Integrand size = 27, antiderivative size = 122 \[ \int \frac {\sin \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{\sqrt [3]{c e+d e x}} \, dx=-\frac {3 b \sqrt [3]{c+d x} \cos (a) \operatorname {CosIntegral}\left (\frac {b}{(c+d x)^{2/3}}\right )}{2 d \sqrt [3]{e (c+d x)}}+\frac {3 (c+d x) \sin \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{2 d \sqrt [3]{e (c+d x)}}+\frac {3 b \sqrt [3]{c+d x} \sin (a) \text {Si}\left (\frac {b}{(c+d x)^{2/3}}\right )}{2 d \sqrt [3]{e (c+d x)}} \]
-3/2*b*(d*x+c)^(1/3)*Ci(b/(d*x+c)^(2/3))*cos(a)/d/(e*(d*x+c))^(1/3)+3/2*b* (d*x+c)^(1/3)*Si(b/(d*x+c)^(2/3))*sin(a)/d/(e*(d*x+c))^(1/3)+3/2*(d*x+c)*s in(a+b/(d*x+c)^(2/3))/d/(e*(d*x+c))^(1/3)
Time = 0.23 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.74 \[ \int \frac {\sin \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{\sqrt [3]{c e+d e x}} \, dx=\frac {3 \left (-b \sqrt [3]{c+d x} \cos (a) \operatorname {CosIntegral}\left (\frac {b}{(c+d x)^{2/3}}\right )+(c+d x) \sin \left (a+\frac {b}{(c+d x)^{2/3}}\right )+b \sqrt [3]{c+d x} \sin (a) \text {Si}\left (\frac {b}{(c+d x)^{2/3}}\right )\right )}{2 d \sqrt [3]{e (c+d x)}} \]
(3*(-(b*(c + d*x)^(1/3)*Cos[a]*CosIntegral[b/(c + d*x)^(2/3)]) + (c + d*x) *Sin[a + b/(c + d*x)^(2/3)] + b*(c + d*x)^(1/3)*Sin[a]*SinIntegral[b/(c + d*x)^(2/3)]))/(2*d*(e*(c + d*x))^(1/3))
Time = 0.58 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.71, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.370, Rules used = {3916, 3862, 3860, 3042, 3778, 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.
\(\displaystyle \int \frac {\sin \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{\sqrt [3]{c e+d e x}} \, dx\) |
\(\Big \downarrow \) 3916 |
\(\displaystyle \frac {\int \frac {\sin \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{\sqrt [3]{e (c+d x)}}d(c+d x)}{d}\) |
\(\Big \downarrow \) 3862 |
\(\displaystyle \frac {\sqrt [3]{c+d x} \int \frac {\sin \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{\sqrt [3]{c+d x}}d(c+d x)}{d \sqrt [3]{e (c+d x)}}\) |
\(\Big \downarrow \) 3860 |
\(\displaystyle -\frac {3 \sqrt [3]{c+d x} \int (c+d x)^{4/3} \sin \left (a+\frac {b}{(c+d x)^{2/3}}\right )d\frac {1}{(c+d x)^{2/3}}}{2 d \sqrt [3]{e (c+d x)}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {3 \sqrt [3]{c+d x} \int (c+d x)^{4/3} \sin \left (a+\frac {b}{(c+d x)^{2/3}}\right )d\frac {1}{(c+d x)^{2/3}}}{2 d \sqrt [3]{e (c+d x)}}\) |
\(\Big \downarrow \) 3778 |
\(\displaystyle -\frac {3 \sqrt [3]{c+d x} \left (b \int (c+d x)^{2/3} \cos \left (a+\frac {b}{(c+d x)^{2/3}}\right )d\frac {1}{(c+d x)^{2/3}}-(c+d x)^{2/3} \sin \left (a+\frac {b}{(c+d x)^{2/3}}\right )\right )}{2 d \sqrt [3]{e (c+d x)}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {3 \sqrt [3]{c+d x} \left (b \int (c+d x)^{2/3} \sin \left (a+\frac {b}{(c+d x)^{2/3}}+\frac {\pi }{2}\right )d\frac {1}{(c+d x)^{2/3}}-(c+d x)^{2/3} \sin \left (a+\frac {b}{(c+d x)^{2/3}}\right )\right )}{2 d \sqrt [3]{e (c+d x)}}\) |
\(\Big \downarrow \) 3784 |
\(\displaystyle -\frac {3 \sqrt [3]{c+d x} \left (b \left (\cos (a) \int (c+d x)^{2/3} \cos \left (\frac {b}{(c+d x)^{2/3}}\right )d\frac {1}{(c+d x)^{2/3}}-\sin (a) \int (c+d x)^{2/3} \sin \left (\frac {b}{(c+d x)^{2/3}}\right )d\frac {1}{(c+d x)^{2/3}}\right )-(c+d x)^{2/3} \sin \left (a+\frac {b}{(c+d x)^{2/3}}\right )\right )}{2 d \sqrt [3]{e (c+d x)}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {3 \sqrt [3]{c+d x} \left (b \left (\cos (a) \int (c+d x)^{2/3} \sin \left (\frac {b}{(c+d x)^{2/3}}+\frac {\pi }{2}\right )d\frac {1}{(c+d x)^{2/3}}-\sin (a) \int (c+d x)^{2/3} \sin \left (\frac {b}{(c+d x)^{2/3}}\right )d\frac {1}{(c+d x)^{2/3}}\right )-(c+d x)^{2/3} \sin \left (a+\frac {b}{(c+d x)^{2/3}}\right )\right )}{2 d \sqrt [3]{e (c+d x)}}\) |
