Integrand size = 27, antiderivative size = 217 \[ \int \frac {\sin \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{(c e+d e x)^{8/3}} \, dx=-\frac {36 \cos \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{b^3 d e^2 (e (c+d x))^{2/3}}+\frac {3 \cos \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{b d e^2 (c+d x)^{2/3} (e (c+d x))^{2/3}}+\frac {72 (c+d x)^{2/3} \cos \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{b^5 d e^2 (e (c+d x))^{2/3}}-\frac {12 \sin \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{b^2 d e^2 \sqrt [3]{c+d x} (e (c+d x))^{2/3}}+\frac {72 \sqrt [3]{c+d x} \sin \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{b^4 d e^2 (e (c+d x))^{2/3}} \]
-36*cos(a+b/(d*x+c)^(1/3))/b^3/d/e^2/(e*(d*x+c))^(2/3)+3*cos(a+b/(d*x+c)^( 1/3))/b/d/e^2/(d*x+c)^(2/3)/(e*(d*x+c))^(2/3)+72*(d*x+c)^(2/3)*cos(a+b/(d* x+c)^(1/3))/b^5/d/e^2/(e*(d*x+c))^(2/3)-12*sin(a+b/(d*x+c)^(1/3))/b^2/d/e^ 2/(d*x+c)^(1/3)/(e*(d*x+c))^(2/3)+72*(d*x+c)^(1/3)*sin(a+b/(d*x+c)^(1/3))/ b^4/d/e^2/(e*(d*x+c))^(2/3)
Time = 0.26 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.52 \[ \int \frac {\sin \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{(c e+d e x)^{8/3}} \, dx=\frac {(c+d x)^{4/3} \left (3 \left (b^4-12 b^2 (c+d x)^{2/3}+24 (c+d x)^{4/3}\right ) \cos \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )+12 b \left (6 c+6 d x-b^2 \sqrt [3]{c+d x}\right ) \sin \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )\right )}{b^5 d (e (c+d x))^{8/3}} \]
((c + d*x)^(4/3)*(3*(b^4 - 12*b^2*(c + d*x)^(2/3) + 24*(c + d*x)^(4/3))*Co s[a + b/(c + d*x)^(1/3)] + 12*b*(6*c + 6*d*x - b^2*(c + d*x)^(1/3))*Sin[a + b/(c + d*x)^(1/3)]))/(b^5*d*(e*(c + d*x))^(8/3))
Time = 0.66 (sec) , antiderivative size = 173, normalized size of antiderivative = 0.80, 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.
| \(\displaystyle \int \frac {\sin \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{(c e+d e x)^{8/3}} \, dx\) |
| \(\Big \downarrow \) 3912 |
| \(\displaystyle -\frac {3 \int \frac {(c+d x)^{4/3} \sin \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{(e (c+d x))^{8/3}}d\frac {1}{\sqrt [3]{c+d x}}}{d}\) |
| \(\Big \downarrow \) 30 |
| \(\displaystyle -\frac {3 (c+d x)^{2/3} \int \frac {\sin \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{(c+d x)^{4/3}}d\frac {1}{\sqrt [3]{c+d x}}}{d e^2 (e (c+d x))^{2/3}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {3 (c+d x)^{2/3} \int \frac {\sin \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{(c+d x)^{4/3}}d\frac {1}{\sqrt [3]{c+d x}}}{d e^2 (e (c+d x))^{2/3}}\) |
| \(\Big \downarrow \) 3777 |
| \(\displaystyle -\frac {3 (c+d x)^{2/3} \left (\frac {4 \int \frac {\cos \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{c+d x}d\frac {1}{\sqrt [3]{c+d x}}}{b}-\frac {\cos \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{b (c+d x)^{4/3}}\right )}{d e^2 (e (c+d x))^{2/3}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {3 (c+d x)^{2/3} \left (\frac {4 \int \frac {\sin \left (a+\frac {b}{\sqrt [3]{c+d x}}+\frac {\pi }{2}\right )}{c+d x}d\frac {1}{\sqrt [3]{c+d x}}}{b}-\frac {\cos \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{b (c+d x)^{4/3}}\right )}{d e^2 (e (c+d x))^{2/3}}\) |
