Integrand size = 27, antiderivative size = 202 \[ \int (c e+d e x)^{2/3} \sin \left (a+b \sqrt [3]{c+d x}\right ) \, dx=\frac {36 (e (c+d x))^{2/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d}-\frac {72 (e (c+d x))^{2/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^5 d (c+d x)^{2/3}}-\frac {3 (c+d x)^{2/3} (e (c+d x))^{2/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b d}-\frac {72 (e (c+d x))^{2/3} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^4 d \sqrt [3]{c+d x}}+\frac {12 \sqrt [3]{c+d x} (e (c+d x))^{2/3} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d} \]
36*(e*(d*x+c))^(2/3)*cos(a+b*(d*x+c)^(1/3))/b^3/d-72*(e*(d*x+c))^(2/3)*cos (a+b*(d*x+c)^(1/3))/b^5/d/(d*x+c)^(2/3)-3*(d*x+c)^(2/3)*(e*(d*x+c))^(2/3)* cos(a+b*(d*x+c)^(1/3))/b/d-72*(e*(d*x+c))^(2/3)*sin(a+b*(d*x+c)^(1/3))/b^4 /d/(d*x+c)^(1/3)+12*(d*x+c)^(1/3)*(e*(d*x+c))^(2/3)*sin(a+b*(d*x+c)^(1/3)) /b^2/d
Time = 0.29 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.55 \[ \int (c e+d e x)^{2/3} \sin \left (a+b \sqrt [3]{c+d x}\right ) \, dx=-\frac {3 (e (c+d x))^{2/3} \left (\left (24-12 b^2 (c+d x)^{2/3}+b^4 (c+d x)^{4/3}\right ) \cos \left (a+b \sqrt [3]{c+d x}\right )-4 b \left (-6 \sqrt [3]{c+d x}+b^2 (c+d x)\right ) \sin \left (a+b \sqrt [3]{c+d x}\right )\right )}{b^5 d (c+d x)^{2/3}} \]
(-3*(e*(c + d*x))^(2/3)*((24 - 12*b^2*(c + d*x)^(2/3) + b^4*(c + d*x)^(4/3 ))*Cos[a + b*(c + d*x)^(1/3)] - 4*b*(-6*(c + d*x)^(1/3) + b^2*(c + d*x))*S in[a + b*(c + d*x)^(1/3)]))/(b^5*d*(c + d*x)^(2/3))
Time = 0.65 (sec) , antiderivative size = 168, normalized size of antiderivative = 0.83, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.519, Rules used = {3912, 30, 3042, 3777, 3042, 3777, 25, 3042, 3777, 3042, 3777, 25, 3042, 3118}
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.
| \(\displaystyle \int (c e+d e x)^{2/3} \sin \left (a+b \sqrt [3]{c+d x}\right ) \, dx\) |
| \(\Big \downarrow \) 3912 |
| \(\displaystyle \frac {3 \int (c+d x)^{2/3} (e (c+d x))^{2/3} \sin \left (a+b \sqrt [3]{c+d x}\right )d\sqrt [3]{c+d x}}{d}\) |
| \(\Big \downarrow \) 30 |
| \(\displaystyle \frac {3 (e (c+d x))^{2/3} \int (c+d x)^{4/3} \sin \left (a+b \sqrt [3]{c+d x}\right )d\sqrt [3]{c+d x}}{d (c+d x)^{2/3}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {3 (e (c+d x))^{2/3} \int (c+d x)^{4/3} \sin \left (a+b \sqrt [3]{c+d x}\right )d\sqrt [3]{c+d x}}{d (c+d x)^{2/3}}\) |
| \(\Big \downarrow \) 3777 |
| \(\displaystyle \frac {3 (e (c+d x))^{2/3} \left (\frac {4 \int (c+d x) \cos \left (a+b \sqrt [3]{c+d x}\right )d\sqrt [3]{c+d x}}{b}-\frac {(c+d x)^{4/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b}\right )}{d (c+d x)^{2/3}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {3 (e (c+d x))^{2/3} \left (\frac {4 \int (c+d x) \sin \left (a+b \sqrt [3]{c+d x}+\frac {\pi }{2}\right )d\sqrt [3]{c+d x}}{b}-\frac {(c+d x)^{4/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b}\right )}{d (c+d x)^{2/3}}\) |
