Integrand size = 25, antiderivative size = 76 \[ \int \frac {\sin ^2\left (a+\sqrt {-\frac {1}{n^2}} \log \left (c x^n\right )\right )}{x^3} \, dx=-\frac {1}{4 x^2}+\frac {e^{2 a \sqrt {-\frac {1}{n^2}} n} \left (c x^n\right )^{-2/n}}{16 x^2}-\frac {e^{-2 a \sqrt {-\frac {1}{n^2}} n} \left (c x^n\right )^{2/n} \log (x)}{4 x^2} \] Output:
-1/4/x^2+1/16*exp(2*a*(-1/n^2)^(1/2)*n)/x^2/((c*x^n)^(2/n))-1/4*(c*x^n)^(2 /n)*ln(x)/exp(2*a*(-1/n^2)^(1/2)*n)/x^2
\[ \int \frac {\sin ^2\left (a+\sqrt {-\frac {1}{n^2}} \log \left (c x^n\right )\right )}{x^3} \, dx=\int \frac {\sin ^2\left (a+\sqrt {-\frac {1}{n^2}} \log \left (c x^n\right )\right )}{x^3} \, dx \] Input:
Integrate[Sin[a + Sqrt[-n^(-2)]*Log[c*x^n]]^2/x^3,x]
Output:
Integrate[Sin[a + Sqrt[-n^(-2)]*Log[c*x^n]]^2/x^3, x]
Time = 0.34 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.18, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, 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.
\(\displaystyle \int \frac {\sin ^2\left (a+\sqrt {-\frac {1}{n^2}} \log \left (c x^n\right )\right )}{x^3} \, dx\) |
\(\Big \downarrow \) 4996 |
\(\displaystyle \frac {\left (c x^n\right )^{2/n} \int \left (c x^n\right )^{-1-\frac {2}{n}} \sin ^2\left (a+\sqrt {-\frac {1}{n^2}} \log \left (c x^n\right )\right )d\left (c x^n\right )}{n x^2}\) |
\(\Big \downarrow \) 4992 |
\(\displaystyle -\frac {\left (c x^n\right )^{2/n} \int \left (\frac {e^{-2 a \sqrt {-\frac {1}{n^2}} n} x^{-n}}{c}-2 \left (c x^n\right )^{-\frac {n+2}{n}}+e^{2 a \sqrt {-\frac {1}{n^2}} n} \left (c x^n\right )^{-\frac {n+4}{n}}\right )d\left (c x^n\right )}{4 n x^2}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {\left (c x^n\right )^{2/n} \left (\frac {1}{4} n e^{2 a \sqrt {-\frac {1}{n^2}} n} \left (c x^n\right )^{-4/n}-e^{-2 a \sqrt {-\frac {1}{n^2}} n} \log \left (c x^n\right )-n \left (c x^n\right )^{-2/n}\right )}{4 n x^2}\) |
Input:
Int[Sin[a + Sqrt[-n^(-2)]*Log[c*x^n]]^2/x^3,x]
Output:
((c*x^n)^(2/n)*((E^(2*a*Sqrt[-n^(-2)]*n)*n)/(4*(c*x^n)^(4/n)) - n/(c*x^n)^ (2/n) - Log[c*x^n]/E^(2*a*Sqrt[-n^(-2)]*n)))/(4*n*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 = 14.92 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.13
method | result | size |
parallelrisch | \(\frac {\left (-2 n -6 \ln \left (c \,x^{n}\right )\right ) \cos \left (2 \sqrt {-\frac {1}{n^{2}}}\, \ln \left (c \,x^{n}\right )+2 a \right )+5 n \left (-\frac {6}{5}+\sqrt {-\frac {1}{n^{2}}}\, \left (n +\frac {6 \ln \left (c \,x^{n}\right )}{5}\right ) \sin \left (2 \sqrt {-\frac {1}{n^{2}}}\, \ln \left (c \,x^{n}\right )+2 a \right )\right )}{24 x^{2} n}\) | \(86\) |
Input:
int(sin(a+(-1/n^2)^(1/2)*ln(c*x^n))^2/x^3,x,method=_RETURNVERBOSE)
Output:
1/24*((-2*n-6*ln(c*x^n))*cos(2*(-1/n^2)^(1/2)*ln(c*x^n)+2*a)+5*n*(-6/5+(-1 /n^2)^(1/2)*(n+6/5*ln(c*x^n))*sin(2*(-1/n^2)^(1/2)*ln(c*x^n)+2*a)))/x^2/n
Result contains complex when optimal does not.
Time = 0.07 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.86 \[ \int \frac {\sin ^2\left (a+\sqrt {-\frac {1}{n^2}} \log \left (c x^n\right )\right )}{x^3} \, dx=-\frac {{\left (4 \, x^{4} \log \left (x\right ) + 4 \, x^{2} e^{\left (\frac {2 \, {\left (i \, a n - \log \left (c\right )\right )}}{n}\right )} - e^{\left (\frac {4 \, {\left (i \, a n - \log \left (c\right )\right )}}{n}\right )}\right )} e^{\left (-\frac {2 \, {\left (i \, a n - \log \left (c\right )\right )}}{n}\right )}}{16 \, x^{4}} \] Input:
integrate(sin(a+(-1/n^2)^(1/2)*log(c*x^n))^2/x^3,x, algorithm="fricas")
Output:
-1/16*(4*x^4*log(x) + 4*x^2*e^(2*(I*a*n - log(c))/n) - e^(4*(I*a*n - log(c ))/n))*e^(-2*(I*a*n - log(c))/n)/x^4
Leaf count of result is larger than twice the leaf count of optimal. 221 vs. \(2 (70) = 140\).
