\(\int \frac {\sin ^3(a+\frac {2}{3} \sqrt {-\frac {1}{n^2}} \log (c x^n))}{x^3} \, dx\) [46]

Optimal result

Integrand size = 28, antiderivative size = 178 \[ \int \frac {\sin ^3\left (a+\frac {2}{3} \sqrt {-\frac {1}{n^2}} \log \left (c x^n\right )\right )}{x^3} \, dx=-\frac {e^{3 a \sqrt {-\frac {1}{n^2}} n} \sqrt {-\frac {1}{n^2}} n \left (c x^n\right )^{-2/n}}{32 x^2}+\frac {9 e^{a \sqrt {-\frac {1}{n^2}} n} \sqrt {-\frac {1}{n^2}} n \left (c x^n\right )^{\left .-\frac {2}{3}\right /n}}{64 x^2}-\frac {9 e^{-a \sqrt {-\frac {1}{n^2}} n} \sqrt {-\frac {1}{n^2}} n \left (c x^n\right )^{\left .\frac {2}{3}\right /n}}{32 x^2}-\frac {e^{-3 a \sqrt {-\frac {1}{n^2}} n} \sqrt {-\frac {1}{n^2}} n \left (c x^n\right )^{2/n} \log (x)}{8 x^2} \]

[Out]

-1/32*exp(3*a*n*(-1/n^2)^(1/2))*n*(-1/n^2)^(1/2)/x^2/((c*x^n)^(2/n))+9/64*exp(a*n*(-1/n^2)^(1/2))*n*(-1/n^2)^(
1/2)/x^2/((c*x^n)^(2/3/n))-9/32*n*(c*x^n)^(2/3/n)*(-1/n^2)^(1/2)/exp(a*n*(-1/n^2)^(1/2))/x^2-1/8*n*(c*x^n)^(2/
n)*ln(x)*(-1/n^2)^(1/2)/exp(3*a*n*(-1/n^2)^(1/2))/x^2

Rubi [A] (verified)

Time = 0.13 (sec) , antiderivative size = 178, 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} \[ \int \frac {\sin ^3\left (a+\frac {2}{3} \sqrt {-\frac {1}{n^2}} \log \left (c x^n\right )\right )}{x^3} \, dx=-\frac {\sqrt {-\frac {1}{n^2}} n e^{3 a \sqrt {-\frac {1}{n^2}} n} \left (c x^n\right )^{-2/n}}{32 x^2}+\frac {9 \sqrt {-\frac {1}{n^2}} n e^{a \sqrt {-\frac {1}{n^2}} n} \left (c x^n\right )^{\left .-\frac {2}{3}\right /n}}{64 x^2}-\frac {9 \sqrt {-\frac {1}{n^2}} n e^{-a \sqrt {-\frac {1}{n^2}} n} \left (c x^n\right )^{\left .\frac {2}{3}\right /n}}{32 x^2}-\frac {\sqrt {-\frac {1}{n^2}} n e^{-3 a \sqrt {-\frac {1}{n^2}} n} \log (x) \left (c x^n\right )^{2/n}}{8 x^2} \]

[In]

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

[Out]

