Integrand size = 24, antiderivative size = 88 \[ \int x^2 \sin \left (a+3 \sqrt {-\frac {1}{n^2}} \log \left (c x^n\right )\right ) \, dx=\frac {1}{12} e^{-a \sqrt {-\frac {1}{n^2}} n} \sqrt {-\frac {1}{n^2}} n x^3 \left (c x^n\right )^{3/n}-\frac {1}{2} e^{a \sqrt {-\frac {1}{n^2}} n} \sqrt {-\frac {1}{n^2}} n x^3 \left (c x^n\right )^{-3/n} \log (x) \]
1/12*n*x^3*(c*x^n)^(3/n)*(-1/n^2)^(1/2)/exp(a*n*(-1/n^2)^(1/2))-1/2*exp(a* n*(-1/n^2)^(1/2))*n*x^3*ln(x)*(-1/n^2)^(1/2)/((c*x^n)^(3/n))
\[ \int x^2 \sin \left (a+3 \sqrt {-\frac {1}{n^2}} \log \left (c x^n\right )\right ) \, dx=\int x^2 \sin \left (a+3 \sqrt {-\frac {1}{n^2}} \log \left (c x^n\right )\right ) \, dx \]
Time = 0.31 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.91, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {4996, 4992, 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.
\(\displaystyle \int x^2 \sin \left (a+3 \sqrt {-\frac {1}{n^2}} \log \left (c x^n\right )\right ) \, dx\) |
\(\Big \downarrow \) 4996 |
\(\displaystyle \frac {x^3 \left (c x^n\right )^{-3/n} \int \left (c x^n\right )^{\frac {3}{n}-1} \sin \left (a+3 \sqrt {-\frac {1}{n^2}} \log \left (c x^n\right )\right )d\left (c x^n\right )}{n}\) |
\(\Big \downarrow \) 4992 |
\(\displaystyle -\frac {1}{2} \sqrt {-\frac {1}{n^2}} x^3 \left (c x^n\right )^{-3/n} \int \left (\frac {e^{a \sqrt {-\frac {1}{n^2}} n} x^{-n}}{c}-e^{-a \sqrt {-\frac {1}{n^2}} n} \left (c x^n\right )^{\frac {6}{n}-1}\right )d\left (c x^n\right )\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {1}{2} \sqrt {-\frac {1}{n^2}} x^3 \left (c x^n\right )^{-3/n} \left (e^{a \sqrt {-\frac {1}{n^2}} n} \log \left (c x^n\right )-\frac {1}{6} n e^{-a \sqrt {-\frac {1}{n^2}} n} \left (c x^n\right )^{6/n}\right )\) |
-1/2*(Sqrt[-n^(-2)]*x^3*(-1/6*(n*(c*x^n)^(6/n))/E^(a*Sqrt[-n^(-2)]*n) + E^ (a*Sqrt[-n^(-2)]*n)*Log[c*x^n]))/(c*x^n)^(3/n)
3.1.27.3.1 Defintions of rubi rules used
Int[((e_.)*(x_))^(m_.)*Sin[((a_.) + Log[x_]*(b_.))*(d_.)]^(p_.), x_Symbol] :> Simp[(m + 1)^p/(2^p*b^p*d^p*p^p) Int[ExpandIntegrand[(e*x)^m*(E^(a*b*d ^2*(p/(m + 1)))/x^((m + 1)/p) - x^((m + 1)/p)/E^(a*b*d^2*(p/(m + 1))))^p, x ], x], x] /; FreeQ[{a, b, d, e, m}, x] && IGtQ[p, 0] && EqQ[b^2*d^2*p^2 + ( m + 1)^2, 0]
Int[((e_.)*(x_))^(m_.)*Sin[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*(d_.)]^(p_ .), x_Symbol] :> Simp[(e*x)^(m + 1)/(e*n*(c*x^n)^((m + 1)/n)) Subst[Int[x ^((m + 1)/n - 1)*Sin[d*(a + b*Log[x])]^p, x], x, c*x^n], x] /; FreeQ[{a, b, c, d, e, m, n, p}, x] && (NeQ[c, 1] || NeQ[n, 1])
Leaf count of result is larger than twice the leaf count of optimal. \(618\) vs. \(2(77)=154\).