Time = 6.57 (sec) , antiderivative size = 221, normalized size of antiderivative = 2.91 \[ \int \frac {\sin ^2\left (a+\sqrt {-\frac {1}{n^2}} \log \left (c x^n\right )\right )}{x^3} \, dx=- \frac {\sin ^{2}{\left (a + \sqrt {- \frac {1}{n^{2}}} \log {\left (c x^{n} \right )} \right )}}{2 x^{2}} + \frac {\log {\left (c x^{n} \right )} \sin ^{2}{\left (a + \sqrt {- \frac {1}{n^{2}}} \log {\left (c x^{n} \right )} \right )}}{4 n x^{2}} - \frac {\log {\left (c x^{n} \right )} \cos ^{2}{\left (a + \sqrt {- \frac {1}{n^{2}}} \log {\left (c x^{n} \right )} \right )}}{4 n x^{2}} + \frac {\sin {\left (a + \sqrt {- \frac {1}{n^{2}}} \log {\left (c x^{n} \right )} \right )} \cos {\left (a + \sqrt {- \frac {1}{n^{2}}} \log {\left (c x^{n} \right )} \right )}}{4 n x^{2} \sqrt {- \frac {1}{n^{2}}}} - \frac {\log {\left (c x^{n} \right )} \sin {\left (a + \sqrt {- \frac {1}{n^{2}}} \log {\left (c x^{n} \right )} \right )} \cos {\left (a + \sqrt {- \frac {1}{n^{2}}} \log {\left (c x^{n} \right )} \right )}}{2 n^{2} x^{2} \sqrt {- \frac {1}{n^{2}}}} \] Input:
integrate(sin(a+(-1/n**2)**(1/2)*ln(c*x**n))**2/x**3,x)
Output:
-sin(a + sqrt(-1/n**2)*log(c*x**n))**2/(2*x**2) + log(c*x**n)*sin(a + sqrt (-1/n**2)*log(c*x**n))**2/(4*n*x**2) - log(c*x**n)*cos(a + sqrt(-1/n**2)*l og(c*x**n))**2/(4*n*x**2) + sin(a + sqrt(-1/n**2)*log(c*x**n))*cos(a + sqr t(-1/n**2)*log(c*x**n))/(4*n*x**2*sqrt(-1/n**2)) - log(c*x**n)*sin(a + sqr t(-1/n**2)*log(c*x**n))*cos(a + sqrt(-1/n**2)*log(c*x**n))/(2*n**2*x**2*sq rt(-1/n**2))
Time = 0.06 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.71 \[ \int \frac {\sin ^2\left (a+\sqrt {-\frac {1}{n^2}} \log \left (c x^n\right )\right )}{x^3} \, dx=-\frac {4 \, c^{\frac {4}{n}} x^{6} \cos \left (2 \, a\right ) \log \left (x\right ) + 4 \, c^{\frac {2}{n}} x^{4} - x^{2} \cos \left (2 \, a\right )}{16 \, c^{\frac {2}{n}} x^{6}} \] Input:
integrate(sin(a+(-1/n^2)^(1/2)*log(c*x^n))^2/x^3,x, algorithm="maxima")
Output:
-1/16*(4*c^(4/n)*x^6*cos(2*a)*log(x) + 4*c^(2/n)*x^4 - x^2*cos(2*a))/(c^(2 /n)*x^6)
\[ \int \frac {\sin ^2\left (a+\sqrt {-\frac {1}{n^2}} \log \left (c x^n\right )\right )}{x^3} \, dx=\int { \frac {\sin \left (\sqrt {-\frac {1}{n^{2}}} \log \left (c x^{n}\right ) + a\right )^{2}}{x^{3}} \,d x } \] Input:
integrate(sin(a+(-1/n^2)^(1/2)*log(c*x^n))^2/x^3,x, algorithm="giac")
Output:
integrate(sin(sqrt(-1/n^2)*log(c*x^n) + a)^2/x^3, x)
Timed out. \[ \int \frac {\sin ^2\left (a+\sqrt {-\frac {1}{n^2}} \log \left (c x^n\right )\right )}{x^3} \, dx=\int \frac {{\sin \left (a+\ln \left (c\,x^n\right )\,\sqrt {-\frac {1}{n^2}}\right )}^2}{x^3} \,d x \] Input:
int(sin(a + log(c*x^n)*(-1/n^2)^(1/2))^2/x^3,x)
Output:
int(sin(a + log(c*x^n)*(-1/n^2)^(1/2))^2/x^3, x)
Time = 0.17 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.93 \[ \int \frac {\sin ^2\left (a+\sqrt {-\frac {1}{n^2}} \log \left (c x^n\right )\right )}{x^3} \, dx=\frac {2 \cos \left (\frac {\mathrm {log}\left (x^{n} c \right ) i +a n}{n}\right ) \mathrm {log}\left (x^{n} c \right ) \sin \left (\frac {\mathrm {log}\left (x^{n} c \right ) i +a n}{n}\right ) i -\cos \left (\frac {\mathrm {log}\left (x^{n} c \right ) i +a n}{n}\right ) \sin \left (\frac {\mathrm {log}\left (x^{n} c \right ) i +a n}{n}\right ) i n +2 \,\mathrm {log}\left (x^{n} c \right ) {\sin \left (\frac {\mathrm {log}\left (x^{n} c \right ) i +a n}{n}\right )}^{2}-\mathrm {log}\left (x^{n} c \right )-2 {\sin \left (\frac {\mathrm {log}\left (x^{n} c \right ) i +a n}{n}\right )}^{2} n}{4 n \,x^{2}} \] Input:
int(sin(a+(-1/n^2)^(1/2)*log(c*x^n))^2/x^3,x)
Output:
(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*x**2)