3.2.0 ∫ xm cos(a + b log(cxn)) dx [126]

Integrand size = 15, antiderivative size = 70 \[ \int x^m \cos \left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {(1+m) x^{1+m} \cos \left (a+b \log \left (c x^n\right )\right )}{(1+m)^2+b^2 n^2}+\frac {b n x^{1+m} \sin \left (a+b \log \left (c x^n\right )\right )}{(1+m)^2+b^2 n^2} \]

(1+m)*x^(1+m)*cos(a+b*ln(c*x^n))/((1+m)^2+b^2*n^2)+b*n*x^(1+m)*sin(a+b*ln(c*x^n))/((1+m)^2+b^2*n^2)

Rubi [A] (veriﬁed)

Time = 0.02 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {4574} \[ \int x^m \cos \left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {b n x^{m+1} \sin \left (a+b \log \left (c x^n\right )\right )}{b^2 n^2+(m+1)^2}+\frac {(m+1) x^{m+1} \cos \left (a+b \log \left (c x^n\right )\right )}{b^2 n^2+(m+1)^2} \]

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

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,

Rubi steps \begin{align*} \text {integral}& = \frac {(1+m) x^{1+m} \cos \left (a+b \log \left (c x^n\right )\right )}{(1+m)^2+b^2 n^2}+\frac {b n x^{1+m} \sin \left (a+b \log \left (c x^n\right )\right )}{(1+m)^2+b^2 n^2} \\ \end{align*}

Mathematica [A] (veriﬁed)

Time = 0.11 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.76 \[ \int x^m \cos \left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {x^{1+m} \left ((1+m) \cos \left (a+b \log \left (c x^n\right )\right )+b n \sin \left (a+b \log \left (c x^n\right )\right )\right )}{1+2 m+m^2+b^2 n^2} \]

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

Maple [A] (veriﬁed)

Time = 0.90 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.90


method	result	size

parallelrisch	\(\frac {x^{1+m} \left (\sin \left (a +b \ln \left (c \,x^{n}\right )\right ) b n +\cos \left (a +b \ln \left (c \,x^{n}\right )\right ) m +\cos \left (a +b \ln \left (c \,x^{n}\right )\right )\right )}{b^{2} n^{2}+m^{2}+2 m +1}\)	\(63\)

x^(1+m)/(b^2*n^2+m^2+2*m+1)*(sin(a+b*ln(c*x^n))*b*n+cos(a+b*ln(c*x^n))*m+cos(a+b*ln(c*x^n)))

Fricas [A] (veriﬁcation not implemented)

Time = 0.24 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.83 \[ \int x^m \cos \left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {b n x x^{m} \sin \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) + {\left (m + 1\right )} x x^{m} \cos \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )}{b^{2} n^{2} + m^{2} + 2 \, m + 1} \]

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

(b*n*x*x^m*sin(b*n*log(x) + b*log(c) + a) + (m + 1)*x*x^m*cos(b*n*log(x) + b*log(c) + a))/(b^2*n^2 + m^2 + 2*m

Sympy [F]

\[ \int x^m \cos \left (a+b \log \left (c x^n\right )\right ) \, dx=\begin {cases} \log {\left (x \right )} \cos {\left (a \right )} & \text {for}\: b = 0 \wedge m = -1 \\\int x^{m} \cos {\left (- a + \frac {i m \log {\left (c x^{n} \right )}}{n} + \frac {i \log {\left (c x^{n} \right )}}{n} \right )}\, dx & \text {for}\: b = - \frac {i \left (m + 1\right )}{n} \\\int x^{m} \cos {\left (a + \frac {i m \log {\left (c x^{n} \right )}}{n} + \frac {i \log {\left (c x^{n} \right )}}{n} \right )}\, dx & \text {for}\: b = \frac {i \left (m + 1\right )}{n} \\\frac {b n x x^{m} \sin {\left (a + b \log {\left (c x^{n} \right )} \right )}}{b^{2} n^{2} + m^{2} + 2 m + 1} + \frac {m x x^{m} \cos {\left (a + b \log {\left (c x^{n} \right )} \right )}}{b^{2} n^{2} + m^{2} + 2 m + 1} + \frac {x x^{m} \cos {\left (a + b \log {\left (c x^{n} \right )} \right )}}{b^{2} n^{2} + m^{2} + 2 m + 1} & \text {otherwise} \end {cases} \]

Piecewise((log(x)*cos(a), Eq(b, 0) & Eq(m, -1)), (Integral(x**m*cos(-a + I*m*log(c*x**n)/n + I*log(c*x**n)/n),

x), Eq(b, -I*(m + 1)/n)), (Integral(x**m*cos(a + I*m*log(c*x**n)/n + I*log(c*x**n)/n), x), Eq(b, I*(m + 1)/n)

), (b*n*x*x**m*sin(a + b*log(c*x**n))/(b**2*n**2 + m**2 + 2*m + 1) + m*x*x**m*cos(a + b*log(c*x**n))/(b**2*n**

2 + m**2 + 2*m + 1) + x*x**m*cos(a + b*log(c*x**n))/(b**2*n**2 + m**2 + 2*m + 1), True))

Maxima [B] (veriﬁcation not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 313 vs. \(2 (70) = 140\).

