Integrand size = 27, antiderivative size = 168 \[ \int \sqrt [3]{c e+d e x} \sin \left (a+\frac {b}{(c+d x)^{2/3}}\right ) \, dx=\frac {3 b \sqrt [3]{c+d x} \sqrt [3]{e (c+d x)} \cos \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{4 d}+\frac {3 b^2 \sqrt [3]{e (c+d x)} \operatorname {CosIntegral}\left (\frac {b}{(c+d x)^{2/3}}\right ) \sin (a)}{4 d \sqrt [3]{c+d x}}+\frac {3 (c+d x) \sqrt [3]{e (c+d x)} \sin \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{4 d}+\frac {3 b^2 \sqrt [3]{e (c+d x)} \cos (a) \text {Si}\left (\frac {b}{(c+d x)^{2/3}}\right )}{4 d \sqrt [3]{c+d x}} \]
3/4*b*(d*x+c)^(1/3)*(e*(d*x+c))^(1/3)*cos(a+b/(d*x+c)^(2/3))/d+3/4*b^2*(e* (d*x+c))^(1/3)*cos(a)*Si(b/(d*x+c)^(2/3))/d/(d*x+c)^(1/3)+3/4*b^2*(e*(d*x+ c))^(1/3)*Ci(b/(d*x+c)^(2/3))*sin(a)/d/(d*x+c)^(1/3)+3/4*(d*x+c)*(e*(d*x+c ))^(1/3)*sin(a+b/(d*x+c)^(2/3))/d
Time = 0.29 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.67 \[ \int \sqrt [3]{c e+d e x} \sin \left (a+\frac {b}{(c+d x)^{2/3}}\right ) \, dx=\frac {3 \sqrt [3]{e (c+d x)} \left (b (c+d x)^{2/3} \cos \left (a+\frac {b}{(c+d x)^{2/3}}\right )+b^2 \operatorname {CosIntegral}\left (\frac {b}{(c+d x)^{2/3}}\right ) \sin (a)+(c+d x)^{4/3} \sin \left (a+\frac {b}{(c+d x)^{2/3}}\right )+b^2 \cos (a) \text {Si}\left (\frac {b}{(c+d x)^{2/3}}\right )\right )}{4 d \sqrt [3]{c+d x}} \]
(3*(e*(c + d*x))^(1/3)*(b*(c + d*x)^(2/3)*Cos[a + b/(c + d*x)^(2/3)] + b^2 *CosIntegral[b/(c + d*x)^(2/3)]*Sin[a] + (c + d*x)^(4/3)*Sin[a + b/(c + d* x)^(2/3)] + b^2*Cos[a]*SinIntegral[b/(c + d*x)^(2/3)]))/(4*d*(c + d*x)^(1/ 3))
Time = 0.70 (sec) , antiderivative size = 120, normalized size of antiderivative = 0.71, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.481, Rules used = {3916, 3862, 3860, 3042, 3778, 3042, 3778, 25, 3042, 3784, 3042, 3780, 3783}
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.
