3.124 \(\int x^m \cos ^3(a+b \log (c x^n)) \, dx\)

Optimal. Leaf size=201 \[ \frac{6 b^3 n^3 x^{m+1} \sin \left (a+b \log \left (c x^n\right )\right )}{\left (b^2 n^2+(m+1)^2\right ) \left (9 b^2 n^2+(m+1)^2\right )}+\frac{(m+1) x^{m+1} \cos ^3\left (a+b \log \left (c x^n\right )\right )}{9 b^2 n^2+(m+1)^2}+\frac{6 b^2 (m+1) n^2 x^{m+1} \cos \left (a+b \log \left (c x^n\right )\right )}{\left (b^2 n^2+(m+1)^2\right ) \left (9 b^2 n^2+(m+1)^2\right )}+\frac{3 b n x^{m+1} \sin \left (a+b \log \left (c x^n\right )\right ) \cos ^2\left (a+b \log \left (c x^n\right )\right )}{9 b^2 n^2+(m+1)^2} \]

[Out]

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

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Rubi [A]  time = 0.0781204, antiderivative size = 201, normalized size of antiderivative = 1., 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 = {4488, 4486} \[ \frac{6 b^3 n^3 x^{m+1} \sin \left (a+b \log \left (c x^n\right )\right )}{\left (b^2 n^2+(m+1)^2\right ) \left (9 b^2 n^2+(m+1)^2\right )}+\frac{(m+1) x^{m+1} \cos ^3\left (a+b \log \left (c x^n\right )\right )}{9 b^2 n^2+(m+1)^2}+\frac{6 b^2 (m+1) n^2 x^{m+1} \cos \left (a+b \log \left (c x^n\right )\right )}{\left (b^2 n^2+(m+1)^2\right ) \left (9 b^2 n^2+(m+1)^2\right )}+\frac{3 b n x^{m+1} \sin \left (a+b \log \left (c x^n\right )\right ) \cos ^2\left (a+b \log \left (c x^n\right )\right )}{9 b^2 n^2+(m+1)^2} \]

Antiderivative was successfully verified.

[In]

Int[x^m*Cos[a + b*Log[c*x^n]]^3,x]

[Out]

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

Rule 4488

Int[Cos[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*(d_.)]^(p_)*((e_.)*(x_))^(m_.), x_Symbol] :> Simp[((m + 1)*(e*x)
^(m + 1)*Cos[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*Cos[d*(a + b*Log[c*x^n])]^(p - 2), x], x] + Simp[(b*d*n*p*(e*x)^(m +
1)*Sin[d*(a + b*Log[c*x^n])]*Cos[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 4486

Int[Cos[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*(d_.)]*((e_.)*(x_))^(m_.), x_Symbol] :> Simp[((m + 1)*(e*x)^(m +
1)*Cos[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)*Sin[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]

Rubi steps

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

Mathematica [A]  time = 1.88161, size = 292, normalized size = 1.45 \[ \frac{1}{4} x^{m+1} \left (-\frac{3 \sin (b n \log (x)) \left ((m+1) \sin \left (a+b \log \left (c x^n\right )-b n \log (x)\right )-b n \cos \left (a+b \log \left (c x^n\right )-b n \log (x)\right )\right )}{b^2 n^2+m^2+2 m+1}+\frac{3 \cos (b n \log (x)) \left ((m+1) \cos \left (a+b \log \left (c x^n\right )-b n \log (x)\right )+b n \sin \left (a+b \log \left (c x^n\right )-b n \log (x)\right )\right )}{b^2 n^2+m^2+2 m+1}-\frac{\sin (3 b n \log (x)) \left ((m+1) \sin \left (3 \left (a+b \log \left (c x^n\right )-b n \log (x)\right )\right )-3 b n \cos \left (3 \left (a+b \log \left (c x^n\right )-b n \log (x)\right )\right )\right )}{9 b^2 n^2+m^2+2 m+1}+\frac{\cos (3 b n \log (x)) \left ((m+1) \cos \left (3 \left (a+b \log \left (c x^n\right )-b n \log (x)\right )\right )+3 b n \sin \left (3 \left (a+b \log \left (c x^n\right )-b n \log (x)\right )\right )\right )}{9 b^2 n^2+m^2+2 m+1}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[x^m*Cos[a + b*Log[c*x^n]]^3,x]

[Out]

