Integrand size = 27, antiderivative size = 91 \[ \int \frac {\sin \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{(c e+d e x)^{5/3}} \, dx=\frac {3 \sqrt [3]{c+d x} \cos \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{b d e (e (c+d x))^{2/3}}-\frac {3 (c+d x)^{2/3} \sin \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{b^2 d e (e (c+d x))^{2/3}} \]
3*(d*x+c)^(1/3)*cos(a+b/(d*x+c)^(1/3))/b/d/e/(e*(d*x+c))^(2/3)-3*(d*x+c)^( 2/3)*sin(a+b/(d*x+c)^(1/3))/b^2/d/e/(e*(d*x+c))^(2/3)
Time = 0.07 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.79 \[ \int \frac {\sin \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{(c e+d e x)^{5/3}} \, dx=\frac {3 (c+d x)^{5/3} \left (\frac {\cos \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{b \sqrt [3]{c+d x}}-\frac {\sin \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{b^2}\right )}{d (e (c+d x))^{5/3}} \]
(3*(c + d*x)^(5/3)*(Cos[a + b/(c + d*x)^(1/3)]/(b*(c + d*x)^(1/3)) - Sin[a + b/(c + d*x)^(1/3)]/b^2))/(d*(e*(c + d*x))^(5/3))
Time = 0.32 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.82, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {3912, 30, 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.
\(\displaystyle \int \frac {\sin \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{(c e+d e x)^{5/3}} \, dx\) |
\(\Big \downarrow \) 3912 |
\(\displaystyle -\frac {3 \int \frac {(c+d x)^{4/3} \sin \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{(e (c+d x))^{5/3}}d\frac {1}{\sqrt [3]{c+d x}}}{d}\) |
\(\Big \downarrow \) 30 |
\(\displaystyle -\frac {3 (c+d x)^{2/3} \int \frac {\sin \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{\sqrt [3]{c+d x}}d\frac {1}{\sqrt [3]{c+d x}}}{d e (e (c+d x))^{2/3}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {3 (c+d x)^{2/3} \int \frac {\sin \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{\sqrt [3]{c+d x}}d\frac {1}{\sqrt [3]{c+d x}}}{d e (e (c+d x))^{2/3}}\) |
\(\Big \downarrow \) 3777 |
\(\displaystyle -\frac {3 (c+d x)^{2/3} \left (\frac {\int \cos \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )d\frac {1}{\sqrt [3]{c+d x}}}{b}-\frac {\cos \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{b \sqrt [3]{c+d x}}\right )}{d e (e (c+d x))^{2/3}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {3 (c+d x)^{2/3} \left (\frac {\int \sin \left (a+\frac {b}{\sqrt [3]{c+d x}}+\frac {\pi }{2}\right )d\frac {1}{\sqrt [3]{c+d x}}}{b}-\frac {\cos \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{b \sqrt [3]{c+d x}}\right )}{d e (e (c+d x))^{2/3}}\) |
\(\Big \downarrow \) 3117 |
\(\displaystyle -\frac {3 (c+d x)^{2/3} \left (\frac {\sin \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{b^2}-\frac {\cos \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{b \sqrt [3]{c+d x}}\right )}{d e (e (c+d x))^{2/3}}\) |
(-3*(c + d*x)^(2/3)*(-(Cos[a + b/(c + d*x)^(1/3)]/(b*(c + d*x)^(1/3))) + S in[a + b/(c + d*x)^(1/3)]/b^2))/(d*e*(e*(c + d*x))^(2/3))
3.3.46.3.1 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]
\[\int \frac {\sin \left (a +\frac {b}{\left (d x +c \right )^{\frac {1}{3}}}\right )}{\left (d e x +c e \right )^{\frac {5}{3}}}d x\]
