3.24 \(\int \frac{\sin ^4(a+b \log (c x^n))}{x^3} \, dx\)

Optimal. Leaf size=210 \[ -\frac{3 b^2 n^2 \sin ^2\left (a+b \log \left (c x^n\right )\right )}{2 x^2 \left (4 b^4 n^4+5 b^2 n^2+1\right )}-\frac{\sin ^4\left (a+b \log \left (c x^n\right )\right )}{2 x^2 \left (4 b^2 n^2+1\right )}-\frac{b n \sin ^3\left (a+b \log \left (c x^n\right )\right ) \cos \left (a+b \log \left (c x^n\right )\right )}{x^2 \left (4 b^2 n^2+1\right )}-\frac{3 b^3 n^3 \sin \left (a+b \log \left (c x^n\right )\right ) \cos \left (a+b \log \left (c x^n\right )\right )}{2 x^2 \left (4 b^4 n^4+5 b^2 n^2+1\right )}-\frac{3 b^4 n^4}{4 x^2 \left (4 b^4 n^4+5 b^2 n^2+1\right )} \]

[Out]

(-3*b^4*n^4)/(4*(1 + 5*b^2*n^2 + 4*b^4*n^4)*x^2) - (3*b^3*n^3*Cos[a + b*Log[c*x^n]]*Sin[a + b*Log[c*x^n]])/(2*
(1 + 5*b^2*n^2 + 4*b^4*n^4)*x^2) - (3*b^2*n^2*Sin[a + b*Log[c*x^n]]^2)/(2*(1 + 5*b^2*n^2 + 4*b^4*n^4)*x^2) - (
b*n*Cos[a + b*Log[c*x^n]]*Sin[a + b*Log[c*x^n]]^3)/((1 + 4*b^2*n^2)*x^2) - Sin[a + b*Log[c*x^n]]^4/(2*(1 + 4*b
^2*n^2)*x^2)

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Rubi [A]  time = 0.0628864, antiderivative size = 210, 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} \[ -\frac{3 b^2 n^2 \sin ^2\left (a+b \log \left (c x^n\right )\right )}{2 x^2 \left (4 b^4 n^4+5 b^2 n^2+1\right )}-\frac{\sin ^4\left (a+b \log \left (c x^n\right )\right )}{2 x^2 \left (4 b^2 n^2+1\right )}-\frac{b n \sin ^3\left (a+b \log \left (c x^n\right )\right ) \cos \left (a+b \log \left (c x^n\right )\right )}{x^2 \left (4 b^2 n^2+1\right )}-\frac{3 b^3 n^3 \sin \left (a+b \log \left (c x^n\right )\right ) \cos \left (a+b \log \left (c x^n\right )\right )}{2 x^2 \left (4 b^4 n^4+5 b^2 n^2+1\right )}-\frac{3 b^4 n^4}{4 x^2 \left (4 b^4 n^4+5 b^2 n^2+1\right )} \]

Antiderivative was successfully verified.

[In]

Int[Sin[a + b*Log[c*x^n]]^4/x^3,x]

[Out]

(-3*b^4*n^4)/(4*(1 + 5*b^2*n^2 + 4*b^4*n^4)*x^2) - (3*b^3*n^3*Cos[a + b*Log[c*x^n]]*Sin[a + b*Log[c*x^n]])/(2*
(1 + 5*b^2*n^2 + 4*b^4*n^4)*x^2) - (3*b^2*n^2*Sin[a + b*Log[c*x^n]]^2)/(2*(1 + 5*b^2*n^2 + 4*b^4*n^4)*x^2) - (
b*n*Cos[a + b*Log[c*x^n]]*Sin[a + b*Log[c*x^n]]^3)/((1 + 4*b^2*n^2)*x^2) - Sin[a + b*Log[c*x^n]]^4/(2*(1 + 4*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 30

