Integrand size = 27, antiderivative size = 95 \[ \int \frac {\sin \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{(c e+d e x)^{7/3}} \, dx=\frac {3 \cos \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{2 b d e^2 \sqrt [3]{c+d x} \sqrt [3]{e (c+d x)}}-\frac {3 \sqrt [3]{c+d x} \sin \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{2 b^2 d e^2 \sqrt [3]{e (c+d x)}} \]
3/2*cos(a+b/(d*x+c)^(2/3))/b/d/e^2/(d*x+c)^(1/3)/(e*(d*x+c))^(1/3)-3/2*(d* x+c)^(1/3)*sin(a+b/(d*x+c)^(2/3))/b^2/d/e^2/(e*(d*x+c))^(1/3)
Time = 0.07 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.76 \[ \int \frac {\sin \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{(c e+d e x)^{7/3}} \, dx=-\frac {3 (c+d x)^{5/3} \left (-b \cos \left (a+\frac {b}{(c+d x)^{2/3}}\right )+(c+d x)^{2/3} \sin \left (a+\frac {b}{(c+d x)^{2/3}}\right )\right )}{2 b^2 d (e (c+d x))^{7/3}} \]
(-3*(c + d*x)^(5/3)*(-(b*Cos[a + b/(c + d*x)^(2/3)]) + (c + d*x)^(2/3)*Sin [a + b/(c + d*x)^(2/3)]))/(2*b^2*d*(e*(c + d*x))^(7/3))
Time = 0.41 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.81, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.259, Rules used = {3916, 3862, 3860, 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.
\(\displaystyle \int \frac {\sin \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{(c e+d e x)^{7/3}} \, dx\) |
\(\Big \downarrow \) 3916 |
\(\displaystyle \frac {\int \frac {\sin \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{(e (c+d x))^{7/3}}d(c+d x)}{d}\) |
\(\Big \downarrow \) 3862 |
\(\displaystyle \frac {\sqrt [3]{c+d x} \int \frac {\sin \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{(c+d x)^{7/3}}d(c+d x)}{d e^2 \sqrt [3]{e (c+d x)}}\) |
\(\Big \downarrow \) 3860 |
\(\displaystyle -\frac {3 \sqrt [3]{c+d x} \int \frac {\sin \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{(c+d x)^{2/3}}d\frac {1}{(c+d x)^{2/3}}}{2 d e^2 \sqrt [3]{e (c+d x)}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {3 \sqrt [3]{c+d x} \int \frac {\sin \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{(c+d x)^{2/3}}d\frac {1}{(c+d x)^{2/3}}}{2 d e^2 \sqrt [3]{e (c+d x)}}\) |
\(\Big \downarrow \) 3777 |
\(\displaystyle -\frac {3 \sqrt [3]{c+d x} \left (\frac {\int \cos \left (a+\frac {b}{(c+d x)^{2/3}}\right )d\frac {1}{(c+d x)^{2/3}}}{b}-\frac {\cos \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{b (c+d x)^{2/3}}\right )}{2 d e^2 \sqrt [3]{e (c+d x)}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {3 \sqrt [3]{c+d x} \left (\frac {\int \sin \left (a+\frac {b}{(c+d x)^{2/3}}+\frac {\pi }{2}\right )d\frac {1}{(c+d x)^{2/3}}}{b}-\frac {\cos \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{b (c+d x)^{2/3}}\right )}{2 d e^2 \sqrt [3]{e (c+d x)}}\) |
\(\Big \downarrow \) 3117 |
\(\displaystyle -\frac {3 \sqrt [3]{c+d x} \left (\frac {\sin \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{b^2}-\frac {\cos \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{b (c+d x)^{2/3}}\right )}{2 d e^2 \sqrt [3]{e (c+d x)}}\) |
(-3*(c + d*x)^(1/3)*(-(Cos[a + b/(c + d*x)^(2/3)]/(b*(c + d*x)^(2/3))) + S in[a + b/(c + d*x)^(2/3)]/b^2))/(2*d*e^2*(e*(c + d*x))^(1/3))
