Integrand size = 23, antiderivative size = 76 \[ \int x \sin ^2\left (a+\sqrt {-\frac {1}{n^2}} \log \left (c x^n\right )\right ) \, dx=\frac {x^2}{4}-\frac {1}{16} e^{-2 a \sqrt {-\frac {1}{n^2}} n} x^2 \left (c x^n\right )^{2/n}-\frac {1}{4} e^{2 a \sqrt {-\frac {1}{n^2}} n} x^2 \left (c x^n\right )^{-2/n} \log (x) \] Output:
1/4*x^2-1/16*x^2*(c*x^n)^(2/n)/exp(2*a*(-1/n^2)^(1/2)*n)-1/4*exp(2*a*(-1/n ^2)^(1/2)*n)*x^2*ln(x)/((c*x^n)^(2/n))
\[ \int x \sin ^2\left (a+\sqrt {-\frac {1}{n^2}} \log \left (c x^n\right )\right ) \, dx=\int x \sin ^2\left (a+\sqrt {-\frac {1}{n^2}} \log \left (c x^n\right )\right ) \, dx \] Input:
Integrate[x*Sin[a + Sqrt[-n^(-2)]*Log[c*x^n]]^2,x]
Output:
Integrate[x*Sin[a + Sqrt[-n^(-2)]*Log[c*x^n]]^2, x]
Time = 0.31 (sec) , antiderivative size = 89, normalized size of antiderivative = 1.17, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, 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.
\(\displaystyle \int x \sin ^2\left (a+\sqrt {-\frac {1}{n^2}} \log \left (c x^n\right )\right ) \, dx\) |
\(\Big \downarrow \) 4996 |
\(\displaystyle \frac {x^2 \left (c x^n\right )^{-2/n} \int \left (c x^n\right )^{\frac {2}{n}-1} \sin ^2\left (a+\sqrt {-\frac {1}{n^2}} \log \left (c x^n\right )\right )d\left (c x^n\right )}{n}\) |
\(\Big \downarrow \) 4992 |
\(\displaystyle -\frac {x^2 \left (c x^n\right )^{-2/n} \int \left (-2 \left (c x^n\right )^{\frac {2}{n}-1}+e^{-2 a \sqrt {-\frac {1}{n^2}} n} \left (c x^n\right )^{\frac {4}{n}-1}+\frac {e^{2 a \sqrt {-\frac {1}{n^2}} n} x^{-n}}{c}\right )d\left (c x^n\right )}{4 n}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {x^2 \left (c x^n\right )^{-2/n} \left (-\frac {1}{4} n e^{-2 a \sqrt {-\frac {1}{n^2}} n} \left (c x^n\right )^{4/n}-e^{2 a \sqrt {-\frac {1}{n^2}} n} \log \left (c x^n\right )+n \left (c x^n\right )^{2/n}\right )}{4 n}\) |
Input:
Int[x*Sin[a + Sqrt[-n^(-2)]*Log[c*x^n]]^2,x]
Output:
(x^2*(n*(c*x^n)^(2/n) - (n*(c*x^n)^(4/n))/(4*E^(2*a*Sqrt[-n^(-2)]*n)) - E^ (2*a*Sqrt[-n^(-2)]*n)*Log[c*x^n]))/(4*n*(c*x^n)^(2/n))
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])
\[\int x {\sin \left (a +\sqrt {-\frac {1}{n^{2}}}\, \ln \left (c \,x^{n}\right )\right )}^{2}d x\]
Input:
int(x*sin(a+(-1/n^2)^(1/2)*ln(c*x^n))^2,x)
Output:
int(x*sin(a+(-1/n^2)^(1/2)*ln(c*x^n))^2,x)
Result contains complex when optimal does not.