| \(\Big \downarrow \) 3777 |
| \(\displaystyle \frac {3 (e (c+d x))^{2/3} \left (\frac {4 \left (\frac {3 \int -(c+d x)^{2/3} \sin \left (a+b \sqrt [3]{c+d x}\right )d\sqrt [3]{c+d x}}{b}+\frac {(c+d x) \sin \left (a+b \sqrt [3]{c+d x}\right )}{b}\right )}{b}-\frac {(c+d x)^{4/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b}\right )}{d (c+d x)^{2/3}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {3 (e (c+d x))^{2/3} \left (\frac {4 \left (\frac {(c+d x) \sin \left (a+b \sqrt [3]{c+d x}\right )}{b}-\frac {3 \int (c+d x)^{2/3} \sin \left (a+b \sqrt [3]{c+d x}\right )d\sqrt [3]{c+d x}}{b}\right )}{b}-\frac {(c+d x)^{4/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b}\right )}{d (c+d x)^{2/3}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {3 (e (c+d x))^{2/3} \left (\frac {4 \left (\frac {(c+d x) \sin \left (a+b \sqrt [3]{c+d x}\right )}{b}-\frac {3 \int (c+d x)^{2/3} \sin \left (a+b \sqrt [3]{c+d x}\right )d\sqrt [3]{c+d x}}{b}\right )}{b}-\frac {(c+d x)^{4/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b}\right )}{d (c+d x)^{2/3}}\) |
| \(\Big \downarrow \) 3777 |
| \(\displaystyle \frac {3 (e (c+d x))^{2/3} \left (\frac {4 \left (\frac {(c+d x) \sin \left (a+b \sqrt [3]{c+d x}\right )}{b}-\frac {3 \left (\frac {2 \int \sqrt [3]{c+d x} \cos \left (a+b \sqrt [3]{c+d x}\right )d\sqrt [3]{c+d x}}{b}-\frac {(c+d x)^{2/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b}\right )}{b}\right )}{b}-\frac {(c+d x)^{4/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b}\right )}{d (c+d x)^{2/3}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {3 (e (c+d x))^{2/3} \left (\frac {4 \left (\frac {(c+d x) \sin \left (a+b \sqrt [3]{c+d x}\right )}{b}-\frac {3 \left (\frac {2 \int \sqrt [3]{c+d x} \sin \left (a+b \sqrt [3]{c+d x}+\frac {\pi }{2}\right )d\sqrt [3]{c+d x}}{b}-\frac {(c+d x)^{2/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b}\right )}{b}\right )}{b}-\frac {(c+d x)^{4/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b}\right )}{d (c+d x)^{2/3}}\) |
| \(\Big \downarrow \) 3777 |
