3.1.27 \(\int x^2 \sin (a+3 \sqrt {-\frac {1}{n^2}} \log (c x^n)) \, dx\) [27]

Optimal. Leaf size=88 \[ \frac {1}{12} e^{-a \sqrt {-\frac {1}{n^2}} n} \sqrt {-\frac {1}{n^2}} n x^3 \left (c x^n\right )^{3/n}-\frac {1}{2} e^{a \sqrt {-\frac {1}{n^2}} n} \sqrt {-\frac {1}{n^2}} n x^3 \left (c x^n\right )^{-3/n} \log (x) \]

[Out]

1/12*n*x^3*(c*x^n)^(3/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^3*ln(x)*(-1/n^
2)^(1/2)/((c*x^n)^(3/n))

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Rubi [A]
time = 0.07, antiderivative size = 88, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {4581, 4577} \begin {gather*} \frac {1}{12} \sqrt {-\frac {1}{n^2}} n x^3 e^{-a \sqrt {-\frac {1}{n^2}} n} \left (c x^n\right )^{3/n}-\frac {1}{2} \sqrt {-\frac {1}{n^2}} n x^3 e^{a \sqrt {-\frac {1}{n^2}} n} \log (x) \left (c x^n\right )^{-3/n} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[-n^(-2)]*n*x^3*(c*x^n)^(3/n))/(12*E^(a*Sqrt[-n^(-2)]*n)) - (E^(a*Sqrt[-n^(-2)]*n)*Sqrt[-n^(-2)]*n*x^3*Lo
g[x])/(2*(c*x^n)^(3/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*} \int x^2 \sin \left (a+3 \sqrt {-\frac {1}{n^2}} \log \left (c x^n\right )\right ) \, dx &=\frac {\left (x^3 \left (c x^n\right )^{-3/n}\right ) \text {Subst}\left (\int x^{-1+\frac {3}{n}} \sin \left (a+3 \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^3 \left (c x^n\right )^{-3/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 {6}{n}}\right ) \, dx,x,c x^n\right )\right )\\ &=\frac {1}{12} e^{-a \sqrt {-\frac {1}{n^2}} n} \sqrt {-\frac {1}{n^2}} n x^3 \left (c x^n\right )^{3/n}-\frac {1}{2} e^{a \sqrt {-\frac {1}{n^2}} n} \sqrt {-\frac {1}{n^2}} n x^3 \left (c x^n\right )^{-3/n} \log (x)\\ \end {align*}

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Mathematica [F]
time = 0.17, size = 0, normalized size = 0.00 \begin {gather*} \int x^2 \sin \left (a+3 \sqrt {-\frac {1}{n^2}} \log \left (c x^n\right )\right ) \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int x^{2} \sin \left (a +3 \ln \left (c \,x^{n}\right ) \sqrt {-\frac {1}{n^{2}}}\right )\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.29, size = 31, normalized size = 0.35 \begin {gather*} \frac {c^{\frac {6}{n}} x^{6} \sin \left (a\right ) + 6 \, \log \left (x\right ) \sin \left (a\right )}{12 \, c^{\frac {3}{n}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*(c^(6/n)*x^6*sin(a) + 6*log(x)*sin(a))/c^(3/n)

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Fricas [C] Result contains complex when optimal does not.
time = 1.16, size = 42, normalized size = 0.48 \begin {gather*} \frac {1}{12} \, {\left (i \, x^{6} - 6 i \, e^{\left (\frac {2 \, {\left (i \, a n - 3 \, \log \left (c\right )\right )}}{n}\right )} \log \left (x\right )\right )} e^{\left (-\frac {i \, a n - 3 \, \log \left (c\right )}{n}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*(I*x^6 - 6*I*e^(2*(I*a*n - 3*log(c))/n)*log(x))*e^(-(I*a*n - 3*log(c))/n)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int x^{2} \sin {\left (a + 3 \sqrt {- \frac {1}{n^{2}}} \log {\left (c x^{n} \right )} \right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]
time = 0.52, size = 1, normalized size = 0.01 \begin {gather*} +\infty \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

+Infinity

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Mupad [B]
time = 3.02, size = 85, normalized size = 0.97 \begin {gather*} -\frac {x^3\,{\mathrm {e}}^{-a\,1{}\mathrm {i}}\,\frac {1}{{\left (c\,x^n\right )}^{\sqrt {-\frac {1}{n^2}}\,3{}\mathrm {i}}}}{6\,n\,\sqrt {-\frac {1}{n^2}}+6{}\mathrm {i}}-\frac {x^3\,{\mathrm {e}}^{a\,1{}\mathrm {i}}\,{\left (c\,x^n\right )}^{\sqrt {-\frac {1}{n^2}}\,3{}\mathrm {i}}}{6\,n\,\sqrt {-\frac {1}{n^2}}-6{}\mathrm {i}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

- (x^3*exp(-a*1i)/(c*x^n)^((-1/n^2)^(1/2)*3i))/(6*n*(-1/n^2)^(1/2) + 6i) - (x^3*exp(a*1i)*(c*x^n)^((-1/n^2)^(1
/2)*3i))/(6*n*(-1/n^2)^(1/2) - 6i)

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