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

Optimal. Leaf size=88 \[ \frac {1}{8} \sqrt {-\frac {1}{n^2}} n x^2 e^{-a \sqrt {-\frac {1}{n^2}} n} \left (c x^n\right )^{2/n}-\frac {1}{2} \sqrt {-\frac {1}{n^2}} n x^2 e^{a \sqrt {-\frac {1}{n^2}} n} \log (x) \left (c x^n\right )^{-2/n} \]

[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))

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Rubi [A]  time = 0.05, antiderivative size = 88, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {4493, 4489} \[ \frac {1}{8} \sqrt {-\frac {1}{n^2}} n x^2 e^{-a \sqrt {-\frac {1}{n^2}} n} \left (c x^n\right )^{2/n}-\frac {1}{2} \sqrt {-\frac {1}{n^2}} n x^2 e^{a \sqrt {-\frac {1}{n^2}} n} \log (x) \left (c x^n\right )^{-2/n} \]

Antiderivative was successfully verified.

[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 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]

Rule 4493

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

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Mathematica [F]  time = 0.16, size = 0, normalized size = 0.00 \[ \int x \sin \left (a+2 \sqrt {-\frac {1}{n^2}} \log \left (c x^n\right )\right ) \, dx \]

Verification is Not applicable to the result.

[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]

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fricas [C]  time = 0.51, size = 42, normalized size = 0.48 \[ \frac {1}{8} \, {\left (i \, x^{4} - 4 i \, e^{\left (\frac {2 \, {\left (i \, a n - 2 \, \log \relax (c)\right )}}{n}\right )} \log \relax (x)\right )} e^{\left (-\frac {i \, a n - 2 \, \log \relax (c)}{n}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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giac [A]  time = 0.50, size = 1, normalized size = 0.01 \[ +\infty \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

+Infinity

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maple [F]  time = 0.03, size = 0, normalized size = 0.00 \[ \int x \sin \left (a +2 \ln \left (c \,x^{n}\right ) \sqrt {-\frac {1}{n^{2}}}\right )\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.36, size = 31, normalized size = 0.35 \[ \frac {c^{\frac {4}{n}} x^{4} \sin \relax (a) + 4 \, \log \relax (x) \sin \relax (a)}{8 \, c^{\frac {2}{n}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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mupad [B]  time = 2.80, size = 85, normalized size = 0.97 \[ -\frac {x^2\,{\mathrm {e}}^{-a\,1{}\mathrm {i}}\,\frac {1}{{\left (c\,x^n\right )}^{\sqrt {-\frac {1}{n^2}}\,2{}\mathrm {i}}}}{4\,n\,\sqrt {-\frac {1}{n^2}}+4{}\mathrm {i}}-\frac {x^2\,{\mathrm {e}}^{a\,1{}\mathrm {i}}\,{\left (c\,x^n\right )}^{\sqrt {-\frac {1}{n^2}}\,2{}\mathrm {i}}}{4\,n\,\sqrt {-\frac {1}{n^2}}-4{}\mathrm {i}} \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(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)

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