3.17 \(\int \frac {\sin ^3(a+b \log (c x^n))}{x^2} \, dx\)
Optimal. Leaf size=158 \[ -\frac {\sin ^3\left (a+b \log \left (c x^n\right )\right )}{x \left (9 b^2 n^2+1\right )}-\frac {3 b n \sin ^2\left (a+b \log \left (c x^n\right )\right ) \cos \left (a+b \log \left (c x^n\right )\right )}{x \left (9 b^2 n^2+1\right )}-\frac {6 b^2 n^2 \sin \left (a+b \log \left (c x^n\right )\right )}{x \left (9 b^4 n^4+10 b^2 n^2+1\right )}-\frac {6 b^3 n^3 \cos \left (a+b \log \left (c x^n\right )\right )}{x \left (9 b^4 n^4+10 b^2 n^2+1\right )} \]
[Out]
-6*b^3*n^3*cos(a+b*ln(c*x^n))/(9*b^4*n^4+10*b^2*n^2+1)/x-6*b^2*n^2*sin(a+b*ln(c*x^n))/(9*b^4*n^4+10*b^2*n^2+1)
/x-3*b*n*cos(a+b*ln(c*x^n))*sin(a+b*ln(c*x^n))^2/(9*b^2*n^2+1)/x-sin(a+b*ln(c*x^n))^3/(9*b^2*n^2+1)/x
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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} \[ -\frac {\sin ^3\left (a+b \log \left (c x^n\right )\right )}{x \left (9 b^2 n^2+1\right )}-\frac {6 b^2 n^2 \sin \left (a+b \log \left (c x^n\right )\right )}{x \left (9 b^4 n^4+10 b^2 n^2+1\right )}-\frac {6 b^3 n^3 \cos \left (a+b \log \left (c x^n\right )\right )}{x \left (9 b^4 n^4+10 b^2 n^2+1\right )}-\frac {3 b n \sin ^2\left (a+b \log \left (c x^n\right )\right ) \cos \left (a+b \log \left (c x^n\right )\right )}{x \left (9 b^2 n^2+1\right )} \]
Antiderivative was successfully verified.
[In]
Int[Sin[a + b*Log[c*x^n]]^3/x^2,x]
[Out]
(-6*b^3*n^3*Cos[a + b*Log[c*x^n]])/((1 + 10*b^2*n^2 + 9*b^4*n^4)*x) - (6*b^2*n^2*Sin[a + b*Log[c*x^n]])/((1 +
10*b^2*n^2 + 9*b^4*n^4)*x) - (3*b*n*Cos[a + b*Log[c*x^n]]*Sin[a + b*Log[c*x^n]]^2)/((1 + 9*b^2*n^2)*x) - Sin[a
+ b*Log[c*x^n]]^3/((1 + 9*b^2*n^2)*x)
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^2} \, 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 (1+9 b^2 n^2\right ) x}-\frac {\sin ^3\left (a+b \log \left (c x^n\right )\right )}{\left (1+9 b^2 n^2\right ) x}+\frac {\left (6 b^2 n^2\right ) \int \frac {\sin \left (a+b \log \left (c x^n\right )\right )}{x^2} \, dx}{1+9 b^2 n^2}\\ &=-\frac {6 b^3 n^3 \cos \left (a+b \log \left (c x^n\right )\right )}{\left (1+10 b^2 n^2+9 b^4 n^4\right ) x}-\frac {6 b^2 n^2 \sin \left (a+b \log \left (c x^n\right )\right )}{\left (1+10 b^2 n^2+9 b^4 n^4\right ) x}-\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 (1+9 b^2 n^2\right ) x}-\frac {\sin ^3\left (a+b \log \left (c x^n\right )\right )}{\left (1+9 b^2 n^2\right ) x}\\ \end {align*}
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Mathematica [A] time = 0.33, size = 125, normalized size = 0.79 \[ \frac {3 \left (b^3 n^3+b n\right ) \cos \left (3 \left (a+b \log \left (c x^n\right )\right )\right )-3 b n \left (9 b^2 n^2+1\right ) \cos \left (a+b \log \left (c x^n\right )\right )+2 \sin \left (a+b \log \left (c x^n\right )\right ) \left (\left (b^2 n^2+1\right ) \cos \left (2 \left (a+b \log \left (c x^n\right )\right )\right )-13 b^2 n^2-1\right )}{4 x \left (9 b^4 n^4+10 b^2 n^2+1\right )} \]
Antiderivative was successfully verified.
