\(\int x \sin (a+2 \sqrt {-\frac {1}{n^2}} \log (c x^n)) \, dx\) [28]

Optimal result

Integrand size = 22, antiderivative size = 88 \[ \int x \sin \left (a+2 \sqrt {-\frac {1}{n^2}} \log \left (c x^n\right )\right ) \, dx=\frac {1}{8} e^{-a \sqrt {-\frac {1}{n^2}} n} \sqrt {-\frac {1}{n^2}} n x^2 \left (c x^n\right )^{2/n}-\frac {1}{2} e^{a \sqrt {-\frac {1}{n^2}} n} \sqrt {-\frac {1}{n^2}} n x^2 \left (c x^n\right )^{-2/n} \log (x) \]

[Out]

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

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, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {4581, 4577} \[ \int x \sin \left (a+2 \sqrt {-\frac {1}{n^2}} \log \left (c x^n\right )\right ) \, dx=\frac {1}{8} \sqrt {-\frac {1}{n^2}} n x^2 e^{-a \sqrt {-\frac {1}{n^2}} n} \left (c x^n\right )^{2/n}-\frac {1}{2} \sqrt {-\frac {1}{n^2}} n x^2 e^{a \sqrt {-\frac {1}{n^2}} n} \log (x) \left (c x^n\right )^{-2/n} \]

[In]

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

[Out]

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

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 (x^2 \left (c x^n\right )^{-2/n}\right ) \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} \\ & = -\left (\frac {1}{2} \left (\sqrt {-\frac {1}{n^2}} x^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^{-1+\frac {4}{n}}\right ) \, dx,x,c x^n\right )\right ) \\ & = \frac {1}{8} e^{-a \sqrt {-\frac {1}{n^2}} n} \sqrt {-\frac {1}{n^2}} n x^2 \left (c x^n\right )^{2/n}-\frac {1}{2} e^{a \sqrt {-\frac {1}{n^2}} n} \sqrt {-\frac {1}{n^2}} n x^2 \left (c x^n\right )^{-2/n} \log (x) \\ \end{align*}

Mathematica [F]

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

[In]

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

[Out]

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(609\) vs. \(2(77)=154\).

Time = 1.82 (sec) , antiderivative size = 610, normalized size of antiderivative = 6.93

method	result	size
parts	\(\frac {2 n x \sqrt {-\frac {1}{n^{2}}}\, {\mathrm e}^{\frac {\ln \left (c \,x^{n}\right )}{n}-\frac {\ln \left (c \right )}{n}} \cos \left (a +2 \ln \left (c \,x^{n}\right ) \sqrt {-\frac {1}{n^{2}}}\right )}{3}-\frac {x \,{\mathrm e}^{\frac {\ln \left (c \,x^{n}\right )}{n}-\frac {\ln \left (c \right )}{n}} \sin \left (a +2 \ln \left (c \,x^{n}\right ) \sqrt {-\frac {1}{n^{2}}}\right )}{3}-\frac {-\frac {n \left (-\frac {c^{-\frac {1}{n}} {\mathrm e}^{\frac {\ln \left (c \,x^{n}\right )-n \ln \left (x \right )}{n}} x^{2}}{4 \sqrt {-\frac {1}{n^{2}}}\, n}+\frac {c^{-\frac {1}{n}} {\mathrm e}^{\frac {\ln \left (c \,x^{n}\right )-n \ln \left (x \right )}{n}} x^{2} \ln \left (x \right )}{2 \sqrt {-\frac {1}{n^{2}}}\, n}+\frac {c^{-\frac {1}{n}} {\mathrm e}^{\frac {\ln \left (c \,x^{n}\right )-n \ln \left (x \right )}{n}} x^{2} {\tan \left (\frac {a}{2}+\ln \left (c \,x^{n}\right ) \sqrt {-\frac {1}{n^{2}}}\right )}^{2}}{4 \sqrt {-\frac {1}{n^{2}}}\, n}+c^{-\frac {1}{n}} {\mathrm e}^{\frac {\ln \left (c \,x^{n}\right )-n \ln \left (x \right )}{n}} x^{2} \ln \left (x \right ) \tan \left (\frac {a}{2}+\ln \left (c \,x^{n}\right ) \sqrt {-\frac {1}{n^{2}}}\right )-\frac {c^{-\frac {1}{n}} {\mathrm e}^{\frac {\ln \left (c \,x^{n}\right )-n \ln \left (x \right )}{n}} x^{2} \ln \left (x \right ) {\tan \left (\frac {a}{2}+\ln \left (c \,x^{n}\right ) \sqrt {-\frac {1}{n^{2}}}\right )}^{2}}{2 \sqrt {-\frac {1}{n^{2}}}\, n}\right )}{1+{\tan \left (\frac {a}{2}+\ln \left (c \,x^{n}\right ) \sqrt {-\frac {1}{n^{2}}}\right )}^{2}}+\frac {2 \sqrt {-\frac {1}{n^{2}}}\, n^{2} \left (-\frac {c^{-\frac {1}{n}} {\mathrm e}^{\frac {\ln \left (c \,x^{n}\right )-n \ln \left (x \right )}{n}} x^{2} \ln \left (x \right ) {\tan \left (\frac {a}{2}+\ln \left (c \,x^{n}\right ) \sqrt {-\frac {1}{n^{2}}}\right )}^{2}}{2}+\frac {c^{-\frac {1}{n}} {\mathrm e}^{\frac {\ln \left (c \,x^{n}\right )-n \ln \left (x \right )}{n}} x^{2} \ln \left (x \right )}{2}-\frac {n \sqrt {-\frac {1}{n^{2}}}\, {\mathrm e}^{\frac {\ln \left (c \,x^{n}\right )-n \ln \left (x \right )}{n}} c^{-\frac {1}{n}} x^{2} \tan \left (\frac {a}{2}+\ln \left (c \,x^{n}\right ) \sqrt {-\frac {1}{n^{2}}}\right )}{2}+c^{-\frac {1}{n}} {\mathrm e}^{\frac {\ln \left (c \,x^{n}\right )-n \ln \left (x \right )}{n}} n \sqrt {-\frac {1}{n^{2}}}\, x^{2} \ln \left (x \right ) \tan \left (\frac {a}{2}+\ln \left (c \,x^{n}\right ) \sqrt {-\frac {1}{n^{2}}}\right )\right )}{1+{\tan \left (\frac {a}{2}+\ln \left (c \,x^{n}\right ) \sqrt {-\frac {1}{n^{2}}}\right )}^{2}}}{3 n}\)	\(610\)