Time = 0.24 (sec) , antiderivative size = 313, normalized size of antiderivative = 4.47 \[ \int x^m \cos \left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {{\left ({\left (\cos \left (2 \, b \log \left (c\right )\right ) \cos \left (b \log \left (c\right )\right ) + \sin \left (2 \, b \log \left (c\right )\right ) \sin \left (b \log \left (c\right )\right ) + \cos \left (b \log \left (c\right )\right )\right )} m + {\left (b \cos \left (b \log \left (c\right )\right ) \sin \left (2 \, b \log \left (c\right )\right ) - b \cos \left (2 \, b \log \left (c\right )\right ) \sin \left (b \log \left (c\right )\right ) + b \sin \left (b \log \left (c\right )\right )\right )} n + \cos \left (2 \, b \log \left (c\right )\right ) \cos \left (b \log \left (c\right )\right ) + \sin \left (2 \, b \log \left (c\right )\right ) \sin \left (b \log \left (c\right )\right ) + \cos \left (b \log \left (c\right )\right )\right )} x x^{m} \cos \left (b \log \left (x^{n}\right ) + a\right ) - {\left ({\left (\cos \left (b \log \left (c\right )\right ) \sin \left (2 \, b \log \left (c\right )\right ) - \cos \left (2 \, b \log \left (c\right )\right ) \sin \left (b \log \left (c\right )\right ) + \sin \left (b \log \left (c\right )\right )\right )} m - {\left (b \cos \left (2 \, b \log \left (c\right )\right ) \cos \left (b \log \left (c\right )\right ) + b \sin \left (2 \, b \log \left (c\right )\right ) \sin \left (b \log \left (c\right )\right ) + b \cos \left (b \log \left (c\right )\right )\right )} n + \cos \left (b \log \left (c\right )\right ) \sin \left (2 \, b \log \left (c\right )\right ) - \cos \left (2 \, b \log \left (c\right )\right ) \sin \left (b \log \left (c\right )\right ) + \sin \left (b \log \left (c\right )\right )\right )} x x^{m} \sin \left (b \log \left (x^{n}\right ) + a\right )}{2 \, {\left ({\left (\cos \left (b \log \left (c\right )\right )^{2} + \sin \left (b \log \left (c\right )\right )^{2}\right )} m^{2} + {\left (b^{2} \cos \left (b \log \left (c\right )\right )^{2} + b^{2} \sin \left (b \log \left (c\right )\right )^{2}\right )} n^{2} + 2 \, {\left (\cos \left (b \log \left (c\right )\right )^{2} + \sin \left (b \log \left (c\right )\right )^{2}\right )} m + \cos \left (b \log \left (c\right )\right )^{2} + \sin \left (b \log \left (c\right )\right )^{2}\right )}} \]

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

1/2*(((cos(2*b*log(c))*cos(b*log(c)) + sin(2*b*log(c))*sin(b*log(c)) + cos(b*log(c)))*m + (b*cos(b*log(c))*sin

(2*b*log(c)) - b*cos(2*b*log(c))*sin(b*log(c)) + b*sin(b*log(c)))*n + cos(2*b*log(c))*cos(b*log(c)) + sin(2*b*

log(c))*sin(b*log(c)) + cos(b*log(c)))*x*x^m*cos(b*log(x^n) + a) - ((cos(b*log(c))*sin(2*b*log(c)) - cos(2*b*l

og(c))*sin(b*log(c)) + sin(b*log(c)))*m - (b*cos(2*b*log(c))*cos(b*log(c)) + b*sin(2*b*log(c))*sin(b*log(c)) +

b*cos(b*log(c)))*n + cos(b*log(c))*sin(2*b*log(c)) - cos(2*b*log(c))*sin(b*log(c)) + sin(b*log(c)))*x*x^m*sin

(b*log(x^n) + a))/((cos(b*log(c))^2 + sin(b*log(c))^2)*m^2 + (b^2*cos(b*log(c))^2 + b^2*sin(b*log(c))^2)*n^2 +

2*(cos(b*log(c))^2 + sin(b*log(c))^2)*m + cos(b*log(c))^2 + sin(b*log(c))^2)

Giac [B] (veriﬁcation not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 5162 vs. \(2 (70) = 140\).

