\(\int \frac {\sin ^2(a+\frac {1}{2} \sqrt {-\frac {1}{n^2}} \log (c x^n))}{x^2} \, dx\) [38]

Optimal result

Integrand size = 28, antiderivative size = 74 \[ \int \frac {\sin ^2\left (a+\frac {1}{2} \sqrt {-\frac {1}{n^2}} \log \left (c x^n\right )\right )}{x^2} \, dx=-\frac {1}{2 x}+\frac {e^{2 a \sqrt {-\frac {1}{n^2}} n} \left (c x^n\right )^{-1/n}}{8 x}-\frac {e^{-2 a \sqrt {-\frac {1}{n^2}} n} \left (c x^n\right )^{\frac {1}{n}} \log (x)}{4 x} \]

[Out]

-1/2/x+1/8*exp(2*a*n*(-1/n^2)^(1/2))/x/((c*x^n)^(1/n))-1/4*(c*x^n)^(1/n)*ln(x)/exp(2*a*n*(-1/n^2)^(1/2))/x

Rubi [A] (verified)

Time = 0.08 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {4581, 4577} \[ \int \frac {\sin ^2\left (a+\frac {1}{2} \sqrt {-\frac {1}{n^2}} \log \left (c x^n\right )\right )}{x^2} \, dx=\frac {e^{2 a \sqrt {-\frac {1}{n^2}} n} \left (c x^n\right )^{-1/n}}{8 x}-\frac {e^{-2 a \sqrt {-\frac {1}{n^2}} n} \log (x) \left (c x^n\right )^{\frac {1}{n}}}{4 x}-\frac {1}{2 x} \]

[In]

Int[Sin[a + (Sqrt[-n^(-2)]*Log[c*x^n])/2]^2/x^2,x]

[Out]

-1/2*1/x + E^(2*a*Sqrt[-n^(-2)]*n)/(8*x*(c*x^n)^n^(-1)) - ((c*x^n)^n^(-1)*Log[x])/(4*E^(2*a*Sqrt[-n^(-2)]*n)*x
)

Rule 4577

Int[((e_.)*(x_))^(m_.)*Sin[((a_.) + Log[x_]*(b_.))*(d_.)]^(p_.), x_Symbol] :> Dist[(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]

Rule 4581

Int[((e_.)*(x_))^(m_.)*Sin[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*(d_.)]^(p_.), x_Symbol] :> Dist[(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])

Rubi steps \begin{align*} \text {integral}& = \frac {\left (c x^n\right )^{\frac {1}{n}} \text {Subst}\left (\int x^{-1-\frac {1}{n}} \sin ^2\left (a+\frac {1}{2} \sqrt {-\frac {1}{n^2}} \log (x)\right ) \, dx,x,c x^n\right )}{n x} \\ & = -\frac {\left (c x^n\right )^{\frac {1}{n}} \text {Subst}\left (\int \left (\frac {e^{-2 a \sqrt {-\frac {1}{n^2}} n}}{x}-2 x^{-\frac {1+n}{n}}+e^{2 a \sqrt {-\frac {1}{n^2}} n} x^{-\frac {2+n}{n}}\right ) \, dx,x,c x^n\right )}{4 n x} \\ & = -\frac {1}{2 x}+\frac {e^{2 a \sqrt {-\frac {1}{n^2}} n} \left (c x^n\right )^{-1/n}}{8 x}-\frac {e^{-2 a \sqrt {-\frac {1}{n^2}} n} \left (c x^n\right )^{\frac {1}{n}} \log (x)}{4 x} \\ \end{align*}

Mathematica [F]

\[ \int \frac {\sin ^2\left (a+\frac {1}{2} \sqrt {-\frac {1}{n^2}} \log \left (c x^n\right )\right )}{x^2} \, dx=\int \frac {\sin ^2\left (a+\frac {1}{2} \sqrt {-\frac {1}{n^2}} \log \left (c x^n\right )\right )}{x^2} \, dx \]

[In]

Integrate[Sin[a + (Sqrt[-n^(-2)]*Log[c*x^n])/2]^2/x^2,x]

[Out]

Integrate[Sin[a + (Sqrt[-n^(-2)]*Log[c*x^n])/2]^2/x^2, x]

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(198\) vs. \(2(64)=128\).

