3.23 \(\int \frac{\sin ^4(a+b \log (c x^n))}{x^2} \, dx\)

Optimal. Leaf size=202 \[ -\frac{12 b^2 n^2 \sin ^2\left (a+b \log \left (c x^n\right )\right )}{x \left (64 b^4 n^4+20 b^2 n^2+1\right )}-\frac{\sin ^4\left (a+b \log \left (c x^n\right )\right )}{x \left (16 b^2 n^2+1\right )}-\frac{4 b n \sin ^3\left (a+b \log \left (c x^n\right )\right ) \cos \left (a+b \log \left (c x^n\right )\right )}{x \left (16 b^2 n^2+1\right )}-\frac{24 b^3 n^3 \sin \left (a+b \log \left (c x^n\right )\right ) \cos \left (a+b \log \left (c x^n\right )\right )}{x \left (64 b^4 n^4+20 b^2 n^2+1\right )}-\frac{24 b^4 n^4}{x \left (64 b^4 n^4+20 b^2 n^2+1\right )} \]

[Out]

(-24*b^4*n^4)/((1 + 20*b^2*n^2 + 64*b^4*n^4)*x) - (24*b^3*n^3*Cos[a + b*Log[c*x^n]]*Sin[a + b*Log[c*x^n]])/((1
+ 20*b^2*n^2 + 64*b^4*n^4)*x) - (12*b^2*n^2*Sin[a + b*Log[c*x^n]]^2)/((1 + 20*b^2*n^2 + 64*b^4*n^4)*x) - (4*b
*n*Cos[a + b*Log[c*x^n]]*Sin[a + b*Log[c*x^n]]^3)/((1 + 16*b^2*n^2)*x) - Sin[a + b*Log[c*x^n]]^4/((1 + 16*b^2*
n^2)*x)

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Rubi [A]  time = 0.0656843, 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} \[ -\frac{12 b^2 n^2 \sin ^2\left (a+b \log \left (c x^n\right )\right )}{x \left (64 b^4 n^4+20 b^2 n^2+1\right )}-\frac{\sin ^4\left (a+b \log \left (c x^n\right )\right )}{x \left (16 b^2 n^2+1\right )}-\frac{4 b n \sin ^3\left (a+b \log \left (c x^n\right )\right ) \cos \left (a+b \log \left (c x^n\right )\right )}{x \left (16 b^2 n^2+1\right )}-\frac{24 b^3 n^3 \sin \left (a+b \log \left (c x^n\right )\right ) \cos \left (a+b \log \left (c x^n\right )\right )}{x \left (64 b^4 n^4+20 b^2 n^2+1\right )}-\frac{24 b^4 n^4}{x \left (64 b^4 n^4+20 b^2 n^2+1\right )} \]

Antiderivative was successfully verified.

[In]

Int[Sin[a + b*Log[c*x^n]]^4/x^2,x]

[Out]

(-24*b^4*n^4)/((1 + 20*b^2*n^2 + 64*b^4*n^4)*x) - (24*b^3*n^3*Cos[a + b*Log[c*x^n]]*Sin[a + b*Log[c*x^n]])/((1
+ 20*b^2*n^2 + 64*b^4*n^4)*x) - (12*b^2*n^2*Sin[a + b*Log[c*x^n]]^2)/((1 + 20*b^2*n^2 + 64*b^4*n^4)*x) - (4*b
*n*Cos[a + b*Log[c*x^n]]*Sin[a + b*Log[c*x^n]]^3)/((1 + 16*b^2*n^2)*x) - Sin[a + b*Log[c*x^n]]^4/((1 + 16*b^2*
n^2)*x)

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^2} \, dx &=-\frac{4 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+16 b^2 n^2\right ) x}-\frac{\sin ^4\left (a+b \log \left (c x^n\right )\right )}{\left (1+16 b^2 n^2\right ) x}+\frac{\left (12 b^2 n^2\right ) \int \frac{\sin ^2\left (a+b \log \left (c x^n\right )\right )}{x^2} \, dx}{1+16 b^2 n^2}\\ &=-\frac{24 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 )}{\left (1+20 b^2 n^2+64 b^4 n^4\right ) x}-\frac{12 b^2 n^2 \sin ^2\left (a+b \log \left (c x^n\right )\right )}{\left (1+20 b^2 n^2+64 b^4 n^4\right ) x}-\frac{4 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+16 b^2 n^2\right ) x}-\frac{\sin ^4\left (a+b \log \left (c x^n\right )\right )}{\left (1+16 b^2 n^2\right ) x}+\frac{\left (24 b^4 n^4\right ) \int \frac{1}{x^2} \, dx}{1+20 b^2 n^2+64 b^4 n^4}\\ &=-\frac{24 b^4 n^4}{\left (1+20 b^2 n^2+64 b^4 n^4\right ) x}-\frac{24 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 )}{\left (1+20 b^2 n^2+64 b^4 n^4\right ) x}-\frac{12 b^2 n^2 \sin ^2\left (a+b \log \left (c x^n\right )\right )}{\left (1+20 b^2 n^2+64 b^4 n^4\right ) x}-\frac{4 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+16 b^2 n^2\right ) x}-\frac{\sin ^4\left (a+b \log \left (c x^n\right )\right )}{\left (1+16 b^2 n^2\right ) x}\\ \end{align*}