Time = 2.67 (sec) , antiderivative size = 619, normalized size of antiderivative = 7.03
method | result | size |
parts | \(\frac {3 n \,x^{2} \sqrt {-\frac {1}{n^{2}}}\, {\mathrm e}^{\frac {\ln \left (c \,x^{n}\right )}{n}-\frac {\ln \left (c \right )}{n}} \cos \left (a +3 \ln \left (c \,x^{n}\right ) \sqrt {-\frac {1}{n^{2}}}\right )}{8}-\frac {x^{2} {\mathrm e}^{\frac {\ln \left (c \,x^{n}\right )}{n}-\frac {\ln \left (c \right )}{n}} \sin \left (a +3 \ln \left (c \,x^{n}\right ) \sqrt {-\frac {1}{n^{2}}}\right )}{8}-\frac {-\frac {n \left (-\frac {\sqrt {-\frac {1}{n^{2}}}\, n \,c^{-\frac {1}{n}} {\mathrm e}^{\frac {\ln \left (c \,x^{n}\right )-n \ln \left (x \right )}{n}} x^{3} \ln \left (x \right )}{2}+\frac {\sqrt {-\frac {1}{n^{2}}}\, n \,c^{-\frac {1}{n}} {\mathrm e}^{\frac {\ln \left (c \,x^{n}\right )-n \ln \left (x \right )}{n}} x^{3}}{6}+c^{-\frac {1}{n}} {\mathrm e}^{\frac {\ln \left (c \,x^{n}\right )-n \ln \left (x \right )}{n}} x^{3} \ln \left (x \right ) \tan \left (\frac {a}{2}+\frac {3 \ln \left (c \,x^{n}\right ) \sqrt {-\frac {1}{n^{2}}}}{2}\right )-\frac {n \sqrt {-\frac {1}{n^{2}}}\, {\mathrm e}^{\frac {\ln \left (c \,x^{n}\right )-n \ln \left (x \right )}{n}} c^{-\frac {1}{n}} x^{3} {\tan \left (\frac {a}{2}+\frac {3 \ln \left (c \,x^{n}\right ) \sqrt {-\frac {1}{n^{2}}}}{2}\right )}^{2}}{6}+\frac {\sqrt {-\frac {1}{n^{2}}}\, n \,c^{-\frac {1}{n}} {\mathrm e}^{\frac {\ln \left (c \,x^{n}\right )-n \ln \left (x \right )}{n}} x^{3} \ln \left (x \right ) {\tan \left (\frac {a}{2}+\frac {3 \ln \left (c \,x^{n}\right ) \sqrt {-\frac {1}{n^{2}}}}{2}\right )}^{2}}{2}\right )}{1+{\tan \left (\frac {a}{2}+\frac {3 \ln \left (c \,x^{n}\right ) \sqrt {-\frac {1}{n^{2}}}}{2}\right )}^{2}}+\frac {3 \sqrt {-\frac {1}{n^{2}}}\, n^{2} \left (\frac {c^{-\frac {1}{n}} {\mathrm e}^{\frac {\ln \left (c \,x^{n}\right )-n \ln \left (x \right )}{n}} x^{3} \ln \left (x \right )}{2}+\frac {c^{-\frac {1}{n}} {\mathrm e}^{\frac {\ln \left (c \,x^{n}\right )-n \ln \left (x \right )}{n}} x^{3} \tan \left (\frac {a}{2}+\frac {3 \ln \left (c \,x^{n}\right ) \sqrt {-\frac {1}{n^{2}}}}{2}\right )}{3 \sqrt {-\frac {1}{n^{2}}}\, n}-\frac {c^{-\frac {1}{n}} {\mathrm e}^{\frac {\ln \left (c \,x^{n}\right )-n \ln \left (x \right )}{n}} x^{3} \ln \left (x \right ) {\tan \left (\frac {a}{2}+\frac {3 \ln \left (c \,x^{n}\right ) \sqrt {-\frac {1}{n^{2}}}}{2}\right )}^{2}}{2}-\frac {c^{-\frac {1}{n}} {\mathrm e}^{\frac {\ln \left (c \,x^{n}\right )-n \ln \left (x \right )}{n}} x^{3} \ln \left (x \right ) \tan \left (\frac {a}{2}+\frac {3 \ln \left (c \,x^{n}\right ) \sqrt {-\frac {1}{n^{2}}}}{2}\right )}{\sqrt {-\frac {1}{n^{2}}}\, n}\right )}{1+{\tan \left (\frac {a}{2}+\frac {3 \ln \left (c \,x^{n}\right ) \sqrt {-\frac {1}{n^{2}}}}{2}\right )}^{2}}}{4 n}\) | \(619\) |