Time = 10.82 (sec) , antiderivative size = 199, normalized size of antiderivative = 2.69

method	result	size
parallelrisch	\(\frac {\left (-8 n -3 \ln \left (c \,x^{n}\right )\right ) {\tan \left (\frac {a}{2}+\sqrt {-\frac {1}{n^{2}}}\, \ln \left (\left (c \,x^{n}\right )^{\frac {1}{4}}\right )\right )}^{4}-20 n \left (n +\frac {3 \ln \left (c \,x^{n}\right )}{5}\right ) \sqrt {-\frac {1}{n^{2}}}\, {\tan \left (\frac {a}{2}+\sqrt {-\frac {1}{n^{2}}}\, \ln \left (\left (c \,x^{n}\right )^{\frac {1}{4}}\right )\right )}^{3}+18 {\tan \left (\frac {a}{2}+\sqrt {-\frac {1}{n^{2}}}\, \ln \left (\left (c \,x^{n}\right )^{\frac {1}{4}}\right )\right )}^{2} \ln \left (c \,x^{n}\right )+20 n \left (n +\frac {3 \ln \left (c \,x^{n}\right )}{5}\right ) \sqrt {-\frac {1}{n^{2}}}\, \tan \left (\frac {a}{2}+\sqrt {-\frac {1}{n^{2}}}\, \ln \left (\left (c \,x^{n}\right )^{\frac {1}{4}}\right )\right )-8 n -3 \ln \left (c \,x^{n}\right )}{12 x n {\left (1+{\tan \left (\frac {a}{2}+\sqrt {-\frac {1}{n^{2}}}\, \ln \left (\left (c \,x^{n}\right )^{\frac {1}{4}}\right )\right )}^{2}\right )}^{2}}\)	\(199\)

[In]

int(sin(a+1/2*ln(c*x^n)*(-1/n^2)^(1/2))^2/x^2,x,method=_RETURNVERBOSE)

[Out]

1/12*((-8*n-3*ln(c*x^n))*tan(1/2*a+(-1/n^2)^(1/2)*ln((c*x^n)^(1/4)))^4-20*n*(n+3/5*ln(c*x^n))*(-1/n^2)^(1/2)*t
an(1/2*a+(-1/n^2)^(1/2)*ln((c*x^n)^(1/4)))^3+18*tan(1/2*a+(-1/n^2)^(1/2)*ln((c*x^n)^(1/4)))^2*ln(c*x^n)+20*n*(
n+3/5*ln(c*x^n))*(-1/n^2)^(1/2)*tan(1/2*a+(-1/n^2)^(1/2)*ln((c*x^n)^(1/4)))-8*n-3*ln(c*x^n))/x/n/(1+tan(1/2*a+
(-1/n^2)^(1/2)*ln((c*x^n)^(1/4)))^2)^2

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.25 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.84 \[ \int \frac {\sin ^2\left (a+\frac {1}{2} \sqrt {-\frac {1}{n^2}} \log \left (c x^n\right )\right )}{x^2} \, dx=-\frac {{\left (2 \, x^{2} \log \left (x\right ) + 4 \, x e^{\left (\frac {2 i \, a n - \log \left (c\right )}{n}\right )} - e^{\left (\frac {2 \, {\left (2 i \, a n - \log \left (c\right )\right )}}{n}\right )}\right )} e^{\left (-\frac {2 i \, a n - \log \left (c\right )}{n}\right )}}{8 \, x^{2}} \]

[In]

integrate(sin(a+1/2*log(c*x^n)*(-1/n^2)^(1/2))^2/x^2,x, algorithm="fricas")

