Integrand size = 23, antiderivative size = 86 \[ \int \frac {\sin \left (a+\sqrt {-\frac {1}{n^2}} \log \left (c x^n\right )\right )}{x^2} \, dx=\frac {e^{a \sqrt {-\frac {1}{n^2}} n} \sqrt {-\frac {1}{n^2}} n \left (c x^n\right )^{-1/n}}{4 x}+\frac {e^{-a \sqrt {-\frac {1}{n^2}} n} \sqrt {-\frac {1}{n^2}} n \left (c x^n\right )^{\frac {1}{n}} \log (x)}{2 x} \] Output:
1/4*exp(a*(-1/n^2)^(1/2)*n)*(-1/n^2)^(1/2)*n/x/((c*x^n)^(1/n))+1/2*(-1/n^2 )^(1/2)*n*(c*x^n)^(1/n)*ln(x)/exp(a*(-1/n^2)^(1/2)*n)/x
\[ \int \frac {\sin \left (a+\sqrt {-\frac {1}{n^2}} \log \left (c x^n\right )\right )}{x^2} \, dx=\int \frac {\sin \left (a+\sqrt {-\frac {1}{n^2}} \log \left (c x^n\right )\right )}{x^2} \, dx \] Input:
Integrate[Sin[a + Sqrt[-n^(-2)]*Log[c*x^n]]/x^2,x]
Output:
Integrate[Sin[a + Sqrt[-n^(-2)]*Log[c*x^n]]/x^2, x]
Time = 0.31 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.91, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, 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.
\(\displaystyle \int \frac {\sin \left (a+\sqrt {-\frac {1}{n^2}} \log \left (c x^n\right )\right )}{x^2} \, dx\) |
\(\Big \downarrow \) 4996 |
\(\displaystyle \frac {\left (c x^n\right )^{\frac {1}{n}} \int \left (c x^n\right )^{-1-\frac {1}{n}} \sin \left (a+\sqrt {-\frac {1}{n^2}} \log \left (c x^n\right )\right )d\left (c x^n\right )}{n x}\) |
\(\Big \downarrow \) 4992 |
\(\displaystyle \frac {\sqrt {-\frac {1}{n^2}} \left (c x^n\right )^{\frac {1}{n}} \int \left (\frac {e^{-a \sqrt {-\frac {1}{n^2}} n} x^{-n}}{c}-e^{a \sqrt {-\frac {1}{n^2}} n} \left (c x^n\right )^{-\frac {n+2}{n}}\right )d\left (c x^n\right )}{2 x}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {\sqrt {-\frac {1}{n^2}} \left (c x^n\right )^{\frac {1}{n}} \left (\frac {1}{2} n e^{a \sqrt {-\frac {1}{n^2}} n} \left (c x^n\right )^{-2/n}+e^{-a \sqrt {-\frac {1}{n^2}} n} \log \left (c x^n\right )\right )}{2 x}\) |
Input:
Int[Sin[a + Sqrt[-n^(-2)]*Log[c*x^n]]/x^2,x]
Output:
(Sqrt[-n^(-2)]*(c*x^n)^n^(-1)*((E^(a*Sqrt[-n^(-2)]*n)*n)/(2*(c*x^n)^(2/n)) + Log[c*x^n]/E^(a*Sqrt[-n^(-2)]*n)))/(2*x)
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])
Time = 1.45 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.79
method | result | size |
parallelrisch | \(\frac {n \sqrt {-\frac {1}{n^{2}}}\, \left (n +\ln \left (c \,x^{n}\right )\right ) \cos \left (a +\sqrt {-\frac {1}{n^{2}}}\, \ln \left (c \,x^{n}\right )\right )+\ln \left (c \,x^{n}\right ) \sin \left (a +\sqrt {-\frac {1}{n^{2}}}\, \ln \left (c \,x^{n}\right )\right )}{2 x n}\) | \(68\) |
Input:
int(sin(a+(-1/n^2)^(1/2)*ln(c*x^n))/x^2,x,method=_RETURNVERBOSE)
Output:
1/2*(n*(-1/n^2)^(1/2)*(n+ln(c*x^n))*cos(a+(-1/n^2)^(1/2)*ln(c*x^n))+ln(c*x ^n)*sin(a+(-1/n^2)^(1/2)*ln(c*x^n)))/x/n
Result contains complex when optimal does not.