Time = 0.48 (sec) , antiderivative size = 5162, normalized size of antiderivative = 73.74 \[ \int x^m \cos \left (a+b \log \left (c x^n\right )\right ) \, dx=\text {Too large to display} \]

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

1/2*(2*b*n*x*abs(x)^m*e^(1/2*pi*b*n*sgn(x) - 1/2*pi*b*n + 1/2*pi*b*sgn(c) - 1/2*pi*b)*tan(1/2*b*n*log(abs(x))

+ 1/2*b*log(abs(c)))^2*tan(1/4*pi*m*sgn(x) - 1/4*pi*m)^2*tan(1/2*a) + 2*b*n*x*abs(x)^m*e^(-1/2*pi*b*n*sgn(x) +

1/2*pi*b*n - 1/2*pi*b*sgn(c) + 1/2*pi*b)*tan(1/2*b*n*log(abs(x)) + 1/2*b*log(abs(c)))^2*tan(1/4*pi*m*sgn(x) -

1/4*pi*m)^2*tan(1/2*a) - 2*b*n*x*abs(x)^m*e^(1/2*pi*b*n*sgn(x) - 1/2*pi*b*n + 1/2*pi*b*sgn(c) - 1/2*pi*b)*tan

(1/2*b*n*log(abs(x)) + 1/2*b*log(abs(c)))^2*tan(1/4*pi*m*sgn(x) - 1/4*pi*m)*tan(1/2*a)^2 + 2*b*n*x*abs(x)^m*e^

(-1/2*pi*b*n*sgn(x) + 1/2*pi*b*n - 1/2*pi*b*sgn(c) + 1/2*pi*b)*tan(1/2*b*n*log(abs(x)) + 1/2*b*log(abs(c)))^2*

tan(1/4*pi*m*sgn(x) - 1/4*pi*m)*tan(1/2*a)^2 + 2*b*n*x*abs(x)^m*e^(1/2*pi*b*n*sgn(x) - 1/2*pi*b*n + 1/2*pi*b*s

gn(c) - 1/2*pi*b)*tan(1/2*b*n*log(abs(x)) + 1/2*b*log(abs(c)))*tan(1/4*pi*m*sgn(x) - 1/4*pi*m)^2*tan(1/2*a)^2

+ 2*b*n*x*abs(x)^m*e^(-1/2*pi*b*n*sgn(x) + 1/2*pi*b*n - 1/2*pi*b*sgn(c) + 1/2*pi*b)*tan(1/2*b*n*log(abs(x)) +

1/2*b*log(abs(c)))*tan(1/4*pi*m*sgn(x) - 1/4*pi*m)^2*tan(1/2*a)^2 - m*x*abs(x)^m*e^(1/2*pi*b*n*sgn(x) - 1/2*pi

*b*n + 1/2*pi*b*sgn(c) - 1/2*pi*b)*tan(1/2*b*n*log(abs(x)) + 1/2*b*log(abs(c)))^2*tan(1/4*pi*m*sgn(x) - 1/4*pi

*m)^2*tan(1/2*a)^2 - m*x*abs(x)^m*e^(-1/2*pi*b*n*sgn(x) + 1/2*pi*b*n - 1/2*pi*b*sgn(c) + 1/2*pi*b)*tan(1/2*b*n

*log(abs(x)) + 1/2*b*log(abs(c)))^2*tan(1/4*pi*m*sgn(x) - 1/4*pi*m)^2*tan(1/2*a)^2 - x*abs(x)^m*e^(1/2*pi*b*n*

sgn(x) - 1/2*pi*b*n + 1/2*pi*b*sgn(c) - 1/2*pi*b)*tan(1/2*b*n*log(abs(x)) + 1/2*b*log(abs(c)))^2*tan(1/4*pi*m*

sgn(x) - 1/4*pi*m)^2*tan(1/2*a)^2 - x*abs(x)^m*e^(-1/2*pi*b*n*sgn(x) + 1/2*pi*b*n - 1/2*pi*b*sgn(c) + 1/2*pi*b

)*tan(1/2*b*n*log(abs(x)) + 1/2*b*log(abs(c)))^2*tan(1/4*pi*m*sgn(x) - 1/4*pi*m)^2*tan(1/2*a)^2 + 2*b*n*x*abs(

x)^m*e^(1/2*pi*b*n*sgn(x) - 1/2*pi*b*n + 1/2*pi*b*sgn(c) - 1/2*pi*b)*tan(1/2*b*n*log(abs(x)) + 1/2*b*log(abs(c

)))^2*tan(1/4*pi*m*sgn(x) - 1/4*pi*m) - 2*b*n*x*abs(x)^m*e^(-1/2*pi*b*n*sgn(x) + 1/2*pi*b*n - 1/2*pi*b*sgn(c)

+ 1/2*pi*b)*tan(1/2*b*n*log(abs(x)) + 1/2*b*log(abs(c)))^2*tan(1/4*pi*m*sgn(x) - 1/4*pi*m) - 2*b*n*x*abs(x)^m*

e^(1/2*pi*b*n*sgn(x) - 1/2*pi*b*n + 1/2*pi*b*sgn(c) - 1/2*pi*b)*tan(1/2*b*n*log(abs(x)) + 1/2*b*log(abs(c)))*t

an(1/4*pi*m*sgn(x) - 1/4*pi*m)^2 - 2*b*n*x*abs(x)^m*e^(-1/2*pi*b*n*sgn(x) + 1/2*pi*b*n - 1/2*pi*b*sgn(c) + 1/2

*pi*b)*tan(1/2*b*n*log(abs(x)) + 1/2*b*log(abs(c)))*tan(1/4*pi*m*sgn(x) - 1/4*pi*m)^2 + m*x*abs(x)^m*e^(1/2*pi

