3.13 \(\int x^2 \sin ^3(a+b \log (c x^n)) \, dx\)

Optimal. Leaf size=160 \[ \frac{x^3 \sin ^3\left (a+b \log \left (c x^n\right )\right )}{3 \left (b^2 n^2+1\right )}+\frac{2 b^2 n^2 x^3 \sin \left (a+b \log \left (c x^n\right )\right )}{b^4 n^4+10 b^2 n^2+9}-\frac{2 b^3 n^3 x^3 \cos \left (a+b \log \left (c x^n\right )\right )}{3 \left (b^4 n^4+10 b^2 n^2+9\right )}-\frac{b n x^3 \sin ^2\left (a+b \log \left (c x^n\right )\right ) \cos \left (a+b \log \left (c x^n\right )\right )}{3 \left (b^2 n^2+1\right )} \]

[Out]

(-2*b^3*n^3*x^3*Cos[a + b*Log[c*x^n]])/(3*(9 + 10*b^2*n^2 + b^4*n^4)) + (2*b^2*n^2*x^3*Sin[a + b*Log[c*x^n]])/
(9 + 10*b^2*n^2 + b^4*n^4) - (b*n*x^3*Cos[a + b*Log[c*x^n]]*Sin[a + b*Log[c*x^n]]^2)/(3*(1 + b^2*n^2)) + (x^3*
Sin[a + b*Log[c*x^n]]^3)/(3*(1 + b^2*n^2))

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Rubi [A]  time = 0.0553367, antiderivative size = 160, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {4487, 4485} \[ \frac{x^3 \sin ^3\left (a+b \log \left (c x^n\right )\right )}{3 \left (b^2 n^2+1\right )}+\frac{2 b^2 n^2 x^3 \sin \left (a+b \log \left (c x^n\right )\right )}{b^4 n^4+10 b^2 n^2+9}-\frac{2 b^3 n^3 x^3 \cos \left (a+b \log \left (c x^n\right )\right )}{3 \left (b^4 n^4+10 b^2 n^2+9\right )}-\frac{b n x^3 \sin ^2\left (a+b \log \left (c x^n\right )\right ) \cos \left (a+b \log \left (c x^n\right )\right )}{3 \left (b^2 n^2+1\right )} \]

Antiderivative was successfully verified.

[In]

Int[x^2*Sin[a + b*Log[c*x^n]]^3,x]

[Out]

(-2*b^3*n^3*x^3*Cos[a + b*Log[c*x^n]])/(3*(9 + 10*b^2*n^2 + b^4*n^4)) + (2*b^2*n^2*x^3*Sin[a + b*Log[c*x^n]])/
(9 + 10*b^2*n^2 + b^4*n^4) - (b*n*x^3*Cos[a + b*Log[c*x^n]]*Sin[a + b*Log[c*x^n]]^2)/(3*(1 + b^2*n^2)) + (x^3*
Sin[a + b*Log[c*x^n]]^3)/(3*(1 + b^2*n^2))

Rule 4487

Int[((e_.)*(x_))^(m_.)*Sin[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*(d_.)]^(p_), x_Symbol] :> Simp[((m + 1)*(e*x)
^(m + 1)*Sin[d*(a + b*Log[c*x^n])]^p)/(b^2*d^2*e*n^2*p^2 + e*(m + 1)^2), x] + (Dist[(b^2*d^2*n^2*p*(p - 1))/(b
^2*d^2*n^2*p^2 + (m + 1)^2), Int[(e*x)^m*Sin[d*(a + b*Log[c*x^n])]^(p - 2), x], x] - Simp[(b*d*n*p*(e*x)^(m +
1)*Cos[d*(a + b*Log[c*x^n])]*Sin[d*(a + b*Log[c*x^n])]^(p - 1))/(b^2*d^2*e*n^2*p^2 + e*(m + 1)^2), x]) /; Free
Q[{a, b, c, d, e, m, n}, x] && IGtQ[p, 1] && NeQ[b^2*d^2*n^2*p^2 + (m + 1)^2, 0]

Rule 4485

Int[((e_.)*(x_))^(m_.)*Sin[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*(d_.)], x_Symbol] :> Simp[((m + 1)*(e*x)^(m +
1)*Sin[d*(a + b*Log[c*x^n])])/(b^2*d^2*e*n^2 + e*(m + 1)^2), x] - Simp[(b*d*n*(e*x)^(m + 1)*Cos[d*(a + b*Log[
c*x^n])])/(b^2*d^2*e*n^2 + e*(m + 1)^2), x] /; FreeQ[{a, b, c, d, e, m, n}, x] && NeQ[b^2*d^2*n^2 + (m + 1)^2,
0]