| \(\displaystyle \int \sqrt [3]{c e+d e x} \sin \left (a+\frac {b}{(c+d x)^{2/3}}\right ) \, dx\) |
| \(\Big \downarrow \) 3916 |
| \(\displaystyle \frac {\int \sqrt [3]{e (c+d x)} \sin \left (a+\frac {b}{(c+d x)^{2/3}}\right )d(c+d x)}{d}\) |
| \(\Big \downarrow \) 3862 |
| \(\displaystyle \frac {\sqrt [3]{e (c+d x)} \int \sqrt [3]{c+d x} \sin \left (a+\frac {b}{(c+d x)^{2/3}}\right )d(c+d x)}{d \sqrt [3]{c+d x}}\) |
| \(\Big \downarrow \) 3860 |
| \(\displaystyle -\frac {3 \sqrt [3]{e (c+d x)} \int (c+d x)^2 \sin \left (a+\frac {b}{(c+d x)^{2/3}}\right )d\frac {1}{(c+d x)^{2/3}}}{2 d \sqrt [3]{c+d x}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {3 \sqrt [3]{e (c+d x)} \int (c+d x)^2 \sin \left (a+\frac {b}{(c+d x)^{2/3}}\right )d\frac {1}{(c+d x)^{2/3}}}{2 d \sqrt [3]{c+d x}}\) |
| \(\Big \downarrow \) 3778 |
| \(\displaystyle -\frac {3 \sqrt [3]{e (c+d x)} \left (\frac {1}{2} b \int (c+d x)^{4/3} \cos \left (a+\frac {b}{(c+d x)^{2/3}}\right )d\frac {1}{(c+d x)^{2/3}}-\frac {1}{2} (c+d x)^{4/3} \sin \left (a+\frac {b}{(c+d x)^{2/3}}\right )\right )}{2 d \sqrt [3]{c+d x}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {3 \sqrt [3]{e (c+d x)} \left (\frac {1}{2} b \int (c+d x)^{4/3} \sin \left (a+\frac {b}{(c+d x)^{2/3}}+\frac {\pi }{2}\right )d\frac {1}{(c+d x)^{2/3}}-\frac {1}{2} (c+d x)^{4/3} \sin \left (a+\frac {b}{(c+d x)^{2/3}}\right )\right )}{2 d \sqrt [3]{c+d x}}\) |
| \(\Big \downarrow \) 3778 |
| \(\displaystyle -\frac {3 \sqrt [3]{e (c+d x)} \left (\frac {1}{2} b \left (b \int -(c+d x)^{2/3} \sin \left (a+\frac {b}{(c+d x)^{2/3}}\right )d\frac {1}{(c+d x)^{2/3}}-(c+d x)^{2/3} \cos \left (a+\frac {b}{(c+d x)^{2/3}}\right )\right )-\frac {1}{2} (c+d x)^{4/3} \sin \left (a+\frac {b}{(c+d x)^{2/3}}\right )\right )}{2 d \sqrt [3]{c+d x}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {3 \sqrt [3]{e (c+d x)} \left (\frac {1}{2} b \left (-b \int (c+d x)^{2/3} \sin \left (a+\frac {b}{(c+d x)^{2/3}}\right )d\frac {1}{(c+d x)^{2/3}}-(c+d x)^{2/3} \cos \left (a+\frac {b}{(c+d x)^{2/3}}\right )\right )-\frac {1}{2} (c+d x)^{4/3} \sin \left (a+\frac {b}{(c+d x)^{2/3}}\right )\right )}{2 d \sqrt [3]{c+d x}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {3 \sqrt [3]{e (c+d x)} \left (\frac {1}{2} b \left (-b \int (c+d x)^{2/3} \sin \left (a+\frac {b}{(c+d x)^{2/3}}\right )d\frac {1}{(c+d x)^{2/3}}-(c+d x)^{2/3} \cos \left (a+\frac {b}{(c+d x)^{2/3}}\right )\right )-\frac {1}{2} (c+d x)^{4/3} \sin \left (a+\frac {b}{(c+d x)^{2/3}}\right )\right )}{2 d \sqrt [3]{c+d x}}\) |
| \(\Big \downarrow \) 3784 |