*b*n*sgn(x) - 1/2*pi*b*n + 1/2*pi*b*sgn(c) - 1/2*pi*b)*tan(1/2*b*n*log(abs(x)) + 1/2*b*log(abs(c)))^2*tan(1/4*

pi*m*sgn(x) - 1/4*pi*m)^2 + m*x*abs(x)^m*e^(-1/2*pi*b*n*sgn(x) + 1/2*pi*b*n - 1/2*pi*b*sgn(c) + 1/2*pi*b)*tan(

1/2*b*n*log(abs(x)) + 1/2*b*log(abs(c)))^2*tan(1/4*pi*m*sgn(x) - 1/4*pi*m)^2 - 2*b*n*x*abs(x)^m*e^(1/2*pi*b*n*

sgn(x) - 1/2*pi*b*n + 1/2*pi*b*sgn(c) - 1/2*pi*b)*tan(1/2*b*n*log(abs(x)) + 1/2*b*log(abs(c)))^2*tan(1/2*a) -

2*b*n*x*abs(x)^m*e^(-1/2*pi*b*n*sgn(x) + 1/2*pi*b*n - 1/2*pi*b*sgn(c) + 1/2*pi*b)*tan(1/2*b*n*log(abs(x)) + 1/

2*b*log(abs(c)))^2*tan(1/2*a) + 8*b*n*x*abs(x)^m*e^(1/2*pi*b*n*sgn(x) - 1/2*pi*b*n + 1/2*pi*b*sgn(c) - 1/2*pi*

b)*tan(1/2*b*n*log(abs(x)) + 1/2*b*log(abs(c)))*tan(1/4*pi*m*sgn(x) - 1/4*pi*m)*tan(1/2*a) - 8*b*n*x*abs(x)^m*

e^(-1/2*pi*b*n*sgn(x) + 1/2*pi*b*n - 1/2*pi*b*sgn(c) + 1/2*pi*b)*tan(1/2*b*n*log(abs(x)) + 1/2*b*log(abs(c)))*

tan(1/4*pi*m*sgn(x) - 1/4*pi*m)*tan(1/2*a) - 4*m*x*abs(x)^m*e^(1/2*pi*b*n*sgn(x) - 1/2*pi*b*n + 1/2*pi*b*sgn(c

) - 1/2*pi*b)*tan(1/2*b*n*log(abs(x)) + 1/2*b*log(abs(c)))^2*tan(1/4*pi*m*sgn(x) - 1/4*pi*m)*tan(1/2*a) + 4*m*

x*abs(x)^m*e^(-1/2*pi*b*n*sgn(x) + 1/2*pi*b*n - 1/2*pi*b*sgn(c) + 1/2*pi*b)*tan(1/2*b*n*log(abs(x)) + 1/2*b*lo

g(abs(c)))^2*tan(1/4*pi*m*sgn(x) - 1/4*pi*m)*tan(1/2*a) - 2*b*n*x*abs(x)^m*e^(1/2*pi*b*n*sgn(x) - 1/2*pi*b*n +

1/2*pi*b*sgn(c) - 1/2*pi*b)*tan(1/4*pi*m*sgn(x) - 1/4*pi*m)^2*tan(1/2*a) - 2*b*n*x*abs(x)^m*e^(-1/2*pi*b*n*sg

n(x) + 1/2*pi*b*n - 1/2*pi*b*sgn(c) + 1/2*pi*b)*tan(1/4*pi*m*sgn(x) - 1/4*pi*m)^2*tan(1/2*a) + 4*m*x*abs(x)^m*

e^(1/2*pi*b*n*sgn(x) - 1/2*pi*b*n + 1/2*pi*b*sgn(c) - 1/2*pi*b)*tan(1/2*b*n*log(abs(x)) + 1/2*b*log(abs(c)))*t

an(1/4*pi*m*sgn(x) - 1/4*pi*m)^2*tan(1/2*a) + 4*m*x*abs(x)^m*e^(-1/2*pi*b*n*sgn(x) + 1/2*pi*b*n - 1/2*pi*b*sgn

b*n*x*abs(x)^m*e^(1/2*pi*b*n*sgn(x) - 1/2*pi*b*n + 1/2*pi*b*sgn(c) - 1/2*pi*b)*tan(1/2*b*n*log(abs(x)) + 1/2*b

*log(abs(c)))*tan(1/2*a)^2 - 2*b*n*x*abs(x)^m*e^(-1/2*pi*b*n*sgn(x) + 1/2*pi*b*n - 1/2*pi*b*sgn(c) + 1/2*pi*b)

*tan(1/2*b*n*log(abs(x)) + 1/2*b*log(abs(c)))*tan(1/2*a)^2 + m*x*abs(x)^m*e^(1/2*pi*b*n*sgn(x) - 1/2*pi*b*n +

1/2*pi*b*sgn(c) - 1/2*pi*b)*tan(1/2*b*n*log(abs(x)) + 1/2*b*log(abs(c)))^2*tan(1/2*a)^2 + m*x*abs(x)^m*e^(-1/2

