3.15 \(\int \sin ^3(a+b \log (c x^n)) \, dx\)

Optimal. Leaf size=149 \[ \frac {x \sin ^3\left (a+b \log \left (c x^n\right )\right )}{9 b^2 n^2+1}-\frac {3 b n x \sin ^2\left (a+b \log \left (c x^n\right )\right ) \cos \left (a+b \log \left (c x^n\right )\right )}{9 b^2 n^2+1}+\frac {6 b^2 n^2 x \sin \left (a+b \log \left (c x^n\right )\right )}{9 b^4 n^4+10 b^2 n^2+1}-\frac {6 b^3 n^3 x \cos \left (a+b \log \left (c x^n\right )\right )}{9 b^4 n^4+10 b^2 n^2+1} \]

[Out]

-6*b^3*n^3*x*cos(a+b*ln(c*x^n))/(9*b^4*n^4+10*b^2*n^2+1)+6*b^2*n^2*x*sin(a+b*ln(c*x^n))/(9*b^4*n^4+10*b^2*n^2+
1)-3*b*n*x*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)

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Rubi [A]  time = 0.04, antiderivative size = 149, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {4477, 4475} \[ \frac {x \sin ^3\left (a+b \log \left (c x^n\right )\right )}{9 b^2 n^2+1}+\frac {6 b^2 n^2 x \sin \left (a+b \log \left (c x^n\right )\right )}{9 b^4 n^4+10 b^2 n^2+1}-\frac {6 b^3 n^3 x \cos \left (a+b \log \left (c x^n\right )\right )}{9 b^4 n^4+10 b^2 n^2+1}-\frac {3 b n x \sin ^2\left (a+b \log \left (c x^n\right )\right ) \cos \left (a+b \log \left (c x^n\right )\right )}{9 b^2 n^2+1} \]

Antiderivative was successfully verified.

[In]

Int[Sin[a + b*Log[c*x^n]]^3,x]

[Out]

(-6*b^3*n^3*x*Cos[a + b*Log[c*x^n]])/(1 + 10*b^2*n^2 + 9*b^4*n^4) + (6*b^2*n^2*x*Sin[a + b*Log[c*x^n]])/(1 + 1
0*b^2*n^2 + 9*b^4*n^4) - (3*b*n*x*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)

Rule 4475

Int[Sin[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*(d_.)], x_Symbol] :> Simp[(x*Sin[d*(a + b*Log[c*x^n])])/(b^2*d^2
*n^2 + 1), x] - Simp[(b*d*n*x*Cos[d*(a + b*Log[c*x^n])])/(b^2*d^2*n^2 + 1), x] /; FreeQ[{a, b, c, d, n}, x] &&
NeQ[b^2*d^2*n^2 + 1, 0]

Rule 4477

Int[Sin[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*(d_.)]^(p_), x_Symbol] :> Simp[(x*Sin[d*(a + b*Log[c*x^n])]^p)/(
b^2*d^2*n^2*p^2 + 1), x] + (Dist[(b^2*d^2*n^2*p*(p - 1))/(b^2*d^2*n^2*p^2 + 1), Int[Sin[d*(a + b*Log[c*x^n])]^
(p - 2), x], x] - Simp[(b*d*n*p*x*Cos[d*(a + b*Log[c*x^n])]*Sin[d*(a + b*Log[c*x^n])]^(p - 1))/(b^2*d^2*n^2*p^
2 + 1), x]) /; FreeQ[{a, b, c, d, n}, x] && IGtQ[p, 1] && NeQ[b^2*d^2*n^2*p^2 + 1, 0]

Rubi steps

\begin {align*} \int \sin ^3\left (a+b \log \left (c x^n\right )\right ) \, dx &=-\frac {3 b n x \cos \left (a+b \log \left (c x^n\right )\right ) \sin ^2\left (a+b \log \left (c x^n\right )\right )}{1+9 b^2 n^2}+\frac {x \sin ^3\left (a+b \log \left (c x^n\right )\right )}{1+9 b^2 n^2}+\frac {\left (6 b^2 n^2\right ) \int \sin \left (a+b \log \left (c x^n\right )\right ) \, dx}{1+9 b^2 n^2}\\ &=-\frac {6 b^3 n^3 x \cos \left (a+b \log \left (c x^n\right )\right )}{1+10 b^2 n^2+9 b^4 n^4}+\frac {6 b^2 n^2 x \sin \left (a+b \log \left (c x^n\right )\right )}{1+10 b^2 n^2+9 b^4 n^4}-\frac {3 b n x \cos \left (a+b \log \left (c x^n\right )\right ) \sin ^2\left (a+b \log \left (c x^n\right )\right )}{1+9 b^2 n^2}+\frac {x \sin ^3\left (a+b \log \left (c x^n\right )\right )}{1+9 b^2 n^2}\\ \end {align*}