Rubi steps

\begin{align*} \int x^2 \sin ^3\left (a+b \log \left (c x^n\right )\right ) \, dx &=-\frac{b n x^3 \cos \left (a+b \log \left (c x^n\right )\right ) \sin ^2\left (a+b \log \left (c x^n\right )\right )}{3 \left (1+b^2 n^2\right )}+\frac{x^3 \sin ^3\left (a+b \log \left (c x^n\right )\right )}{3 \left (1+b^2 n^2\right )}+\frac{\left (2 b^2 n^2\right ) \int x^2 \sin \left (a+b \log \left (c x^n\right )\right ) \, dx}{3 \left (1+b^2 n^2\right )}\\ &=-\frac{2 b^3 n^3 x^3 \cos \left (a+b \log \left (c x^n\right )\right )}{3 \left (9+10 b^2 n^2+b^4 n^4\right )}+\frac{2 b^2 n^2 x^3 \sin \left (a+b \log \left (c x^n\right )\right )}{9+10 b^2 n^2+b^4 n^4}-\frac{b n x^3 \cos \left (a+b \log \left (c x^n\right )\right ) \sin ^2\left (a+b \log \left (c x^n\right )\right )}{3 \left (1+b^2 n^2\right )}+\frac{x^3 \sin ^3\left (a+b \log \left (c x^n\right )\right )}{3 \left (1+b^2 n^2\right )}\\ \end{align*}

Mathematica [A]  time = 0.507012, size = 122, normalized size = 0.76 \[ \frac{x^3 \left (-9 b n \left (b^2 n^2+1\right ) \cos \left (a+b \log \left (c x^n\right )\right )+b n \left (b^2 n^2+9\right ) \cos \left (3 \left (a+b \log \left (c x^n\right )\right )\right )-2 \sin \left (a+b \log \left (c x^n\right )\right ) \left (\left (b^2 n^2+9\right ) \cos \left (2 \left (a+b \log \left (c x^n\right )\right )\right )-13 b^2 n^2-9\right )\right )}{12 \left (b^4 n^4+10 b^2 n^2+9\right )} \]

Antiderivative was successfully verified.

[In]

Integrate[x^2*Sin[a + b*Log[c*x^n]]^3,x]

[Out]

(x^3*(-9*b*n*(1 + b^2*n^2)*Cos[a + b*Log[c*x^n]] + b*n*(9 + b^2*n^2)*Cos[3*(a + b*Log[c*x^n])] - 2*(-9 - 13*b^
2*n^2 + (9 + b^2*n^2)*Cos[2*(a + b*Log[c*x^n])])*Sin[a + b*Log[c*x^n]]))/(12*(9 + 10*b^2*n^2 + b^4*n^4))

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Maple [F]  time = 0.065, size = 0, normalized size = 0. \begin{align*} \int{x}^{2} \left ( \sin \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) \right ) ^{3}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*sin(a+b*ln(c*x^n))^3,x)

[Out]

int(x^2*sin(a+b*ln(c*x^n))^3,x)