| \(\displaystyle \frac {3 (e (c+d x))^{2/3} \left (\frac {4 \left (\frac {(c+d x) \sin \left (a+b \sqrt [3]{c+d x}\right )}{b}-\frac {3 \left (\frac {2 \left (\frac {\int -\sin \left (a+b \sqrt [3]{c+d x}\right )d\sqrt [3]{c+d x}}{b}+\frac {\sqrt [3]{c+d x} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b}\right )}{b}-\frac {(c+d x)^{2/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b}\right )}{b}\right )}{b}-\frac {(c+d x)^{4/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b}\right )}{d (c+d x)^{2/3}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {3 (e (c+d x))^{2/3} \left (\frac {4 \left (\frac {(c+d x) \sin \left (a+b \sqrt [3]{c+d x}\right )}{b}-\frac {3 \left (\frac {2 \left (\frac {\sqrt [3]{c+d x} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b}-\frac {\int \sin \left (a+b \sqrt [3]{c+d x}\right )d\sqrt [3]{c+d x}}{b}\right )}{b}-\frac {(c+d x)^{2/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b}\right )}{b}\right )}{b}-\frac {(c+d x)^{4/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b}\right )}{d (c+d x)^{2/3}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {3 (e (c+d x))^{2/3} \left (\frac {4 \left (\frac {(c+d x) \sin \left (a+b \sqrt [3]{c+d x}\right )}{b}-\frac {3 \left (\frac {2 \left (\frac {\sqrt [3]{c+d x} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b}-\frac {\int \sin \left (a+b \sqrt [3]{c+d x}\right )d\sqrt [3]{c+d x}}{b}\right )}{b}-\frac {(c+d x)^{2/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b}\right )}{b}\right )}{b}-\frac {(c+d x)^{4/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b}\right )}{d (c+d x)^{2/3}}\) |
| \(\Big \downarrow \) 3118 |
| \(\displaystyle \frac {3 (e (c+d x))^{2/3} \left (\frac {4 \left (\frac {(c+d x) \sin \left (a+b \sqrt [3]{c+d x}\right )}{b}-\frac {3 \left (\frac {2 \left (\frac {\cos \left (a+b \sqrt [3]{c+d x}\right )}{b^2}+\frac {\sqrt [3]{c+d x} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b}\right )}{b}-\frac {(c+d x)^{2/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b}\right )}{b}\right )}{b}-\frac {(c+d x)^{4/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b}\right )}{d (c+d x)^{2/3}}\) |
(3*(e*(c + d*x))^(2/3)*(-(((c + d*x)^(4/3)*Cos[a + b*(c + d*x)^(1/3)])/b) + (4*(((c + d*x)*Sin[a + b*(c + d*x)^(1/3)])/b - (3*(-(((c + d*x)^(2/3)*Co s[a + b*(c + d*x)^(1/3)])/b) + (2*(Cos[a + b*(c + d*x)^(1/3)]/b^2 + ((c + d*x)^(1/3)*Sin[a + b*(c + d*x)^(1/3)])/b))/b))/b))/b))/(d*(c + d*x)^(2/3))
3.3.28.3.1 Defintions of rubi rules used
Int[(u_.)*((a_.)*(x_))^(m_.)*((b_.)*(x_)^(i_.))^(p_), x_Symbol] :> Simp[b^I ntPart[p]*((b*x^i)^FracPart[p]/(a^(i*IntPart[p])*(a*x)^(i*FracPart[p]))) Int[u*(a*x)^(m + i*p), x], x] /; FreeQ[{a, b, i, m, p}, x] && IntegerQ[i] & & !IntegerQ[p]
Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[( -(c + d*x)^m)*(Cos[e + f*x]/f), x] + Simp[d*(m/f) Int[(c + d*x)^(m - 1)*C os[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]
Int[((g_.) + (h_.)*(x_))^(m_.)*((a_.) + (b_.)*Sin[(c_.) + (d_.)*((e_.) + (f _.)*(x_))^(n_)])^(p_.), x_Symbol] :> Simp[1/(n*f) Subst[Int[ExpandIntegra nd[(a + b*Sin[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]