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Mathematica [A]  time = 0.47, size = 121, normalized size = 0.81 \[ -\frac {x \left (-3 \left (b^3 n^3+b n\right ) \cos \left (3 \left (a+b \log \left (c x^n\right )\right )\right )+3 b n \left (9 b^2 n^2+1\right ) \cos \left (a+b \log \left (c x^n\right )\right )+2 \sin \left (a+b \log \left (c x^n\right )\right ) \left (\left (b^2 n^2+1\right ) \cos \left (2 \left (a+b \log \left (c x^n\right )\right )\right )-13 b^2 n^2-1\right )\right )}{36 b^4 n^4+40 b^2 n^2+4} \]

Antiderivative was successfully verified.

[In]

Integrate[Sin[a + b*Log[c*x^n]]^3,x]

[Out]

-((x*(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 + 40*b^2*n^2 + 36*b^4*n^4))

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fricas [A]  time = 0.52, size = 130, normalized size = 0.87 \[ \frac {3 \, {\left (b^{3} n^{3} + b n\right )} x \cos \left (b n \log \relax (x) + b \log \relax (c) + a\right )^{3} - 3 \, {\left (3 \, b^{3} n^{3} + b n\right )} x \cos \left (b n \log \relax (x) + b \log \relax (c) + a\right ) - {\left ({\left (b^{2} n^{2} + 1\right )} x \cos \left (b n \log \relax (x) + b \log \relax (c) + a\right )^{2} - {\left (7 \, b^{2} n^{2} + 1\right )} x\right )} \sin \left (b n \log \relax (x) + b \log \relax (c) + a\right )}{9 \, b^{4} n^{4} + 10 \, b^{2} n^{2} + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(3*(b^3*n^3 + b*n)*x*cos(b*n*log(x) + b*log(c) + a)^3 - 3*(3*b^3*n^3 + b*n)*x*cos(b*n*log(x) + b*log(c) + a) -
((b^2*n^2 + 1)*x*cos(b*n*log(x) + b*log(c) + a)^2 - (7*b^2*n^2 + 1)*x)*sin(b*n*log(x) + b*log(c) + a))/(9*b^4
*n^4 + 10*b^2*n^2 + 1)

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giac [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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maple [F]  time = 0.07, size = 0, normalized size = 0.00 \[ \int \sin ^{3}\left (a +b \ln \left (c \,x^{n}\right )\right )\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [B]  time = 0.39, size = 990, normalized size = 6.64 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(a+b*log(c*x^n))^3,x, 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)))*x*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 - 9*(b^2*cos(3*b*log(c))*sin(4*b*log(c)) - b^2*cos(4*b*log(c))*sin(3*b*l
og(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))*c
os(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*
log(c)) + cos(3*b*log(c))*sin(2*b*log(c)))*x*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 + (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)))*x*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*c
os(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)
) + cos(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)))*x*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*log(c))^2)*n^2 + cos(3*b*log(c))^2 + sin(3*b*log(c))^2)

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mupad [B]  time = 2.89, size = 114, normalized size = 0.77 \[ -\frac {x\,{\mathrm {e}}^{-a\,1{}\mathrm {i}}\,\frac {1}{{\left (c\,x^n\right )}^{b\,1{}\mathrm {i}}}\,3{}\mathrm {i}}{-8+b\,n\,8{}\mathrm {i}}-\frac {3\,x\,{\mathrm {e}}^{a\,1{}\mathrm {i}}\,{\left (c\,x^n\right )}^{b\,1{}\mathrm {i}}}{8\,b\,n-8{}\mathrm {i}}+\frac {x\,{\mathrm {e}}^{-a\,3{}\mathrm {i}}\,\frac {1}{{\left (c\,x^n\right )}^{b\,3{}\mathrm {i}}}\,1{}\mathrm {i}}{-8+b\,n\,24{}\mathrm {i}}+\frac {x\,{\mathrm {e}}^{a\,3{}\mathrm {i}}\,{\left (c\,x^n\right )}^{b\,3{}\mathrm {i}}}{24\,b\,n-8{}\mathrm {i}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(a + b*log(c*x^n))^3,x)