Mathematica [A]  time = 0.500479, size = 170, normalized size = 0.84 \[ -\frac{128 b^3 n^3 \sin \left (2 \left (a+b \log \left (c x^n\right )\right )\right )-16 b^3 n^3 \sin \left (4 \left (a+b \log \left (c x^n\right )\right )\right )-4 \left (16 b^2 n^2+1\right ) \cos \left (2 \left (a+b \log \left (c x^n\right )\right )\right )+\left (4 b^2 n^2+1\right ) \cos \left (4 \left (a+b \log \left (c x^n\right )\right )\right )+8 b n \sin \left (2 \left (a+b \log \left (c x^n\right )\right )\right )-4 b n \sin \left (4 \left (a+b \log \left (c x^n\right )\right )\right )+192 b^4 n^4+60 b^2 n^2+3}{8 x \left (64 b^4 n^4+20 b^2 n^2+1\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(3 + 60*b^2*n^2 + 192*b^4*n^4 - 4*(1 + 16*b^2*n^2)*Cos[2*(a + b*Log[c*x^n])] + (1 + 4*b^2*n^2)*Cos[4*(a + b*L
og[c*x^n])] + 8*b*n*Sin[2*(a + b*Log[c*x^n])] + 128*b^3*n^3*Sin[2*(a + b*Log[c*x^n])] - 4*b*n*Sin[4*(a + b*Log
[c*x^n])] - 16*b^3*n^3*Sin[4*(a + b*Log[c*x^n])])/(8*(1 + 20*b^2*n^2 + 64*b^4*n^4)*x)

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Maple [F]  time = 0.075, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( \sin \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) \right ) ^{4}}{{x}^{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B]  time = 1.3707, size = 1465, normalized size = 7.25 \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^2,x, algorithm="maxima")

[Out]

-1/16*(384*(b^4*cos(4*b*log(c))^2 + b^4*sin(4*b*log(c))^2)*n^4 + 120*(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 - (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 - 4*(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 + 4*(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 - 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*(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 - 16*(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*lo
g(c)))*n^2 + 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 - (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*l
og(c)))*n^3 + 4*(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 + 4*(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 + co
s(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
*(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 + 16*(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 + 2*(b*c
os(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 + 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))/((64*(b^4*cos(4*b*log(c))^
2 + b^4*sin(4*b*log(c))^2)*n^4 + 20*(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)

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Fricas [A]  time = 0.523396, size = 420, normalized size = 2.08 \begin{align*} -\frac{24 \, b^{4} n^{4} +{\left (4 \, b^{2} n^{2} + 1\right )} \cos \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{4} + 16 \, b^{2} n^{2} - 2 \,{\left (10 \, b^{2} n^{2} + 1\right )} \cos \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{2} - 4 \,{\left ({\left (4 \, b^{3} n^{3} + b n\right )} \cos \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{3} -{\left (10 \, b^{3} n^{3} + 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 ) + 1}{{\left (64 \, b^{4} n^{4} + 20 \, b^{2} n^{2} + 1\right )} x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(24*b^4*n^4 + (4*b^2*n^2 + 1)*cos(b*n*log(x) + b*log(c) + a)^4 + 16*b^2*n^2 - 2*(10*b^2*n^2 + 1)*cos(b*n*log(
x) + b*log(c) + a)^2 - 4*((4*b^3*n^3 + b*n)*cos(b*n*log(x) + b*log(c) + a)^3 - (10*b^3*n^3 + b*n)*cos(b*n*log(
x) + b*log(c) + a))*sin(b*n*log(x) + b*log(c) + a) + 1)/((64*b^4*n^4 + 20*b^2*n^2 + 1)*x)

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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**2,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^{2}}\,{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^2,x, algorithm="giac")

[Out]

integrate(sin(b*log(c*x^n) + a)^4/x^2, x)