| \(\Big \downarrow \) 3777 |
| \(\displaystyle -\frac {3 (c+d x)^{2/3} \left (\frac {4 \left (\frac {3 \int -\frac {\sin \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{(c+d x)^{2/3}}d\frac {1}{\sqrt [3]{c+d x}}}{b}+\frac {\sin \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{b (c+d x)}\right )}{b}-\frac {\cos \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{b (c+d x)^{4/3}}\right )}{d e^2 (e (c+d x))^{2/3}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {3 (c+d x)^{2/3} \left (\frac {4 \left (\frac {\sin \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{b (c+d x)}-\frac {3 \int \frac {\sin \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{(c+d x)^{2/3}}d\frac {1}{\sqrt [3]{c+d x}}}{b}\right )}{b}-\frac {\cos \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{b (c+d x)^{4/3}}\right )}{d e^2 (e (c+d x))^{2/3}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {3 (c+d x)^{2/3} \left (\frac {4 \left (\frac {\sin \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{b (c+d x)}-\frac {3 \int \frac {\sin \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{(c+d x)^{2/3}}d\frac {1}{\sqrt [3]{c+d x}}}{b}\right )}{b}-\frac {\cos \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{b (c+d x)^{4/3}}\right )}{d e^2 (e (c+d x))^{2/3}}\) |
| \(\Big \downarrow \) 3777 |
| \(\displaystyle -\frac {3 (c+d x)^{2/3} \left (\frac {4 \left (\frac {\sin \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{b (c+d x)}-\frac {3 \left (\frac {2 \int \frac {\cos \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{\sqrt [3]{c+d x}}d\frac {1}{\sqrt [3]{c+d x}}}{b}-\frac {\cos \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{b (c+d x)^{2/3}}\right )}{b}\right )}{b}-\frac {\cos \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{b (c+d x)^{4/3}}\right )}{d e^2 (e (c+d x))^{2/3}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {3 (c+d x)^{2/3} \left (\frac {4 \left (\frac {\sin \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{b (c+d x)}-\frac {3 \left (\frac {2 \int \frac {\sin \left (a+\frac {b}{\sqrt [3]{c+d x}}+\frac {\pi }{2}\right )}{\sqrt [3]{c+d x}}d\frac {1}{\sqrt [3]{c+d x}}}{b}-\frac {\cos \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{b (c+d x)^{2/3}}\right )}{b}\right )}{b}-\frac {\cos \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{b (c+d x)^{4/3}}\right )}{d e^2 (e (c+d x))^{2/3}}\) |
| \(\Big \downarrow \) 3777 |