| \(\displaystyle -\frac {3 \sqrt [3]{e (c+d x)} \left (\frac {1}{2} b \left (-b \left (\sin (a) \int (c+d x)^{2/3} \cos \left (\frac {b}{(c+d x)^{2/3}}\right )d\frac {1}{(c+d x)^{2/3}}+\cos (a) \int (c+d x)^{2/3} \sin \left (\frac {b}{(c+d x)^{2/3}}\right )d\frac {1}{(c+d x)^{2/3}}\right )-(c+d x)^{2/3} \cos \left (a+\frac {b}{(c+d x)^{2/3}}\right )\right )-\frac {1}{2} (c+d x)^{4/3} \sin \left (a+\frac {b}{(c+d x)^{2/3}}\right )\right )}{2 d \sqrt [3]{c+d x}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {3 \sqrt [3]{e (c+d x)} \left (\frac {1}{2} b \left (-b \left (\sin (a) \int (c+d x)^{2/3} \sin \left (\frac {b}{(c+d x)^{2/3}}+\frac {\pi }{2}\right )d\frac {1}{(c+d x)^{2/3}}+\cos (a) \int (c+d x)^{2/3} \sin \left (\frac {b}{(c+d x)^{2/3}}\right )d\frac {1}{(c+d x)^{2/3}}\right )-(c+d x)^{2/3} \cos \left (a+\frac {b}{(c+d x)^{2/3}}\right )\right )-\frac {1}{2} (c+d x)^{4/3} \sin \left (a+\frac {b}{(c+d x)^{2/3}}\right )\right )}{2 d \sqrt [3]{c+d x}}\) |
| \(\Big \downarrow \) 3780 |
| \(\displaystyle -\frac {3 \sqrt [3]{e (c+d x)} \left (\frac {1}{2} b \left (-b \left (\sin (a) \int (c+d x)^{2/3} \sin \left (\frac {b}{(c+d x)^{2/3}}+\frac {\pi }{2}\right )d\frac {1}{(c+d x)^{2/3}}+\cos (a) \text {Si}\left (\frac {b}{(c+d x)^{2/3}}\right )\right )-(c+d x)^{2/3} \cos \left (a+\frac {b}{(c+d x)^{2/3}}\right )\right )-\frac {1}{2} (c+d x)^{4/3} \sin \left (a+\frac {b}{(c+d x)^{2/3}}\right )\right )}{2 d \sqrt [3]{c+d x}}\) |
| \(\Big \downarrow \) 3783 |
| \(\displaystyle -\frac {3 \sqrt [3]{e (c+d x)} \left (\frac {1}{2} b \left (-b \left (\sin (a) \operatorname {CosIntegral}\left (\frac {b}{(c+d x)^{2/3}}\right )+\cos (a) \text {Si}\left (\frac {b}{(c+d x)^{2/3}}\right )\right )-(c+d x)^{2/3} \cos \left (a+\frac {b}{(c+d x)^{2/3}}\right )\right )-\frac {1}{2} (c+d x)^{4/3} \sin \left (a+\frac {b}{(c+d x)^{2/3}}\right )\right )}{2 d \sqrt [3]{c+d x}}\) |
(-3*(e*(c + d*x))^(1/3)*(-1/2*((c + d*x)^(4/3)*Sin[a + b/(c + d*x)^(2/3)]) + (b*(-((c + d*x)^(2/3)*Cos[a + b/(c + d*x)^(2/3)]) - b*(CosIntegral[b/(c + d*x)^(2/3)]*Sin[a] + Cos[a]*SinIntegral[b/(c + d*x)^(2/3)])))/2))/(2*d* (c + d*x)^(1/3))
3.3.51.3.1 Defintions of rubi rules used
Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[(c + d*x)^(m + 1)*(Sin[e + f*x]/(d*(m + 1))), x] - Simp[f/(d*(m + 1)) Int[( c + d*x)^(m + 1)*Cos[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && LtQ[m, - 1]
Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[SinInte gral[e + f*x]/d, x] /; FreeQ[{c, d, e, f}, x] && EqQ[d*e - c*f, 0]
Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[CosInte gral[e - Pi/2 + f*x]/d, x] /; FreeQ[{c, d, e, f}, x] && EqQ[d*(e - Pi/2) - c*f, 0]
Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[Cos[(d* e - c*f)/d] Int[Sin[c*(f/d) + f*x]/(c + d*x), x], x] + Simp[Sin[(d*e - c* f)/d] Int[Cos[c*(f/d) + f*x]/(c + d*x), x], x] /; FreeQ[{c, d, e, f}, x] && NeQ[d*e - c*f, 0]
Int[(x_)^(m_.)*((a_.) + (b_.)*Sin[(c_.) + (d_.)*(x_)^(n_)])^(p_.), x_Symbol ] :> Simp[1/n Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*Sin[c + d*x])^ p, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p}, x] && IntegerQ[Simplify[ (m + 1)/n]] && (EqQ[p, 1] || EqQ[m, n - 1] || (IntegerQ[p] && GtQ[Simplify[ (m + 1)/n], 0]))