\[\int \left (d e x +c e \right )^{\frac {2}{3}} \sin \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )d x\]
Time = 0.70 (sec) , antiderivative size = 143, normalized size of antiderivative = 0.71 \[ \int (c e+d e x)^{2/3} \sin \left (a+b \sqrt [3]{c+d x}\right ) \, dx=\frac {3 \, {\left ({\left (12 \, b^{2} d x + 12 \, b^{2} c - {\left (b^{4} d x + b^{4} c\right )} {\left (d x + c\right )}^{\frac {2}{3}} - 24 \, {\left (d x + c\right )}^{\frac {1}{3}}\right )} {\left (d e x + c e\right )}^{\frac {2}{3}} \cos \left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right ) - 4 \, {\left (d e x + c e\right )}^{\frac {2}{3}} {\left (6 \, {\left (d x + c\right )}^{\frac {2}{3}} b - {\left (b^{3} d x + b^{3} c\right )} {\left (d x + c\right )}^{\frac {1}{3}}\right )} \sin \left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )\right )}}{b^{5} d^{2} x + b^{5} c d} \]
3*((12*b^2*d*x + 12*b^2*c - (b^4*d*x + b^4*c)*(d*x + c)^(2/3) - 24*(d*x + c)^(1/3))*(d*e*x + c*e)^(2/3)*cos((d*x + c)^(1/3)*b + a) - 4*(d*e*x + c*e) ^(2/3)*(6*(d*x + c)^(2/3)*b - (b^3*d*x + b^3*c)*(d*x + c)^(1/3))*sin((d*x + c)^(1/3)*b + a))/(b^5*d^2*x + b^5*c*d)
\[ \int (c e+d e x)^{2/3} \sin \left (a+b \sqrt [3]{c+d x}\right ) \, dx=\int \left (e \left (c + d x\right )\right )^{\frac {2}{3}} \sin {\left (a + b \sqrt [3]{c + d x} \right )}\, dx \]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.37 (sec) , antiderivative size = 193, normalized size of antiderivative = 0.96 \[ \int (c e+d e x)^{2/3} \sin \left (a+b \sqrt [3]{c+d x}\right ) \, dx=-\frac {3 \, {\left ({\left (b^{4} d x + b^{4} c\right )} {\left (d x + c\right )}^{\frac {1}{3}} e^{\frac {2}{3}} \cos \left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right ) + {\left (3 \, {\left (\Gamma \left (3, i \, b \overline {{\left (d x + c\right )}^{\frac {1}{3}}}\right ) + \Gamma \left (3, -i \, b \overline {{\left (d x + c\right )}^{\frac {1}{3}}}\right ) + \Gamma \left (3, i \, {\left (d x + c\right )}^{\frac {1}{3}} b\right ) + \Gamma \left (3, -i \, {\left (d x + c\right )}^{\frac {1}{3}} b\right )\right )} \cos \left (a\right ) - 4 \, {\left (b^{3} d x + b^{3} c\right )} \sin \left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right ) - 3 \, {\left (i \, \Gamma \left (3, i \, b \overline {{\left (d x + c\right )}^{\frac {1}{3}}}\right ) - i \, \Gamma \left (3, -i \, b \overline {{\left (d x + c\right )}^{\frac {1}{3}}}\right ) + i \, \Gamma \left (3, i \, {\left (d x + c\right )}^{\frac {1}{3}} b\right ) - i \, \Gamma \left (3, -i \, {\left (d x + c\right )}^{\frac {1}{3}} b\right )\right )} \sin \left (a\right )\right )} e^{\frac {2}{3}}\right )}}{b^{5} d} \]
