3.18 \(\int \frac {\sin ^3(a+b \log (c x^n))}{x^3} \, dx\)
Optimal. Leaf size=158 \[ -\frac {2 \sin ^3\left (a+b \log \left (c x^n\right )\right )}{x^2 \left (9 b^2 n^2+4\right )}-\frac {3 b n \sin ^2\left (a+b \log \left (c x^n\right )\right ) \cos \left (a+b \log \left (c x^n\right )\right )}{x^2 \left (9 b^2 n^2+4\right )}-\frac {12 b^2 n^2 \sin \left (a+b \log \left (c x^n\right )\right )}{x^2 \left (9 b^4 n^4+40 b^2 n^2+16\right )}-\frac {6 b^3 n^3 \cos \left (a+b \log \left (c x^n\right )\right )}{x^2 \left (9 b^4 n^4+40 b^2 n^2+16\right )} \]
[Out]
-6*b^3*n^3*cos(a+b*ln(c*x^n))/(9*b^4*n^4+40*b^2*n^2+16)/x^2-12*b^2*n^2*sin(a+b*ln(c*x^n))/(9*b^4*n^4+40*b^2*n^
2+16)/x^2-3*b*n*cos(a+b*ln(c*x^n))*sin(a+b*ln(c*x^n))^2/(9*b^2*n^2+4)/x^2-2*sin(a+b*ln(c*x^n))^3/(9*b^2*n^2+4)
/x^2
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Rubi [A] time = 0.05, antiderivative size = 158, normalized size of antiderivative = 1.00,
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 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]
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]
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*}
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Mathematica [A] time = 0.39, 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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fricas [A] time = 0.70, size = 129, normalized size = 0.82 \[ \frac {3 \, {\left (b^{3} n^{3} + 4 \, b n\right )} \cos \left (b n \log \relax (x) + b \log \relax (c) + a\right )^{3} - 3 \, {\left (3 \, b^{3} n^{3} + 4 \, b n\right )} \cos \left (b n \log \relax (x) + b \log \relax (c) + a\right ) - 2 \, {\left (7 \, b^{2} n^{2} - {\left (b^{2} n^{2} + 4\right )} \cos \left (b n \log \relax (x) + b \log \relax (c) + a\right )^{2} + 4\right )} \sin \left (b n \log \relax (x) + b \log \relax (c) + a\right )}{{\left (9 \, b^{4} n^{4} + 40 \, b^{2} n^{2} + 16\right )} x^{2}} \]
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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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sin \left (b \log \left (c x^{n}\right ) + a\right )^{3}}{x^{3}}\,{d x} \]
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)
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maple [F] time = 0.07, size = 0, normalized size = 0.00 \[ \int \frac {\sin ^{3}\left (a +b \ln \left (c \,x^{n}\right )\right )}{x^{3}}\, dx \]
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 = 0.41, size = 1007, normalized size = 6.37 \[ \text {result too large to display} \]
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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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\sin \left (a+b\,\ln \left (c\,x^n\right )\right )}^3}{x^3} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(sin(a + b*log(c*x^n))^3/x^3,x)
