Integrand size = 27, antiderivative size = 160 \[ \int \sqrt [3]{c e+d e x} \sin \left (a+b \sqrt [3]{c+d x}\right ) \, dx=\frac {18 \sqrt [3]{e (c+d x)} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d}-\frac {3 (c+d x)^{2/3} \sqrt [3]{e (c+d x)} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b d}-\frac {18 \sqrt [3]{e (c+d x)} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^4 d \sqrt [3]{c+d x}}+\frac {9 \sqrt [3]{c+d x} \sqrt [3]{e (c+d x)} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d} \]
18*(e*(d*x+c))^(1/3)*cos(a+b*(d*x+c)^(1/3))/b^3/d-3*(d*x+c)^(2/3)*(e*(d*x+ c))^(1/3)*cos(a+b*(d*x+c)^(1/3))/b/d-18*(e*(d*x+c))^(1/3)*sin(a+b*(d*x+c)^ (1/3))/b^4/d/(d*x+c)^(1/3)+9*(d*x+c)^(1/3)*(e*(d*x+c))^(1/3)*sin(a+b*(d*x+ c)^(1/3))/b^2/d
Time = 0.23 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.61 \[ \int \sqrt [3]{c e+d e x} \sin \left (a+b \sqrt [3]{c+d x}\right ) \, dx=-\frac {3 \sqrt [3]{e (c+d x)} \left (\left (-6 b \sqrt [3]{c+d x}+b^3 (c+d x)\right ) \cos \left (a+b \sqrt [3]{c+d x}\right )-3 \left (-2+b^2 (c+d x)^{2/3}\right ) \sin \left (a+b \sqrt [3]{c+d x}\right )\right )}{b^4 d \sqrt [3]{c+d x}} \]
(-3*(e*(c + d*x))^(1/3)*((-6*b*(c + d*x)^(1/3) + b^3*(c + d*x))*Cos[a + b* (c + d*x)^(1/3)] - 3*(-2 + b^2*(c + d*x)^(2/3))*Sin[a + b*(c + d*x)^(1/3)] ))/(b^4*d*(c + d*x)^(1/3))
Time = 0.53 (sec) , antiderivative size = 135, normalized size of antiderivative = 0.84, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.407, Rules used = {3912, 30, 3042, 3777, 3042, 3777, 25, 3042, 3777, 3042, 3117}
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.
| \(\displaystyle \int \sqrt [3]{c e+d e x} \sin \left (a+b \sqrt [3]{c+d x}\right ) \, dx\) |
| \(\Big \downarrow \) 3912 |
| \(\displaystyle \frac {3 \int (c+d x)^{2/3} \sqrt [3]{e (c+d x)} \sin \left (a+b \sqrt [3]{c+d x}\right )d\sqrt [3]{c+d x}}{d}\) |
| \(\Big \downarrow \) 30 |
| \(\displaystyle \frac {3 \sqrt [3]{e (c+d x)} \int (c+d x) \sin \left (a+b \sqrt [3]{c+d x}\right )d\sqrt [3]{c+d x}}{d \sqrt [3]{c+d x}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {3 \sqrt [3]{e (c+d x)} \int (c+d x) \sin \left (a+b \sqrt [3]{c+d x}\right )d\sqrt [3]{c+d x}}{d \sqrt [3]{c+d x}}\) |
| \(\Big \downarrow \) 3777 |
| \(\displaystyle \frac {3 \sqrt [3]{e (c+d x)} \left (\frac {3 \int (c+d x)^{2/3} \cos \left (a+b \sqrt [3]{c+d x}\right )d\sqrt [3]{c+d x}}{b}-\frac {(c+d x) \cos \left (a+b \sqrt [3]{c+d x}\right )}{b}\right )}{d \sqrt [3]{c+d x}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {3 \sqrt [3]{e (c+d x)} \left (\frac {3 \int (c+d x)^{2/3} \sin \left (a+b \sqrt [3]{c+d x}+\frac {\pi }{2}\right )d\sqrt [3]{c+d x}}{b}-\frac {(c+d x) \cos \left (a+b \sqrt [3]{c+d x}\right )}{b}\right )}{d \sqrt [3]{c+d x}}\) |
| \(\Big \downarrow \) 3777 |
| \(\displaystyle \frac {3 \sqrt [3]{e (c+d x)} \left (\frac {3 \left (\frac {2 \int -\sqrt [3]{c+d x} \sin \left (a+b \sqrt [3]{c+d x}\right )d\sqrt [3]{c+d x}}{b}+\frac {(c+d x)^{2/3} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b}\right )}{b}-\frac {(c+d x) \cos \left (a+b \sqrt [3]{c+d x}\right )}{b}\right )}{d \sqrt [3]{c+d x}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {3 \sqrt [3]{e (c+d x)} \left (\frac {3 \left (\frac {(c+d x)^{2/3} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b}-\frac {2 \int \sqrt [3]{c+d x} \sin \left (a+b \sqrt [3]{c+d x}\right )d\sqrt [3]{c+d x}}{b}\right )}{b}-\frac {(c+d x) \cos \left (a+b \sqrt [3]{c+d x}\right )}{b}\right )}{d \sqrt [3]{c+d x}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {3 \sqrt [3]{e (c+d x)} \left (\frac {3 \left (\frac {(c+d x)^{2/3} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b}-\frac {2 \int \sqrt [3]{c+d x} \sin \left (a+b \sqrt [3]{c+d x}\right )d\sqrt [3]{c+d x}}{b}\right )}{b}-\frac {(c+d x) \cos \left (a+b \sqrt [3]{c+d x}\right )}{b}\right )}{d \sqrt [3]{c+d x}}\) |
| \(\Big \downarrow \) 3777 |
| \(\displaystyle \frac {3 \sqrt [3]{e (c+d x)} \left (\frac {3 \left (\frac {(c+d x)^{2/3} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b}-\frac {2 \left (\frac {\int \cos \left (a+b \sqrt [3]{c+d x}\right )d\sqrt [3]{c+d x}}{b}-\frac {\sqrt [3]{c+d x} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b}\right )}{b}\right )}{b}-\frac {(c+d x) \cos \left (a+b \sqrt [3]{c+d x}\right )}{b}\right )}{d \sqrt [3]{c+d x}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {3 \sqrt [3]{e (c+d x)} \left (\frac {3 \left (\frac {(c+d x)^{2/3} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b}-\frac {2 \left (\frac {\int \sin \left (a+b \sqrt [3]{c+d x}+\frac {\pi }{2}\right )d\sqrt [3]{c+d x}}{b}-\frac {\sqrt [3]{c+d x} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b}\right )}{b}\right )}{b}-\frac {(c+d x) \cos \left (a+b \sqrt [3]{c+d x}\right )}{b}\right )}{d \sqrt [3]{c+d x}}\) |
| \(\Big \downarrow \) 3117 |
| \(\displaystyle \frac {3 \sqrt [3]{e (c+d x)} \left (\frac {3 \left (\frac {(c+d x)^{2/3} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b}-\frac {2 \left (\frac {\sin \left (a+b \sqrt [3]{c+d x}\right )}{b^2}-\frac {\sqrt [3]{c+d x} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b}\right )}{b}\right )}{b}-\frac {(c+d x) \cos \left (a+b \sqrt [3]{c+d x}\right )}{b}\right )}{d \sqrt [3]{c+d x}}\) |
(3*(e*(c + d*x))^(1/3)*(-(((c + d*x)*Cos[a + b*(c + d*x)^(1/3)])/b) + (3*( ((c + d*x)^(2/3)*Sin[a + b*(c + d*x)^(1/3)])/b - (2*(-(((c + d*x)^(1/3)*Co s[a + b*(c + d*x)^(1/3)])/b) + Sin[a + b*(c + d*x)^(1/3)]/b^2))/b))/b))/(d *(c + d*x)^(1/3))
3.3.29.3.1 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[sin[Pi/2 + (c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Sin[c + d*x]/d, x] /; FreeQ[{c, d}, x]
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]
\[\int \left (d e x +c e \right )^{\frac {1}{3}} \sin \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )d x\]
Time = 0.66 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.80 \[ \int \sqrt [3]{c e+d e x} \sin \left (a+b \sqrt [3]{c+d x}\right ) \, dx=\frac {3 \, {\left ({\left (6 \, b d x + 6 \, b c - {\left (b^{3} d x + b^{3} c\right )} {\left (d x + c\right )}^{\frac {2}{3}}\right )} {\left (d e x + c e\right )}^{\frac {1}{3}} \cos \left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right ) + 3 \, {\left (d e x + c e\right )}^{\frac {1}{3}} {\left ({\left (b^{2} d x + b^{2} c\right )} {\left (d x + c\right )}^{\frac {1}{3}} - 2 \, {\left (d x + c\right )}^{\frac {2}{3}}\right )} \sin \left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )\right )}}{b^{4} d^{2} x + b^{4} c d} \]
