\(\int \frac {\sin ^3(a+\frac {1}{3} \sqrt {-\frac {1}{n^2}} \log (c x^n))}{x^2} \, dx\) [45]

Optimal result

Integrand size = 28, antiderivative size = 176 \[ \int \frac {\sin ^3\left (a+\frac {1}{3} \sqrt {-\frac {1}{n^2}} \log \left (c x^n\right )\right )}{x^2} \, dx=-\frac {e^{3 a \sqrt {-\frac {1}{n^2}} n} \sqrt {-\frac {1}{n^2}} n \left (c x^n\right )^{-1/n}}{16 x}+\frac {9 e^{a \sqrt {-\frac {1}{n^2}} n} \sqrt {-\frac {1}{n^2}} n \left (c x^n\right )^{\left .-\frac {1}{3}\right /n}}{32 x}-\frac {9 e^{-a \sqrt {-\frac {1}{n^2}} n} \sqrt {-\frac {1}{n^2}} n \left (c x^n\right )^{\left .\frac {1}{3}\right /n}}{16 x}-\frac {e^{-3 a \sqrt {-\frac {1}{n^2}} n} \sqrt {-\frac {1}{n^2}} n \left (c x^n\right )^{\frac {1}{n}} \log (x)}{8 x} \]

[Out]

-1/16*exp(3*a*n*(-1/n^2)^(1/2))*n*(-1/n^2)^(1/2)/x/((c*x^n)^(1/n))+9/32*exp(a*n*(-1/n^2)^(1/2))*n*(-1/n^2)^(1/
2)/x/((c*x^n)^(1/3/n))-9/16*n*(c*x^n)^(1/3/n)*(-1/n^2)^(1/2)/exp(a*n*(-1/n^2)^(1/2))/x-1/8*n*(c*x^n)^(1/n)*ln(
x)*(-1/n^2)^(1/2)/exp(3*a*n*(-1/n^2)^(1/2))/x

Rubi [A] (verified)

Time = 0.14 (sec) , antiderivative size = 176, 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} \[ \int \frac {\sin ^3\left (a+\frac {1}{3} \sqrt {-\frac {1}{n^2}} \log \left (c x^n\right )\right )}{x^2} \, dx=-\frac {\sqrt {-\frac {1}{n^2}} n e^{3 a \sqrt {-\frac {1}{n^2}} n} \left (c x^n\right )^{-1/n}}{16 x}+\frac {9 \sqrt {-\frac {1}{n^2}} n e^{a \sqrt {-\frac {1}{n^2}} n} \left (c x^n\right )^{\left .-\frac {1}{3}\right /n}}{32 x}-\frac {9 \sqrt {-\frac {1}{n^2}} n e^{-a \sqrt {-\frac {1}{n^2}} n} \left (c x^n\right )^{\left .\frac {1}{3}\right /n}}{16 x}-\frac {\sqrt {-\frac {1}{n^2}} n e^{-3 a \sqrt {-\frac {1}{n^2}} n} \log (x) \left (c x^n\right )^{\frac {1}{n}}}{8 x} \]

[In]

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

[Out]

-1/16*(E^(3*a*Sqrt[-n^(-2)]*n)*Sqrt[-n^(-2)]*n)/(x*(c*x^n)^n^(-1)) + (9*E^(a*Sqrt[-n^(-2)]*n)*Sqrt[-n^(-2)]*n)
/(32*x*(c*x^n)^(1/(3*n))) - (9*Sqrt[-n^(-2)]*n*(c*x^n)^(1/(3*n)))/(16*E^(a*Sqrt[-n^(-2)]*n)*x) - (Sqrt[-n^(-2)
]*n*(c*x^n)^n^(-1)*Log[x])/(8*E^(3*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 ^3\left (a+\frac {1}{3} \sqrt {-\frac {1}{n^2}} \log (x)\right ) \, dx,x,c x^n\right )}{n x} \\ & = -\frac {\left (\sqrt {-\frac {1}{n^2}} \left (c x^n\right )^{\frac {1}{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 {4}{3 n}}-3 e^{-a \sqrt {-\frac {1}{n^2}} n} x^{-1-\frac {2}{3 n}}-e^{3 a \sqrt {-\frac {1}{n^2}} n} x^{-\frac {2+n}{n}}\right ) \, dx,x,c x^n\right )}{8 x} \\ & = -\frac {e^{3 a \sqrt {-\frac {1}{n^2}} n} \sqrt {-\frac {1}{n^2}} n \left (c x^n\right )^{-1/n}}{16 x}+\frac {9 e^{a \sqrt {-\frac {1}{n^2}} n} \sqrt {-\frac {1}{n^2}} n \left (c x^n\right )^{\left .-\frac {1}{3}\right /n}}{32 x}-\frac {9 e^{-a \sqrt {-\frac {1}{n^2}} n} \sqrt {-\frac {1}{n^2}} n \left (c x^n\right )^{\left .\frac {1}{3}\right /n}}{16 x}-\frac {e^{-3 a \sqrt {-\frac {1}{n^2}} n} \sqrt {-\frac {1}{n^2}} n \left (c x^n\right )^{\frac {1}{n}} \log (x)}{8 x} \\ \end{align*}