3/8*n*x^2*(-1/n^2)^(1/2)*exp(1/n*ln(c*x^n)-1/n*ln(c))*cos(a+3*ln(c*x^n)*(- 1/n^2)^(1/2))-1/8*x^2*exp(1/n*ln(c*x^n)-1/n*ln(c))*sin(a+3*ln(c*x^n)*(-1/n ^2)^(1/2))-1/4/n*(-n*(-1/2*(-1/n^2)^(1/2)*n/(c^(1/n))*exp(1/n*(ln(c*x^n)-n *ln(x)))*x^3*ln(x)+1/6*(-1/n^2)^(1/2)*n/(c^(1/n))*exp(1/n*(ln(c*x^n)-n*ln( x)))*x^3+1/(c^(1/n))*exp(1/n*(ln(c*x^n)-n*ln(x)))*x^3*ln(x)*tan(1/2*a+3/2* ln(c*x^n)*(-1/n^2)^(1/2))-1/6*n*(-1/n^2)^(1/2)*exp(1/n*(ln(c*x^n)-n*ln(x)) )/(c^(1/n))*x^3*tan(1/2*a+3/2*ln(c*x^n)*(-1/n^2)^(1/2))^2+1/2*(-1/n^2)^(1/ 2)*n/(c^(1/n))*exp(1/n*(ln(c*x^n)-n*ln(x)))*x^3*ln(x)*tan(1/2*a+3/2*ln(c*x ^n)*(-1/n^2)^(1/2))^2)/(1+tan(1/2*a+3/2*ln(c*x^n)*(-1/n^2)^(1/2))^2)+3*(-1 /n^2)^(1/2)*n^2*(1/2/(c^(1/n))*exp(1/n*(ln(c*x^n)-n*ln(x)))*x^3*ln(x)+1/3/ (-1/n^2)^(1/2)/(c^(1/n))/n*exp(1/n*(ln(c*x^n)-n*ln(x)))*x^3*tan(1/2*a+3/2* ln(c*x^n)*(-1/n^2)^(1/2))-1/2/(c^(1/n))*exp(1/n*(ln(c*x^n)-n*ln(x)))*x^3*l n(x)*tan(1/2*a+3/2*ln(c*x^n)*(-1/n^2)^(1/2))^2-1/(-1/n^2)^(1/2)/(c^(1/n))/ n*exp(1/n*(ln(c*x^n)-n*ln(x)))*x^3*ln(x)*tan(1/2*a+3/2*ln(c*x^n)*(-1/n^2)^ (1/2)))/(1+tan(1/2*a+3/2*ln(c*x^n)*(-1/n^2)^(1/2))^2))
Result contains complex when optimal does not.
Time = 0.26 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.48 \[ \int x^2 \sin \left (a+3 \sqrt {-\frac {1}{n^2}} \log \left (c x^n\right )\right ) \, dx=\frac {1}{12} \, {\left (i \, x^{6} - 6 i \, e^{\left (\frac {2 \, {\left (i \, a n - 3 \, \log \left (c\right )\right )}}{n}\right )} \log \left (x\right )\right )} e^{\left (-\frac {i \, a n - 3 \, \log \left (c\right )}{n}\right )} \]
\[ \int x^2 \sin \left (a+3 \sqrt {-\frac {1}{n^2}} \log \left (c x^n\right )\right ) \, dx=\int x^{2} \sin {\left (a + 3 \sqrt {- \frac {1}{n^{2}}} \log {\left (c x^{n} \right )} \right )}\, dx \]
Time = 0.22 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.35 \[ \int x^2 \sin \left (a+3 \sqrt {-\frac {1}{n^2}} \log \left (c x^n\right )\right ) \, dx=\frac {c^{\frac {6}{n}} x^{6} \sin \left (a\right ) + 6 \, \log \left (x\right ) \sin \left (a\right )}{12 \, c^{\frac {3}{n}}} \]
Time = 0.43 (sec) , antiderivative size = 1, normalized size of antiderivative = 0.01 \[ \int x^2 \sin \left (a+3 \sqrt {-\frac {1}{n^2}} \log \left (c x^n\right )\right ) \, dx=+\infty \]
Time = 27.83 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.97 \[ \int x^2 \sin \left (a+3 \sqrt {-\frac {1}{n^2}} \log \left (c x^n\right )\right ) \, dx=-\frac {x^3\,{\mathrm {e}}^{-a\,1{}\mathrm {i}}\,\frac {1}{{\left (c\,x^n\right )}^{\sqrt {-\frac {1}{n^2}}\,3{}\mathrm {i}}}}{6\,n\,\sqrt {-\frac {1}{n^2}}+6{}\mathrm {i}}-\frac {x^3\,{\mathrm {e}}^{a\,1{}\mathrm {i}}\,{\left (c\,x^n\right )}^{\sqrt {-\frac {1}{n^2}}\,3{}\mathrm {i}}}{6\,n\,\sqrt {-\frac {1}{n^2}}-6{}\mathrm {i}} \]