\(\Big \downarrow \) 3780 |
\(\displaystyle -\frac {3 \sqrt [3]{c+d x} \left (b \left (\cos (a) \int (c+d x)^{2/3} \sin \left (\frac {b}{(c+d x)^{2/3}}+\frac {\pi }{2}\right )d\frac {1}{(c+d x)^{2/3}}-\sin (a) \text {Si}\left (\frac {b}{(c+d x)^{2/3}}\right )\right )-(c+d x)^{2/3} \sin \left (a+\frac {b}{(c+d x)^{2/3}}\right )\right )}{2 d \sqrt [3]{e (c+d x)}}\) |
\(\Big \downarrow \) 3783 |
\(\displaystyle -\frac {3 \sqrt [3]{c+d x} \left (b \left (\cos (a) \operatorname {CosIntegral}\left (\frac {b}{(c+d x)^{2/3}}\right )-\sin (a) \text {Si}\left (\frac {b}{(c+d x)^{2/3}}\right )\right )-(c+d x)^{2/3} \sin \left (a+\frac {b}{(c+d x)^{2/3}}\right )\right )}{2 d \sqrt [3]{e (c+d x)}}\) |
(-3*(c + d*x)^(1/3)*(-((c + d*x)^(2/3)*Sin[a + b/(c + d*x)^(2/3)]) + b*(Co s[a]*CosIntegral[b/(c + d*x)^(2/3)] - Sin[a]*SinIntegral[b/(c + d*x)^(2/3) ])))/(2*d*(e*(c + d*x))^(1/3))
3.3.52.3.1 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]
\[\int \frac {\sin \left (a +\frac {b}{\left (d x +c \right )^{\frac {2}{3}}}\right )}{\left (d e x +c e \right )^{\frac {1}{3}}}d x\]
\[ \int \frac {\sin \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{\sqrt [3]{c e+d e x}} \, dx=\int { \frac {\sin \left (a + \frac {b}{{\left (d x + c\right )}^{\frac {2}{3}}}\right )}{{\left (d e x + c e\right )}^{\frac {1}{3}}} \,d x } \]
\[ \int \frac {\sin \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{\sqrt [3]{c e+d e x}} \, dx=\int \frac {\sin {\left (a + \frac {b}{\left (c + d x\right )^{\frac {2}{3}}} \right )}}{\sqrt [3]{e \left (c + d x\right )}}\, dx \]
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.37 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.03 \[ \int \frac {\sin \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{\sqrt [3]{c e+d e x}} \, dx=-\frac {3 \, {\left ({\left (\Gamma \left (-1, i \, b \overline {\frac {1}{{\left (d x + c\right )}^{\frac {2}{3}}}}\right ) + \Gamma \left (-1, -i \, b \overline {\frac {1}{{\left (d x + c\right )}^{\frac {2}{3}}}}\right ) + \Gamma \left (-1, \frac {i \, b}{{\left (d x + c\right )}^{\frac {2}{3}}}\right ) + \Gamma \left (-1, -\frac {i \, b}{{\left (d x + c\right )}^{\frac {2}{3}}}\right )\right )} \cos \left (a\right ) + {\left (-i \, \Gamma \left (-1, i \, b \overline {\frac {1}{{\left (d x + c\right )}^{\frac {2}{3}}}}\right ) + i \, \Gamma \left (-1, -i \, b \overline {\frac {1}{{\left (d x + c\right )}^{\frac {2}{3}}}}\right ) - i \, \Gamma \left (-1, \frac {i \, b}{{\left (d x + c\right )}^{\frac {2}{3}}}\right ) + i \, \Gamma \left (-1, -\frac {i \, b}{{\left (d x + c\right )}^{\frac {2}{3}}}\right )\right )} \sin \left (a\right )\right )} b}{8 \, d e^{\frac {1}{3}}} \]
-3/8*((gamma(-1, I*b*conjugate((d*x + c)^(-2/3))) + gamma(-1, -I*b*conjuga te((d*x + c)^(-2/3))) + gamma(-1, I*b/(d*x + c)^(2/3)) + gamma(-1, -I*b/(d *x + c)^(2/3)))*cos(a) + (-I*gamma(-1, I*b*conjugate((d*x + c)^(-2/3))) + I*gamma(-1, -I*b*conjugate((d*x + c)^(-2/3))) - I*gamma(-1, I*b/(d*x + c)^ (2/3)) + I*gamma(-1, -I*b/(d*x + c)^(2/3)))*sin(a))*b/(d*e^(1/3))
\[ \int \frac {\sin \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{\sqrt [3]{c e+d e x}} \, dx=\int { \frac {\sin \left (a + \frac {b}{{\left (d x + c\right )}^{\frac {2}{3}}}\right )}{{\left (d e x + c e\right )}^{\frac {1}{3}}} \,d x } \]
Timed out. \[ \int \frac {\sin \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{\sqrt [3]{c e+d e x}} \, dx=\int \frac {\sin \left (a+\frac {b}{{\left (c+d\,x\right )}^{2/3}}\right )}{{\left (c\,e+d\,e\,x\right )}^{1/3}} \,d x \]