-1/32*(E^(3*a*Sqrt[-n^(-2)]*n)*Sqrt[-n^(-2)]*n)/(x^2*(c*x^n)^(2/n)) + (9*E^(a*Sqrt[-n^(-2)]*n)*Sqrt[-n^(-2)]*n
)/(64*x^2*(c*x^n)^(2/(3*n))) - (9*Sqrt[-n^(-2)]*n*(c*x^n)^(2/(3*n)))/(32*E^(a*Sqrt[-n^(-2)]*n)*x^2) - (Sqrt[-n
^(-2)]*n*(c*x^n)^(2/n)*Log[x])/(8*E^(3*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 ^3\left (a+\frac {2}{3} \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^{-3 a \sqrt {-\frac {1}{n^2}} n}}{x}+3 e^{a \sqrt {-\frac {1}{n^2}} n} x^{-1-\frac {8}{3 n}}-3 e^{-a \sqrt {-\frac {1}{n^2}} n} x^{-1-\frac {4}{3 n}}-e^{3 a \sqrt {-\frac {1}{n^2}} n} x^{-\frac {4+n}{n}}\right ) \, dx,x,c x^n\right )}{8 x^2} \\ & = -\frac {e^{3 a \sqrt {-\frac {1}{n^2}} n} \sqrt {-\frac {1}{n^2}} n \left (c x^n\right )^{-2/n}}{32 x^2}+\frac {9 e^{a \sqrt {-\frac {1}{n^2}} n} \sqrt {-\frac {1}{n^2}} n \left (c x^n\right )^{\left .-\frac {2}{3}\right /n}}{64 x^2}-\frac {9 e^{-a \sqrt {-\frac {1}{n^2}} n} \sqrt {-\frac {1}{n^2}} n \left (c x^n\right )^{\left .\frac {2}{3}\right /n}}{32 x^2}-\frac {e^{-3 a \sqrt {-\frac {1}{n^2}} n} \sqrt {-\frac {1}{n^2}} n \left (c x^n\right )^{2/n} \log (x)}{8 x^2} \\ \end{align*}

Mathematica [F]

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

[In]

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

[Out]

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

Maple [A] (verified)

Time = 88.06 (sec) , antiderivative size = 136, normalized size of antiderivative = 0.76

method	result	size
parallelrisch	\(\frac {-47 n \left (n +\frac {40 \ln \left (c \,x^{n}\right )}{47}\right ) \sqrt {-\frac {1}{n^{2}}}\, \cos \left (2 \ln \left (c \,x^{n}\right ) \sqrt {-\frac {1}{n^{2}}}+3 a \right )+\left (-27 n -40 \ln \left (c \,x^{n}\right )\right ) \sin \left (2 \ln \left (c \,x^{n}\right ) \sqrt {-\frac {1}{n^{2}}}+3 a \right )-45 n \left (\cos \left (a +\sqrt {-\frac {1}{n^{2}}}\, \ln \left (\left (c \,x^{n}\right )^{\frac {2}{3}}\right )\right ) n \sqrt {-\frac {1}{n^{2}}}+3 \sin \left (a +\sqrt {-\frac {1}{n^{2}}}\, \ln \left (\left (c \,x^{n}\right )^{\frac {2}{3}}\right )\right )\right )}{320 x^{2} n}\)	\(136\)

[In]

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

[Out]

1/320*(-47*n*(n+40/47*ln(c*x^n))*(-1/n^2)^(1/2)*cos(2*ln(c*x^n)*(-1/n^2)^(1/2)+3*a)+(-27*n-40*ln(c*x^n))*sin(2
*ln(c*x^n)*(-1/n^2)^(1/2)+3*a)-45*n*(cos(a+(-1/n^2)^(1/2)*ln((c*x^n)^(2/3)))*n*(-1/n^2)^(1/2)+3*sin(a+(-1/n^2)
^(1/2)*ln((c*x^n)^(2/3)))))/x^2/n

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.25 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.49 \[ \int \frac {\sin ^3\left (a+\frac {2}{3} \sqrt {-\frac {1}{n^2}} \log \left (c x^n\right )\right )}{x^3} \, dx=\frac {{\left (-24 i \, x^{4} \log \left (x^{\frac {1}{3}}\right ) - 18 i \, x^{\frac {8}{3}} e^{\left (\frac {2 \, {\left (3 i \, a n - 2 \, \log \left (c\right )\right )}}{3 \, n}\right )} + 9 i \, x^{\frac {4}{3}} e^{\left (\frac {4 \, {\left (3 i \, a n - 2 \, \log \left (c\right )\right )}}{3 \, n}\right )} - 2 i \, e^{\left (\frac {2 \, {\left (3 i \, a n - 2 \, \log \left (c\right )\right )}}{n}\right )}\right )} e^{\left (-\frac {3 i \, a n - 2 \, \log \left (c\right )}{n}\right )}}{64 \, x^{4}} \]

[In]

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

[Out]