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

Rubi steps

\begin{align*} \int \frac{\sin ^4\left (a+b \log \left (c x^n\right )\right )}{x^3} \, dx &=-\frac{b n \cos \left (a+b \log \left (c x^n\right )\right ) \sin ^3\left (a+b \log \left (c x^n\right )\right )}{\left (1+4 b^2 n^2\right ) x^2}-\frac{\sin ^4\left (a+b \log \left (c x^n\right )\right )}{2 \left (1+4 b^2 n^2\right ) x^2}+\frac{\left (3 b^2 n^2\right ) \int \frac{\sin ^2\left (a+b \log \left (c x^n\right )\right )}{x^3} \, dx}{1+4 b^2 n^2}\\ &=-\frac{3 b^3 n^3 \cos \left (a+b \log \left (c x^n\right )\right ) \sin \left (a+b \log \left (c x^n\right )\right )}{2 \left (1+5 b^2 n^2+4 b^4 n^4\right ) x^2}-\frac{3 b^2 n^2 \sin ^2\left (a+b \log \left (c x^n\right )\right )}{2 \left (1+5 b^2 n^2+4 b^4 n^4\right ) x^2}-\frac{b n \cos \left (a+b \log \left (c x^n\right )\right ) \sin ^3\left (a+b \log \left (c x^n\right )\right )}{\left (1+4 b^2 n^2\right ) x^2}-\frac{\sin ^4\left (a+b \log \left (c x^n\right )\right )}{2 \left (1+4 b^2 n^2\right ) x^2}+\frac{\left (3 b^4 n^4\right ) \int \frac{1}{x^3} \, dx}{2 \left (1+5 b^2 n^2+4 b^4 n^4\right )}\\ &=-\frac{3 b^4 n^4}{4 \left (1+5 b^2 n^2+4 b^4 n^4\right ) x^2}-\frac{3 b^3 n^3 \cos \left (a+b \log \left (c x^n\right )\right ) \sin \left (a+b \log \left (c x^n\right )\right )}{2 \left (1+5 b^2 n^2+4 b^4 n^4\right ) x^2}-\frac{3 b^2 n^2 \sin ^2\left (a+b \log \left (c x^n\right )\right )}{2 \left (1+5 b^2 n^2+4 b^4 n^4\right ) x^2}-\frac{b n \cos \left (a+b \log \left (c x^n\right )\right ) \sin ^3\left (a+b \log \left (c x^n\right )\right )}{\left (1+4 b^2 n^2\right ) x^2}-\frac{\sin ^4\left (a+b \log \left (c x^n\right )\right )}{2 \left (1+4 b^2 n^2\right ) x^2}\\ \end{align*}

Mathematica [A]  time = 0.456564, size = 169, normalized size = 0.8 \[ -\frac{16 b^3 n^3 \sin \left (2 \left (a+b \log \left (c x^n\right )\right )\right )-2 b^3 n^3 \sin \left (4 \left (a+b \log \left (c x^n\right )\right )\right )-4 \left (4 b^2 n^2+1\right ) \cos \left (2 \left (a+b \log \left (c x^n\right )\right )\right )+\left (b^2 n^2+1\right ) \cos \left (4 \left (a+b \log \left (c x^n\right )\right )\right )+4 b n \sin \left (2 \left (a+b \log \left (c x^n\right )\right )\right )-2 b n \sin \left (4 \left (a+b \log \left (c x^n\right )\right )\right )+12 b^4 n^4+15 b^2 n^2+3}{16 x^2 \left (4 b^4 n^4+5 b^2 n^2+1\right )} \]

Antiderivative was successfully verified.

[In]

Integrate[Sin[a + b*Log[c*x^n]]^4/x^3,x]

[Out]

-(3 + 15*b^2*n^2 + 12*b^4*n^4 - 4*(1 + 4*b^2*n^2)*Cos[2*(a + b*Log[c*x^n])] + (1 + b^2*n^2)*Cos[4*(a + b*Log[c
*x^n])] + 4*b*n*Sin[2*(a + b*Log[c*x^n])] + 16*b^3*n^3*Sin[2*(a + b*Log[c*x^n])] - 2*b*n*Sin[4*(a + b*Log[c*x^
n])] - 2*b^3*n^3*Sin[4*(a + b*Log[c*x^n])])/(16*(1 + 5*b^2*n^2 + 4*b^4*n^4)*x^2)

