left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )}^{2} a + 60 \, {\left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )} a^{2} - 20 \, a^{3} - 120 \, {\left (d x + c\right )}^{\frac {1}{3}} b\right )} \sin \left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )}{b^{5}}\right )}}{b d^{2}} \]

[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)

Mupad [F(-1)]

Timed out. \[ \int x \cos \left (a+b \sqrt [3]{c+d x}\right ) \, dx=\int x\,\cos \left (a+b\,{\left (c+d\,x\right )}^{1/3}\right ) \,d x \]

[In]

int(x*cos(a + b*(c + d*x)^(1/3)),x)

[Out]

int(x*cos(a + b*(c + d*x)^(1/3)), x)