| \(\displaystyle -\frac {3 (c+d x)^{2/3} \left (\frac {4 \left (\frac {\sin \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{b (c+d x)}-\frac {3 \left (\frac {2 \left (\frac {\int -\sin \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )d\frac {1}{\sqrt [3]{c+d x}}}{b}+\frac {\sin \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{b \sqrt [3]{c+d x}}\right )}{b}-\frac {\cos \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{b (c+d x)^{2/3}}\right )}{b}\right )}{b}-\frac {\cos \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{b (c+d x)^{4/3}}\right )}{d e^2 (e (c+d x))^{2/3}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {3 (c+d x)^{2/3} \left (\frac {4 \left (\frac {\sin \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{b (c+d x)}-\frac {3 \left (\frac {2 \left (\frac {\sin \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{b \sqrt [3]{c+d x}}-\frac {\int \sin \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )d\frac {1}{\sqrt [3]{c+d x}}}{b}\right )}{b}-\frac {\cos \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{b (c+d x)^{2/3}}\right )}{b}\right )}{b}-\frac {\cos \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{b (c+d x)^{4/3}}\right )}{d e^2 (e (c+d x))^{2/3}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {3 (c+d x)^{2/3} \left (\frac {4 \left (\frac {\sin \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{b (c+d x)}-\frac {3 \left (\frac {2 \left (\frac {\sin \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{b \sqrt [3]{c+d x}}-\frac {\int \sin \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )d\frac {1}{\sqrt [3]{c+d x}}}{b}\right )}{b}-\frac {\cos \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{b (c+d x)^{2/3}}\right )}{b}\right )}{b}-\frac {\cos \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{b (c+d x)^{4/3}}\right )}{d e^2 (e (c+d x))^{2/3}}\) |
| \(\Big \downarrow \) 3118 |
| \(\displaystyle -\frac {3 (c+d x)^{2/3} \left (\frac {4 \left (\frac {\sin \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{b (c+d x)}-\frac {3 \left (\frac {2 \left (\frac {\cos \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{b^2}+\frac {\sin \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{b \sqrt [3]{c+d x}}\right )}{b}-\frac {\cos \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{b (c+d x)^{2/3}}\right )}{b}\right )}{b}-\frac {\cos \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{b (c+d x)^{4/3}}\right )}{d e^2 (e (c+d x))^{2/3}}\) |
(-3*(c + d*x)^(2/3)*(-(Cos[a + b/(c + d*x)^(1/3)]/(b*(c + d*x)^(4/3))) + ( 4*(Sin[a + b/(c + d*x)^(1/3)]/(b*(c + d*x)) - (3*(-(Cos[a + b/(c + d*x)^(1 /3)]/(b*(c + d*x)^(2/3))) + (2*(Cos[a + b/(c + d*x)^(1/3)]/b^2 + Sin[a + b /(c + d*x)^(1/3)]/(b*(c + d*x)^(1/3))))/b))/b))/b))/(d*e^2*(e*(c + d*x))^( 2/3))
3.3.48.3.1 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]
\[\int \frac {\sin \left (a +\frac {b}{\left (d x +c \right )^{\frac {1}{3}}}\right )}{\left (d e x +c e \right )^{\frac {8}{3}}}d x\]
Time = 0.78 (sec) , antiderivative size = 181, normalized size of antiderivative = 0.83 \[ \int \frac {\sin \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{(c e+d e x)^{8/3}} \, dx=\frac {3 \, {\left ({\left ({\left (d x + c\right )}^{\frac {1}{3}} b^{4} - 12 \, b^{2} d x - 12 \, b^{2} c + 24 \, {\left (d x + c\right )}^{\frac {5}{3}}\right )} {\left (d e x + c e\right )}^{\frac {1}{3}} \cos \left (\frac {a d x + a c + {\left (d x + c\right )}^{\frac {2}{3}} b}{d x + c}\right ) - 4 \, {\left ({\left (d x + c\right )}^{\frac {2}{3}} b^{3} - 6 \, {\left (b d x + b c\right )} {\left (d x + c\right )}^{\frac {1}{3}}\right )} {\left (d e x + c e\right )}^{\frac {1}{3}} \sin \left (\frac {a d x + a c + {\left (d x + c\right )}^{\frac {2}{3}} b}{d x + c}\right )\right )}}{b^{5} d^{3} e^{3} x^{2} + 2 \, b^{5} c d^{2} e^{3} x + b^{5} c^{2} d e^{3}} \]
