3.19 \(\int x^2 \sin ^4(a+b \log (c x^n)) \, dx\)

Optimal. Leaf size=202 \[ \frac{36 b^2 n^2 x^3 \sin ^2\left (a+b \log \left (c x^n\right )\right )}{64 b^4 n^4+180 b^2 n^2+81}+\frac{3 x^3 \sin ^4\left (a+b \log \left (c x^n\right )\right )}{16 b^2 n^2+9}-\frac{4 b n x^3 \sin ^3\left (a+b \log \left (c x^n\right )\right ) \cos \left (a+b \log \left (c x^n\right )\right )}{16 b^2 n^2+9}-\frac{24 b^3 n^3 x^3 \sin \left (a+b \log \left (c x^n\right )\right ) \cos \left (a+b \log \left (c x^n\right )\right )}{64 b^4 n^4+180 b^2 n^2+81}+\frac{8 b^4 n^4 x^3}{64 b^4 n^4+180 b^2 n^2+81} \]

[Out]

(8*b^4*n^4*x^3)/(81 + 180*b^2*n^2 + 64*b^4*n^4) - (24*b^3*n^3*x^3*Cos[a + b*Log[c*x^n]]*Sin[a + b*Log[c*x^n]])
/(81 + 180*b^2*n^2 + 64*b^4*n^4) + (36*b^2*n^2*x^3*Sin[a + b*Log[c*x^n]]^2)/(81 + 180*b^2*n^2 + 64*b^4*n^4) -
(4*b*n*x^3*Cos[a + b*Log[c*x^n]]*Sin[a + b*Log[c*x^n]]^3)/(9 + 16*b^2*n^2) + (3*x^3*Sin[a + b*Log[c*x^n]]^4)/(
9 + 16*b^2*n^2)

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Rubi [A]  time = 0.0783835, antiderivative size = 202, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {4487, 30} \[ \frac{36 b^2 n^2 x^3 \sin ^2\left (a+b \log \left (c x^n\right )\right )}{64 b^4 n^4+180 b^2 n^2+81}+\frac{3 x^3 \sin ^4\left (a+b \log \left (c x^n\right )\right )}{16 b^2 n^2+9}-\frac{4 b n x^3 \sin ^3\left (a+b \log \left (c x^n\right )\right ) \cos \left (a+b \log \left (c x^n\right )\right )}{16 b^2 n^2+9}-\frac{24 b^3 n^3 x^3 \sin \left (a+b \log \left (c x^n\right )\right ) \cos \left (a+b \log \left (c x^n\right )\right )}{64 b^4 n^4+180 b^2 n^2+81}+\frac{8 b^4 n^4 x^3}{64 b^4 n^4+180 b^2 n^2+81} \]

Antiderivative was successfully verified.

[In]

Int[x^2*Sin[a + b*Log[c*x^n]]^4,x]

[Out]

(8*b^4*n^4*x^3)/(81 + 180*b^2*n^2 + 64*b^4*n^4) - (24*b^3*n^3*x^3*Cos[a + b*Log[c*x^n]]*Sin[a + b*Log[c*x^n]])
/(81 + 180*b^2*n^2 + 64*b^4*n^4) + (36*b^2*n^2*x^3*Sin[a + b*Log[c*x^n]]^2)/(81 + 180*b^2*n^2 + 64*b^4*n^4) -
(4*b*n*x^3*Cos[a + b*Log[c*x^n]]*Sin[a + b*Log[c*x^n]]^3)/(9 + 16*b^2*n^2) + (3*x^3*Sin[a + b*Log[c*x^n]]^4)/(
9 + 16*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 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rubi steps

\begin{align*} \int x^2 \sin ^4\left (a+b \log \left (c x^n\right )\right ) \, dx &=-\frac{4 b n x^3 \cos \left (a+b \log \left (c x^n\right )\right ) \sin ^3\left (a+b \log \left (c x^n\right )\right )}{9+16 b^2 n^2}+\frac{3 x^3 \sin ^4\left (a+b \log \left (c x^n\right )\right )}{9+16 b^2 n^2}+\frac{\left (12 b^2 n^2\right ) \int x^2 \sin ^2\left (a+b \log \left (c x^n\right )\right ) \, dx}{9+16 b^2 n^2}\\ &=-\frac{24 b^3 n^3 x^3 \cos \left (a+b \log \left (c x^n\right )\right ) \sin \left (a+b \log \left (c x^n\right )\right )}{81+180 b^2 n^2+64 b^4 n^4}+\frac{36 b^2 n^2 x^3 \sin ^2\left (a+b \log \left (c x^n\right )\right )}{81+180 b^2 n^2+64 b^4 n^4}-\frac{4 b n x^3 \cos \left (a+b \log \left (c x^n\right )\right ) \sin ^3\left (a+b \log \left (c x^n\right )\right )}{9+16 b^2 n^2}+\frac{3 x^3 \sin ^4\left (a+b \log \left (c x^n\right )\right )}{9+16 b^2 n^2}+\frac{\left (24 b^4 n^4\right ) \int x^2 \, dx}{81+180 b^2 n^2+64 b^4 n^4}\\ &=\frac{8 b^4 n^4 x^3}{81+180 b^2 n^2+64 b^4 n^4}-\frac{24 b^3 n^3 x^3 \cos \left (a+b \log \left (c x^n\right )\right ) \sin \left (a+b \log \left (c x^n\right )\right )}{81+180 b^2 n^2+64 b^4 n^4}+\frac{36 b^2 n^2 x^3 \sin ^2\left (a+b \log \left (c x^n\right )\right )}{81+180 b^2 n^2+64 b^4 n^4}-\frac{4 b n x^3 \cos \left (a+b \log \left (c x^n\right )\right ) \sin ^3\left (a+b \log \left (c x^n\right )\right )}{9+16 b^2 n^2}+\frac{3 x^3 \sin ^4\left (a+b \log \left (c x^n\right )\right )}{9+16 b^2 n^2}\\ \end{align*}