*pi*b*n*sgn(x) + 1/2*pi*b*n - 1/2*pi*b*sgn(c) + 1/2*pi*b)*tan(1/2*b*n*log(abs(x)) + 1/2*b*log(abs(c)))^2*tan(1

/2*a)^2 + 2*b*n*x*abs(x)^m*e^(1/2*pi*b*n*sgn(x) - 1/2*pi*b*n + 1/2*pi*b*sgn(c) - 1/2*pi*b)*tan(1/4*pi*m*sgn(x)

- 1/4*pi*m)*tan(1/2*a)^2 - 2*b*n*x*abs(x)^m*e^(-1/2*pi*b*n*sgn(x) + 1/2*pi*b*n - 1/2*pi*b*sgn(c) + 1/2*pi*b)*

tan(1/4*pi*m*sgn(x) - 1/4*pi*m)*tan(1/2*a)^2 - 4*m*x*abs(x)^m*e^(1/2*pi*b*n*sgn(x) - 1/2*pi*b*n + 1/2*pi*b*sgn

m*x*abs(x)^m*e^(-1/2*pi*b*n*sgn(x) + 1/2*pi*b*n - 1/2*pi*b*sgn(c) + 1/2*pi*b)*tan(1/2*b*n*log(abs(x)) + 1/2*b*

log(abs(c)))*tan(1/4*pi*m*sgn(x) - 1/4*pi*m)*tan(1/2*a)^2 + m*x*abs(x)^m*e^(1/2*pi*b*n*sgn(x) - 1/2*pi*b*n + 1

/2*pi*b*sgn(c) - 1/2*pi*b)*tan(1/4*pi*m*sgn(x) - 1/4*pi*m)^2*tan(1/2*a)^2 + m*x*abs(x)^m*e^(-1/2*pi*b*n*sgn(x)

+ 1/2*pi*b*n - 1/2*pi*b*sgn(c) + 1/2*pi*b)*tan(1/4*pi*m*sgn(x) - 1/4*pi*m)^2*tan(1/2*a)^2 + x*abs(x)^m*e^(1/2

*pi*b*n*sgn(x) - 1/2*pi*b*n + 1/2*pi*b*sgn(c) - 1/2*pi*b)*tan(1/2*b*n*log(abs(x)) + 1/2*b*log(abs(c)))^2*tan(1

/4*pi*m*sgn(x) - 1/4*pi*m)^2 + x*abs(x)^m*e^(-1/2*pi*b*n*sgn(x) + 1/2*pi*b*n - 1/2*pi*b*sgn(c) + 1/2*pi*b)*tan

(1/2*b*n*log(abs(x)) + 1/2*b*log(abs(c)))^2*tan(1/4*pi*m*sgn(x) - 1/4*pi*m)^2 - 4*x*abs(x)^m*e^(1/2*pi*b*n*sgn

(x) - 1/2*pi*b*n + 1/2*pi*b*sgn(c) - 1/2*pi*b)*tan(1/2*b*n*log(abs(x)) + 1/2*b*log(abs(c)))^2*tan(1/4*pi*m*sgn

(x) - 1/4*pi*m)*tan(1/2*a) + 4*x*abs(x)^m*e^(-1/2*pi*b*n*sgn(x) + 1/2*pi*b*n - 1/2*pi*b*sgn(c) + 1/2*pi*b)*tan

(1/2*b*n*log(abs(x)) + 1/2*b*log(abs(c)))^2*tan(1/4*pi*m*sgn(x) - 1/4*pi*m)*tan(1/2*a) + 4*x*abs(x)^m*e^(1/2*p

i*b*n*sgn(x) - 1/2*pi*b*n + 1/2*pi*b*sgn(c) - 1/2*pi*b)*tan(1/2*b*n*log(abs(x)) + 1/2*b*log(abs(c)))*tan(1/4*p

i*m*sgn(x) - 1/4*pi*m)^2*tan(1/2*a) + 4*x*abs(x)^m*e^(-1/2*pi*b*n*sgn(x) + 1/2*pi*b*n - 1/2*pi*b*sgn(c) + 1/2*

pi*b)*tan(1/2*b*n*log(abs(x)) + 1/2*b*log(abs(c)))*tan(1/4*pi*m*sgn(x) - 1/4*pi*m)^2*tan(1/2*a) + x*abs(x)^m*e

^(1/2*pi*b*n*sgn(x) - 1/2*pi*b*n + 1/2*pi*b*sgn(c) - 1/2*pi*b)*tan(1/2*b*n*log(abs(x)) + 1/2*b*log(abs(c)))^2*

tan(1/2*a)^2 + x*abs(x)^m*e^(-1/2*pi*b*n*sgn(x) + 1/2*pi*b*n - 1/2*pi*b*sgn(c) + 1/2*pi*b)*tan(1/2*b*n*log(abs

(x)) + 1/2*b*log(abs(c)))^2*tan(1/2*a)^2 - 4*x*abs(x)^m*e^(1/2*pi*b*n*sgn(x) - 1/2*pi*b*n + 1/2*pi*b*sgn(c) -

