Integrand size = 24, antiderivative size = 88 \[ \int \frac {\sin \left (a+2 \sqrt {-\frac {1}{n^2}} \log \left (c x^n\right )\right )}{x^3} \, dx=\frac {e^{a \sqrt {-\frac {1}{n^2}} n} \sqrt {-\frac {1}{n^2}} n \left (c x^n\right )^{-2/n}}{8 x^2}+\frac {e^{-a \sqrt {-\frac {1}{n^2}} n} \sqrt {-\frac {1}{n^2}} n \left (c x^n\right )^{2/n} \log (x)}{2 x^2} \] Output:
1/8*exp(a*(-1/n^2)^(1/2)*n)*(-1/n^2)^(1/2)*n/x^2/((c*x^n)^(2/n))+1/2*(-1/n ^2)^(1/2)*n*(c*x^n)^(2/n)*ln(x)/exp(a*(-1/n^2)^(1/2)*n)/x^2
\[ \int \frac {\sin \left (a+2 \sqrt {-\frac {1}{n^2}} \log \left (c x^n\right )\right )}{x^3} \, dx=\int \frac {\sin \left (a+2 \sqrt {-\frac {1}{n^2}} \log \left (c x^n\right )\right )}{x^3} \, dx \] Input:
Integrate[Sin[a + 2*Sqrt[-n^(-2)]*Log[c*x^n]]/x^3,x]
Output:
Integrate[Sin[a + 2*Sqrt[-n^(-2)]*Log[c*x^n]]/x^3, x]
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.
\(\displaystyle \int \frac {\sin \left (a+2 \sqrt {-\frac {1}{n^2}} \log \left (c x^n\right )\right )}{x^3} \, dx\) |
\(\Big \downarrow \) 4996 |
\(\displaystyle \frac {\left (c x^n\right )^{2/n} \int \left (c x^n\right )^{-1-\frac {2}{n}} \sin \left (a+2 \sqrt {-\frac {1}{n^2}} \log \left (c x^n\right )\right )d\left (c x^n\right )}{n x^2}\) |
\(\Big \downarrow \) 4992 |
\(\displaystyle \frac {\sqrt {-\frac {1}{n^2}} \left (c x^n\right )^{2/n} \int \left (\frac {e^{-a \sqrt {-\frac {1}{n^2}} n} x^{-n}}{c}-e^{a \sqrt {-\frac {1}{n^2}} n} \left (c x^n\right )^{-\frac {n+4}{n}}\right )d\left (c x^n\right )}{2 x^2}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {\sqrt {-\frac {1}{n^2}} \left (c x^n\right )^{2/n} \left (\frac {1}{4} n e^{a \sqrt {-\frac {1}{n^2}} n} \left (c x^n\right )^{-4/n}+e^{-a \sqrt {-\frac {1}{n^2}} n} \log \left (c x^n\right )\right )}{2 x^2}\) |
Input:
Int[Sin[a + 2*Sqrt[-n^(-2)]*Log[c*x^n]]/x^3,x]
Output:
(Sqrt[-n^(-2)]*(c*x^n)^(2/n)*((E^(a*Sqrt[-n^(-2)]*n)*n)/(4*(c*x^n)^(4/n)) + Log[c*x^n]/E^(a*Sqrt[-n^(-2)]*n)))/(2*x^2)
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 = 2.79 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.83
method | result | size |
parallelrisch | \(\frac {n \sqrt {-\frac {1}{n^{2}}}\, \left (n +2 \ln \left (c \,x^{n}\right )\right ) \cos \left (a +2 \sqrt {-\frac {1}{n^{2}}}\, \ln \left (c \,x^{n}\right )\right )+2 \ln \left (c \,x^{n}\right ) \sin \left (a +2 \sqrt {-\frac {1}{n^{2}}}\, \ln \left (c \,x^{n}\right )\right )}{4 x^{2} n}\) | \(73\) |
Input:
int(sin(a+2*(-1/n^2)^(1/2)*ln(c*x^n))/x^3,x,method=_RETURNVERBOSE)
Output:
1/4*(n*(-1/n^2)^(1/2)*(n+2*ln(c*x^n))*cos(a+2*(-1/n^2)^(1/2)*ln(c*x^n))+2* ln(c*x^n)*sin(a+2*(-1/n^2)^(1/2)*ln(c*x^n)))/x^2/n
Result contains complex when optimal does not.
