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

Optimal. Leaf size=74 \[ \frac {e^{2 a \sqrt {-\frac {1}{n^2}} n} \left (c x^n\right )^{-1/n}}{8 x}-\frac {e^{-2 a \sqrt {-\frac {1}{n^2}} n} \log (x) \left (c x^n\right )^{\frac {1}{n}}}{4 x}-\frac {1}{2 x} \]

[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

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Rubi [A]  time = 0.07, antiderivative size = 74, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {4493, 4489} \[ \frac {e^{2 a \sqrt {-\frac {1}{n^2}} n} \left (c x^n\right )^{-1/n}}{8 x}-\frac {e^{-2 a \sqrt {-\frac {1}{n^2}} n} \log (x) \left (c x^n\right )^{\frac {1}{n}}}{4 x}-\frac {1}{2 x} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-1/(2*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 4489

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 4493

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*} \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 (c x^n\right )^{\frac {1}{n}} \operatorname {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}} \operatorname {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*}

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Mathematica [F]  time = 0.21, size = 0, normalized size = 0.00 \[ \int \frac {\sin ^2\left (a+\frac {1}{2} \sqrt {-\frac {1}{n^2}} \log \left (c x^n\right )\right )}{x^2} \, dx \]

Verification is Not applicable to the result.

[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]

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fricas [C]  time = 0.43, size = 62, normalized size = 0.84 \[ -\frac {{\left (2 \, x^{2} \log \relax (x) + 4 \, x e^{\left (\frac {2 i \, a n - \log \relax (c)}{n}\right )} - e^{\left (\frac {2 \, {\left (2 i \, a n - \log \relax (c)\right )}}{n}\right )}\right )} e^{\left (-\frac {2 i \, a n - \log \relax (c)}{n}\right )}}{8 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[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

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sin \left (\frac {1}{2} \, \sqrt {-\frac {1}{n^{2}}} \log \left (c x^{n}\right ) + a\right )^{2}}{x^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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maple [F]  time = 0.07, size = 0, normalized size = 0.00 \[ \int \frac {\sin ^{2}\left (a +\frac {\ln \left (c \,x^{n}\right ) \sqrt {-\frac {1}{n^{2}}}}{2}\right )}{x^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.36, size = 48, normalized size = 0.65 \[ -\frac {2 \, c^{\frac {2}{n}} x^{3} \cos \left (2 \, a\right ) \log \relax (x) + 4 \, c^{\left (\frac {1}{n}\right )} x^{2} - x \cos \left (2 \, a\right )}{8 \, c^{\left (\frac {1}{n}\right )} x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\sin \left (a+\frac {\ln \left (c\,x^n\right )\,\sqrt {-\frac {1}{n^2}}}{2}\right )}^2}{x^2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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sympy [C]  time = 28.49, size = 240, normalized size = 3.24 \[ \frac {i n \sqrt {\frac {1}{n^{2}}} \log {\relax (x )} \sin {\left (2 a + i n \sqrt {\frac {1}{n^{2}}} \log {\relax (x )} + i \sqrt {\frac {1}{n^{2}}} \log {\relax (c )} \right )}}{4 x} + \frac {i n \sqrt {\frac {1}{n^{2}}} \sin {\left (2 a + i n \sqrt {\frac {1}{n^{2}}} \log {\relax (x )} + i \sqrt {\frac {1}{n^{2}}} \log {\relax (c )} \right )}}{4 x} + \frac {i \sqrt {\frac {1}{n^{2}}} \log {\relax (c )} \sin {\left (2 a + i n \sqrt {\frac {1}{n^{2}}} \log {\relax (x )} + i \sqrt {\frac {1}{n^{2}}} \log {\relax (c )} \right )}}{4 x} - \frac {\log {\relax (x )} \cos {\left (2 a + i n \sqrt {\frac {1}{n^{2}}} \log {\relax (x )} + i \sqrt {\frac {1}{n^{2}}} \log {\relax (c )} \right )}}{4 x} - \frac {1}{2 x} - \frac {\log {\relax (c )} \cos {\left (2 a + i n \sqrt {\frac {1}{n^{2}}} \log {\relax (x )} + i \sqrt {\frac {1}{n^{2}}} \log {\relax (c )} \right )}}{4 n x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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