Time = 0.07 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.52 \[ \int \frac {\sin \left (a+\sqrt {-\frac {1}{n^2}} \log \left (c x^n\right )\right )}{x^2} \, dx=\frac {{\left (2 i \, x^{2} \log \left (x\right ) + i \, e^{\left (\frac {2 \, {\left (i \, a n - \log \left (c\right )\right )}}{n}\right )}\right )} e^{\left (-\frac {i \, a n - \log \left (c\right )}{n}\right )}}{4 \, x^{2}} \] Input:
integrate(sin(a+(-1/n^2)^(1/2)*log(c*x^n))/x^2,x, algorithm="fricas")
Output:
1/4*(2*I*x^2*log(x) + I*e^(2*(I*a*n - log(c))/n))*e^(-(I*a*n - log(c))/n)/ x^2
Time = 1.96 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.10 \[ \int \frac {\sin \left (a+\sqrt {-\frac {1}{n^2}} \log \left (c x^n\right )\right )}{x^2} \, dx=\frac {\sqrt {- \frac {1}{n^{2}}} \log {\left (c x^{n} \right )} \cos {\left (a + \sqrt {- \frac {1}{n^{2}}} \log {\left (c x^{n} \right )} \right )}}{2 x} - \frac {\sin {\left (a + \sqrt {- \frac {1}{n^{2}}} \log {\left (c x^{n} \right )} \right )}}{2 x} + \frac {\log {\left (c x^{n} \right )} \sin {\left (a + \sqrt {- \frac {1}{n^{2}}} \log {\left (c x^{n} \right )} \right )}}{2 n x} \] Input:
integrate(sin(a+(-1/n**2)**(1/2)*ln(c*x**n))/x**2,x)
Output:
sqrt(-1/n**2)*log(c*x**n)*cos(a + sqrt(-1/n**2)*log(c*x**n))/(2*x) - sin(a + sqrt(-1/n**2)*log(c*x**n))/(2*x) + log(c*x**n)*sin(a + sqrt(-1/n**2)*lo g(c*x**n))/(2*n*x)
Time = 0.07 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.38 \[ \int \frac {\sin \left (a+\sqrt {-\frac {1}{n^2}} \log \left (c x^n\right )\right )}{x^2} \, dx=\frac {2 \, c^{\frac {2}{n}} x^{2} \log \left (x\right ) \sin \left (a\right ) - \sin \left (a\right )}{4 \, c^{\left (\frac {1}{n}\right )} x^{2}} \] Input:
integrate(sin(a+(-1/n^2)^(1/2)*log(c*x^n))/x^2,x, algorithm="maxima")
Output:
1/4*(2*c^(2/n)*x^2*log(x)*sin(a) - sin(a))/(c^(1/n)*x^2)
\[ \int \frac {\sin \left (a+\sqrt {-\frac {1}{n^2}} \log \left (c x^n\right )\right )}{x^2} \, dx=\int { \frac {\sin \left (\sqrt {-\frac {1}{n^{2}}} \log \left (c x^{n}\right ) + a\right )}{x^{2}} \,d x } \] Input:
integrate(sin(a+(-1/n^2)^(1/2)*log(c*x^n))/x^2,x, algorithm="giac")
Output:
integrate(sin(sqrt(-1/n^2)*log(c*x^n) + a)/x^2, x)
Timed out. \[ \int \frac {\sin \left (a+\sqrt {-\frac {1}{n^2}} \log \left (c x^n\right )\right )}{x^2} \, dx=\int \frac {\sin \left (a+\ln \left (c\,x^n\right )\,\sqrt {-\frac {1}{n^2}}\right )}{x^2} \,d x \] Input:
int(sin(a + log(c*x^n)*(-1/n^2)^(1/2))/x^2,x)
Output:
int(sin(a + log(c*x^n)*(-1/n^2)^(1/2))/x^2, x)
Time = 0.17 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.91 \[ \int \frac {\sin \left (a+\sqrt {-\frac {1}{n^2}} \log \left (c x^n\right )\right )}{x^2} \, dx=\frac {\cos \left (\frac {\mathrm {log}\left (x^{n} c \right ) i +a n}{n}\right ) \mathrm {log}\left (x^{n} c \right ) i +\cos \left (\frac {\mathrm {log}\left (x^{n} c \right ) i +a n}{n}\right ) i n +\mathrm {log}\left (x^{n} c \right ) \sin \left (\frac {\mathrm {log}\left (x^{n} c \right ) i +a n}{n}\right )}{2 n x} \] Input:
int(sin(a+(-1/n^2)^(1/2)*log(c*x^n))/x^2,x)
Output:
(cos((log(x**n*c)*i + a*n)/n)*log(x**n*c)*i + cos((log(x**n*c)*i + a*n)/n) *i*n + log(x**n*c)*sin((log(x**n*c)*i + a*n)/n))/(2*n*x)