Int[((e_)*(x_))^(m_)*((a_.) + (b_.)*Sin[(c_.) + (d_.)*(x_)^(n_)])^(p_.), x_ Symbol] :> Simp[e^IntPart[m]*((e*x)^FracPart[m]/x^FracPart[m]) Int[x^m*(a + b*Sin[c + d*x^n])^p, x], x] /; FreeQ[{a, b, c, d, e, m, n, p}, x] && Int egerQ[Simplify[(m + 1)/n]]
Int[((g_.) + (h_.)*(x_))^(m_.)*((a_.) + (b_.)*Sin[(c_.) + (d_.)*((e_.) + (f _.)*(x_))^(n_)])^(p_.), x_Symbol] :> Simp[1/f Subst[Int[(h*(x/f))^m*(a + b*Sin[c + d*x^n])^p, x], x, e + f*x], x] /; FreeQ[{a, b, c, d, e, f, g, h, m}, x] && IGtQ[p, 0] && EqQ[f*g - e*h, 0]
\[\int \left (d e x +c e \right )^{\frac {1}{3}} \sin \left (a +\frac {b}{\left (d x +c \right )^{\frac {2}{3}}}\right )d x\]
\[ \int \sqrt [3]{c e+d e x} \sin \left (a+\frac {b}{(c+d x)^{2/3}}\right ) \, dx=\int { {\left (d e x + c e\right )}^{\frac {1}{3}} \sin \left (a + \frac {b}{{\left (d x + c\right )}^{\frac {2}{3}}}\right ) \,d x } \]
\[ \int \sqrt [3]{c e+d e x} \sin \left (a+\frac {b}{(c+d x)^{2/3}}\right ) \, dx=\int \sqrt [3]{e \left (c + d x\right )} \sin {\left (a + \frac {b}{\left (c + d x\right )^{\frac {2}{3}}} \right )}\, dx \]
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.40 (sec) , antiderivative size = 129, normalized size of antiderivative = 0.77 \[ \int \sqrt [3]{c e+d e x} \sin \left (a+\frac {b}{(c+d x)^{2/3}}\right ) \, dx=\frac {3 \, {\left ({\left (-i \, \Gamma \left (-2, i \, b \overline {\frac {1}{{\left (d x + c\right )}^{\frac {2}{3}}}}\right ) + i \, \Gamma \left (-2, -i \, b \overline {\frac {1}{{\left (d x + c\right )}^{\frac {2}{3}}}}\right ) - i \, \Gamma \left (-2, \frac {i \, b}{{\left (d x + c\right )}^{\frac {2}{3}}}\right ) + i \, \Gamma \left (-2, -\frac {i \, b}{{\left (d x + c\right )}^{\frac {2}{3}}}\right )\right )} \cos \left (a\right ) - {\left (\Gamma \left (-2, i \, b \overline {\frac {1}{{\left (d x + c\right )}^{\frac {2}{3}}}}\right ) + \Gamma \left (-2, -i \, b \overline {\frac {1}{{\left (d x + c\right )}^{\frac {2}{3}}}}\right ) + \Gamma \left (-2, \frac {i \, b}{{\left (d x + c\right )}^{\frac {2}{3}}}\right ) + \Gamma \left (-2, -\frac {i \, b}{{\left (d x + c\right )}^{\frac {2}{3}}}\right )\right )} \sin \left (a\right )\right )} b^{2} e^{\frac {1}{3}}}{8 \, d} \]
3/8*((-I*gamma(-2, I*b*conjugate((d*x + c)^(-2/3))) + I*gamma(-2, -I*b*con jugate((d*x + c)^(-2/3))) - I*gamma(-2, I*b/(d*x + c)^(2/3)) + I*gamma(-2, -I*b/(d*x + c)^(2/3)))*cos(a) - (gamma(-2, I*b*conjugate((d*x + c)^(-2/3) )) + gamma(-2, -I*b*conjugate((d*x + c)^(-2/3))) + gamma(-2, I*b/(d*x + c) ^(2/3)) + gamma(-2, -I*b/(d*x + c)^(2/3)))*sin(a))*b^2*e^(1/3)/d
\[ \int \sqrt [3]{c e+d e x} \sin \left (a+\frac {b}{(c+d x)^{2/3}}\right ) \, dx=\int { {\left (d e x + c e\right )}^{\frac {1}{3}} \sin \left (a + \frac {b}{{\left (d x + c\right )}^{\frac {2}{3}}}\right ) \,d x } \]
Timed out. \[ \int \sqrt [3]{c e+d e x} \sin \left (a+\frac {b}{(c+d x)^{2/3}}\right ) \, dx=\int \sin \left (a+\frac {b}{{\left (c+d\,x\right )}^{2/3}}\right )\,{\left (c\,e+d\,e\,x\right )}^{1/3} \,d x \]