Integrand size = 22, antiderivative size = 88 \[ \int x \sin \left (a+2 \sqrt {-\frac {1}{n^2}} \log \left (c x^n\right )\right ) \, dx=\frac {1}{8} e^{-a \sqrt {-\frac {1}{n^2}} n} \sqrt {-\frac {1}{n^2}} n x^2 \left (c x^n\right )^{2/n}-\frac {1}{2} e^{a \sqrt {-\frac {1}{n^2}} n} \sqrt {-\frac {1}{n^2}} n x^2 \left (c x^n\right )^{-2/n} \log (x) \]
1/8*n*x^2*(c*x^n)^(2/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^2*ln(x)*(-1/n^2)^(1/2)/((c*x^n)^(2/n))
\[ \int x \sin \left (a+2 \sqrt {-\frac {1}{n^2}} \log \left (c x^n\right )\right ) \, dx=\int x \sin \left (a+2 \sqrt {-\frac {1}{n^2}} \log \left (c x^n\right )\right ) \, dx \]
Time = 0.29 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.91, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, 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.
\(\displaystyle \int x \sin \left (a+2 \sqrt {-\frac {1}{n^2}} \log \left (c x^n\right )\right ) \, dx\) |
\(\Big \downarrow \) 4996 |
\(\displaystyle \frac {x^2 \left (c x^n\right )^{-2/n} \int \left (c x^n\right )^{\frac {2}{n}-1} \sin \left (a+2 \sqrt {-\frac {1}{n^2}} \log \left (c x^n\right )\right )d\left (c x^n\right )}{n}\) |
\(\Big \downarrow \) 4992 |
\(\displaystyle -\frac {1}{2} \sqrt {-\frac {1}{n^2}} x^2 \left (c x^n\right )^{-2/n} \int \left (\frac {e^{a \sqrt {-\frac {1}{n^2}} n} x^{-n}}{c}-e^{-a \sqrt {-\frac {1}{n^2}} n} \left (c x^n\right )^{\frac {4}{n}-1}\right )d\left (c x^n\right )\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {1}{2} \sqrt {-\frac {1}{n^2}} x^2 \left (c x^n\right )^{-2/n} \left (e^{a \sqrt {-\frac {1}{n^2}} n} \log \left (c x^n\right )-\frac {1}{4} n e^{-a \sqrt {-\frac {1}{n^2}} n} \left (c x^n\right )^{4/n}\right )\) |
-1/2*(Sqrt[-n^(-2)]*x^2*(-1/4*(n*(c*x^n)^(4/n))/E^(a*Sqrt[-n^(-2)]*n) + E^ (a*Sqrt[-n^(-2)]*n)*Log[c*x^n]))/(c*x^n)^(2/n)
3.1.28.3.1 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. \(609\) vs. \(2(77)=154\).
Time = 1.82 (sec) , antiderivative size = 610, normalized size of antiderivative = 6.93
method | result | size |
parts | \(\frac {2 n x \sqrt {-\frac {1}{n^{2}}}\, {\mathrm e}^{\frac {\ln \left (c \,x^{n}\right )}{n}-\frac {\ln \left (c \right )}{n}} \cos \left (a +2 \ln \left (c \,x^{n}\right ) \sqrt {-\frac {1}{n^{2}}}\right )}{3}-\frac {x \,{\mathrm e}^{\frac {\ln \left (c \,x^{n}\right )}{n}-\frac {\ln \left (c \right )}{n}} \sin \left (a +2 \ln \left (c \,x^{n}\right ) \sqrt {-\frac {1}{n^{2}}}\right )}{3}-\frac {-\frac {n \left (-\frac {c^{-\frac {1}{n}} {\mathrm e}^{\frac {\ln \left (c \,x^{n}\right )-n \ln \left (x \right )}{n}} x^{2}}{4 \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^{2} \ln \left (x \right )}{2 \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^{2} {\tan \left (\frac {a}{2}+\ln \left (c \,x^{n}\right ) \sqrt {-\frac {1}{n^{2}}}\right )}^{2}}{4 \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^{2} \ln \left (x \right ) \tan \left (\frac {a}{2}+\ln \left (c \,x^{n}\right ) \sqrt {-\frac {1}{n^{2}}}\right )-\frac {c^{-\frac {1}{n}} {\mathrm e}^{\frac {\ln \left (c \,x^{n}\right )-n \ln \left (x \right )}{n}} x^{2} \ln \left (x \right ) {\tan \left (\frac {a}{2}+\ln \left (c \,x^{n}\right ) \sqrt {-\frac {1}{n^{2}}}\right )}^{2}}{2 \sqrt {-\frac {1}{n^{2}}}\, n}\right )}{1+{\tan \left (\frac {a}{2}+\ln \left (c \,x^{n}\right ) \sqrt {-\frac {1}{n^{2}}}\right )}^{2}}+\frac {2 \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^{2} \ln \left (x \right ) {\tan \left (\frac {a}{2}+\ln \left (c \,x^{n}\right ) \sqrt {-\frac {1}{n^{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^{2} \ln \left (x \right )}{2}-\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^{2} \tan \left (\frac {a}{2}+\ln \left (c \,x^{n}\right ) \sqrt {-\frac {1}{n^{2}}}\right )}{2}+c^{-\frac {1}{n}} {\mathrm e}^{\frac {\ln \left (c \,x^{n}\right )-n \ln \left (x \right )}{n}} n \sqrt {-\frac {1}{n^{2}}}\, x^{2} \ln \left (x \right ) \tan \left (\frac {a}{2}+\ln \left (c \,x^{n}\right ) \sqrt {-\frac {1}{n^{2}}}\right )\right )}{1+{\tan \left (\frac {a}{2}+\ln \left (c \,x^{n}\right ) \sqrt {-\frac {1}{n^{2}}}\right )}^{2}}}{3 n}\) | \(610\) |
2/3*n*x*(-1/n^2)^(1/2)*exp(1/n*ln(c*x^n)-1/n*ln(c))*cos(a+2*ln(c*x^n)*(-1/ n^2)^(1/2))-1/3*x*exp(1/n*ln(c*x^n)-1/n*ln(c))*sin(a+2*ln(c*x^n)*(-1/n^2)^ (1/2))-1/3/n*(-n*(-1/4/(-1/n^2)^(1/2)/(c^(1/n))/n*exp(1/n*(ln(c*x^n)-n*ln( x)))*x^2+1/2/(-1/n^2)^(1/2)/(c^(1/n))/n*exp(1/n*(ln(c*x^n)-n*ln(x)))*x^2*l n(x)+1/4/(-1/n^2)^(1/2)/(c^(1/n))/n*exp(1/n*(ln(c*x^n)-n*ln(x)))*x^2*tan(1 /2*a+ln(c*x^n)*(-1/n^2)^(1/2))^2+1/(c^(1/n))*exp(1/n*(ln(c*x^n)-n*ln(x)))* x^2*ln(x)*tan(1/2*a+ln(c*x^n)*(-1/n^2)^(1/2))-1/2/(-1/n^2)^(1/2)/(c^(1/n)) /n*exp(1/n*(ln(c*x^n)-n*ln(x)))*x^2*ln(x)*tan(1/2*a+ln(c*x^n)*(-1/n^2)^(1/ 2))^2)/(1+tan(1/2*a+ln(c*x^n)*(-1/n^2)^(1/2))^2)+2*(-1/n^2)^(1/2)*n^2*(-1/ 2/(c^(1/n))*exp(1/n*(ln(c*x^n)-n*ln(x)))*x^2*ln(x)*tan(1/2*a+ln(c*x^n)*(-1 /n^2)^(1/2))^2+1/2/(c^(1/n))*exp(1/n*(ln(c*x^n)-n*ln(x)))*x^2*ln(x)-1/2*n* (-1/n^2)^(1/2)*exp(1/n*(ln(c*x^n)-n*ln(x)))/(c^(1/n))*x^2*tan(1/2*a+ln(c*x ^n)*(-1/n^2)^(1/2))+1/(c^(1/n))*exp(1/n*(ln(c*x^n)-n*ln(x)))*n*(-1/n^2)^(1 /2)*x^2*ln(x)*tan(1/2*a+ln(c*x^n)*(-1/n^2)^(1/2)))/(1+tan(1/2*a+ln(c*x^n)* (-1/n^2)^(1/2))^2))
Result contains complex when optimal does not.
Time = 0.25 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.48 \[ \int x \sin \left (a+2 \sqrt {-\frac {1}{n^2}} \log \left (c x^n\right )\right ) \, dx=\frac {1}{8} \, {\left (i \, x^{4} - 4 i \, e^{\left (\frac {2 \, {\left (i \, a n - 2 \, \log \left (c\right )\right )}}{n}\right )} \log \left (x\right )\right )} e^{\left (-\frac {i \, a n - 2 \, \log \left (c\right )}{n}\right )} \]
\[ \int x \sin \left (a+2 \sqrt {-\frac {1}{n^2}} \log \left (c x^n\right )\right ) \, dx=\int x \sin {\left (a + 2 \sqrt {- \frac {1}{n^{2}}} \log {\left (c x^{n} \right )} \right )}\, dx \]
Time = 0.21 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.35 \[ \int x \sin \left (a+2 \sqrt {-\frac {1}{n^2}} \log \left (c x^n\right )\right ) \, dx=\frac {c^{\frac {4}{n}} x^{4} \sin \left (a\right ) + 4 \, \log \left (x\right ) \sin \left (a\right )}{8 \, c^{\frac {2}{n}}} \]
Time = 0.42 (sec) , antiderivative size = 1, normalized size of antiderivative = 0.01 \[ \int x \sin \left (a+2 \sqrt {-\frac {1}{n^2}} \log \left (c x^n\right )\right ) \, dx=+\infty \]
Time = 27.34 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.97 \[ \int x \sin \left (a+2 \sqrt {-\frac {1}{n^2}} \log \left (c x^n\right )\right ) \, dx=-\frac {x^2\,{\mathrm {e}}^{-a\,1{}\mathrm {i}}\,\frac {1}{{\left (c\,x^n\right )}^{\sqrt {-\frac {1}{n^2}}\,2{}\mathrm {i}}}}{4\,n\,\sqrt {-\frac {1}{n^2}}+4{}\mathrm {i}}-\frac {x^2\,{\mathrm {e}}^{a\,1{}\mathrm {i}}\,{\left (c\,x^n\right )}^{\sqrt {-\frac {1}{n^2}}\,2{}\mathrm {i}}}{4\,n\,\sqrt {-\frac {1}{n^2}}-4{}\mathrm {i}} \]