Time = 0.07 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.51 \[ \int \frac {\sin \left (a+2 \sqrt {-\frac {1}{n^2}} \log \left (c x^n\right )\right )}{x^3} \, dx=\frac {{\left (4 i \, x^{4} \log \left (x\right ) + i \, e^{\left (\frac {2 \, {\left (i \, a n - 2 \, \log \left (c\right )\right )}}{n}\right )}\right )} e^{\left (-\frac {i \, a n - 2 \, \log \left (c\right )}{n}\right )}}{8 \, x^{4}} \] Input:
integrate(sin(a+2*(-1/n^2)^(1/2)*log(c*x^n))/x^3,x, algorithm="fricas")
Output:
1/8*(4*I*x^4*log(x) + I*e^(2*(I*a*n - 2*log(c))/n))*e^(-(I*a*n - 2*log(c)) /n)/x^4
Time = 6.34 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.33 \[ \int \frac {\sin \left (a+2 \sqrt {-\frac {1}{n^2}} \log \left (c x^n\right )\right )}{x^3} \, dx=\frac {n \sqrt {- \frac {1}{n^{2}}} \cos {\left (a + 2 \sqrt {- \frac {1}{n^{2}}} \log {\left (c x^{n} \right )} \right )}}{4 x^{2}} + \frac {\sqrt {- \frac {1}{n^{2}}} \log {\left (c x^{n} \right )} \cos {\left (a + 2 \sqrt {- \frac {1}{n^{2}}} \log {\left (c x^{n} \right )} \right )}}{2 x^{2}} + \frac {\log {\left (c x^{n} \right )} \sin {\left (a + 2 \sqrt {- \frac {1}{n^{2}}} \log {\left (c x^{n} \right )} \right )}}{2 n x^{2}} \] Input:
integrate(sin(a+2*(-1/n**2)**(1/2)*ln(c*x**n))/x**3,x)
Output:
n*sqrt(-1/n**2)*cos(a + 2*sqrt(-1/n**2)*log(c*x**n))/(4*x**2) + sqrt(-1/n* *2)*log(c*x**n)*cos(a + 2*sqrt(-1/n**2)*log(c*x**n))/(2*x**2) + log(c*x**n )*sin(a + 2*sqrt(-1/n**2)*log(c*x**n))/(2*n*x**2)
Time = 0.06 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.40 \[ \int \frac {\sin \left (a+2 \sqrt {-\frac {1}{n^2}} \log \left (c x^n\right )\right )}{x^3} \, dx=\frac {4 \, c^{\frac {4}{n}} x^{4} \log \left (x\right ) \sin \left (a\right ) - \sin \left (a\right )}{8 \, c^{\frac {2}{n}} x^{4}} \] Input:
integrate(sin(a+2*(-1/n^2)^(1/2)*log(c*x^n))/x^3,x, algorithm="maxima")
Output:
1/8*(4*c^(4/n)*x^4*log(x)*sin(a) - sin(a))/(c^(2/n)*x^4)
\[ \int \frac {\sin \left (a+2 \sqrt {-\frac {1}{n^2}} \log \left (c x^n\right )\right )}{x^3} \, dx=\int { \frac {\sin \left (2 \, \sqrt {-\frac {1}{n^{2}}} \log \left (c x^{n}\right ) + a\right )}{x^{3}} \,d x } \] Input:
integrate(sin(a+2*(-1/n^2)^(1/2)*log(c*x^n))/x^3,x, algorithm="giac")
Output:
integrate(sin(2*sqrt(-1/n^2)*log(c*x^n) + a)/x^3, x)
Timed out. \[ \int \frac {\sin \left (a+2 \sqrt {-\frac {1}{n^2}} \log \left (c x^n\right )\right )}{x^3} \, dx=\int \frac {\sin \left (a+2\,\ln \left (c\,x^n\right )\,\sqrt {-\frac {1}{n^2}}\right )}{x^3} \,d x \] Input:
int(sin(a + 2*log(c*x^n)*(-1/n^2)^(1/2))/x^3,x)
Output:
int(sin(a + 2*log(c*x^n)*(-1/n^2)^(1/2))/x^3, x)
Time = 0.17 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.94 \[ \int \frac {\sin \left (a+2 \sqrt {-\frac {1}{n^2}} \log \left (c x^n\right )\right )}{x^3} \, dx=\frac {2 \cos \left (\frac {2 \,\mathrm {log}\left (x^{n} c \right ) i +a n}{n}\right ) \mathrm {log}\left (x^{n} c \right ) i +\cos \left (\frac {2 \,\mathrm {log}\left (x^{n} c \right ) i +a n}{n}\right ) i n +2 \,\mathrm {log}\left (x^{n} c \right ) \sin \left (\frac {2 \,\mathrm {log}\left (x^{n} c \right ) i +a n}{n}\right )}{4 n \,x^{2}} \] Input:
int(sin(a+2*(-1/n^2)^(1/2)*log(c*x^n))/x^3,x)
Output:
(2*cos((2*log(x**n*c)*i + a*n)/n)*log(x**n*c)*i + cos((2*log(x**n*c)*i + a *n)/n)*i*n + 2*log(x**n*c)*sin((2*log(x**n*c)*i + a*n)/n))/(4*n*x**2)