[In]
Integrate[Sin[a + b*Log[c*x^n]]^3/x^2,x]
[Out]
(-3*b*n*(1 + 9*b^2*n^2)*Cos[a + b*Log[c*x^n]] + 3*(b*n + b^3*n^3)*Cos[3*(a + b*Log[c*x^n])] + 2*(-1 - 13*b^2*n
^2 + (1 + b^2*n^2)*Cos[2*(a + b*Log[c*x^n])])*Sin[a + b*Log[c*x^n]])/(4*(1 + 10*b^2*n^2 + 9*b^4*n^4)*x)
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fricas [A] time = 0.91, size = 127, normalized size = 0.80 \[ \frac {3 \, {\left (b^{3} n^{3} + b n\right )} \cos \left (b n \log \relax (x) + b \log \relax (c) + a\right )^{3} - 3 \, {\left (3 \, b^{3} n^{3} + b n\right )} \cos \left (b n \log \relax (x) + b \log \relax (c) + a\right ) - {\left (7 \, b^{2} n^{2} - {\left (b^{2} n^{2} + 1\right )} \cos \left (b n \log \relax (x) + b \log \relax (c) + a\right )^{2} + 1\right )} \sin \left (b n \log \relax (x) + b \log \relax (c) + a\right )}{{\left (9 \, b^{4} n^{4} + 10 \, b^{2} n^{2} + 1\right )} x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(sin(a+b*log(c*x^n))^3/x^2,x, algorithm="fricas")
[Out]
(3*(b^3*n^3 + b*n)*cos(b*n*log(x) + b*log(c) + a)^3 - 3*(3*b^3*n^3 + b*n)*cos(b*n*log(x) + b*log(c) + a) - (7*
b^2*n^2 - (b^2*n^2 + 1)*cos(b*n*log(x) + b*log(c) + a)^2 + 1)*sin(b*n*log(x) + b*log(c) + a))/((9*b^4*n^4 + 10
*b^2*n^2 + 1)*x)
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sin \left (b \log \left (c x^{n}\right ) + a\right )^{3}}{x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(sin(a+b*log(c*x^n))^3/x^2,x, algorithm="giac")
[Out]
integrate(sin(b*log(c*x^n) + a)^3/x^2, x)
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maple [F] time = 0.07, size = 0, normalized size = 0.00 \[ \int \frac {\sin ^{3}\left (a +b \ln \left (c \,x^{n}\right )\right )}{x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(sin(a+b*ln(c*x^n))^3/x^2,x)
[Out]
int(sin(a+b*ln(c*x^n))^3/x^2,x)
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maxima [B] time = 0.40, size = 995, normalized size = 6.30 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(sin(a+b*log(c*x^n))^3/x^2,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
+ (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 + 3*(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 + cos(3*b*log(c))*
sin(6*b*log(c)) - cos(6*b*log(c))*sin(3*b*log(c)) + 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*s
in(3*b*log(c))*sin(2*b*log(c)))*n^3 + 9*(b^2*cos(3*b*log(c))*sin(4*b*log(c)) - b^2*cos(4*b*log(c))*sin(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 + (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*log(c))*sin(2
*b*log(c)))*n + cos(3*b*log(c))*sin(4*b*log(c)) - cos(4*b*log(c))*sin(3*b*log(c)) + cos(2*b*log(c))*sin(3*b*lo
g(c)) - 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*c
os(6*b*log(c))*sin(3*b*log(c)) + b^3*sin(3*b*log(c)))*n^3 - (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 + 3*(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 - cos(6*b*log(c))*cos(3*b*log(c)) - sin(6*b*log(c))*sin(3*b*log(c)) -
