3.1.42 \(\int x \sin ^3(a+\frac {2}{3} \sqrt {-\frac {1}{n^2}} \log (c x^n)) \, dx\) [42]

Optimal. Leaf size=178 \[ -\frac {9}{32} e^{a \sqrt {-\frac {1}{n^2}} n} \sqrt {-\frac {1}{n^2}} n x^2 \left (c x^n\right )^{\left .-\frac {2}{3}\right /n}+\frac {9}{64} e^{-a \sqrt {-\frac {1}{n^2}} n} \sqrt {-\frac {1}{n^2}} n x^2 \left (c x^n\right )^{\left .\frac {2}{3}\right /n}-\frac {1}{32} e^{-3 a \sqrt {-\frac {1}{n^2}} n} \sqrt {-\frac {1}{n^2}} n x^2 \left (c x^n\right )^{2/n}+\frac {1}{8} e^{3 a \sqrt {-\frac {1}{n^2}} n} \sqrt {-\frac {1}{n^2}} n x^2 \left (c x^n\right )^{-2/n} \log (x) \]

[Out]

-9/32*exp(a*n*(-1/n^2)^(1/2))*n*x^2*(-1/n^2)^(1/2)/((c*x^n)^(2/3/n))+9/64*n*x^2*(c*x^n)^(2/3/n)*(-1/n^2)^(1/2)
/exp(a*n*(-1/n^2)^(1/2))-1/32*n*x^2*(c*x^n)^(2/n)*(-1/n^2)^(1/2)/exp(3*a*n*(-1/n^2)^(1/2))+1/8*exp(3*a*n*(-1/n
^2)^(1/2))*n*x^2*ln(x)*(-1/n^2)^(1/2)/((c*x^n)^(2/n))

________________________________________________________________________________________

Rubi [A]
time = 0.08, antiderivative size = 178, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {4581, 4577} \begin {gather*} -\frac {9}{32} \sqrt {-\frac {1}{n^2}} n x^2 e^{a \sqrt {-\frac {1}{n^2}} n} \left (c x^n\right )^{\left .-\frac {2}{3}\right /n}+\frac {9}{64} \sqrt {-\frac {1}{n^2}} n x^2 e^{-a \sqrt {-\frac {1}{n^2}} n} \left (c x^n\right )^{\left .\frac {2}{3}\right /n}-\frac {1}{32} \sqrt {-\frac {1}{n^2}} n x^2 e^{-3 a \sqrt {-\frac {1}{n^2}} n} \left (c x^n\right )^{2/n}+\frac {1}{8} \sqrt {-\frac {1}{n^2}} n x^2 e^{3 a \sqrt {-\frac {1}{n^2}} n} \log (x) \left (c x^n\right )^{-2/n} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-9*E^(a*Sqrt[-n^(-2)]*n)*Sqrt[-n^(-2)]*n*x^2)/(32*(c*x^n)^(2/(3*n))) + (9*Sqrt[-n^(-2)]*n*x^2*(c*x^n)^(2/(3*n
)))/(64*E^(a*Sqrt[-n^(-2)]*n)) - (Sqrt[-n^(-2)]*n*x^2*(c*x^n)^(2/n))/(32*E^(3*a*Sqrt[-n^(-2)]*n)) + (E^(3*a*Sq
rt[-n^(-2)]*n)*Sqrt[-n^(-2)]*n*x^2*Log[x])/(8*(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*} \int x \sin ^3\left (a+\frac {2}{3} \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 ) \text {Subst}\left (\int x^{-1+\frac {2}{n}} \sin ^3\left (a+\frac {2}{3} \sqrt {-\frac {1}{n^2}} \log (x)\right ) \, dx,x,c x^n\right )}{n}\\ &=\frac {1}{8} \left (\sqrt {-\frac {1}{n^2}} x^2 \left (c x^n\right )^{-2/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 {8}{3 n}}-e^{-3 a \sqrt {-\frac {1}{n^2}} n} x^{-1+\frac {4}{n}}\right ) \, dx,x,c x^n\right )\\ &=-\frac {9}{32} e^{a \sqrt {-\frac {1}{n^2}} n} \sqrt {-\frac {1}{n^2}} n x^2 \left (c x^n\right )^{\left .-\frac {2}{3}\right /n}+\frac {9}{64} e^{-a \sqrt {-\frac {1}{n^2}} n} \sqrt {-\frac {1}{n^2}} n x^2 \left (c x^n\right )^{\left .\frac {2}{3}\right /n}-\frac {1}{32} e^{-3 a \sqrt {-\frac {1}{n^2}} n} \sqrt {-\frac {1}{n^2}} n x^2 \left (c x^n\right )^{2/n}+\frac {1}{8} e^{3 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]
time = 0.29, size = 0, normalized size = 0.00 \begin {gather*} \int x \sin ^3\left (a+\frac {2}{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*Sin[a + (2*Sqrt[-n^(-2)]*Log[c*x^n])/3]^3,x]

[Out]