1/2*pi*b)*tan(1/2*b*n*log(abs(x)) + 1/2*b*log(abs(c)))*tan(1/4*pi*m*sgn(x) - 1/4*pi*m)*tan(1/2*a)^2 + 4*x*abs(

x)^m*e^(-1/2*pi*b*n*sgn(x) + 1/2*pi*b*n - 1/2*pi*b*sgn(c) + 1/2*pi*b)*tan(1/2*b*n*log(abs(x)) + 1/2*b*log(abs(

c)))*tan(1/4*pi*m*sgn(x) - 1/4*pi*m)*tan(1/2*a)^2 + x*abs(x)^m*e^(1/2*pi*b*n*sgn(x) - 1/2*pi*b*n + 1/2*pi*b*sg

n(c) - 1/2*pi*b)*tan(1/4*pi*m*sgn(x) - 1/4*pi*m)^2*tan(1/2*a)^2 + x*abs(x)^m*e^(-1/2*pi*b*n*sgn(x) + 1/2*pi*b*

n - 1/2*pi*b*sgn(c) + 1/2*pi*b)*tan(1/4*pi*m*sgn(x) - 1/4*pi*m)^2*tan(1/2*a)^2 + 2*b*n*x*abs(x)^m*e^(1/2*pi*b*

n*sgn(x) - 1/2*pi*b*n + 1/2*pi*b*sgn(c) - 1/2*pi*b)*tan(1/2*b*n*log(abs(x)) + 1/2*b*log(abs(c))) + 2*b*n*x*abs

(x)^m*e^(-1/2*pi*b*n*sgn(x) + 1/2*pi*b*n - 1/2*pi*b*sgn(c) + 1/2*pi*b)*tan(1/2*b*n*log(abs(x)) + 1/2*b*log(abs

(c))) - m*x*abs(x)^m*e^(1/2*pi*b*n*sgn(x) - 1/2*pi*b*n + 1/2*pi*b*sgn(c) - 1/2*pi*b)*tan(1/2*b*n*log(abs(x)) +

1/2*b*log(abs(c)))^2 - m*x*abs(x)^m*e^(-1/2*pi*b*n*sgn(x) + 1/2*pi*b*n - 1/2*pi*b*sgn(c) + 1/2*pi*b)*tan(1/2*

b*n*log(abs(x)) + 1/2*b*log(abs(c)))^2 - 2*b*n*x*abs(x)^m*e^(1/2*pi*b*n*sgn(x) - 1/2*pi*b*n + 1/2*pi*b*sgn(c)

- 1/2*pi*b)*tan(1/4*pi*m*sgn(x) - 1/4*pi*m) + 2*b*n*x*abs(x)^m*e^(-1/2*pi*b*n*sgn(x) + 1/2*pi*b*n - 1/2*pi*b*s

gn(c) + 1/2*pi*b)*tan(1/4*pi*m*sgn(x) - 1/4*pi*m) + 4*m*x*abs(x)^m*e^(1/2*pi*b*n*sgn(x) - 1/2*pi*b*n + 1/2*pi*

b*sgn(c) - 1/2*pi*b)*tan(1/2*b*n*log(abs(x)) + 1/2*b*log(abs(c)))*tan(1/4*pi*m*sgn(x) - 1/4*pi*m) - 4*m*x*abs(

x)^m*e^(-1/2*pi*b*n*sgn(x) + 1/2*pi*b*n - 1/2*pi*b*sgn(c) + 1/2*pi*b)*tan(1/2*b*n*log(abs(x)) + 1/2*b*log(abs(

c)))*tan(1/4*pi*m*sgn(x) - 1/4*pi*m) - m*x*abs(x)^m*e^(1/2*pi*b*n*sgn(x) - 1/2*pi*b*n + 1/2*pi*b*sgn(c) - 1/2*

pi*b)*tan(1/4*pi*m*sgn(x) - 1/4*pi*m)^2 - m*x*abs(x)^m*e^(-1/2*pi*b*n*sgn(x) + 1/2*pi*b*n - 1/2*pi*b*sgn(c) +

1/2*pi*b)*tan(1/4*pi*m*sgn(x) - 1/4*pi*m)^2 + 2*b*n*x*abs(x)^m*e^(1/2*pi*b*n*sgn(x) - 1/2*pi*b*n + 1/2*pi*b*sg

n(c) - 1/2*pi*b)*tan(1/2*a) + 2*b*n*x*abs(x)^m*e^(-1/2*pi*b*n*sgn(x) + 1/2*pi*b*n - 1/2*pi*b*sgn(c) + 1/2*pi*b

)*tan(1/2*a) - 4*m*x*abs(x)^m*e^(1/2*pi*b*n*sgn(x) - 1/2*pi*b*n + 1/2*pi*b*sgn(c) - 1/2*pi*b)*tan(1/2*b*n*log(

abs(x)) + 1/2*b*log(abs(c)))*tan(1/2*a) - 4*m*x*abs(x)^m*e^(-1/2*pi*b*n*sgn(x) + 1/2*pi*b*n - 1/2*pi*b*sgn(c)