[Out]

-1/8*(2*x^2*log(x) + 4*x*e^((2*I*a*n - log(c))/n) - e^(2*(2*I*a*n - log(c))/n))*e^(-(2*I*a*n - log(c))/n)/x^2

Sympy [A] (verification not implemented)

Time = 11.56 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.42 \[ \int \frac {\sin ^2\left (a+\frac {1}{2} \sqrt {-\frac {1}{n^2}} \log \left (c x^n\right )\right )}{x^2} \, dx=\frac {\sqrt {- \frac {1}{n^{2}}} \log {\left (c x^{n} \right )} \sin {\left (2 a + \sqrt {- \frac {1}{n^{2}}} \log {\left (c x^{n} \right )} \right )}}{4 x} + \frac {\cos {\left (2 a + \sqrt {- \frac {1}{n^{2}}} \log {\left (c x^{n} \right )} \right )}}{4 x} - \frac {1}{2 x} - \frac {\log {\left (c x^{n} \right )} \cos {\left (2 a + \sqrt {- \frac {1}{n^{2}}} \log {\left (c x^{n} \right )} \right )}}{4 n x} \]

[In]

integrate(sin(a+1/2*ln(c*x**n)*(-1/n**2)**(1/2))**2/x**2,x)

[Out]

sqrt(-1/n**2)*log(c*x**n)*sin(2*a + sqrt(-1/n**2)*log(c*x**n))/(4*x) + cos(2*a + sqrt(-1/n**2)*log(c*x**n))/(4
*x) - 1/(2*x) - log(c*x**n)*cos(2*a + sqrt(-1/n**2)*log(c*x**n))/(4*n*x)

Maxima [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.65 \[ \int \frac {\sin ^2\left (a+\frac {1}{2} \sqrt {-\frac {1}{n^2}} \log \left (c x^n\right )\right )}{x^2} \, dx=-\frac {2 \, c^{\frac {2}{n}} x^{3} \cos \left (2 \, a\right ) \log \left (x\right ) + 4 \, c^{\left (\frac {1}{n}\right )} x^{2} - x \cos \left (2 \, a\right )}{8 \, c^{\left (\frac {1}{n}\right )} x^{3}} \]

[In]

integrate(sin(a+1/2*log(c*x^n)*(-1/n^2)^(1/2))^2/x^2,x, algorithm="maxima")

[Out]

-1/8*(2*c^(2/n)*x^3*cos(2*a)*log(x) + 4*c^(1/n)*x^2 - x*cos(2*a))/(c^(1/n)*x^3)

Giac [F]

\[ \int \frac {\sin ^2\left (a+\frac {1}{2} \sqrt {-\frac {1}{n^2}} \log \left (c x^n\right )\right )}{x^2} \, dx=\int { \frac {\sin \left (\frac {1}{2} \, \sqrt {-\frac {1}{n^{2}}} \log \left (c x^{n}\right ) + a\right )^{2}}{x^{2}} \,d x } \]

[In]

integrate(sin(a+1/2*log(c*x^n)*(-1/n^2)^(1/2))^2/x^2,x, algorithm="giac")

[Out]

integrate(sin(1/2*sqrt(-1/n^2)*log(c*x^n) + a)^2/x^2, x)

Mupad [F(-1)]

Timed out. \[ \int \frac {\sin ^2\left (a+\frac {1}{2} \sqrt {-\frac {1}{n^2}} \log \left (c x^n\right )\right )}{x^2} \, dx=\int \frac {{\sin \left (a+\frac {\ln \left (c\,x^n\right )\,\sqrt {-\frac {1}{n^2}}}{2}\right )}^2}{x^2} \,d x \]

[In]

int(sin(a + (log(c*x^n)*(-1/n^2)^(1/2))/2)^2/x^2,x)

[Out]

int(sin(a + (log(c*x^n)*(-1/n^2)^(1/2))/2)^2/x^2, x)