3.3.56.3.1 Defintions of rubi rules used
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[(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]
\[\int \frac {\sin \left (a +\frac {b}{\left (d x +c \right )^{\frac {2}{3}}}\right )}{\left (d e x +c e \right )^{\frac {7}{3}}}d x\]
Time = 0.72 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.40 \[ \int \frac {\sin \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{(c e+d e x)^{7/3}} \, dx=\frac {3 \, {\left ({\left (d e x + c e\right )}^{\frac {2}{3}} {\left (d x + c\right )}^{\frac {2}{3}} b \cos \left (\frac {a d x + a c + {\left (d x + c\right )}^{\frac {1}{3}} b}{d x + c}\right ) - {\left (d e x + c e\right )}^{\frac {2}{3}} {\left (d x + c\right )}^{\frac {4}{3}} \sin \left (\frac {a d x + a c + {\left (d x + c\right )}^{\frac {1}{3}} b}{d x + c}\right )\right )}}{2 \, {\left (b^{2} d^{3} e^{3} x^{2} + 2 \, b^{2} c d^{2} e^{3} x + b^{2} c^{2} d e^{3}\right )}} \]
3/2*((d*e*x + c*e)^(2/3)*(d*x + c)^(2/3)*b*cos((a*d*x + a*c + (d*x + c)^(1 /3)*b)/(d*x + c)) - (d*e*x + c*e)^(2/3)*(d*x + c)^(4/3)*sin((a*d*x + a*c + (d*x + c)^(1/3)*b)/(d*x + c)))/(b^2*d^3*e^3*x^2 + 2*b^2*c*d^2*e^3*x + b^2 *c^2*d*e^3)
Timed out. \[ \int \frac {\sin \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{(c e+d e x)^{7/3}} \, dx=\text {Timed out} \]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.38 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.36 \[ \int \frac {\sin \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{(c e+d e x)^{7/3}} \, dx=\frac {3 \, {\left ({\left (-i \, \Gamma \left (2, i \, b \overline {\frac {1}{{\left (d x + c\right )}^{\frac {2}{3}}}}\right ) + i \, \Gamma \left (2, -i \, b \overline {\frac {1}{{\left (d x + c\right )}^{\frac {2}{3}}}}\right ) - i \, \Gamma \left (2, \frac {i \, b}{{\left (d x + c\right )}^{\frac {2}{3}}}\right ) + i \, \Gamma \left (2, -\frac {i \, b}{{\left (d x + c\right )}^{\frac {2}{3}}}\right )\right )} \cos \left (a\right ) - {\left (\Gamma \left (2, i \, b \overline {\frac {1}{{\left (d x + c\right )}^{\frac {2}{3}}}}\right ) + \Gamma \left (2, -i \, b \overline {\frac {1}{{\left (d x + c\right )}^{\frac {2}{3}}}}\right ) + \Gamma \left (2, \frac {i \, b}{{\left (d x + c\right )}^{\frac {2}{3}}}\right ) + \Gamma \left (2, -\frac {i \, b}{{\left (d x + c\right )}^{\frac {2}{3}}}\right )\right )} \sin \left (a\right )\right )}}{8 \, b^{2} d e^{\frac {7}{3}}} \]
3/8*((-I*gamma(2, I*b*conjugate((d*x + c)^(-2/3))) + I*gamma(2, -I*b*conju gate((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*d*e^(7/3))
\[ \int \frac {\sin \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{(c e+d e x)^{7/3}} \, dx=\int { \frac {\sin \left (a + \frac {b}{{\left (d x + c\right )}^{\frac {2}{3}}}\right )}{{\left (d e x + c e\right )}^{\frac {7}{3}}} \,d x } \]
Timed out. \[ \int \frac {\sin \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{(c e+d e x)^{7/3}} \, dx=\int \frac {\sin \left (a+\frac {b}{{\left (c+d\,x\right )}^{2/3}}\right )}{{\left (c\,e+d\,e\,x\right )}^{7/3}} \,d x \]