[In]

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

[Out]

2/3*n*x*(-1/n^2)^(1/2)*exp(1/n*ln(c*x^n)-1/n*ln(c))*cos(a+2*ln(c*x^n)*(-1/n^2)^(1/2))-1/3*x*exp(1/n*ln(c*x^n)-
1/n*ln(c))*sin(a+2*ln(c*x^n)*(-1/n^2)^(1/2))-1/3/n*(-n*(-1/4/(-1/n^2)^(1/2)/(c^(1/n))/n*exp(1/n*(ln(c*x^n)-n*l
n(x)))*x^2+1/2/(-1/n^2)^(1/2)/(c^(1/n))/n*exp(1/n*(ln(c*x^n)-n*ln(x)))*x^2*ln(x)+1/4/(-1/n^2)^(1/2)/(c^(1/n))/
n*exp(1/n*(ln(c*x^n)-n*ln(x)))*x^2*tan(1/2*a+ln(c*x^n)*(-1/n^2)^(1/2))^2+1/(c^(1/n))*exp(1/n*(ln(c*x^n)-n*ln(x
)))*x^2*ln(x)*tan(1/2*a+ln(c*x^n)*(-1/n^2)^(1/2))-1/2/(-1/n^2)^(1/2)/(c^(1/n))/n*exp(1/n*(ln(c*x^n)-n*ln(x)))*
x^2*ln(x)*tan(1/2*a+ln(c*x^n)*(-1/n^2)^(1/2))^2)/(1+tan(1/2*a+ln(c*x^n)*(-1/n^2)^(1/2))^2)+2*(-1/n^2)^(1/2)*n^
2*(-1/2/(c^(1/n))*exp(1/n*(ln(c*x^n)-n*ln(x)))*x^2*ln(x)*tan(1/2*a+ln(c*x^n)*(-1/n^2)^(1/2))^2+1/2/(c^(1/n))*e
xp(1/n*(ln(c*x^n)-n*ln(x)))*x^2*ln(x)-1/2*n*(-1/n^2)^(1/2)*exp(1/n*(ln(c*x^n)-n*ln(x)))/(c^(1/n))*x^2*tan(1/2*
a+ln(c*x^n)*(-1/n^2)^(1/2))+1/(c^(1/n))*exp(1/n*(ln(c*x^n)-n*ln(x)))*n*(-1/n^2)^(1/2)*x^2*ln(x)*tan(1/2*a+ln(c
*x^n)*(-1/n^2)^(1/2)))/(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 = 0.25 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.48 \[ \int x \sin \left (a+2 \sqrt {-\frac {1}{n^2}} \log \left (c x^n\right )\right ) \, dx=\frac {1}{8} \, {\left (i \, x^{4} - 4 i \, e^{\left (\frac {2 \, {\left (i \, a n - 2 \, \log \left (c\right )\right )}}{n}\right )} \log \left (x\right )\right )} e^{\left (-\frac {i \, a n - 2 \, \log \left (c\right )}{n}\right )} \]

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.35 \[ \int x \sin \left (a+2 \sqrt {-\frac {1}{n^2}} \log \left (c x^n\right )\right ) \, dx=\frac {c^{\frac {4}{n}} x^{4} \sin \left (a\right ) + 4 \, \log \left (x\right ) \sin \left (a\right )}{8 \, c^{\frac {2}{n}}} \]

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.42 (sec) , antiderivative size = 1, normalized size of antiderivative = 0.01 \[ \int x \sin \left (a+2 \sqrt {-\frac {1}{n^2}} \log \left (c x^n\right )\right ) \, dx=+\infty \]

[In]

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

[Out]

+Infinity

Mupad [B] (verification not implemented)

Time = 27.34 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.97 \[ \int x \sin \left (a+2 \sqrt {-\frac {1}{n^2}} \log \left (c x^n\right )\right ) \, dx=-\frac {x^2\,{\mathrm {e}}^{-a\,1{}\mathrm {i}}\,\frac {1}{{\left (c\,x^n\right )}^{\sqrt {-\frac {1}{n^2}}\,2{}\mathrm {i}}}}{4\,n\,\sqrt {-\frac {1}{n^2}}+4{}\mathrm {i}}-\frac {x^2\,{\mathrm {e}}^{a\,1{}\mathrm {i}}\,{\left (c\,x^n\right )}^{\sqrt {-\frac {1}{n^2}}\,2{}\mathrm {i}}}{4\,n\,\sqrt {-\frac {1}{n^2}}-4{}\mathrm {i}} \]

[In]

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

[Out]

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