3.18 \(\int \frac{\sin ^3(a+b \log (c x^n))}{x^3} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0477091, antiderivative size = 158, 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} \[ -\frac{2 \sin ^3\left (a+b \log \left (c x^n\right )\right )}{x^2 \left (9 b^2 n^2+4\right )}-\frac{12 b^2 n^2 \sin \left (a+b \log \left (c x^n\right )\right )}{x^2 \left (9 b^4 n^4+40 b^2 n^2+16\right )}-\frac{6 b^3 n^3 \cos \left (a+b \log \left (c x^n\right )\right )}{x^2 \left (9 b^4 n^4+40 b^2 n^2+16\right )}-\frac{3 b n \sin ^2\left (a+b \log \left (c x^n\right )\right ) \cos \left (a+b \log \left (c x^n\right )\right )}{x^2 \left (9 b^2 n^2+4\right )} \]

Antiderivative was successfully verified.

[In]

Int[Sin[a + b*Log[c*x^n]]^3/x^3,x]

[Out]

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

Mathematica [A]  time = 0.384536, size = 125, normalized size = 0.79 \[ \frac{-3 b n \left (9 b^2 n^2+4\right ) \cos \left (a+b \log \left (c x^n\right )\right )+3 b n \left (b^2 n^2+4\right ) \cos \left (3 \left (a+b \log \left (c x^n\right )\right )\right )+4 \sin \left (a+b \log \left (c x^n\right )\right ) \left (\left (b^2 n^2+4\right ) \cos \left (2 \left (a+b \log \left (c x^n\right )\right )\right )-13 b^2 n^2-4\right )}{4 x^2 \left (9 b^4 n^4+40 b^2 n^2+16\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-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 - 13*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)*x^2)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B]  time = 1.24026, size = 1359, normalized size = 8.6 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(a+b*log(c*x^n))^3/x^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)))*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*log(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))*s
in(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*l
og(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)))*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)))*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*co
s(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)))*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)*x^2)

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Fricas [A]  time = 0.509654, size = 333, normalized size = 2.11 \begin{align*} \frac{3 \,{\left (b^{3} n^{3} + 4 \, b n\right )} \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 )} \cos \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) - 2 \,{\left (7 \, b^{2} n^{2} -{\left (b^{2} n^{2} + 4\right )} \cos \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{2} + 4\right )} \sin \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )}{{\left (9 \, b^{4} n^{4} + 40 \, b^{2} n^{2} + 16\right )} x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sin \left (b \log \left (c x^{n}\right ) + a\right )^{3}}{x^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sin(b*log(c*x^n) + a)^3/x^3, x)