+ 1/2*pi*b)*tan(1/2*b*n*log(abs(x)) + 1/2*b*log(abs(c)))*tan(1/2*a) + 4*m*x*abs(x)^m*e^(1/2*pi*b*n*sgn(x) - 1/

2*pi*b*n + 1/2*pi*b*sgn(c) - 1/2*pi*b)*tan(1/4*pi*m*sgn(x) - 1/4*pi*m)*tan(1/2*a) - 4*m*x*abs(x)^m*e^(-1/2*pi*

b*n*sgn(x) + 1/2*pi*b*n - 1/2*pi*b*sgn(c) + 1/2*pi*b)*tan(1/4*pi*m*sgn(x) - 1/4*pi*m)*tan(1/2*a) - m*x*abs(x)^

m*e^(1/2*pi*b*n*sgn(x) - 1/2*pi*b*n + 1/2*pi*b*sgn(c) - 1/2*pi*b)*tan(1/2*a)^2 - m*x*abs(x)^m*e^(-1/2*pi*b*n*s

gn(x) + 1/2*pi*b*n - 1/2*pi*b*sgn(c) + 1/2*pi*b)*tan(1/2*a)^2 - x*abs(x)^m*e^(1/2*pi*b*n*sgn(x) - 1/2*pi*b*n +

1/2*pi*b*sgn(c) - 1/2*pi*b)*tan(1/2*b*n*log(abs(x)) + 1/2*b*log(abs(c)))^2 - x*abs(x)^m*e^(-1/2*pi*b*n*sgn(x)

+ 1/2*pi*b*n - 1/2*pi*b*sgn(c) + 1/2*pi*b)*tan(1/2*b*n*log(abs(x)) + 1/2*b*log(abs(c)))^2 + 4*x*abs(x)^m*e^(1

/2*pi*b*n*sgn(x) - 1/2*pi*b*n + 1/2*pi*b*sgn(c) - 1/2*pi*b)*tan(1/2*b*n*log(abs(x)) + 1/2*b*log(abs(c)))*tan(1

/4*pi*m*sgn(x) - 1/4*pi*m) - 4*x*abs(x)^m*e^(-1/2*pi*b*n*sgn(x) + 1/2*pi*b*n - 1/2*pi*b*sgn(c) + 1/2*pi*b)*tan

(1/2*b*n*log(abs(x)) + 1/2*b*log(abs(c)))*tan(1/4*pi*m*sgn(x) - 1/4*pi*m) - x*abs(x)^m*e^(1/2*pi*b*n*sgn(x) -

1/2*pi*b*n + 1/2*pi*b*sgn(c) - 1/2*pi*b)*tan(1/4*pi*m*sgn(x) - 1/4*pi*m)^2 - x*abs(x)^m*e^(-1/2*pi*b*n*sgn(x)

+ 1/2*pi*b*n - 1/2*pi*b*sgn(c) + 1/2*pi*b)*tan(1/4*pi*m*sgn(x) - 1/4*pi*m)^2 - 4*x*abs(x)^m*e^(1/2*pi*b*n*sgn(

x) - 1/2*pi*b*n + 1/2*pi*b*sgn(c) - 1/2*pi*b)*tan(1/2*b*n*log(abs(x)) + 1/2*b*log(abs(c)))*tan(1/2*a) - 4*x*ab

s(x)^m*e^(-1/2*pi*b*n*sgn(x) + 1/2*pi*b*n - 1/2*pi*b*sgn(c) + 1/2*pi*b)*tan(1/2*b*n*log(abs(x)) + 1/2*b*log(ab

s(c)))*tan(1/2*a) + 4*x*abs(x)^m*e^(1/2*pi*b*n*sgn(x) - 1/2*pi*b*n + 1/2*pi*b*sgn(c) - 1/2*pi*b)*tan(1/4*pi*m*

sgn(x) - 1/4*pi*m)*tan(1/2*a) - 4*x*abs(x)^m*e^(-1/2*pi*b*n*sgn(x) + 1/2*pi*b*n - 1/2*pi*b*sgn(c) + 1/2*pi*b)*

tan(1/4*pi*m*sgn(x) - 1/4*pi*m)*tan(1/2*a) - x*abs(x)^m*e^(1/2*pi*b*n*sgn(x) - 1/2*pi*b*n + 1/2*pi*b*sgn(c) -

1/2*pi*b)*tan(1/2*a)^2 - x*abs(x)^m*e^(-1/2*pi*b*n*sgn(x) + 1/2*pi*b*n - 1/2*pi*b*sgn(c) + 1/2*pi*b)*tan(1/2*a

)^2 + m*x*abs(x)^m*e^(1/2*pi*b*n*sgn(x) - 1/2*pi*b*n + 1/2*pi*b*sgn(c) - 1/2*pi*b) + m*x*abs(x)^m*e^(-1/2*pi*b

*n*sgn(x) + 1/2*pi*b*n - 1/2*pi*b*sgn(c) + 1/2*pi*b) + x*abs(x)^m*e^(1/2*pi*b*n*sgn(x) - 1/2*pi*b*n + 1/2*pi*b