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Maple [F]  time = 0.069, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( \sin \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) \right ) ^{4}}{{x}^{3}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(a+b*ln(c*x^n))^4/x^3,x)

[Out]

int(sin(a+b*ln(c*x^n))^4/x^3,x)

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Maxima [B]  time = 1.32137, size = 1461, normalized size = 6.96 \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))^4/x^3,x, algorithm="maxima")

[Out]

-1/32*(24*(b^4*cos(4*b*log(c))^2 + b^4*sin(4*b*log(c))^2)*n^4 + 30*(b^2*cos(4*b*log(c))^2 + b^2*sin(4*b*log(c)
)^2)*n^2 + 6*cos(4*b*log(c))^2 - (2*(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 - (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 + 2*(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*l
og(c)))*n - cos(8*b*log(c))*cos(4*b*log(c)) - sin(8*b*log(c))*sin(4*b*log(c)) - cos(4*b*log(c)))*cos(4*b*log(x
^n) + 4*a) + 4*(4*(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 - 4*(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 + (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 - cos(6*b*log(c))*cos(4*b*log(c)) - cos(4*b*log(c))*cos(2*b*log(c))
- sin(6*b*log(c))*sin(4*b*log(c)) - sin(4*b*log(c))*sin(2*b*log(c)))*cos(2*b*log(x^n) + 2*a) + 6*sin(4*b*log(c
))^2 - (2*(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 + (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 + 2*
(b*cos(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 + cos(4*b*log(c)
)*sin(8*b*log(c)) - cos(8*b*log(c))*sin(4*b*log(c)) + sin(4*b*log(c)))*sin(4*b*log(x^n) + 4*a) + 4*(4*(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 + 4*(b^2*cos(4*b*log(c))*sin(6*b*log(c)) - b^2*cos(6*b*log(c))*sin(4*b*l
og(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 + (b*cos(6*b*log(c))*c
os(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 + cos(4*b*log(c))*sin(6*b*log(c)) - cos(6*b*log(c))*sin(4*b*log(c)) + cos(2*b*log(c))*sin(4*b*
log(c)) - cos(4*b*log(c))*sin(2*b*log(c)))*sin(2*b*log(x^n) + 2*a))/((4*(b^4*cos(4*b*log(c))^2 + b^4*sin(4*b*l
og(c))^2)*n^4 + 5*(b^2*cos(4*b*log(c))^2 + b^2*sin(4*b*log(c))^2)*n^2 + cos(4*b*log(c))^2 + sin(4*b*log(c))^2)
*x^2)

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Fricas [A]  time = 0.529785, size = 423, normalized size = 2.01 \begin{align*} -\frac{3 \, b^{4} n^{4} + 2 \,{\left (b^{2} n^{2} + 1\right )} \cos \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{4} + 8 \, b^{2} n^{2} - 2 \,{\left (5 \, b^{2} n^{2} + 2\right )} \cos \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{2} - 2 \,{\left (2 \,{\left (b^{3} n^{3} + b n\right )} \cos \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{3} -{\left (5 \, b^{3} n^{3} + 2 \, b n\right )} \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 ) + 2}{4 \,{\left (4 \, b^{4} n^{4} + 5 \, b^{2} n^{2} + 1\right )} x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(a+b*log(c*x^n))^4/x^3,x, algorithm="fricas")

[Out]

-1/4*(3*b^4*n^4 + 2*(b^2*n^2 + 1)*cos(b*n*log(x) + b*log(c) + a)^4 + 8*b^2*n^2 - 2*(5*b^2*n^2 + 2)*cos(b*n*log
(x) + b*log(c) + a)^2 - 2*(2*(b^3*n^3 + b*n)*cos(b*n*log(x) + b*log(c) + a)^3 - (5*b^3*n^3 + 2*b*n)*cos(b*n*lo
g(x) + b*log(c) + a))*sin(b*n*log(x) + b*log(c) + a) + 2)/((4*b^4*n^4 + 5*b^2*n^2 + 1)*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))**4/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 )^{4}}{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))^4/x^3,x, algorithm="giac")

[Out]

integrate(sin(b*log(c*x^n) + a)^4/x^3, x)