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

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

[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)

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Rubi [A]  time = 0.0365887, antiderivative size = 149, normalized size of antiderivative = 1., 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 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]

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]

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*}

Mathematica [A]  time = 0.482902, size = 121, normalized size = 0.81 \[ -\frac{x \left (3 b n \left (9 b^2 n^2+1\right ) \cos \left (a+b \log \left (c x^n\right )\right )-3 \left (b^3 n^3+b n\right ) \cos \left (3 \left (a+b \log \left (c x^n\right )\right )\right )+2 \sin \left (a+b \log \left (c x^n\right )\right ) \left (\left (b^2 n^2+1\right ) \cos \left (2 \left (a+b \log \left (c x^n\right )\right )\right )-13 b^2 n^2-1\right )\right )}{36 b^4 n^4+40 b^2 n^2+4} \]

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

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 = 1.24488, size = 1337, normalized size = 8.97 \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))^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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Fricas [A]  time = 0.504358, size = 329, normalized size = 2.21 \begin{align*} \frac{3 \,{\left (b^{3} n^{3} + b n\right )} x \cos \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{3} - 3 \,{\left (3 \, b^{3} n^{3} + b n\right )} x \cos \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) -{\left ({\left (b^{2} n^{2} + 1\right )} x \cos \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{2} -{\left (7 \, b^{2} n^{2} + 1\right )} x\right )} \sin \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )}{9 \, b^{4} n^{4} + 10 \, b^{2} n^{2} + 1} \end{align*}

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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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))**3,x)

[Out]

Timed out

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Giac [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*log(c*x^n))^3,x, algorithm="giac")

[Out]

Timed out