(x^(1 + m)*((-3*Sin[b*n*Log[x]]*(-(b*n*Cos[a - b*n*Log[x] + b*Log[c*x^n]]) + (1 + m)*Sin[a - b*n*Log[x] + b*Lo
g[c*x^n]]))/(1 + 2*m + m^2 + b^2*n^2) + (3*Cos[b*n*Log[x]]*((1 + m)*Cos[a - b*n*Log[x] + b*Log[c*x^n]] + b*n*S
in[a - b*n*Log[x] + b*Log[c*x^n]]))/(1 + 2*m + m^2 + b^2*n^2) - (Sin[3*b*n*Log[x]]*(-3*b*n*Cos[3*(a - b*n*Log[
x] + b*Log[c*x^n])] + (1 + m)*Sin[3*(a - b*n*Log[x] + b*Log[c*x^n])]))/(1 + 2*m + m^2 + 9*b^2*n^2) + (Cos[3*b*
n*Log[x]]*((1 + m)*Cos[3*(a - b*n*Log[x] + b*Log[c*x^n])] + 3*b*n*Sin[3*(a - b*n*Log[x] + b*Log[c*x^n])]))/(1
+ 2*m + m^2 + 9*b^2*n^2)))/4

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Maple [F]  time = 0.082, size = 0, normalized size = 0. \begin{align*} \int{x}^{m} \left ( \cos \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(x^m*cos(a+b*ln(c*x^n))^3,x)

[Out]

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

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Maxima [B]  time = 1.5955, size = 3175, normalized size = 15.8 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^m*cos(a+b*log(c*x^n))^3,x, algorithm="maxima")

[Out]

1/8*(((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)))*m^3 + 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 + 3*(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)))*m^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)) + (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)))*m)*n^2 + 3*(cos(6*b*log(c))*cos(3*b*log(c)) + sin(6*b*l
og(c))*sin(3*b*log(c)) + cos(3*b*log(c)))*m + 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)))*m^2 + b*cos(3*b*log(c))*sin(6*b*log(c)) - b*cos(6*b*log(c))*sin(3*b*log(c)) + 2
*(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)))*m + 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*x^m*cos(3*b*l
og(x^n) + 3*a) + 3*((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)))*m^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 + 3*(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)))*m^2 + 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*s
in(4*b*log(c))*sin(3*b*log(c)) + b^2*sin(3*b*log(c))*sin(2*b*log(c)) + (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)
))*m)*n^2 + 3*(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)))*m + ((b*cos(3*b*log(c))*sin(4*b*log(c)) - b*cos(4*b*log(c))*sin(3*b*lo
g(c)) + b*cos(2*b*log(c))*sin(3*b*log(c)) - b*cos(3*b*log(c))*sin(2*b*log(c)))*m^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*l
og(c)) + 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)))*m)*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*x^m*cos(b*log(x^n) + a) - ((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)))*m^3 - 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 + 3*(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)))*m^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)) + (b^2*cos(3*b*log(c))*sin(6*b*log(c)) - b^2*cos(6*b*lo
g(c))*sin(3*b*log(c)) + b^2*sin(3*b*log(c)))*m)*n^2 + 3*(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)))*m - 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)))*m^2 + b*cos(6*b*log(c))*cos(3*b*log(c)) + b*sin(6*b*log(c))*sin(3*b*log(c)) + 2*(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)))*m + 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*x^m*sin(3*b*log(x^n) +
3*a) - 3*((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)))*m^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 + 3*(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)))*m^2 + 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)) + (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)))*m)*n^2
+ 3*(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)))*m - ((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)))*m^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)) + 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)))*m)*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*x^m*sin(b*log(x^n) + a))/((cos(3*b*log(c)
)^2 + sin(3*b*log(c))^2)*m^4 + 9*(b^4*cos(3*b*log(c))^2 + b^4*sin(3*b*log(c))^2)*n^4 + 4*(cos(3*b*log(c))^2 +
sin(3*b*log(c))^2)*m^3 + 6*(cos(3*b*log(c))^2 + sin(3*b*log(c))^2)*m^2 + 10*(b^2*cos(3*b*log(c))^2 + b^2*sin(3
*b*log(c))^2 + (b^2*cos(3*b*log(c))^2 + b^2*sin(3*b*log(c))^2)*m^2 + 2*(b^2*cos(3*b*log(c))^2 + b^2*sin(3*b*lo
g(c))^2)*m)*n^2 + 4*(cos(3*b*log(c))^2 + sin(3*b*log(c))^2)*m + cos(3*b*log(c))^2 + sin(3*b*log(c))^2)

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Fricas [A]  time = 0.53167, size = 467, normalized size = 2.32 \begin{align*} \frac{3 \,{\left (2 \, b^{3} n^{3} x +{\left (b^{3} n^{3} +{\left (b m^{2} + 2 \, b m + b\right )} n\right )} x \cos \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{2}\right )} x^{m} \sin \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) +{\left (6 \,{\left (b^{2} m + b^{2}\right )} n^{2} x \cos \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) +{\left (m^{3} +{\left (b^{2} m + b^{2}\right )} n^{2} + 3 \, m^{2} + 3 \, m + 1\right )} x \cos \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{3}\right )} x^{m}}{9 \, b^{4} n^{4} + m^{4} + 4 \, m^{3} + 10 \,{\left (b^{2} m^{2} + 2 \, b^{2} m + b^{2}\right )} n^{2} + 6 \, m^{2} + 4 \, m + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Timed out

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Giac [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: NotImplementedError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: NotImplementedError