[Out]

(x*exp(-a*3i)/(c*x^n)^(b*3i)*1i)/(b*n*24i - 8) - (3*x*exp(a*1i)*(c*x^n)^(b*1i))/(8*b*n - 8i) - (x*exp(-a*1i)/(
c*x^n)^(b*1i)*3i)/(b*n*8i - 8) + (x*exp(a*3i)*(c*x^n)^(b*3i))/(24*b*n - 8i)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \begin {cases} \int \sin ^{3}{\left (a - \frac {i \log {\left (c x^{n} \right )}}{n} \right )}\, dx & \text {for}\: b = - \frac {i}{n} \\\int \sin ^{3}{\left (a - \frac {i \log {\left (c x^{n} \right )}}{3 n} \right )}\, dx & \text {for}\: b = - \frac {i}{3 n} \\\int \sin ^{3}{\left (a + \frac {i \log {\left (c x^{n} \right )}}{3 n} \right )}\, dx & \text {for}\: b = \frac {i}{3 n} \\\int \sin ^{3}{\left (a + \frac {i \log {\left (c x^{n} \right )}}{n} \right )}\, dx & \text {for}\: b = \frac {i}{n} \\- \frac {9 b^{3} n^{3} x \sin ^{2}{\left (a + b n \log {\relax (x )} + b \log {\relax (c )} \right )} \cos {\left (a + b n \log {\relax (x )} + b \log {\relax (c )} \right )}}{9 b^{4} n^{4} + 10 b^{2} n^{2} + 1} - \frac {6 b^{3} n^{3} x \cos ^{3}{\left (a + b n \log {\relax (x )} + b \log {\relax (c )} \right )}}{9 b^{4} n^{4} + 10 b^{2} n^{2} + 1} + \frac {7 b^{2} n^{2} x \sin ^{3}{\left (a + b n \log {\relax (x )} + b \log {\relax (c )} \right )}}{9 b^{4} n^{4} + 10 b^{2} n^{2} + 1} + \frac {6 b^{2} n^{2} x \sin {\left (a + b n \log {\relax (x )} + b \log {\relax (c )} \right )} \cos ^{2}{\left (a + b n \log {\relax (x )} + b \log {\relax (c )} \right )}}{9 b^{4} n^{4} + 10 b^{2} n^{2} + 1} - \frac {3 b n x \sin ^{2}{\left (a + b n \log {\relax (x )} + b \log {\relax (c )} \right )} \cos {\left (a + b n \log {\relax (x )} + b \log {\relax (c )} \right )}}{9 b^{4} n^{4} + 10 b^{2} n^{2} + 1} + \frac {x \sin ^{3}{\left (a + b n \log {\relax (x )} + b \log {\relax (c )} \right )}}{9 b^{4} n^{4} + 10 b^{2} n^{2} + 1} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((Integral(sin(a - I*log(c*x**n)/n)**3, x), Eq(b, -I/n)), (Integral(sin(a - I*log(c*x**n)/(3*n))**3,
x), Eq(b, -I/(3*n))), (Integral(sin(a + I*log(c*x**n)/(3*n))**3, x), Eq(b, I/(3*n))), (Integral(sin(a + I*log(
c*x**n)/n)**3, x), Eq(b, I/n)), (-9*b**3*n**3*x*sin(a + b*n*log(x) + b*log(c))**2*cos(a + b*n*log(x) + b*log(c
))/(9*b**4*n**4 + 10*b**2*n**2 + 1) - 6*b**3*n**3*x*cos(a + b*n*log(x) + b*log(c))**3/(9*b**4*n**4 + 10*b**2*n
**2 + 1) + 7*b**2*n**2*x*sin(a + b*n*log(x) + b*log(c))**3/(9*b**4*n**4 + 10*b**2*n**2 + 1) + 6*b**2*n**2*x*si
n(a + b*n*log(x) + b*log(c))*cos(a + b*n*log(x) + b*log(c))**2/(9*b**4*n**4 + 10*b**2*n**2 + 1) - 3*b*n*x*sin(
a + b*n*log(x) + b*log(c))**2*cos(a + b*n*log(x) + b*log(c))/(9*b**4*n**4 + 10*b**2*n**2 + 1) + x*sin(a + b*n*
log(x) + b*log(c))**3/(9*b**4*n**4 + 10*b**2*n**2 + 1), True))

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