Integrand size = 16, antiderivative size = 261 \[ \int x \cos \left (a+b \sqrt [3]{c+d x}\right ) \, dx=\frac {360 \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^6 d^2}-\frac {6 c \sqrt [3]{c+d x} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^2}-\frac {180 (c+d x)^{2/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^4 d^2}+\frac {15 (c+d x)^{4/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d^2}+\frac {6 c \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d^2}+\frac {360 \sqrt [3]{c+d x} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^5 d^2}-\frac {3 c (c+d x)^{2/3} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b d^2}-\frac {60 (c+d x) \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d^2}+\frac {3 (c+d x)^{5/3} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b d^2} \] Output:
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
Time = 0.55 (sec) , antiderivative size = 117, normalized size of antiderivative = 0.45 \[ \int x \cos \left (a+b \sqrt [3]{c+d x}\right ) \, dx=\frac {3 \left (\left (120-60 b^2 (c+d x)^{2/3}+b^4 \sqrt [3]{c+d x} (3 c+5 d x)\right ) \cos \left (a+b \sqrt [3]{c+d x}\right )+b \left (120 \sqrt [3]{c+d x}+b^4 d x (c+d x)^{2/3}-2 b^2 (9 c+10 d x)\right ) \sin \left (a+b \sqrt [3]{c+d x}\right )\right )}{b^6 d^2} \] Input:
Integrate[x*Cos[a + b*(c + d*x)^(1/3)],x]
Output:
(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)
Time = 0.41 (sec) , antiderivative size = 265, normalized size of antiderivative = 1.02, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {3913, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int x \cos \left (a+b \sqrt [3]{c+d x}\right ) \, dx\) |
| \(\Big \downarrow \) 3913 |
| \(\displaystyle \frac {3 \int \left (\frac {(c+d x)^{5/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{d}-\frac {c (c+d x)^{2/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{d}\right )d\sqrt [3]{c+d x}}{d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {3 \left (\frac {120 \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^6 d}+\frac {120 \sqrt [3]{c+d x} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^5 d}-\frac {60 (c+d x)^{2/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^4 d}-\frac {20 (c+d x) \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d}+\frac {2 c \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d}+\frac {5 (c+d x)^{4/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d}-\frac {2 c \sqrt [3]{c+d x} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d}+\frac {(c+d x)^{5/3} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b d}-\frac {c (c+d x)^{2/3} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b d}\right )}{d}\) |
Input:
Int[x*Cos[a + b*(c + d*x)^(1/3)],x]
Output:
(3*((120*Cos[a + b*(c + d*x)^(1/3)])/(b^6*d) - (2*c*(c + d*x)^(1/3)*Cos[a + b*(c + d*x)^(1/3)])/(b^2*d) - (60*(c + d*x)^(2/3)*Cos[a + b*(c + d*x)^(1 /3)])/(b^4*d) + (5*(c + d*x)^(4/3)*Cos[a + b*(c + d*x)^(1/3)])/(b^2*d) + ( 2*c*Sin[a + b*(c + d*x)^(1/3)])/(b^3*d) + (120*(c + d*x)^(1/3)*Sin[a + b*( c + d*x)^(1/3)])/(b^5*d) - (c*(c + d*x)^(2/3)*Sin[a + b*(c + d*x)^(1/3)])/ (b*d) - (20*(c + d*x)*Sin[a + b*(c + d*x)^(1/3)])/(b^3*d) + ((c + d*x)^(5/ 3)*Sin[a + b*(c + d*x)^(1/3)])/(b*d)))/d
Int[((a_.) + Cos[(c_.) + (d_.)*((e_.) + (f_.)*(x_))^(n_)]*(b_.))^(p_.)*((g_ .) + (h_.)*(x_))^(m_.), x_Symbol] :> Simp[1/(n*f) Subst[Int[ExpandIntegra nd[(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]
Leaf count of result is larger than twice the leaf count of optimal. \(654\) vs. \(2(231)=462\).
Time = 1.73 (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 | \(\text {Expression too large to display}\) | \(1213\) |
Input:
int(x*cos(a+b*(d*x+c)^(1/3)),x,method=_RETURNVERBOSE)
Output:
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))))
Time = 0.09 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.42 \[ \int x \cos \left (a+b \sqrt [3]{c+d x}\right ) \, dx=-\frac {3 \, {\left ({\left (60 \, {\left (d x + c\right )}^{\frac {2}{3}} b^{2} - {\left (5 \, b^{4} d x + 3 \, b^{4} c\right )} {\left (d x + c\right )}^{\frac {1}{3}} - 120\right )} \cos \left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right ) - {\left ({\left (d x + c\right )}^{\frac {2}{3}} b^{5} d x - 20 \, b^{3} d x - 18 \, b^{3} c + 120 \, {\left (d x + c\right )}^{\frac {1}{3}} b\right )} \sin \left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )\right )}}{b^{6} d^{2}} \] Input:
integrate(x*cos(a+b*(d*x+c)^(1/3)),x, algorithm="fricas")
Output:
-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)
\[ \int x \cos \left (a+b \sqrt [3]{c+d x}\right ) \, dx=\int x \cos {\left (a + b \sqrt [3]{c + d x} \right )}\, dx \] Input:
integrate(x*cos(a+b*(d*x+c)**(1/3)),x)
Output:
Integral(x*cos(a + b*(c + d*x)**(1/3)), x)
Leaf count of result is larger than twice the leaf count of optimal. 523 vs. \(2 (231) = 462\).
