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

Optimal. Leaf size=82 \[ \frac {1}{4} \sqrt {-\frac {1}{n^2}} n x e^{-a \sqrt {-\frac {1}{n^2}} n} \left (c x^n\right )^{\frac {1}{n}}-\frac {1}{2} \sqrt {-\frac {1}{n^2}} n x e^{a \sqrt {-\frac {1}{n^2}} n} \log (x) \left (c x^n\right )^{-1/n} \]

[Out]

1/4*n*x*(c*x^n)^(1/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*ln(x)*(-1/n^2)^(1
/2)/((c*x^n)^(1/n))

________________________________________________________________________________________

Rubi [A]  time = 0.05, antiderivative size = 82, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {4483, 4489} \[ \frac {1}{4} \sqrt {-\frac {1}{n^2}} n x e^{-a \sqrt {-\frac {1}{n^2}} n} \left (c x^n\right )^{\frac {1}{n}}-\frac {1}{2} \sqrt {-\frac {1}{n^2}} n x e^{a \sqrt {-\frac {1}{n^2}} n} \log (x) \left (c x^n\right )^{-1/n} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[-n^(-2)]*n*x*(c*x^n)^n^(-1))/(4*E^(a*Sqrt[-n^(-2)]*n)) - (E^(a*Sqrt[-n^(-2)]*n)*Sqrt[-n^(-2)]*n*x*Log[x]
)/(2*(c*x^n)^n^(-1))

Rule 4483

Int[Sin[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*(d_.)]^(p_.), x_Symbol] :> Dist[x/(n*(c*x^n)^(1/n)), Subst[Int[x
^(1/n - 1)*Sin[d*(a + b*Log[x])]^p, x], x, c*x^n], x] /; FreeQ[{a, b, c, d, n, p}, x] && (NeQ[c, 1] || NeQ[n,
1])

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]

Rubi steps

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

________________________________________________________________________________________

Mathematica [F]  time = 0.11, size = 0, normalized size = 0.00 \[ \int \sin \left (a+\sqrt {-\frac {1}{n^2}} \log \left (c x^n\right )\right ) \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

________________________________________________________________________________________

fricas [C]  time = 0.57, size = 42, normalized size = 0.51 \[ \frac {1}{4} \, {\left (i \, x^{2} - 2 i \, e^{\left (\frac {2 \, {\left (i \, a n - \log \relax (c)\right )}}{n}\right )} \log \relax (x)\right )} e^{\left (-\frac {i \, a n - \log \relax (c)}{n}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

giac [A]  time = 0.43, size = 1, normalized size = 0.01 \[ +\infty \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

+Infinity

________________________________________________________________________________________

maple [F]  time = 0.02, size = 0, normalized size = 0.00 \[ \int \sin \left (a +\ln \left (c \,x^{n}\right ) \sqrt {-\frac {1}{n^{2}}}\right )\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maxima [A]  time = 0.36, size = 29, normalized size = 0.35 \[ \frac {c^{\frac {2}{n}} x^{2} \sin \relax (a) + 2 \, \log \relax (x) \sin \relax (a)}{4 \, c^{\left (\frac {1}{n}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

mupad [B]  time = 2.73, size = 81, normalized size = 0.99 \[ -\frac {x\,{\mathrm {e}}^{-a\,1{}\mathrm {i}}\,\frac {1}{{\left (c\,x^n\right )}^{\sqrt {-\frac {1}{n^2}}\,1{}\mathrm {i}}}}{2\,n\,\sqrt {-\frac {1}{n^2}}+2{}\mathrm {i}}-\frac {x\,{\mathrm {e}}^{a\,1{}\mathrm {i}}\,{\left (c\,x^n\right )}^{\sqrt {-\frac {1}{n^2}}\,1{}\mathrm {i}}}{2\,n\,\sqrt {-\frac {1}{n^2}}-2{}\mathrm {i}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________