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

________________________________________________________________________________________

Maple [F]
time = 0.06, size = 0, normalized size = 0.00 \[\int x \left (\sin ^{3}\left (a +\frac {2 \ln \left (c \,x^{n}\right ) \sqrt {-\frac {1}{n^{2}}}}{3}\right )\right )\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [A]
time = 0.30, size = 112, normalized size = 0.63 \begin {gather*} \frac {9 \, c^{\frac {10}{3 \, n}} x^{2} {\left (x^{n}\right )}^{\frac {4}{3 \, n}} \sin \left (a\right ) - 8 \, c^{\frac {2}{3 \, n}} {\left (x^{n}\right )}^{\frac {2}{3 \, n}} \log \left (x\right ) \sin \left (3 \, a\right ) + 18 \, c^{\frac {2}{n}} x^{2} \sin \left (a\right ) - 2 \, c^{\frac {14}{3 \, n}} e^{\left (\frac {2 \, \log \left (x^{n}\right )}{3 \, n} + 4 \, \log \left (x\right )\right )} \sin \left (3 \, a\right )}{64 \, c^{\frac {8}{3 \, n}} {\left (x^{n}\right )}^{\frac {2}{3 \, n}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/64*(9*c^(10/3/n)*x^2*(x^n)^(4/3/n)*sin(a) - 8*c^(2/3/n)*(x^n)^(2/3/n)*log(x)*sin(3*a) + 18*c^(2/n)*x^2*sin(a
) - 2*c^(14/3/n)*e^(2/3*log(x^n)/n + 4*log(x))*sin(3*a))/(c^(8/3/n)*(x^n)^(2/3/n))

________________________________________________________________________________________

Fricas [C] Result contains complex when optimal does not.
time = 1.92, size = 84, normalized size = 0.47 \begin {gather*} \frac {1}{64} \, {\left (-2 i \, x^{4} + 9 i \, x^{\frac {8}{3}} e^{\left (\frac {2 \, {\left (3 i \, a n - 2 \, \log \left (c\right )\right )}}{3 \, n}\right )} - 18 i \, x^{\frac {4}{3}} e^{\left (\frac {4 \, {\left (3 i \, a n - 2 \, \log \left (c\right )\right )}}{3 \, n}\right )} + 24 i \, e^{\left (\frac {2 \, {\left (3 i \, a n - 2 \, \log \left (c\right )\right )}}{n}\right )} \log \left (x^{\frac {1}{3}}\right )\right )} e^{\left (-\frac {3 i \, a n - 2 \, \log \left (c\right )}{n}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/64*(-2*I*x^4 + 9*I*x^(8/3)*e^(2/3*(3*I*a*n - 2*log(c))/n) - 18*I*x^(4/3)*e^(4/3*(3*I*a*n - 2*log(c))/n) + 24
*I*e^(2*(3*I*a*n - 2*log(c))/n)*log(x^(1/3)))*e^(-(3*I*a*n - 2*log(c))/n)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: NotImplementedError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: NotImplementedError >> Unable to parse Giac output: ((-9*i)*sageVARn^4*sageVARx^2*exp((-3*i)
*sageVARa)*exp((2*sageVARn*abs(sageVARn)*ln(sageVARx)+2*abs(sageVARn)*ln(sageVARc))/sageVARn^2)+27*i*sageVARn^
4*sageVARx^2*exp((-i)

________________________________________________________________________________________

Mupad [B]
time = 3.32, size = 163, normalized size = 0.92 \begin {gather*} -x^2\,{\mathrm {e}}^{-a\,1{}\mathrm {i}}\,\frac {1}{{\left (c\,x^n\right )}^{\frac {\sqrt {-\frac {1}{n^2}}\,2{}\mathrm {i}}{3}}}\,\left (\frac {9\,n\,\sqrt {-\frac {1}{n^2}}}{128}-\frac {27}{128}{}\mathrm {i}\right )-x^2\,{\mathrm {e}}^{a\,1{}\mathrm {i}}\,{\left (c\,x^n\right )}^{\frac {\sqrt {-\frac {1}{n^2}}\,2{}\mathrm {i}}{3}}\,\left (\frac {9\,n\,\sqrt {-\frac {1}{n^2}}}{128}+\frac {27}{128}{}\mathrm {i}\right )+\frac {x^2\,{\mathrm {e}}^{-a\,3{}\mathrm {i}}\,\frac {1}{{\left (c\,x^n\right )}^{\sqrt {-\frac {1}{n^2}}\,2{}\mathrm {i}}}}{16\,n\,\sqrt {-\frac {1}{n^2}}+16{}\mathrm {i}}+\frac {x^2\,{\mathrm {e}}^{a\,3{}\mathrm {i}}\,{\left (c\,x^n\right )}^{\sqrt {-\frac {1}{n^2}}\,2{}\mathrm {i}}}{16\,n\,\sqrt {-\frac {1}{n^2}}-16{}\mathrm {i}} \end {gather*}

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))/3)^3,x)

[Out]

(x^2*exp(-a*3i)/(c*x^n)^((-1/n^2)^(1/2)*2i))/(16*n*(-1/n^2)^(1/2) + 16i) - x^2*exp(a*1i)*(c*x^n)^(((-1/n^2)^(1
/2)*2i)/3)*((9*n*(-1/n^2)^(1/2))/128 + 27i/128) - x^2*exp(-a*1i)/(c*x^n)^(((-1/n^2)^(1/2)*2i)/3)*((9*n*(-1/n^2
)^(1/2))/128 - 27i/128) + (x^2*exp(a*3i)*(c*x^n)^((-1/n^2)^(1/2)*2i))/(16*n*(-1/n^2)^(1/2) - 16i)

________________________________________________________________________________________