Mathematica [F]

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

[In]

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

[Out]

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

Maple [A] (verified)

Time = 45.93 (sec) , antiderivative size = 284, normalized size of antiderivative = 1.61

method	result	size
parallelrisch	\(\frac {12 n \sqrt {-\frac {1}{n^{2}}}\, \left (n +\frac {5 \ln \left (c \,x^{n}\right )}{12}\right ) {\tan \left (\frac {a}{2}+\sqrt {-\frac {1}{n^{2}}}\, \ln \left (\left (c \,x^{n}\right )^{\frac {1}{6}}\right )\right )}^{6}+\left (-30 \ln \left (c \,x^{n}\right )-42 n \right ) {\tan \left (\frac {a}{2}+\sqrt {-\frac {1}{n^{2}}}\, \ln \left (\left (c \,x^{n}\right )^{\frac {1}{6}}\right )\right )}^{5}-75 \sqrt {-\frac {1}{n^{2}}}\, {\tan \left (\frac {a}{2}+\sqrt {-\frac {1}{n^{2}}}\, \ln \left (\left (c \,x^{n}\right )^{\frac {1}{6}}\right )\right )}^{4} \ln \left (c \,x^{n}\right ) n +\left (100 \ln \left (c \,x^{n}\right )-220 n \right ) {\tan \left (\frac {a}{2}+\sqrt {-\frac {1}{n^{2}}}\, \ln \left (\left (c \,x^{n}\right )^{\frac {1}{6}}\right )\right )}^{3}+75 \sqrt {-\frac {1}{n^{2}}}\, {\tan \left (\frac {a}{2}+\sqrt {-\frac {1}{n^{2}}}\, \ln \left (\left (c \,x^{n}\right )^{\frac {1}{6}}\right )\right )}^{2} \ln \left (c \,x^{n}\right ) n +\left (-30 \ln \left (c \,x^{n}\right )-42 n \right ) \tan \left (\frac {a}{2}+\sqrt {-\frac {1}{n^{2}}}\, \ln \left (\left (c \,x^{n}\right )^{\frac {1}{6}}\right )\right )-12 n \sqrt {-\frac {1}{n^{2}}}\, \left (n +\frac {5 \ln \left (c \,x^{n}\right )}{12}\right )}{40 x n {\left (1+{\tan \left (\frac {a}{2}+\sqrt {-\frac {1}{n^{2}}}\, \ln \left (\left (c \,x^{n}\right )^{\frac {1}{6}}\right )\right )}^{2}\right )}^{3}}\)	\(284\)

[In]

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

[Out]

1/40*(12*n*(-1/n^2)^(1/2)*(n+5/12*ln(c*x^n))*tan(1/2*a+(-1/n^2)^(1/2)*ln((c*x^n)^(1/6)))^6+(-30*ln(c*x^n)-42*n
)*tan(1/2*a+(-1/n^2)^(1/2)*ln((c*x^n)^(1/6)))^5-75*(-1/n^2)^(1/2)*tan(1/2*a+(-1/n^2)^(1/2)*ln((c*x^n)^(1/6)))^
4*ln(c*x^n)*n+(100*ln(c*x^n)-220*n)*tan(1/2*a+(-1/n^2)^(1/2)*ln((c*x^n)^(1/6)))^3+75*(-1/n^2)^(1/2)*tan(1/2*a+
(-1/n^2)^(1/2)*ln((c*x^n)^(1/6)))^2*ln(c*x^n)*n+(-30*ln(c*x^n)-42*n)*tan(1/2*a+(-1/n^2)^(1/2)*ln((c*x^n)^(1/6)
))-12*n*(-1/n^2)^(1/2)*(n+5/12*ln(c*x^n)))/x/n/(1+tan(1/2*a+(-1/n^2)^(1/2)*ln((c*x^n)^(1/6)))^2)^3