Time = 0.68 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.27 \[ \int \frac {\sin \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{(c e+d e x)^{5/3}} \, dx=\frac {3 \, {\left ({\left (d e x + c e\right )}^{\frac {1}{3}} {\left (d x + c\right )}^{\frac {1}{3}} b \cos \left (\frac {a d x + a c + {\left (d x + c\right )}^{\frac {2}{3}} b}{d x + c}\right ) - {\left (d e x + c e\right )}^{\frac {1}{3}} {\left (d x + c\right )}^{\frac {2}{3}} \sin \left (\frac {a d x + a c + {\left (d x + c\right )}^{\frac {2}{3}} b}{d x + c}\right )\right )}}{b^{2} d^{2} e^{2} x + b^{2} c d e^{2}} \]
3*((d*e*x + c*e)^(1/3)*(d*x + c)^(1/3)*b*cos((a*d*x + a*c + (d*x + c)^(2/3 )*b)/(d*x + c)) - (d*e*x + c*e)^(1/3)*(d*x + c)^(2/3)*sin((a*d*x + a*c + ( d*x + c)^(2/3)*b)/(d*x + c)))/(b^2*d^2*e^2*x + b^2*c*d*e^2)
\[ \int \frac {\sin \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{(c e+d e x)^{5/3}} \, dx=\int \frac {\sin {\left (a + \frac {b}{\sqrt [3]{c + d x}} \right )}}{\left (e \left (c + d x\right )\right )^{\frac {5}{3}}}\, dx \]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.38 (sec) , antiderivative size = 171, normalized size of antiderivative = 1.88 \[ \int \frac {\sin \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{(c e+d e x)^{5/3}} \, dx=-\frac {3 \, {\left (4 \, b^{2} \sin \left (\frac {{\left (d x + c\right )}^{\frac {1}{3}} a + b}{{\left (d x + c\right )}^{\frac {1}{3}}}\right ) - {\left (d x + c\right )}^{\frac {2}{3}} {\left ({\left (-i \, \Gamma \left (3, i \, b \overline {\frac {1}{{\left (d x + c\right )}^{\frac {1}{3}}}}\right ) + i \, \Gamma \left (3, -i \, b \overline {\frac {1}{{\left (d x + c\right )}^{\frac {1}{3}}}}\right ) - i \, \Gamma \left (3, \frac {i \, b}{{\left (d x + c\right )}^{\frac {1}{3}}}\right ) + i \, \Gamma \left (3, -\frac {i \, b}{{\left (d x + c\right )}^{\frac {1}{3}}}\right )\right )} \cos \left (a\right ) - {\left (\Gamma \left (3, i \, b \overline {\frac {1}{{\left (d x + c\right )}^{\frac {1}{3}}}}\right ) + \Gamma \left (3, -i \, b \overline {\frac {1}{{\left (d x + c\right )}^{\frac {1}{3}}}}\right ) + \Gamma \left (3, \frac {i \, b}{{\left (d x + c\right )}^{\frac {1}{3}}}\right ) + \Gamma \left (3, -\frac {i \, b}{{\left (d x + c\right )}^{\frac {1}{3}}}\right )\right )} \sin \left (a\right )\right )}\right )}}{8 \, {\left (d x + c\right )}^{\frac {2}{3}} b^{2} d e^{\frac {5}{3}}} \]
-3/8*(4*b^2*sin(((d*x + c)^(1/3)*a + b)/(d*x + c)^(1/3)) - (d*x + c)^(2/3) *((-I*gamma(3, I*b*conjugate((d*x + c)^(-1/3))) + I*gamma(3, -I*b*conjugat e((d*x + c)^(-1/3))) - I*gamma(3, I*b/(d*x + c)^(1/3)) + I*gamma(3, -I*b/( d*x + c)^(1/3)))*cos(a) - (gamma(3, I*b*conjugate((d*x + c)^(-1/3))) + gam ma(3, -I*b*conjugate((d*x + c)^(-1/3))) + gamma(3, I*b/(d*x + c)^(1/3)) + gamma(3, -I*b/(d*x + c)^(1/3)))*sin(a)))/((d*x + c)^(2/3)*b^2*d*e^(5/3))
\[ \int \frac {\sin \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{(c e+d e x)^{5/3}} \, dx=\int { \frac {\sin \left (a + \frac {b}{{\left (d x + c\right )}^{\frac {1}{3}}}\right )}{{\left (d e x + c e\right )}^{\frac {5}{3}}} \,d x } \]
Timed out. \[ \int \frac {\sin \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{(c e+d e x)^{5/3}} \, dx=\int \frac {\sin \left (a+\frac {b}{{\left (c+d\,x\right )}^{1/3}}\right )}{{\left (c\,e+d\,e\,x\right )}^{5/3}} \,d x \]