3*(((d*x + c)^(1/3)*b^4 - 12*b^2*d*x - 12*b^2*c + 24*(d*x + c)^(5/3))*(d*e *x + c*e)^(1/3)*cos((a*d*x + a*c + (d*x + c)^(2/3)*b)/(d*x + c)) - 4*((d*x + c)^(2/3)*b^3 - 6*(b*d*x + b*c)*(d*x + c)^(1/3))*(d*e*x + c*e)^(1/3)*sin ((a*d*x + a*c + (d*x + c)^(2/3)*b)/(d*x + c)))/(b^5*d^3*e^3*x^2 + 2*b^5*c* d^2*e^3*x + b^5*c^2*d*e^3)
Timed out. \[ \int \frac {\sin \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{(c e+d e x)^{8/3}} \, dx=\text {Timed out} \]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 1.67 (sec) , antiderivative size = 1943, normalized size of antiderivative = 8.95 \[ \int \frac {\sin \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{(c e+d e x)^{8/3}} \, dx=\text {Too large to display} \]
-3/20*(2*((cos(a)^2 + sin(a)^2)*b^5*sin(((d*x + c)^(1/3)*a + b)/(d*x + c)^ (1/3)) - (b^5*cos(((d*x + c)^(1/3)*a + b)/(d*x + c)^(1/3))^2*sin(a) + b^5* sin(a)*sin(((d*x + c)^(1/3)*a + b)/(d*x + c)^(1/3))^2)*cos((2*(d*x + c)^(1 /3)*a + b)/(d*x + c)^(1/3)) + (b^5*cos(a)*cos(((d*x + c)^(1/3)*a + b)/(d*x + c)^(1/3))^2 + b^5*cos(a)*sin(((d*x + c)^(1/3)*a + b)/(d*x + c)^(1/3))^2 )*sin((2*(d*x + c)^(1/3)*a + b)/(d*x + c)^(1/3)))*(d*x + c)^(1/3)*e^(1/3) - ((((gamma(6, I*b*conjugate((d*x + c)^(-1/3))) + gamma(6, -I*b*conjugate( (d*x + c)^(-1/3))) + gamma(6, I*b/(d*x + c)^(1/3)) + gamma(6, -I*b/(d*x + c)^(1/3)))*cos(a)^3 + (-I*gamma(6, I*b*conjugate((d*x + c)^(-1/3))) + I*ga mma(6, -I*b*conjugate((d*x + c)^(-1/3))) - I*gamma(6, I*b/(d*x + c)^(1/3)) + I*gamma(6, -I*b/(d*x + c)^(1/3)))*cos(a)^2*sin(a) + (gamma(6, I*b*conju gate((d*x + c)^(-1/3))) + gamma(6, -I*b*conjugate((d*x + c)^(-1/3))) + gam ma(6, I*b/(d*x + c)^(1/3)) + gamma(6, -I*b/(d*x + c)^(1/3)))*cos(a)*sin(a) ^2 + (-I*gamma(6, I*b*conjugate((d*x + c)^(-1/3))) + I*gamma(6, -I*b*conju gate((d*x + c)^(-1/3))) - I*gamma(6, I*b/(d*x + c)^(1/3)) + I*gamma(6, -I* b/(d*x + c)^(1/3)))*sin(a)^3)*d^2*x^2 + 2*((gamma(6, I*b*conjugate((d*x + c)^(-1/3))) + gamma(6, -I*b*conjugate((d*x + c)^(-1/3))) + gamma(6, I*b/(d *x + c)^(1/3)) + gamma(6, -I*b/(d*x + c)^(1/3)))*cos(a)^3 + (-I*gamma(6, I *b*conjugate((d*x + c)^(-1/3))) + I*gamma(6, -I*b*conjugate((d*x + c)^(-1/ 3))) - I*gamma(6, I*b/(d*x + c)^(1/3)) + I*gamma(6, -I*b/(d*x + c)^(1/3...
\[ \int \frac {\sin \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{(c e+d e x)^{8/3}} \, dx=\int { \frac {\sin \left (a + \frac {b}{{\left (d x + c\right )}^{\frac {1}{3}}}\right )}{{\left (d e x + c e\right )}^{\frac {8}{3}}} \,d x } \]
Timed out. \[ \int \frac {\sin \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{(c e+d e x)^{8/3}} \, dx=\int \frac {\sin \left (a+\frac {b}{{\left (c+d\,x\right )}^{1/3}}\right )}{{\left (c\,e+d\,e\,x\right )}^{8/3}} \,d x \]