Time = 0.05 (sec) , antiderivative size = 523, normalized size of antiderivative = 2.00 \[ \int x \cos \left (a+b \sqrt [3]{c+d x}\right ) \, dx =\text {Too large to display} \] Input:
integrate(x*cos(a+b*(d*x+c)^(1/3)),x, algorithm="maxima")
Output:
-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)
Time = 0.36 (sec) , antiderivative size = 370, normalized size of antiderivative = 1.42 \[ \int x \cos \left (a+b \sqrt [3]{c+d x}\right ) \, dx=-\frac {3 \, {\left (\frac {{\left (2 \, {\left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )} b^{3} c - 2 \, a b^{3} c - 5 \, {\left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )}^{4} + 20 \, {\left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )}^{3} a - 30 \, {\left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )}^{2} a^{2} + 20 \, {\left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )} a^{3} - 5 \, a^{4} + 60 \, {\left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )}^{2} - 120 \, {\left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )} a + 60 \, a^{2} - 120\right )} \cos \left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )}{b^{5}} + \frac {{\left ({\left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )}^{2} b^{3} c - 2 \, {\left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )} a b^{3} c + a^{2} b^{3} c - {\left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )}^{5} + 5 \, {\left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )}^{4} a - 10 \, {\left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )}^{3} a^{2} + 10 \, {\left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )}^{2} a^{3} - 5 \, {\left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )} a^{4} + a^{5} - 2 \, b^{3} c + 20 \, {\left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )}^{3} - 60 \, {\left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )}^{2} a + 60 \, {\left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )} a^{2} - 20 \, a^{3} - 120 \, {\left (d x + c\right )}^{\frac {1}{3}} b\right )} \sin \left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )}{b^{5}}\right )}}{b d^{2}} \] Input:
integrate(x*cos(a+b*(d*x+c)^(1/3)),x, algorithm="giac")
Output:
-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 - 1 20*((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 - 1 0*((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)
Timed out. \[ \int x \cos \left (a+b \sqrt [3]{c+d x}\right ) \, dx=\int x\,\cos \left (a+b\,{\left (c+d\,x\right )}^{1/3}\right ) \,d x \] Input:
int(x*cos(a + b*(c + d*x)^(1/3)),x)
Output:
int(x*cos(a + b*(c + d*x)^(1/3)), x)
Time = 0.19 (sec) , antiderivative size = 182, normalized size of antiderivative = 0.70 \[ \int x \cos \left (a+b \sqrt [3]{c+d x}\right ) \, dx=\frac {-180 \left (d x +c \right )^{\frac {2}{3}} \cos \left (\left (d x +c \right )^{\frac {1}{3}} b +a \right ) b^{2}+9 \left (d x +c \right )^{\frac {1}{3}} \cos \left (\left (d x +c \right )^{\frac {1}{3}} b +a \right ) b^{4} c +15 \left (d x +c \right )^{\frac {1}{3}} \cos \left (\left (d x +c \right )^{\frac {1}{3}} b +a \right ) b^{4} d x +360 \cos \left (\left (d x +c \right )^{\frac {1}{3}} b +a \right )+3 \left (d x +c \right )^{\frac {2}{3}} \sin \left (\left (d x +c \right )^{\frac {1}{3}} b +a \right ) b^{5} d x +360 \left (d x +c \right )^{\frac {1}{3}} \sin \left (\left (d x +c \right )^{\frac {1}{3}} b +a \right ) b -54 \sin \left (\left (d x +c \right )^{\frac {1}{3}} b +a \right ) b^{3} c -60 \sin \left (\left (d x +c \right )^{\frac {1}{3}} b +a \right ) b^{3} d x}{b^{6} d^{2}} \] Input:
int(x*cos(a+b*(d*x+c)^(1/3)),x)
Output:
(3*( - 60*(c + d*x)**(2/3)*cos((c + d*x)**(1/3)*b + a)*b**2 + 3*(c + d*x)* *(1/3)*cos((c + d*x)**(1/3)*b + a)*b**4*c + 5*(c + d*x)**(1/3)*cos((c + d* x)**(1/3)*b + a)*b**4*d*x + 120*cos((c + d*x)**(1/3)*b + a) + (c + d*x)**( 2/3)*sin((c + d*x)**(1/3)*b + a)*b**5*d*x + 120*(c + d*x)**(1/3)*sin((c + d*x)**(1/3)*b + a)*b - 18*sin((c + d*x)**(1/3)*b + a)*b**3*c - 20*sin((c + d*x)**(1/3)*b + a)*b**3*d*x))/(b**6*d**2)