-3*((b^4*d*x + b^4*c)*(d*x + c)^(1/3)*e^(2/3)*cos((d*x + c)^(1/3)*b + a) + (3*(gamma(3, I*b*conjugate((d*x + c)^(1/3))) + gamma(3, -I*b*conjugate((d *x + c)^(1/3))) + gamma(3, I*(d*x + c)^(1/3)*b) + gamma(3, -I*(d*x + c)^(1 /3)*b))*cos(a) - 4*(b^3*d*x + b^3*c)*sin((d*x + c)^(1/3)*b + a) - 3*(I*gam ma(3, I*b*conjugate((d*x + c)^(1/3))) - I*gamma(3, -I*b*conjugate((d*x + c )^(1/3))) + I*gamma(3, I*(d*x + c)^(1/3)*b) - I*gamma(3, -I*(d*x + c)^(1/3 )*b))*sin(a))*e^(2/3))/(b^5*d)
Time = 0.34 (sec) , antiderivative size = 286, normalized size of antiderivative = 1.42 \[ \int (c e+d e x)^{2/3} \sin \left (a+b \sqrt [3]{c+d x}\right ) \, dx=-\frac {3 \, {\left (c {\left (\frac {{\left (d e x + c e\right )}^{\frac {1}{3}} e \cos \left (\frac {a e + {\left (d e x + c e\right )}^{\frac {1}{3}} b {\left | e \right |}^{\frac {2}{3}}}{e}\right )}{b {\left | e \right |}^{\frac {2}{3}}} - \frac {e^{2} \sin \left (\frac {a e + {\left (d e x + c e\right )}^{\frac {1}{3}} b {\left | e \right |}^{\frac {2}{3}}}{e}\right )}{b^{2} {\left | e \right |}^{\frac {4}{3}}}\right )} - \frac {\frac {{\left ({\left (d e x + c e\right )}^{\frac {1}{3}} b^{4} c e^{4} {\left | e \right |}^{\frac {2}{3}} - {\left (d e x + c e\right )}^{\frac {4}{3}} b^{4} e^{3} {\left | e \right |}^{\frac {2}{3}} + 12 \, {\left (d e x + c e\right )}^{\frac {2}{3}} b^{2} e^{3} {\left | e \right |}^{\frac {4}{3}} - 24 \, e^{5}\right )} \cos \left (\frac {a e + {\left (d e x + c e\right )}^{\frac {1}{3}} b {\left | e \right |}^{\frac {2}{3}}}{e}\right )}{b^{5} e^{2} {\left | e \right |}^{\frac {4}{3}}} - \frac {{\left (b^{3} c e^{5} - 4 \, {\left (d e x + c e\right )} b^{3} e^{4} + 24 \, {\left (d e x + c e\right )}^{\frac {1}{3}} b e^{4} {\left | e \right |}^{\frac {2}{3}}\right )} \sin \left (\frac {a e + {\left (d e x + c e\right )}^{\frac {1}{3}} b {\left | e \right |}^{\frac {2}{3}}}{e}\right )}{b^{5} e^{2} {\left | e \right |}^{\frac {4}{3}}}}{e}\right )}}{d} \]
-3*(c*((d*e*x + c*e)^(1/3)*e*cos((a*e + (d*e*x + c*e)^(1/3)*b*abs(e)^(2/3) )/e)/(b*abs(e)^(2/3)) - e^2*sin((a*e + (d*e*x + c*e)^(1/3)*b*abs(e)^(2/3)) /e)/(b^2*abs(e)^(4/3))) - (((d*e*x + c*e)^(1/3)*b^4*c*e^4*abs(e)^(2/3) - ( d*e*x + c*e)^(4/3)*b^4*e^3*abs(e)^(2/3) + 12*(d*e*x + c*e)^(2/3)*b^2*e^3*a bs(e)^(4/3) - 24*e^5)*cos((a*e + (d*e*x + c*e)^(1/3)*b*abs(e)^(2/3))/e)/(b ^5*e^2*abs(e)^(4/3)) - (b^3*c*e^5 - 4*(d*e*x + c*e)*b^3*e^4 + 24*(d*e*x + c*e)^(1/3)*b*e^4*abs(e)^(2/3))*sin((a*e + (d*e*x + c*e)^(1/3)*b*abs(e)^(2/ 3))/e)/(b^5*e^2*abs(e)^(4/3)))/e)/d
Timed out. \[ \int (c e+d e x)^{2/3} \sin \left (a+b \sqrt [3]{c+d x}\right ) \, dx=\int \sin \left (a+b\,{\left (c+d\,x\right )}^{1/3}\right )\,{\left (c\,e+d\,e\,x\right )}^{2/3} \,d x \]