Integrand size = 20, antiderivative size = 217 \[ \int x^m \left (c \sin ^3\left (a+b x^n\right )\right )^{2/3} \, dx=\frac {x^{1+m} \csc ^2\left (a+b x^n\right ) \left (c \sin ^3\left (a+b x^n\right )\right )^{2/3}}{2 (1+m)}+\frac {2^{-\frac {1+m+2 n}{n}} e^{2 i a} x^{1+m} \left (-i b x^n\right )^{-\frac {1+m}{n}} \csc ^2\left (a+b x^n\right ) \Gamma \left (\frac {1+m}{n},-2 i b x^n\right ) \left (c \sin ^3\left (a+b x^n\right )\right )^{2/3}}{n}+\frac {2^{-\frac {1+m+2 n}{n}} e^{-2 i a} x^{1+m} \left (i b x^n\right )^{-\frac {1+m}{n}} \csc ^2\left (a+b x^n\right ) \Gamma \left (\frac {1+m}{n},2 i b x^n\right ) \left (c \sin ^3\left (a+b x^n\right )\right )^{2/3}}{n} \]
1/2*x^(1+m)*csc(a+b*x^n)^2*(c*sin(a+b*x^n)^3)^(2/3)/(1+m)+exp(2*I*a)*x^(1+ m)*csc(a+b*x^n)^2*GAMMA((1+m)/n,-2*I*b*x^n)*(c*sin(a+b*x^n)^3)^(2/3)/(2^(( 1+m+2*n)/n))/n/((-I*b*x^n)^((1+m)/n))+x^(1+m)*csc(a+b*x^n)^2*GAMMA((1+m)/n ,2*I*b*x^n)*(c*sin(a+b*x^n)^3)^(2/3)/(2^((1+m+2*n)/n))/exp(2*I*a)/n/((I*b* x^n)^((1+m)/n))
Time = 0.73 (sec) , antiderivative size = 194, normalized size of antiderivative = 0.89 \[ \int x^m \left (c \sin ^3\left (a+b x^n\right )\right )^{2/3} \, dx=\frac {2^{-\frac {1+m+2 n}{n}} e^{-2 i a} x^{1+m} \left (b^2 x^{2 n}\right )^{-\frac {1+m}{n}} \csc ^2\left (a+b x^n\right ) \left (2^{\frac {1+m+n}{n}} e^{2 i a} n \left (b^2 x^{2 n}\right )^{\frac {1+m}{n}}+e^{4 i a} (1+m) \left (i b x^n\right )^{\frac {1+m}{n}} \Gamma \left (\frac {1+m}{n},-2 i b x^n\right )+(1+m) \left (-i b x^n\right )^{\frac {1+m}{n}} \Gamma \left (\frac {1+m}{n},2 i b x^n\right )\right ) \left (c \sin ^3\left (a+b x^n\right )\right )^{2/3}}{(1+m) n} \]
(x^(1 + m)*Csc[a + b*x^n]^2*(2^((1 + m + n)/n)*E^((2*I)*a)*n*(b^2*x^(2*n)) ^((1 + m)/n) + E^((4*I)*a)*(1 + m)*(I*b*x^n)^((1 + m)/n)*Gamma[(1 + m)/n, (-2*I)*b*x^n] + (1 + m)*((-I)*b*x^n)^((1 + m)/n)*Gamma[(1 + m)/n, (2*I)*b* x^n])*(c*Sin[a + b*x^n]^3)^(2/3))/(2^((1 + m + 2*n)/n)*E^((2*I)*a)*(1 + m) *n*(b^2*x^(2*n))^((1 + m)/n))
Time = 0.62 (sec) , antiderivative size = 166, normalized size of antiderivative = 0.76, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {7271, 3906, 2009}
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.
\(\displaystyle \int x^m \left (c \sin ^3\left (a+b x^n\right )\right )^{2/3} \, dx\) |
\(\Big \downarrow \) 7271 |
\(\displaystyle \csc ^2\left (a+b x^n\right ) \left (c \sin ^3\left (a+b x^n\right )\right )^{2/3} \int x^m \sin ^2\left (b x^n+a\right )dx\) |
\(\Big \downarrow \) 3906 |
\(\displaystyle \csc ^2\left (a+b x^n\right ) \left (c \sin ^3\left (a+b x^n\right )\right )^{2/3} \int \left (\frac {x^m}{2}-\frac {1}{2} x^m \cos \left (2 b x^n+2 a\right )\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \csc ^2\left (a+b x^n\right ) \left (c \sin ^3\left (a+b x^n\right )\right )^{2/3} \left (\frac {e^{2 i a} 2^{-\frac {m+2 n+1}{n}} x^{m+1} \left (-i b x^n\right )^{-\frac {m+1}{n}} \Gamma \left (\frac {m+1}{n},-2 i b x^n\right )}{n}+\frac {e^{-2 i a} 2^{-\frac {m+2 n+1}{n}} x^{m+1} \left (i b x^n\right )^{-\frac {m+1}{n}} \Gamma \left (\frac {m+1}{n},2 i b x^n\right )}{n}+\frac {x^{m+1}}{2 (m+1)}\right )\) |
Csc[a + b*x^n]^2*(x^(1 + m)/(2*(1 + m)) + (E^((2*I)*a)*x^(1 + m)*Gamma[(1 + m)/n, (-2*I)*b*x^n])/(2^((1 + m + 2*n)/n)*n*((-I)*b*x^n)^((1 + m)/n)) + (x^(1 + m)*Gamma[(1 + m)/n, (2*I)*b*x^n])/(2^((1 + m + 2*n)/n)*E^((2*I)*a) *n*(I*b*x^n)^((1 + m)/n)))*(c*Sin[a + b*x^n]^3)^(2/3)
3.4.50.3.1 Defintions of rubi rules used
Int[((e_.)*(x_))^(m_.)*((a_.) + (b_.)*Sin[(c_.) + (d_.)*(x_)^(n_)])^(p_), x _Symbol] :> Int[ExpandTrigReduce[(e*x)^m, (a + b*Sin[c + d*x^n])^p, x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && IGtQ[p, 0]
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])
\[\int x^{m} {\left (c \left (\sin ^{3}\left (a +b \,x^{n}\right )\right )\right )}^{\frac {2}{3}}d x\]
\[ \int x^m \left (c \sin ^3\left (a+b x^n\right )\right )^{2/3} \, dx=\int { \left (c \sin \left (b x^{n} + a\right )^{3}\right )^{\frac {2}{3}} x^{m} \,d x } \]
Timed out. \[ \int x^m \left (c \sin ^3\left (a+b x^n\right )\right )^{2/3} \, dx=\text {Timed out} \]
\[ \int x^m \left (c \sin ^3\left (a+b x^n\right )\right )^{2/3} \, dx=\int { \left (c \sin \left (b x^{n} + a\right )^{3}\right )^{\frac {2}{3}} x^{m} \,d x } \]
\[ \int x^m \left (c \sin ^3\left (a+b x^n\right )\right )^{2/3} \, dx=\int { \left (c \sin \left (b x^{n} + a\right )^{3}\right )^{\frac {2}{3}} x^{m} \,d x } \]
Timed out. \[ \int x^m \left (c \sin ^3\left (a+b x^n\right )\right )^{2/3} \, dx=\int x^m\,{\left (c\,{\sin \left (a+b\,x^n\right )}^3\right )}^{2/3} \,d x \]