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

Optimal. Leaf size=158 \[ \frac{2 x^2 \sin ^3\left (a+b \log \left (c x^n\right )\right )}{9 b^2 n^2+4}+\frac{12 b^2 n^2 x^2 \sin \left (a+b \log \left (c x^n\right )\right )}{9 b^4 n^4+40 b^2 n^2+16}-\frac{6 b^3 n^3 x^2 \cos \left (a+b \log \left (c x^n\right )\right )}{9 b^4 n^4+40 b^2 n^2+16}-\frac{3 b n x^2 \sin ^2\left (a+b \log \left (c x^n\right )\right ) \cos \left (a+b \log \left (c x^n\right )\right )}{9 b^2 n^2+4} \]

[Out]

(-6*b^3*n^3*x^2*Cos[a + b*Log[c*x^n]])/(16 + 40*b^2*n^2 + 9*b^4*n^4) + (12*b^2*n^2*x^2*Sin[a + b*Log[c*x^n]])/
(16 + 40*b^2*n^2 + 9*b^4*n^4) - (3*b*n*x^2*Cos[a + b*Log[c*x^n]]*Sin[a + b*Log[c*x^n]]^2)/(4 + 9*b^2*n^2) + (2
*x^2*Sin[a + b*Log[c*x^n]]^3)/(4 + 9*b^2*n^2)

________________________________________________________________________________________

Rubi [A]  time = 0.0448578, antiderivative size = 158, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {4487, 4485} \[ \frac{2 x^2 \sin ^3\left (a+b \log \left (c x^n\right )\right )}{9 b^2 n^2+4}+\frac{12 b^2 n^2 x^2 \sin \left (a+b \log \left (c x^n\right )\right )}{9 b^4 n^4+40 b^2 n^2+16}-\frac{6 b^3 n^3 x^2 \cos \left (a+b \log \left (c x^n\right )\right )}{9 b^4 n^4+40 b^2 n^2+16}-\frac{3 b n x^2 \sin ^2\left (a+b \log \left (c x^n\right )\right ) \cos \left (a+b \log \left (c x^n\right )\right )}{9 b^2 n^2+4} \]

Antiderivative was successfully verified.

[In]

Int[x*Sin[a + b*Log[c*x^n]]^3,x]

[Out]

(-6*b^3*n^3*x^2*Cos[a + b*Log[c*x^n]])/(16 + 40*b^2*n^2 + 9*b^4*n^4) + (12*b^2*n^2*x^2*Sin[a + b*Log[c*x^n]])/
(16 + 40*b^2*n^2 + 9*b^4*n^4) - (3*b*n*x^2*Cos[a + b*Log[c*x^n]]*Sin[a + b*Log[c*x^n]]^2)/(4 + 9*b^2*n^2) + (2
*x^2*Sin[a + b*Log[c*x^n]]^3)/(4 + 9*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 \sin ^3\left (a+b \log \left (c x^n\right )\right ) \, dx &=-\frac{3 b n x^2 \cos \left (a+b \log \left (c x^n\right )\right ) \sin ^2\left (a+b \log \left (c x^n\right )\right )}{4+9 b^2 n^2}+\frac{2 x^2 \sin ^3\left (a+b \log \left (c x^n\right )\right )}{4+9 b^2 n^2}+\frac{\left (6 b^2 n^2\right ) \int x \sin \left (a+b \log \left (c x^n\right )\right ) \, dx}{4+9 b^2 n^2}\\ &=-\frac{6 b^3 n^3 x^2 \cos \left (a+b \log \left (c x^n\right )\right )}{16+40 b^2 n^2+9 b^4 n^4}+\frac{12 b^2 n^2 x^2 \sin \left (a+b \log \left (c x^n\right )\right )}{16+40 b^2 n^2+9 b^4 n^4}-\frac{3 b n x^2 \cos \left (a+b \log \left (c x^n\right )\right ) \sin ^2\left (a+b \log \left (c x^n\right )\right )}{4+9 b^2 n^2}+\frac{2 x^2 \sin ^3\left (a+b \log \left (c x^n\right )\right )}{4+9 b^2 n^2}\\ \end{align*}

