\(\int \frac {\sin (a+2 \sqrt {-\frac {1}{n^2}} \log (c x^n))}{x^3} \, dx\) [32]

Optimal result

Integrand size = 24, antiderivative size = 88 \[ \int \frac {\sin \left (a+2 \sqrt {-\frac {1}{n^2}} \log \left (c x^n\right )\right )}{x^3} \, dx=\frac {e^{a \sqrt {-\frac {1}{n^2}} n} \sqrt {-\frac {1}{n^2}} n \left (c x^n\right )^{-2/n}}{8 x^2}+\frac {e^{-a \sqrt {-\frac {1}{n^2}} n} \sqrt {-\frac {1}{n^2}} n \left (c x^n\right )^{2/n} \log (x)}{2 x^2} \]

[Out]

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

Rubi [A] (verified)

Time = 0.07 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {4581, 4577} \[ \int \frac {\sin \left (a+2 \sqrt {-\frac {1}{n^2}} \log \left (c x^n\right )\right )}{x^3} \, dx=\frac {\sqrt {-\frac {1}{n^2}} n e^{a \sqrt {-\frac {1}{n^2}} n} \left (c x^n\right )^{-2/n}}{8 x^2}+\frac {\sqrt {-\frac {1}{n^2}} n e^{-a \sqrt {-\frac {1}{n^2}} n} \log (x) \left (c x^n\right )^{2/n}}{2 x^2} \]

[In]

Int[Sin[a + 2*Sqrt[-n^(-2)]*Log[c*x^n]]/x^3,x]

[Out]

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

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

Mathematica [F]

\[ \int \frac {\sin \left (a+2 \sqrt {-\frac {1}{n^2}} \log \left (c x^n\right )\right )}{x^3} \, dx=\int \frac {\sin \left (a+2 \sqrt {-\frac {1}{n^2}} \log \left (c x^n\right )\right )}{x^3} \, dx \]

[In]

Integrate[Sin[a + 2*Sqrt[-n^(-2)]*Log[c*x^n]]/x^3,x]

[Out]

Integrate[Sin[a + 2*Sqrt[-n^(-2)]*Log[c*x^n]]/x^3, x]

Maple [A] (verified)

Time = 4.85 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.35

method	result	size
parallelrisch	\(\frac {-\sqrt {-\frac {1}{n^{2}}}\, {\tan \left (\frac {a}{2}+\ln \left (c \,x^{n}\right ) \sqrt {-\frac {1}{n^{2}}}\right )}^{2} \ln \left (c \,x^{n}\right ) n +\left (-n +2 \ln \left (c \,x^{n}\right )\right ) \tan \left (\frac {a}{2}+\ln \left (c \,x^{n}\right ) \sqrt {-\frac {1}{n^{2}}}\right )+\sqrt {-\frac {1}{n^{2}}}\, \ln \left (c \,x^{n}\right ) n}{2 x^{2} n \left (1+{\tan \left (\frac {a}{2}+\ln \left (c \,x^{n}\right ) \sqrt {-\frac {1}{n^{2}}}\right )}^{2}\right )}\)	\(119\)

[In]

int(sin(a+2*ln(c*x^n)*(-1/n^2)^(1/2))/x^3,x,method=_RETURNVERBOSE)

[Out]

1/2*(-(-1/n^2)^(1/2)*tan(1/2*a+ln(c*x^n)*(-1/n^2)^(1/2))^2*ln(c*x^n)*n+(-n+2*ln(c*x^n))*tan(1/2*a+ln(c*x^n)*(-
1/n^2)^(1/2))+(-1/n^2)^(1/2)*ln(c*x^n)*n)/x^2/n/(1+tan(1/2*a+ln(c*x^n)*(-1/n^2)^(1/2))^2)

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.24 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.51 \[ \int \frac {\sin \left (a+2 \sqrt {-\frac {1}{n^2}} \log \left (c x^n\right )\right )}{x^3} \, dx=\frac {{\left (4 i \, x^{4} \log \left (x\right ) + i \, e^{\left (\frac {2 \, {\left (i \, a n - 2 \, \log \left (c\right )\right )}}{n}\right )}\right )} e^{\left (-\frac {i \, a n - 2 \, \log \left (c\right )}{n}\right )}}{8 \, x^{4}} \]

[In]

integrate(sin(a+2*log(c*x^n)*(-1/n^2)^(1/2))/x^3,x, algorithm="fricas")

[Out]

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

Sympy [A] (verification not implemented)

Time = 5.41 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.33 \[ \int \frac {\sin \left (a+2 \sqrt {-\frac {1}{n^2}} \log \left (c x^n\right )\right )}{x^3} \, dx=\frac {n \sqrt {- \frac {1}{n^{2}}} \cos {\left (a + 2 \sqrt {- \frac {1}{n^{2}}} \log {\left (c x^{n} \right )} \right )}}{4 x^{2}} + \frac {\sqrt {- \frac {1}{n^{2}}} \log {\left (c x^{n} \right )} \cos {\left (a + 2 \sqrt {- \frac {1}{n^{2}}} \log {\left (c x^{n} \right )} \right )}}{2 x^{2}} + \frac {\log {\left (c x^{n} \right )} \sin {\left (a + 2 \sqrt {- \frac {1}{n^{2}}} \log {\left (c x^{n} \right )} \right )}}{2 n x^{2}} \]

[In]

integrate(sin(a+2*ln(c*x**n)*(-1/n**2)**(1/2))/x**3,x)

[Out]

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)

Maxima [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.40 \[ \int \frac {\sin \left (a+2 \sqrt {-\frac {1}{n^2}} \log \left (c x^n\right )\right )}{x^3} \, dx=\frac {4 \, c^{\frac {4}{n}} x^{4} \log \left (x\right ) \sin \left (a\right ) - \sin \left (a\right )}{8 \, c^{\frac {2}{n}} x^{4}} \]

[In]

integrate(sin(a+2*log(c*x^n)*(-1/n^2)^(1/2))/x^3,x, algorithm="maxima")

[Out]

1/8*(4*c^(4/n)*x^4*log(x)*sin(a) - sin(a))/(c^(2/n)*x^4)

Giac [F]

\[ \int \frac {\sin \left (a+2 \sqrt {-\frac {1}{n^2}} \log \left (c x^n\right )\right )}{x^3} \, dx=\int { \frac {\sin \left (2 \, \sqrt {-\frac {1}{n^{2}}} \log \left (c x^{n}\right ) + a\right )}{x^{3}} \,d x } \]

[In]

integrate(sin(a+2*log(c*x^n)*(-1/n^2)^(1/2))/x^3,x, algorithm="giac")

[Out]

integrate(sin(2*sqrt(-1/n^2)*log(c*x^n) + a)/x^3, x)

Mupad [F(-1)]

Timed out. \[ \int \frac {\sin \left (a+2 \sqrt {-\frac {1}{n^2}} \log \left (c x^n\right )\right )}{x^3} \, dx=\int \frac {\sin \left (a+2\,\ln \left (c\,x^n\right )\,\sqrt {-\frac {1}{n^2}}\right )}{x^3} \,d x \]

[In]

int(sin(a + 2*log(c*x^n)*(-1/n^2)^(1/2))/x^3,x)

[Out]

int(sin(a + 2*log(c*x^n)*(-1/n^2)^(1/2))/x^3, x)