Time = 0.07 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.79 \[ \int x \sin ^2\left (a+\sqrt {-\frac {1}{n^2}} \log \left (c x^n\right )\right ) \, dx=-\frac {1}{16} \, {\left (x^{4} - 4 \, x^{2} e^{\left (\frac {2 \, {\left (i \, a n - \log \left (c\right )\right )}}{n}\right )} + 4 \, e^{\left (\frac {4 \, {\left (i \, a n - \log \left (c\right )\right )}}{n}\right )} \log \left (x\right )\right )} e^{\left (-\frac {2 \, {\left (i \, a n - \log \left (c\right )\right )}}{n}\right )} \] Input:
integrate(x*sin(a+(-1/n^2)^(1/2)*log(c*x^n))^2,x, algorithm="fricas")
Output:
-1/16*(x^4 - 4*x^2*e^(2*(I*a*n - log(c))/n) + 4*e^(4*(I*a*n - log(c))/n)*l og(x))*e^(-2*(I*a*n - log(c))/n)
\[ \int x \sin ^2\left (a+\sqrt {-\frac {1}{n^2}} \log \left (c x^n\right )\right ) \, dx=\int x \sin ^{2}{\left (a + \sqrt {- \frac {1}{n^{2}}} \log {\left (c x^{n} \right )} \right )}\, dx \] Input:
integrate(x*sin(a+(-1/n**2)**(1/2)*ln(c*x**n))**2,x)
Output:
Integral(x*sin(a + sqrt(-1/n**2)*log(c*x**n))**2, x)
Time = 0.05 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.62 \[ \int x \sin ^2\left (a+\sqrt {-\frac {1}{n^2}} \log \left (c x^n\right )\right ) \, dx=-\frac {c^{\frac {4}{n}} x^{4} \cos \left (2 \, a\right ) - 4 \, c^{\frac {2}{n}} x^{2} + 4 \, \cos \left (2 \, a\right ) \log \left (x\right )}{16 \, c^{\frac {2}{n}}} \] Input:
integrate(x*sin(a+(-1/n^2)^(1/2)*log(c*x^n))^2,x, algorithm="maxima")
Output:
-1/16*(c^(4/n)*x^4*cos(2*a) - 4*c^(2/n)*x^2 + 4*cos(2*a)*log(x))/c^(2/n)
Time = 0.58 (sec) , antiderivative size = 1, normalized size of antiderivative = 0.01 \[ \int x \sin ^2\left (a+\sqrt {-\frac {1}{n^2}} \log \left (c x^n\right )\right ) \, dx=+\infty \] Input:
integrate(x*sin(a+(-1/n^2)^(1/2)*log(c*x^n))^2,x, algorithm="giac")
Output:
+Infinity
Time = 20.47 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.21 \[ \int x \sin ^2\left (a+\sqrt {-\frac {1}{n^2}} \log \left (c x^n\right )\right ) \, dx=\frac {x^2}{4}-\frac {x^2\,{\mathrm {e}}^{-a\,2{}\mathrm {i}}\,\frac {1}{{\left (c\,x^n\right )}^{\sqrt {-\frac {1}{n^2}}\,2{}\mathrm {i}}}\,1{}\mathrm {i}}{8\,n\,\sqrt {-\frac {1}{n^2}}+8{}\mathrm {i}}+\frac {x^2\,{\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\left (c\,x^n\right )}^{\sqrt {-\frac {1}{n^2}}\,2{}\mathrm {i}}\,1{}\mathrm {i}}{8\,n\,\sqrt {-\frac {1}{n^2}}-8{}\mathrm {i}} \] Input:
int(x*sin(a + log(c*x^n)*(-1/n^2)^(1/2))^2,x)
Output:
x^2/4 - (x^2*exp(-a*2i)/(c*x^n)^((-1/n^2)^(1/2)*2i)*1i)/(8*n*(-1/n^2)^(1/2 ) + 8i) + (x^2*exp(a*2i)*(c*x^n)^((-1/n^2)^(1/2)*2i)*1i)/(8*n*(-1/n^2)^(1/ 2) - 8i)
Time = 0.17 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.93 \[ \int x \sin ^2\left (a+\sqrt {-\frac {1}{n^2}} \log \left (c x^n\right )\right ) \, dx=\frac {x^{2} \left (-2 \cos \left (\frac {\mathrm {log}\left (x^{n} c \right ) i +a n}{n}\right ) \mathrm {log}\left (x^{n} c \right ) \sin \left (\frac {\mathrm {log}\left (x^{n} c \right ) i +a n}{n}\right ) i -\cos \left (\frac {\mathrm {log}\left (x^{n} c \right ) i +a n}{n}\right ) \sin \left (\frac {\mathrm {log}\left (x^{n} c \right ) i +a n}{n}\right ) i n +2 \,\mathrm {log}\left (x^{n} c \right ) {\sin \left (\frac {\mathrm {log}\left (x^{n} c \right ) i +a n}{n}\right )}^{2}-\mathrm {log}\left (x^{n} c \right )+2 {\sin \left (\frac {\mathrm {log}\left (x^{n} c \right ) i +a n}{n}\right )}^{2} n \right )}{4 n} \] Input:
int(x*sin(a+(-1/n^2)^(1/2)*log(c*x^n))^2,x)
Output:
(x**2*( - 2*cos((log(x**n*c)*i + a*n)/n)*log(x**n*c)*sin((log(x**n*c)*i + a*n)/n)*i - cos((log(x**n*c)*i + a*n)/n)*sin((log(x**n*c)*i + a*n)/n)*i*n + 2*log(x**n*c)*sin((log(x**n*c)*i + a*n)/n)**2 - log(x**n*c) + 2*sin((log (x**n*c)*i + a*n)/n)**2*n))/(4*n)