Mathematica [A]  time = 0.495872, size = 171, normalized size = 0.85 \[ \frac{x^3 \left (-128 b^3 n^3 \sin \left (2 \left (a+b \log \left (c x^n\right )\right )\right )+16 b^3 n^3 \sin \left (4 \left (a+b \log \left (c x^n\right )\right )\right )-12 \left (16 b^2 n^2+9\right ) \cos \left (2 \left (a+b \log \left (c x^n\right )\right )\right )+3 \left (4 b^2 n^2+9\right ) \cos \left (4 \left (a+b \log \left (c x^n\right )\right )\right )-72 b n \sin \left (2 \left (a+b \log \left (c x^n\right )\right )\right )+36 b n \sin \left (4 \left (a+b \log \left (c x^n\right )\right )\right )+64 b^4 n^4+180 b^2 n^2+81\right )}{8 \left (64 b^4 n^4+180 b^2 n^2+81\right )} \]

Antiderivative was successfully verified.

[In]

Integrate[x^2*Sin[a + b*Log[c*x^n]]^4,x]

[Out]

(x^3*(81 + 180*b^2*n^2 + 64*b^4*n^4 - 12*(9 + 16*b^2*n^2)*Cos[2*(a + b*Log[c*x^n])] + 3*(9 + 4*b^2*n^2)*Cos[4*
(a + b*Log[c*x^n])] - 72*b*n*Sin[2*(a + b*Log[c*x^n])] - 128*b^3*n^3*Sin[2*(a + b*Log[c*x^n])] + 36*b*n*Sin[4*
(a + b*Log[c*x^n])] + 16*b^3*n^3*Sin[4*(a + b*Log[c*x^n])]))/(8*(81 + 180*b^2*n^2 + 64*b^4*n^4))

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Maple [F]  time = 0.098, size = 0, normalized size = 0. \begin{align*} \int{x}^{2} \left ( \sin \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) \right ) ^{4}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*sin(a+b*ln(c*x^n))^4,x)

[Out]

int(x^2*sin(a+b*ln(c*x^n))^4,x)

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Maxima [B]  time = 1.2689, size = 1494, normalized size = 7.4 \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))^4,x, algorithm="maxima")

[Out]