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 \[ \int \frac {\sin ^3\left (a+\frac {1}{3} \sqrt {-\frac {1}{n^2}} \log \left (c x^n\right )\right )}{x^2} \, dx=\frac {{\left (-12 i \, x^{2} \log \left (x^{\frac {1}{3}}\right ) - 18 i \, x^{\frac {4}{3}} e^{\left (\frac {2 \, {\left (3 i \, a n - \log \left (c\right )\right )}}{3 \, n}\right )} + 9 i \, x^{\frac {2}{3}} e^{\left (\frac {4 \, {\left (3 i \, a n - \log \left (c\right )\right )}}{3 \, n}\right )} - 2 i \, e^{\left (\frac {2 \, {\left (3 i \, a n - \log \left (c\right )\right )}}{n}\right )}\right )} e^{\left (-\frac {3 i \, a n - \log \left (c\right )}{n}\right )}}{32 \, x^{2}} \]

[In]

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

[Out]

1/32*(-12*I*x^2*log(x^(1/3)) - 18*I*x^(4/3)*e^(2/3*(3*I*a*n - log(c))/n) + 9*I*x^(2/3)*e^(4/3*(3*I*a*n - log(c
))/n) - 2*I*e^(2*(3*I*a*n - log(c))/n))*e^(-(3*I*a*n - log(c))/n)/x^2

Sympy [A] (verification not implemented)

Time = 44.82 (sec) , antiderivative size = 167, normalized size of antiderivative = 0.95 \[ \int \frac {\sin ^3\left (a+\frac {1}{3} \sqrt {-\frac {1}{n^2}} \log \left (c x^n\right )\right )}{x^2} \, dx=- \frac {9 n \sqrt {- \frac {1}{n^{2}}} \cos {\left (a + \frac {\sqrt {- \frac {1}{n^{2}}} \log {\left (c x^{n} \right )}}{3} \right )}}{32 x} - \frac {\sqrt {- \frac {1}{n^{2}}} \log {\left (c x^{n} \right )} \cos {\left (3 a + \sqrt {- \frac {1}{n^{2}}} \log {\left (c x^{n} \right )} \right )}}{8 x} - \frac {27 \sin {\left (a + \frac {\sqrt {- \frac {1}{n^{2}}} \log {\left (c x^{n} \right )}}{3} \right )}}{32 x} + \frac {\sin {\left (3 a + \sqrt {- \frac {1}{n^{2}}} \log {\left (c x^{n} \right )} \right )}}{8 x} - \frac {\log {\left (c x^{n} \right )} \sin {\left (3 a + \sqrt {- \frac {1}{n^{2}}} \log {\left (c x^{n} \right )} \right )}}{8 n x} \]

[In]

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

[Out]

-9*n*sqrt(-1/n**2)*cos(a + sqrt(-1/n**2)*log(c*x**n)/3)/(32*x) - sqrt(-1/n**2)*log(c*x**n)*cos(3*a + sqrt(-1/n
**2)*log(c*x**n))/(8*x) - 27*sin(a + sqrt(-1/n**2)*log(c*x**n)/3)/(32*x) + sin(3*a + sqrt(-1/n**2)*log(c*x**n)
)/(8*x) - log(c*x**n)*sin(3*a + sqrt(-1/n**2)*log(c*x**n))/(8*n*x)

Maxima [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 122, normalized size of antiderivative = 0.69 \[ \int \frac {\sin ^3\left (a+\frac {1}{3} \sqrt {-\frac {1}{n^2}} \log \left (c x^n\right )\right )}{x^2} \, dx=-\frac {{\left (4 \, c^{\frac {7}{3 \, n}} x e^{\left (\frac {\log \left (x^{n}\right )}{3 \, n} + 2 \, \log \left (x\right )\right )} \log \left (x\right ) \sin \left (3 \, a\right ) - 2 \, c^{\frac {1}{3 \, n}} x {\left (x^{n}\right )}^{\frac {1}{3 \, n}} \sin \left (3 \, a\right ) + 9 \, c^{\left (\frac {1}{n}\right )} x^{2} \sin \left (a\right ) + 18 \, c^{\frac {5}{3 \, n}} e^{\left (\frac {2 \, \log \left (x^{n}\right )}{3 \, n} + 2 \, \log \left (x\right )\right )} \sin \left (a\right )\right )} e^{\left (-\frac {\log \left (x^{n}\right )}{3 \, n} - 2 \, \log \left (x\right )\right )}}{32 \, c^{\frac {4}{3 \, n}} x} \]

[In]

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

[Out]

-1/32*(4*c^(7/3/n)*x*e^(1/3*log(x^n)/n + 2*log(x))*log(x)*sin(3*a) - 2*c^(1/3/n)*x*(x^n)^(1/3/n)*sin(3*a) + 9*
c^(1/n)*x^2*sin(a) + 18*c^(5/3/n)*e^(2/3*log(x^n)/n + 2*log(x))*sin(a))*e^(-1/3*log(x^n)/n - 2*log(x))/(c^(4/3
/n)*x)

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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