1/64*(-24*I*x^4*log(x^(1/3)) - 18*I*x^(8/3)*e^(2/3*(3*I*a*n - 2*log(c))/n) + 9*I*x^(4/3)*e^(4/3*(3*I*a*n - 2*l
og(c))/n) - 2*I*e^(2*(3*I*a*n - 2*log(c))/n))*e^(-(3*I*a*n - 2*log(c))/n)/x^4

Sympy [A] (verification not implemented)

Time = 54.03 (sec) , antiderivative size = 184, normalized size of antiderivative = 1.03 \[ \int \frac {\sin ^3\left (a+\frac {2}{3} \sqrt {-\frac {1}{n^2}} \log \left (c x^n\right )\right )}{x^3} \, dx=- \frac {9 n \sqrt {- \frac {1}{n^{2}}} \cos {\left (a + \frac {2 \sqrt {- \frac {1}{n^{2}}} \log {\left (c x^{n} \right )}}{3} \right )}}{64 x^{2}} - \frac {\sqrt {- \frac {1}{n^{2}}} \log {\left (c x^{n} \right )} \cos {\left (3 a + 2 \sqrt {- \frac {1}{n^{2}}} \log {\left (c x^{n} \right )} \right )}}{8 x^{2}} - \frac {27 \sin {\left (a + \frac {2 \sqrt {- \frac {1}{n^{2}}} \log {\left (c x^{n} \right )}}{3} \right )}}{64 x^{2}} + \frac {\sin {\left (3 a + 2 \sqrt {- \frac {1}{n^{2}}} \log {\left (c x^{n} \right )} \right )}}{16 x^{2}} - \frac {\log {\left (c x^{n} \right )} \sin {\left (3 a + 2 \sqrt {- \frac {1}{n^{2}}} \log {\left (c x^{n} \right )} \right )}}{8 n x^{2}} \]

[In]

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

[Out]

-9*n*sqrt(-1/n**2)*cos(a + 2*sqrt(-1/n**2)*log(c*x**n)/3)/(64*x**2) - sqrt(-1/n**2)*log(c*x**n)*cos(3*a + 2*sq
rt(-1/n**2)*log(c*x**n))/(8*x**2) - 27*sin(a + 2*sqrt(-1/n**2)*log(c*x**n)/3)/(64*x**2) + sin(3*a + 2*sqrt(-1/
n**2)*log(c*x**n))/(16*x**2) - log(c*x**n)*sin(3*a + 2*sqrt(-1/n**2)*log(c*x**n))/(8*n*x**2)

Maxima [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.72 \[ \int \frac {\sin ^3\left (a+\frac {2}{3} \sqrt {-\frac {1}{n^2}} \log \left (c x^n\right )\right )}{x^3} \, dx=-\frac {{\left (8 \, c^{\frac {14}{3 \, n}} x^{2} e^{\left (\frac {2 \, \log \left (x^{n}\right )}{3 \, n} + 4 \, \log \left (x\right )\right )} \log \left (x\right ) \sin \left (3 \, a\right ) + 9 \, c^{\frac {2}{n}} x^{4} \sin \left (a\right ) - 2 \, c^{\frac {2}{3 \, n}} x^{2} {\left (x^{n}\right )}^{\frac {2}{3 \, n}} \sin \left (3 \, a\right ) + 18 \, c^{\frac {10}{3 \, n}} e^{\left (\frac {4 \, \log \left (x^{n}\right )}{3 \, n} + 4 \, \log \left (x\right )\right )} \sin \left (a\right )\right )} e^{\left (-\frac {2 \, \log \left (x^{n}\right )}{3 \, n} - 4 \, \log \left (x\right )\right )}}{64 \, c^{\frac {8}{3 \, n}} x^{2}} \]

[In]

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

[Out]

-1/64*(8*c^(14/3/n)*x^2*e^(2/3*log(x^n)/n + 4*log(x))*log(x)*sin(3*a) + 9*c^(2/n)*x^4*sin(a) - 2*c^(2/3/n)*x^2
*(x^n)^(2/3/n)*sin(3*a) + 18*c^(10/3/n)*e^(4/3*log(x^n)/n + 4*log(x))*sin(a))*e^(-2/3*log(x^n)/n - 4*log(x))/(
c^(8/3/n)*x^2)

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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