*sgn(c) - 1/2*pi*b) + x*abs(x)^m*e^(-1/2*pi*b*n*sgn(x) + 1/2*pi*b*n - 1/2*pi*b*sgn(c) + 1/2*pi*b))/(b^2*n^2*ta

n(1/2*b*n*log(abs(x)) + 1/2*b*log(abs(c)))^2*tan(1/4*pi*m*sgn(x) - 1/4*pi*m)^2*tan(1/2*a)^2 + b^2*n^2*tan(1/2*

b*n*log(abs(x)) + 1/2*b*log(abs(c)))^2*tan(1/4*pi*m*sgn(x) - 1/4*pi*m)^2 + b^2*n^2*tan(1/2*b*n*log(abs(x)) + 1

/2*b*log(abs(c)))^2*tan(1/2*a)^2 + b^2*n^2*tan(1/4*pi*m*sgn(x) - 1/4*pi*m)^2*tan(1/2*a)^2 + m^2*tan(1/2*b*n*lo

g(abs(x)) + 1/2*b*log(abs(c)))^2*tan(1/4*pi*m*sgn(x) - 1/4*pi*m)^2*tan(1/2*a)^2 + 2*m*tan(1/2*b*n*log(abs(x))

+ 1/2*b*log(abs(c)))^2*tan(1/4*pi*m*sgn(x) - 1/4*pi*m)^2*tan(1/2*a)^2 + b^2*n^2*tan(1/2*b*n*log(abs(x)) + 1/2*

b*log(abs(c)))^2 + b^2*n^2*tan(1/4*pi*m*sgn(x) - 1/4*pi*m)^2 + m^2*tan(1/2*b*n*log(abs(x)) + 1/2*b*log(abs(c))

)^2*tan(1/4*pi*m*sgn(x) - 1/4*pi*m)^2 + b^2*n^2*tan(1/2*a)^2 + m^2*tan(1/2*b*n*log(abs(x)) + 1/2*b*log(abs(c))

)^2*tan(1/2*a)^2 + m^2*tan(1/4*pi*m*sgn(x) - 1/4*pi*m)^2*tan(1/2*a)^2 + tan(1/2*b*n*log(abs(x)) + 1/2*b*log(ab

s(c)))^2*tan(1/4*pi*m*sgn(x) - 1/4*pi*m)^2*tan(1/2*a)^2 + 2*m*tan(1/2*b*n*log(abs(x)) + 1/2*b*log(abs(c)))^2*t

an(1/4*pi*m*sgn(x) - 1/4*pi*m)^2 + 2*m*tan(1/2*b*n*log(abs(x)) + 1/2*b*log(abs(c)))^2*tan(1/2*a)^2 + 2*m*tan(1

/4*pi*m*sgn(x) - 1/4*pi*m)^2*tan(1/2*a)^2 + b^2*n^2 + m^2*tan(1/2*b*n*log(abs(x)) + 1/2*b*log(abs(c)))^2 + m^2

*tan(1/4*pi*m*sgn(x) - 1/4*pi*m)^2 + tan(1/2*b*n*log(abs(x)) + 1/2*b*log(abs(c)))^2*tan(1/4*pi*m*sgn(x) - 1/4*

pi*m)^2 + m^2*tan(1/2*a)^2 + tan(1/2*b*n*log(abs(x)) + 1/2*b*log(abs(c)))^2*tan(1/2*a)^2 + tan(1/4*pi*m*sgn(x)

- 1/4*pi*m)^2*tan(1/2*a)^2 + 2*m*tan(1/2*b*n*log(abs(x)) + 1/2*b*log(abs(c)))^2 + 2*m*tan(1/4*pi*m*sgn(x) - 1

/4*pi*m)^2 + 2*m*tan(1/2*a)^2 + m^2 + tan(1/2*b*n*log(abs(x)) + 1/2*b*log(abs(c)))^2 + tan(1/4*pi*m*sgn(x) - 1

Mupad [B] (veriﬁcation not implemented)

Time = 27.08 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.00 \[ \int x^m \cos \left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {x\,x^m\,{\mathrm {e}}^{a\,1{}\mathrm {i}}\,{\left (c\,x^n\right )}^{b\,1{}\mathrm {i}}}{2\,m+2+b\,n\,2{}\mathrm {i}}+\frac {x\,x^m\,{\mathrm {e}}^{-a\,1{}\mathrm {i}}\,\frac {1}{{\left (c\,x^n\right )}^{b\,1{}\mathrm {i}}}\,1{}\mathrm {i}}{m\,2{}\mathrm {i}+2\,b\,n+2{}\mathrm {i}} \]

(x*x^m*exp(a*1i)*(c*x^n)^(b*1i))/(2*m + b*n*2i + 2) + (x*x^m*exp(-a*1i)/(c*x^n)^(b*1i)*1i)/(m*2i + 2*b*n + 2i)

\(\int x^m \cos (a+b \log (c x^n)) \, dx\) [126]

Optimal result