[Out]
int(sin(a + b*log(c*x^n))^3/x^3, x)
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sympy [B] time = 160.48, size = 1197, normalized size = 7.58 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(sin(a+b*ln(c*x**n))**3/x**3,x)
[Out]
Piecewise((-3*log(x)*sin(-a + 2*I*log(x) + 2*I*log(c)/n)/(8*x**2) - 3*I*log(x)*cos(-a + 2*I*log(x) + 2*I*log(c
)/n)/(8*x**2) + sin(-3*a + 6*I*log(x) + 6*I*log(c)/n)/(64*x**2) + 3*sin(-a + 2*I*log(x) + 2*I*log(c)/n)/(16*x*
*2) + 3*I*cos(-3*a + 6*I*log(x) + 6*I*log(c)/n)/(64*x**2) - 3*log(c)*sin(-a + 2*I*log(x) + 2*I*log(c)/n)/(8*n*
x**2) - 3*I*log(c)*cos(-a + 2*I*log(x) + 2*I*log(c)/n)/(8*n*x**2), Eq(b, -2*I/n)), (log(x)*sin(-3*a + 2*I*log(
x) + 2*I*log(c)/n)/(8*x**2) + I*log(x)*cos(-3*a + 2*I*log(x) + 2*I*log(c)/n)/(8*x**2) - sin(-3*a + 2*I*log(x)
+ 2*I*log(c)/n)/(16*x**2) + 27*sin(-a + 2*I*log(x)/3 + 2*I*log(c)/(3*n))/(64*x**2) + 9*I*cos(-a + 2*I*log(x)/3
+ 2*I*log(c)/(3*n))/(64*x**2) + log(c)*sin(-3*a + 2*I*log(x) + 2*I*log(c)/n)/(8*n*x**2) + I*log(c)*cos(-3*a +
2*I*log(x) + 2*I*log(c)/n)/(8*n*x**2), Eq(b, -2*I/(3*n))), (-log(x)*sin(3*a + 2*I*log(x) + 2*I*log(c)/n)/(8*x
**2) - I*log(x)*cos(3*a + 2*I*log(x) + 2*I*log(c)/n)/(8*x**2) - 27*sin(a + 2*I*log(x)/3 + 2*I*log(c)/(3*n))/(6
4*x**2) + sin(3*a + 2*I*log(x) + 2*I*log(c)/n)/(16*x**2) - 9*I*cos(a + 2*I*log(x)/3 + 2*I*log(c)/(3*n))/(64*x*
*2) - log(c)*sin(3*a + 2*I*log(x) + 2*I*log(c)/n)/(8*n*x**2) - I*log(c)*cos(3*a + 2*I*log(x) + 2*I*log(c)/n)/(
8*n*x**2), Eq(b, 2*I/(3*n))), (3*log(x)*sin(a + 2*I*log(x) + 2*I*log(c)/n)/(8*x**2) + 3*I*log(x)*cos(a + 2*I*l
og(x) + 2*I*log(c)/n)/(8*x**2) - 3*sin(a + 2*I*log(x) + 2*I*log(c)/n)/(16*x**2) - sin(3*a + 6*I*log(x) + 6*I*l
og(c)/n)/(64*x**2) - 3*I*cos(3*a + 6*I*log(x) + 6*I*log(c)/n)/(64*x**2) + 3*log(c)*sin(a + 2*I*log(x) + 2*I*lo
g(c)/n)/(8*n*x**2) + 3*I*log(c)*cos(a + 2*I*log(x) + 2*I*log(c)/n)/(8*n*x**2), Eq(b, 2*I/n)), (-9*b**3*n**3*si
n(a + b*n*log(x) + b*log(c))**2*cos(a + b*n*log(x) + b*log(c))/(9*b**4*n**4*x**2 + 40*b**2*n**2*x**2 + 16*x**2
) - 6*b**3*n**3*cos(a + b*n*log(x) + b*log(c))**3/(9*b**4*n**4*x**2 + 40*b**2*n**2*x**2 + 16*x**2) - 14*b**2*n
**2*sin(a + b*n*log(x) + b*log(c))**3/(9*b**4*n**4*x**2 + 40*b**2*n**2*x**2 + 16*x**2) - 12*b**2*n**2*sin(a +
b*n*log(x) + b*log(c))*cos(a + b*n*log(x) + b*log(c))**2/(9*b**4*n**4*x**2 + 40*b**2*n**2*x**2 + 16*x**2) - 12
*b*n*sin(a + b*n*log(x) + b*log(c))**2*cos(a + b*n*log(x) + b*log(c))/(9*b**4*n**4*x**2 + 40*b**2*n**2*x**2 +
16*x**2) - 8*sin(a + b*n*log(x) + b*log(c))**3/(9*b**4*n**4*x**2 + 40*b**2*n**2*x**2 + 16*x**2), True))
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