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 - 9*(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 + (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 - cos(4*b*log(c))*cos(3*b*log(c)) - co
s(3*b*log(c))*cos(2*b*log(c)) - sin(4*b*log(c))*sin(3*b*log(c)) - 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 + 10*(b^2*cos(3*b*log(c))^2 + b^2*sin(3*b*lo
g(c))^2)*n^2 + cos(3*b*log(c))^2 + sin(3*b*log(c))^2)*x)
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\sin \left (a+b\,\ln \left (c\,x^n\right )\right )}^3}{x^2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(sin(a + b*log(c*x^n))^3/x^2,x)
[Out]
int(sin(a + b*log(c*x^n))^3/x^2, x)
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sympy [B] time = 125.90, size = 1020, normalized size = 6.46 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(sin(a+b*ln(c*x**n))**3/x**2,x)
[Out]
Piecewise((-3*log(x)*sin(-a + I*log(x) + I*log(c)/n)/(8*x) - 3*I*log(x)*cos(-a + I*log(x) + I*log(c)/n)/(8*x)
+ sin(-3*a + 3*I*log(x) + 3*I*log(c)/n)/(32*x) + 3*sin(-a + I*log(x) + I*log(c)/n)/(8*x) + 3*I*cos(-3*a + 3*I*
log(x) + 3*I*log(c)/n)/(32*x) - 3*log(c)*sin(-a + I*log(x) + I*log(c)/n)/(8*n*x) - 3*I*log(c)*cos(-a + I*log(x
) + I*log(c)/n)/(8*n*x), Eq(b, -I/n)), (log(x)*sin(-3*a + I*log(x) + I*log(c)/n)/(8*x) + I*log(x)*cos(-3*a + I
*log(x) + I*log(c)/n)/(8*x) - sin(-3*a + I*log(x) + I*log(c)/n)/(8*x) + 27*sin(-a + I*log(x)/3 + I*log(c)/(3*n
))/(32*x) + 9*I*cos(-a + I*log(x)/3 + I*log(c)/(3*n))/(32*x) + log(c)*sin(-3*a + I*log(x) + I*log(c)/n)/(8*n*x
) + I*log(c)*cos(-3*a + I*log(x) + I*log(c)/n)/(8*n*x), Eq(b, -I/(3*n))), (-log(x)*sin(3*a + I*log(x) + I*log(
c)/n)/(8*x) - I*log(x)*cos(3*a + I*log(x) + I*log(c)/n)/(8*x) - 27*sin(a + I*log(x)/3 + I*log(c)/(3*n))/(32*x)
+ sin(3*a + I*log(x) + I*log(c)/n)/(8*x) - 9*I*cos(a + I*log(x)/3 + I*log(c)/(3*n))/(32*x) - log(c)*sin(3*a +
I*log(x) + I*log(c)/n)/(8*n*x) - I*log(c)*cos(3*a + I*log(x) + I*log(c)/n)/(8*n*x), Eq(b, I/(3*n))), (3*log(x
)*sin(a + I*log(x) + I*log(c)/n)/(8*x) + 3*I*log(x)*cos(a + I*log(x) + I*log(c)/n)/(8*x) - sin(3*a + 3*I*log(x
) + 3*I*log(c)/n)/(32*x) + 3*I*cos(a + I*log(x) + I*log(c)/n)/(8*x) - 3*I*cos(3*a + 3*I*log(x) + 3*I*log(c)/n)
/(32*x) + 3*log(c)*sin(a + I*log(x) + I*log(c)/n)/(8*n*x) + 3*I*log(c)*cos(a + I*log(x) + I*log(c)/n)/(8*n*x),
Eq(b, I/n)), (-9*b**3*n**3*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 +
10*b**2*n**2*x + x) - 6*b**3*n**3*cos(a + b*n*log(x) + b*log(c))**3/(9*b**4*n**4*x + 10*b**2*n**2*x + x) - 7*b
**2*n**2*sin(a + b*n*log(x) + b*log(c))**3/(9*b**4*n**4*x + 10*b**2*n**2*x + x) - 6*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 + 10*b**2*n**2*x + x) - 3*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 + 10*b**2*n**2*x + x) - sin(a + b*n*log(x) + b
*log(c))**3/(9*b**4*n**4*x + 10*b**2*n**2*x + x), True))
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