1/16*((16*(b^3*cos(4*b*log(c))*sin(8*b*log(c)) - b^3*cos(8*b*log(c))*sin(4*b*log(c)) + b^3*sin(4*b*log(c)))*n^
3 + 12*(b^2*cos(8*b*log(c))*cos(4*b*log(c)) + b^2*sin(8*b*log(c))*sin(4*b*log(c)) + b^2*cos(4*b*log(c)))*n^2 +
36*(b*cos(4*b*log(c))*sin(8*b*log(c)) - b*cos(8*b*log(c))*sin(4*b*log(c)) + b*sin(4*b*log(c)))*n + 27*cos(8*b
*log(c))*cos(4*b*log(c)) + 27*sin(8*b*log(c))*sin(4*b*log(c)) + 27*cos(4*b*log(c)))*x^3*cos(4*b*log(x^n) + 4*a
) - 4*(32*(b^3*cos(4*b*log(c))*sin(6*b*log(c)) - b^3*cos(6*b*log(c))*sin(4*b*log(c)) + b^3*cos(2*b*log(c))*sin
(4*b*log(c)) - b^3*cos(4*b*log(c))*sin(2*b*log(c)))*n^3 + 48*(b^2*cos(6*b*log(c))*cos(4*b*log(c)) + b^2*cos(4*
b*log(c))*cos(2*b*log(c)) + b^2*sin(6*b*log(c))*sin(4*b*log(c)) + b^2*sin(4*b*log(c))*sin(2*b*log(c)))*n^2 + 1
8*(b*cos(4*b*log(c))*sin(6*b*log(c)) - b*cos(6*b*log(c))*sin(4*b*log(c)) + b*cos(2*b*log(c))*sin(4*b*log(c)) -
b*cos(4*b*log(c))*sin(2*b*log(c)))*n + 27*cos(6*b*log(c))*cos(4*b*log(c)) + 27*cos(4*b*log(c))*cos(2*b*log(c)
) + 27*sin(6*b*log(c))*sin(4*b*log(c)) + 27*sin(4*b*log(c))*sin(2*b*log(c)))*x^3*cos(2*b*log(x^n) + 2*a) + (16
*(b^3*cos(8*b*log(c))*cos(4*b*log(c)) + b^3*sin(8*b*log(c))*sin(4*b*log(c)) + b^3*cos(4*b*log(c)))*n^3 - 12*(b
^2*cos(4*b*log(c))*sin(8*b*log(c)) - b^2*cos(8*b*log(c))*sin(4*b*log(c)) + b^2*sin(4*b*log(c)))*n^2 + 36*(b*co
s(8*b*log(c))*cos(4*b*log(c)) + b*sin(8*b*log(c))*sin(4*b*log(c)) + b*cos(4*b*log(c)))*n - 27*cos(4*b*log(c))*
sin(8*b*log(c)) + 27*cos(8*b*log(c))*sin(4*b*log(c)) - 27*sin(4*b*log(c)))*x^3*sin(4*b*log(x^n) + 4*a) - 4*(32
*(b^3*cos(6*b*log(c))*cos(4*b*log(c)) + b^3*cos(4*b*log(c))*cos(2*b*log(c)) + b^3*sin(6*b*log(c))*sin(4*b*log(
c)) + b^3*sin(4*b*log(c))*sin(2*b*log(c)))*n^3 - 48*(b^2*cos(4*b*log(c))*sin(6*b*log(c)) - b^2*cos(6*b*log(c))
*sin(4*b*log(c)) + b^2*cos(2*b*log(c))*sin(4*b*log(c)) - b^2*cos(4*b*log(c))*sin(2*b*log(c)))*n^2 + 18*(b*cos(
6*b*log(c))*cos(4*b*log(c)) + b*cos(4*b*log(c))*cos(2*b*log(c)) + b*sin(6*b*log(c))*sin(4*b*log(c)) + b*sin(4*
b*log(c))*sin(2*b*log(c)))*n - 27*cos(4*b*log(c))*sin(6*b*log(c)) + 27*cos(6*b*log(c))*sin(4*b*log(c)) - 27*co
s(2*b*log(c))*sin(4*b*log(c)) + 27*cos(4*b*log(c))*sin(2*b*log(c)))*x^3*sin(2*b*log(x^n) + 2*a) + 2*(64*(b^4*c
os(4*b*log(c))^2 + b^4*sin(4*b*log(c))^2)*n^4 + 180*(b^2*cos(4*b*log(c))^2 + b^2*sin(4*b*log(c))^2)*n^2 + 81*c
os(4*b*log(c))^2 + 81*sin(4*b*log(c))^2)*x^3)/(64*(b^4*cos(4*b*log(c))^2 + b^4*sin(4*b*log(c))^2)*n^4 + 180*(b
^2*cos(4*b*log(c))^2 + b^2*sin(4*b*log(c))^2)*n^2 + 81*cos(4*b*log(c))^2 + 81*sin(4*b*log(c))^2)

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Fricas [A]  time = 0.517694, size = 454, normalized size = 2.25 \begin{align*} \frac{3 \,{\left (4 \, b^{2} n^{2} + 9\right )} x^{3} \cos \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{4} - 6 \,{\left (10 \, 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 (8 \, b^{4} n^{4} + 48 \, b^{2} n^{2} + 27\right )} x^{3} + 4 \,{\left ({\left (4 \, 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} -{\left (10 \, 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 )\right )} \sin \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )}{64 \, b^{4} n^{4} + 180 \, b^{2} n^{2} + 81} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*sin(a+b*log(c*x^n))^4,x, algorithm="fricas")

[Out]

(3*(4*b^2*n^2 + 9)*x^3*cos(b*n*log(x) + b*log(c) + a)^4 - 6*(10*b^2*n^2 + 9)*x^3*cos(b*n*log(x) + b*log(c) + a
)^2 + (8*b^4*n^4 + 48*b^2*n^2 + 27)*x^3 + 4*((4*b^3*n^3 + 9*b*n)*x^3*cos(b*n*log(x) + b*log(c) + a)^3 - (10*b^
3*n^3 + 9*b*n)*x^3*cos(b*n*log(x) + b*log(c) + a))*sin(b*n*log(x) + b*log(c) + a))/(64*b^4*n^4 + 180*b^2*n^2 +
81)

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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))**4,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))^4,x, algorithm="giac")

[Out]

Timed out