Mathematica [A]  time = 0.47611, size = 125, normalized size = 0.79 \[ \frac{x^2 \left (-3 b n \left (9 b^2 n^2+4\right ) \cos \left (a+b \log \left (c x^n\right )\right )+3 b n \left (b^2 n^2+4\right ) \cos \left (3 \left (a+b \log \left (c x^n\right )\right )\right )-4 \sin \left (a+b \log \left (c x^n\right )\right ) \left (\left (b^2 n^2+4\right ) \cos \left (2 \left (a+b \log \left (c x^n\right )\right )\right )-13 b^2 n^2-4\right )\right )}{4 \left (9 b^4 n^4+40 b^2 n^2+16\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(x^2*(-3*b*n*(4 + 9*b^2*n^2)*Cos[a + b*Log[c*x^n]] + 3*b*n*(4 + b^2*n^2)*Cos[3*(a + b*Log[c*x^n])] - 4*(-4 - 1
3*b^2*n^2 + (4 + b^2*n^2)*Cos[2*(a + b*Log[c*x^n])])*Sin[a + b*Log[c*x^n]]))/(4*(16 + 40*b^2*n^2 + 9*b^4*n^4))

________________________________________________________________________________________

Maple [F]  time = 0.063, size = 0, normalized size = 0. \begin{align*} \int x \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*sin(a+b*ln(c*x^n))^3,x)

[Out]

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

________________________________________________________________________________________

Maxima [B]  time = 1.4297, size = 1372, normalized size = 8.68 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*((3*(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
- 2*(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 + 12
*(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 - 8*cos(3*b*log
(c))*sin(6*b*log(c)) + 8*cos(6*b*log(c))*sin(3*b*log(c)) - 8*sin(3*b*log(c)))*x^2*cos(3*b*log(x^n) + 3*a) - 3*
(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*lo
g(c)) + b^3*sin(3*b*log(c))*sin(2*b*log(c)))*n^3 - 18*(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 + 4*(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 - 8*cos(3*b*log(c))*sin(4*b*log(c)) + 8*cos(4*b*log(c))*sin(3*b*log(c)) - 8*cos(
2*b*log(c))*sin(3*b*log(c)) + 8*cos(3*b*log(c))*sin(2*b*log(c)))*x^2*cos(b*log(x^n) + a) - (3*(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 + 2*(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 + 12*(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 + 8*cos(6*b*log(c))*cos(3*b*log(c)) + 8
*sin(6*b*log(c))*sin(3*b*log(c)) + 8*cos(3*b*log(c)))*x^2*sin(3*b*log(x^n) + 3*a) + 3*(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 + 18*(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 + 4*(b*cos(3*b*log(c))*sin(4*b*log
(c)) - b*cos(4*b*log(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 + 8*cos(4*b*log(c))*cos(3*b*log(c)) + 8*cos(3*b*log(c))*cos(2*b*log(c)) + 8*sin(4*b*log(c))*sin(3*b*log(
c)) + 8*sin(3*b*log(c))*sin(2*b*log(c)))*x^2*sin(b*log(x^n) + a))/(9*(b^4*cos(3*b*log(c))^2 + b^4*sin(3*b*log(
c))^2)*n^4 + 40*(b^2*cos(3*b*log(c))^2 + b^2*sin(3*b*log(c))^2)*n^2 + 16*cos(3*b*log(c))^2 + 16*sin(3*b*log(c)
)^2)

________________________________________________________________________________________

Fricas [A]  time = 0.52004, size = 350, normalized size = 2.22 \begin{align*} \frac{3 \,{\left (b^{3} n^{3} + 4 \, b n\right )} x^{2} \cos \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{3} - 3 \,{\left (3 \, b^{3} n^{3} + 4 \, b n\right )} x^{2} \cos \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) - 2 \,{\left ({\left (b^{2} n^{2} + 4\right )} x^{2} \cos \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{2} -{\left (7 \, b^{2} n^{2} + 4\right )} x^{2}\right )} \sin \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )}{9 \, b^{4} n^{4} + 40 \, b^{2} n^{2} + 16} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(3*(b^3*n^3 + 4*b*n)*x^2*cos(b*n*log(x) + b*log(c) + a)^3 - 3*(3*b^3*n^3 + 4*b*n)*x^2*cos(b*n*log(x) + b*log(c
) + a) - 2*((b^2*n^2 + 4)*x^2*cos(b*n*log(x) + b*log(c) + a)^2 - (7*b^2*n^2 + 4)*x^2)*sin(b*n*log(x) + b*log(c
) + a))/(9*b^4*n^4 + 40*b^2*n^2 + 16)

________________________________________________________________________________________

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

[Out]

Timed out

________________________________________________________________________________________

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

[Out]

Timed out