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Maxima [B]  time = 1.22961, size = 1361, normalized size = 8.51 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/24*(((b^3*cos(6*b*log(c))*cos(3*b*log(c)) + b^3*sin(6*b*log(c))*sin(3*b*log(c)) + b^3*cos(3*b*log(c)))*n^3 -
(b^2*cos(3*b*log(c))*sin(6*b*log(c)) - b^2*cos(6*b*log(c))*sin(3*b*log(c)) + b^2*sin(3*b*log(c)))*n^2 + 9*(b*
cos(6*b*log(c))*cos(3*b*log(c)) + b*sin(6*b*log(c))*sin(3*b*log(c)) + b*cos(3*b*log(c)))*n - 9*cos(3*b*log(c))
*sin(6*b*log(c)) + 9*cos(6*b*log(c))*sin(3*b*log(c)) - 9*sin(3*b*log(c)))*x^3*cos(3*b*log(x^n) + 3*a) - 9*((b^
3*cos(4*b*log(c))*cos(3*b*log(c)) + b^3*cos(3*b*log(c))*cos(2*b*log(c)) + b^3*sin(4*b*log(c))*sin(3*b*log(c))
+ b^3*sin(3*b*log(c))*sin(2*b*log(c)))*n^3 - 3*(b^2*cos(3*b*log(c))*sin(4*b*log(c)) - b^2*cos(4*b*log(c))*sin(
3*b*log(c)) + b^2*cos(2*b*log(c))*sin(3*b*log(c)) - b^2*cos(3*b*log(c))*sin(2*b*log(c)))*n^2 + (b*cos(4*b*log(
c))*cos(3*b*log(c)) + b*cos(3*b*log(c))*cos(2*b*log(c)) + b*sin(4*b*log(c))*sin(3*b*log(c)) + b*sin(3*b*log(c)
)*sin(2*b*log(c)))*n - 3*cos(3*b*log(c))*sin(4*b*log(c)) + 3*cos(4*b*log(c))*sin(3*b*log(c)) - 3*cos(2*b*log(c
))*sin(3*b*log(c)) + 3*cos(3*b*log(c))*sin(2*b*log(c)))*x^3*cos(b*log(x^n) + a) - ((b^3*cos(3*b*log(c))*sin(6*
b*log(c)) - b^3*cos(6*b*log(c))*sin(3*b*log(c)) + b^3*sin(3*b*log(c)))*n^3 + (b^2*cos(6*b*log(c))*cos(3*b*log(
c)) + b^2*sin(6*b*log(c))*sin(3*b*log(c)) + b^2*cos(3*b*log(c)))*n^2 + 9*(b*cos(3*b*log(c))*sin(6*b*log(c)) -
b*cos(6*b*log(c))*sin(3*b*log(c)) + b*sin(3*b*log(c)))*n + 9*cos(6*b*log(c))*cos(3*b*log(c)) + 9*sin(6*b*log(c
))*sin(3*b*log(c)) + 9*cos(3*b*log(c)))*x^3*sin(3*b*log(x^n) + 3*a) + 9*((b^3*cos(3*b*log(c))*sin(4*b*log(c))
- b^3*cos(4*b*log(c))*sin(3*b*log(c)) + b^3*cos(2*b*log(c))*sin(3*b*log(c)) - b^3*cos(3*b*log(c))*sin(2*b*log(
c)))*n^3 + 3*(b^2*cos(4*b*log(c))*cos(3*b*log(c)) + b^2*cos(3*b*log(c))*cos(2*b*log(c)) + b^2*sin(4*b*log(c))*
sin(3*b*log(c)) + b^2*sin(3*b*log(c))*sin(2*b*log(c)))*n^2 + (b*cos(3*b*log(c))*sin(4*b*log(c)) - b*cos(4*b*lo
g(c))*sin(3*b*log(c)) + b*cos(2*b*log(c))*sin(3*b*log(c)) - b*cos(3*b*log(c))*sin(2*b*log(c)))*n + 3*cos(4*b*l
og(c))*cos(3*b*log(c)) + 3*cos(3*b*log(c))*cos(2*b*log(c)) + 3*sin(4*b*log(c))*sin(3*b*log(c)) + 3*sin(3*b*log
(c))*sin(2*b*log(c)))*x^3*sin(b*log(x^n) + a))/((b^4*cos(3*b*log(c))^2 + b^4*sin(3*b*log(c))^2)*n^4 + 10*(b^2*
cos(3*b*log(c))^2 + b^2*sin(3*b*log(c))^2)*n^2 + 9*cos(3*b*log(c))^2 + 9*sin(3*b*log(c))^2)

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Fricas [A]  time = 0.509526, size = 343, normalized size = 2.14 \begin{align*} \frac{{\left (b^{3} n^{3} + 9 \, b n\right )} x^{3} \cos \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{3} - 3 \,{\left (b^{3} n^{3} + 3 \, b n\right )} x^{3} \cos \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) -{\left ({\left (b^{2} n^{2} + 9\right )} x^{3} \cos \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{2} -{\left (7 \, b^{2} n^{2} + 9\right )} x^{3}\right )} \sin \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )}{3 \,{\left (b^{4} n^{4} + 10 \, b^{2} n^{2} + 9\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*((b^3*n^3 + 9*b*n)*x^3*cos(b*n*log(x) + b*log(c) + a)^3 - 3*(b^3*n^3 + 3*b*n)*x^3*cos(b*n*log(x) + b*log(c
) + a) - ((b^2*n^2 + 9)*x^3*cos(b*n*log(x) + b*log(c) + a)^2 - (7*b^2*n^2 + 9)*x^3)*sin(b*n*log(x) + b*log(c)
+ a))/(b^4*n^4 + 10*b^2*n^2 + 9)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**2*sin(a+b*ln(c*x**n))**3,x)

[Out]

Timed out

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Giac [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out