3*((6*b*d*x + 6*b*c - (b^3*d*x + b^3*c)*(d*x + c)^(2/3))*(d*e*x + c*e)^(1/ 3)*cos((d*x + c)^(1/3)*b + a) + 3*(d*e*x + c*e)^(1/3)*((b^2*d*x + b^2*c)*( d*x + c)^(1/3) - 2*(d*x + c)^(2/3))*sin((d*x + c)^(1/3)*b + a))/(b^4*d^2*x + b^4*c*d)
\[ \int \sqrt [3]{c e+d e x} \sin \left (a+b \sqrt [3]{c+d x}\right ) \, dx=\int \sqrt [3]{e \left (c + d x\right )} \sin {\left (a + b \sqrt [3]{c + d x} \right )}\, dx \]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.38 (sec) , antiderivative size = 156, normalized size of antiderivative = 0.98 \[ \int \sqrt [3]{c e+d e x} \sin \left (a+b \sqrt [3]{c+d x}\right ) \, dx=-\frac {3 \, {\left (4 \, {\left (b^{3} d x + b^{3} c\right )} \cos \left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right ) - 3 \, {\left (-i \, \Gamma \left (3, i \, b \overline {{\left (d x + c\right )}^{\frac {1}{3}}}\right ) + i \, \Gamma \left (3, -i \, b \overline {{\left (d x + c\right )}^{\frac {1}{3}}}\right ) - i \, \Gamma \left (3, i \, {\left (d x + c\right )}^{\frac {1}{3}} b\right ) + i \, \Gamma \left (3, -i \, {\left (d x + c\right )}^{\frac {1}{3}} b\right )\right )} \cos \left (a\right ) + 3 \, {\left (\Gamma \left (3, i \, b \overline {{\left (d x + c\right )}^{\frac {1}{3}}}\right ) + \Gamma \left (3, -i \, b \overline {{\left (d x + c\right )}^{\frac {1}{3}}}\right ) + \Gamma \left (3, i \, {\left (d x + c\right )}^{\frac {1}{3}} b\right ) + \Gamma \left (3, -i \, {\left (d x + c\right )}^{\frac {1}{3}} b\right )\right )} \sin \left (a\right )\right )} e^{\frac {1}{3}}}{4 \, b^{4} d} \]
-3/4*(4*(b^3*d*x + b^3*c)*cos((d*x + c)^(1/3)*b + a) - 3*(-I*gamma(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))*cos( 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))*sin(a))*e^(1/3)/(b^4*d)
Time = 0.34 (sec) , antiderivative size = 192, normalized size of antiderivative = 1.20 \[ \int \sqrt [3]{c e+d e x} \sin \left (a+b \sqrt [3]{c+d x}\right ) \, dx=-\frac {3 \, {\left (\frac {c e \cos \left (\frac {a e + {\left (d e x + c e\right )}^{\frac {1}{3}} b {\left | e \right |}^{\frac {2}{3}}}{e}\right )}{b {\left | e \right |}^{\frac {2}{3}}} - \frac {\frac {{\left (b^{3} c e^{4} - {\left (d e x + c e\right )} b^{3} e^{3} + 6 \, {\left (d e x + c e\right )}^{\frac {1}{3}} b e^{3} {\left | e \right |}^{\frac {2}{3}}\right )} \cos \left (\frac {a e + {\left (d e x + c e\right )}^{\frac {1}{3}} b {\left | e \right |}^{\frac {2}{3}}}{e}\right )}{b^{4} e^{2} {\left | e \right |}^{\frac {2}{3}}} + \frac {3 \, {\left ({\left (d e x + c e\right )}^{\frac {2}{3}} b^{2} e^{2} {\left | e \right |}^{\frac {4}{3}} - 2 \, e^{4}\right )} \sin \left (\frac {a e + {\left (d e x + c e\right )}^{\frac {1}{3}} b {\left | e \right |}^{\frac {2}{3}}}{e}\right )}{b^{4} e^{2} {\left | e \right |}^{\frac {2}{3}}}}{e}\right )}}{d} \]
-3*(c*e*cos((a*e + (d*e*x + c*e)^(1/3)*b*abs(e)^(2/3))/e)/(b*abs(e)^(2/3)) - ((b^3*c*e^4 - (d*e*x + c*e)*b^3*e^3 + 6*(d*e*x + c*e)^(1/3)*b*e^3*abs(e )^(2/3))*cos((a*e + (d*e*x + c*e)^(1/3)*b*abs(e)^(2/3))/e)/(b^4*e^2*abs(e) ^(2/3)) + 3*((d*e*x + c*e)^(2/3)*b^2*e^2*abs(e)^(4/3) - 2*e^4)*sin((a*e + (d*e*x + c*e)^(1/3)*b*abs(e)^(2/3))/e)/(b^4*e^2*abs(e)^(2/3)))/e)/d
Timed out. \[ \int \sqrt [3]{c e+d e x} \sin \left (a+b \sqrt [3]{c+d x}\right ) \, dx=\int \sin \left (a+b\,{\left (c+d\,x\right )}^{1/3}\right )\,{\left (c\,e+d\,e\,x\right )}^{1/3} \,d x \]