Integrand size = 20, antiderivative size = 139 \[ \int \frac {\sqrt [3]{c \sin ^3\left (a+b x^n\right )}}{x^2} \, dx=\frac {i e^{i a} \left (-i b x^n\right )^{\frac {1}{n}} \csc \left (a+b x^n\right ) \Gamma \left (-\frac {1}{n},-i b x^n\right ) \sqrt [3]{c \sin ^3\left (a+b x^n\right )}}{2 n x}-\frac {i e^{-i a} \left (i b x^n\right )^{\frac {1}{n}} \csc \left (a+b x^n\right ) \Gamma \left (-\frac {1}{n},i b x^n\right ) \sqrt [3]{c \sin ^3\left (a+b x^n\right )}}{2 n x} \]
1/2*I*exp(I*a)*(-I*b*x^n)^(1/n)*csc(a+b*x^n)*GAMMA(-1/n,-I*b*x^n)*(c*sin(a +b*x^n)^3)^(1/3)/n/x-1/2*I*(I*b*x^n)^(1/n)*csc(a+b*x^n)*GAMMA(-1/n,I*b*x^n )*(c*sin(a+b*x^n)^3)^(1/3)/exp(I*a)/n/x
Time = 0.20 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.79 \[ \int \frac {\sqrt [3]{c \sin ^3\left (a+b x^n\right )}}{x^2} \, dx=\frac {i \csc \left (a+b x^n\right ) \left (-\left (i b x^n\right )^{\frac {1}{n}} \Gamma \left (-\frac {1}{n},i b x^n\right ) (\cos (a)-i \sin (a))+\left (-i b x^n\right )^{\frac {1}{n}} \Gamma \left (-\frac {1}{n},-i b x^n\right ) (\cos (a)+i \sin (a))\right ) \sqrt [3]{c \sin ^3\left (a+b x^n\right )}}{2 n x} \]
((I/2)*Csc[a + b*x^n]*(-((I*b*x^n)^n^(-1)*Gamma[-n^(-1), I*b*x^n]*(Cos[a] - I*Sin[a])) + ((-I)*b*x^n)^n^(-1)*Gamma[-n^(-1), (-I)*b*x^n]*(Cos[a] + I* Sin[a]))*(c*Sin[a + b*x^n]^3)^(1/3))/(n*x)
Time = 0.42 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.83, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {7271, 3904, 2648}
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.
\(\displaystyle \int \frac {\sqrt [3]{c \sin ^3\left (a+b x^n\right )}}{x^2} \, dx\) |
\(\Big \downarrow \) 7271 |
\(\displaystyle \csc \left (a+b x^n\right ) \sqrt [3]{c \sin ^3\left (a+b x^n\right )} \int \frac {\sin \left (b x^n+a\right )}{x^2}dx\) |
\(\Big \downarrow \) 3904 |
\(\displaystyle \csc \left (a+b x^n\right ) \sqrt [3]{c \sin ^3\left (a+b x^n\right )} \left (\frac {1}{2} i \int \frac {e^{-i b x^n-i a}}{x^2}dx-\frac {1}{2} i \int \frac {e^{i b x^n+i a}}{x^2}dx\right )\) |
\(\Big \downarrow \) 2648 |
\(\displaystyle \csc \left (a+b x^n\right ) \left (\frac {i e^{i a} \left (-i b x^n\right )^{\frac {1}{n}} \Gamma \left (-\frac {1}{n},-i b x^n\right )}{2 n x}-\frac {i e^{-i a} \left (i b x^n\right )^{\frac {1}{n}} \Gamma \left (-\frac {1}{n},i b x^n\right )}{2 n x}\right ) \sqrt [3]{c \sin ^3\left (a+b x^n\right )}\) |
Csc[a + b*x^n]*(((I/2)*E^(I*a)*((-I)*b*x^n)^n^(-1)*Gamma[-n^(-1), (-I)*b*x ^n])/(n*x) - ((I/2)*(I*b*x^n)^n^(-1)*Gamma[-n^(-1), I*b*x^n])/(E^(I*a)*n*x ))*(c*Sin[a + b*x^n]^3)^(1/3)
3.4.32.3.1 Defintions of rubi rules used
Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*((e_.) + (f_.)*(x_))^(m_ .), x_Symbol] :> Simp[(-F^a)*((e + f*x)^(m + 1)/(f*n*((-b)*(c + d*x)^n*Log[ F])^((m + 1)/n)))*Gamma[(m + 1)/n, (-b)*(c + d*x)^n*Log[F]], x] /; FreeQ[{F , a, b, c, d, e, f, m, n}, x] && EqQ[d*e - c*f, 0]
Int[((e_.)*(x_))^(m_.)*Sin[(c_.) + (d_.)*(x_)^(n_)], x_Symbol] :> Simp[I/2 Int[(e*x)^m*E^((-c)*I - d*I*x^n), x], x] - Simp[I/2 Int[(e*x)^m*E^(c*I + d*I*x^n), x], x] /; FreeQ[{c, d, e, m, n}, x]
Int[(u_.)*((a_.)*(v_)^(m_.))^(p_), x_Symbol] :> Simp[a^IntPart[p]*((a*v^m)^ FracPart[p]/v^(m*FracPart[p])) Int[u*v^(m*p), x], x] /; FreeQ[{a, m, p}, x] && !IntegerQ[p] && !FreeQ[v, x] && !(EqQ[a, 1] && EqQ[m, 1]) && !(Eq Q[v, x] && EqQ[m, 1])
\[\int \frac {{\left (c \left (\sin ^{3}\left (a +b \,x^{n}\right )\right )\right )}^{\frac {1}{3}}}{x^{2}}d x\]
\[ \int \frac {\sqrt [3]{c \sin ^3\left (a+b x^n\right )}}{x^2} \, dx=\int { \frac {\left (c \sin \left (b x^{n} + a\right )^{3}\right )^{\frac {1}{3}}}{x^{2}} \,d x } \]
\[ \int \frac {\sqrt [3]{c \sin ^3\left (a+b x^n\right )}}{x^2} \, dx=\int \frac {\sqrt [3]{c \sin ^{3}{\left (a + b x^{n} \right )}}}{x^{2}}\, dx \]
\[ \int \frac {\sqrt [3]{c \sin ^3\left (a+b x^n\right )}}{x^2} \, dx=\int { \frac {\left (c \sin \left (b x^{n} + a\right )^{3}\right )^{\frac {1}{3}}}{x^{2}} \,d x } \]
\[ \int \frac {\sqrt [3]{c \sin ^3\left (a+b x^n\right )}}{x^2} \, dx=\int { \frac {\left (c \sin \left (b x^{n} + a\right )^{3}\right )^{\frac {1}{3}}}{x^{2}} \,d x } \]
Timed out. \[ \int \frac {\sqrt [3]{c \sin ^3\left (a+b x^n\right )}}{x^2} \, dx=\int \frac {{\left (c\,{\sin \left (a+b\,x^n\right )}^3\right )}^{1/3}}{x^2} \,d x \]