\(\int \cos (a+b \log (c x^n)) \, dx\) [88]

Optimal result

Integrand size = 11, antiderivative size = 51 \[ \int \cos \left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {x \cos \left (a+b \log \left (c x^n\right )\right )}{1+b^2 n^2}+\frac {b n x \sin \left (a+b \log \left (c x^n\right )\right )}{1+b^2 n^2} \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {4564} \[ \int \cos \left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {b n x \sin \left (a+b \log \left (c x^n\right )\right )}{b^2 n^2+1}+\frac {x \cos \left (a+b \log \left (c x^n\right )\right )}{b^2 n^2+1} \]

[In]

Int[Cos[a + b*Log[c*x^n]],x]

[Out]

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

Rule 4564

Int[Cos[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*(d_.)], x_Symbol] :> Simp[x*(Cos[d*(a + b*Log[c*x^n])]/(b^2*d^2*
n^2 + 1)), x] + Simp[b*d*n*x*(Sin[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*} \text {integral}& = \frac {x \cos \left (a+b \log \left (c x^n\right )\right )}{1+b^2 n^2}+\frac {b n x \sin \left (a+b \log \left (c x^n\right )\right )}{1+b^2 n^2} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.76 \[ \int \cos \left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {x \left (\cos \left (a+b \log \left (c x^n\right )\right )+b n \sin \left (a+b \log \left (c x^n\right )\right )\right )}{1+b^2 n^2} \]

[In]

Integrate[Cos[a + b*Log[c*x^n]],x]

[Out]

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

Maple [A] (verified)

Time = 1.10 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.78

method	result	size
parallelrisch	\(\frac {x \left (\sin \left (a +b \ln \left (c \,x^{n}\right )\right ) b n +\cos \left (a +b \ln \left (c \,x^{n}\right )\right )\right )}{b^{2} n^{2}+1}\)	\(40\)
default	\(\frac {\frac {{\mathrm e}^{\frac {\ln \left (c \,x^{n}\right )}{n}-\frac {\ln \left (c \right )}{n}} \cos \left (a +b \ln \left (c \,x^{n}\right )\right )}{n \left (\frac {1}{n^{2}}+b^{2}\right )}+\frac {b \,{\mathrm e}^{\frac {\ln \left (c \,x^{n}\right )}{n}-\frac {\ln \left (c \right )}{n}} \sin \left (a +b \ln \left (c \,x^{n}\right )\right )}{\frac {1}{n^{2}}+b^{2}}}{n}\)	\(90\)

[In]

int(cos(a+b*ln(c*x^n)),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.84 \[ \int \cos \left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {b n x \sin \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) + x \cos \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )}{b^{2} n^{2} + 1} \]

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

Piecewise((Integral(cos(a - I*log(c*x**n)/n), x), Eq(b, -I/n)), (Integral(cos(a + I*log(c*x**n)/n), x), Eq(b,
I/n)), (b*n*x*sin(a + b*log(c*x**n))/(b**2*n**2 + 1) + x*cos(a + b*log(c*x**n))/(b**2*n**2 + 1), True))

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 205 vs. \(2 (51) = 102\).

Time = 0.23 (sec) , antiderivative size = 205, normalized size of antiderivative = 4.02 \[ \int \cos \left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {{\left ({\left (b \cos \left (b \log \left (c\right )\right ) \sin \left (2 \, b \log \left (c\right )\right ) - b \cos \left (2 \, b \log \left (c\right )\right ) \sin \left (b \log \left (c\right )\right ) + b \sin \left (b \log \left (c\right )\right )\right )} n + \cos \left (2 \, b \log \left (c\right )\right ) \cos \left (b \log \left (c\right )\right ) + \sin \left (2 \, b \log \left (c\right )\right ) \sin \left (b \log \left (c\right )\right ) + \cos \left (b \log \left (c\right )\right )\right )} x \cos \left (b \log \left (x^{n}\right ) + a\right ) + {\left ({\left (b \cos \left (2 \, b \log \left (c\right )\right ) \cos \left (b \log \left (c\right )\right ) + b \sin \left (2 \, b \log \left (c\right )\right ) \sin \left (b \log \left (c\right )\right ) + b \cos \left (b \log \left (c\right )\right )\right )} n - \cos \left (b \log \left (c\right )\right ) \sin \left (2 \, b \log \left (c\right )\right ) + \cos \left (2 \, b \log \left (c\right )\right ) \sin \left (b \log \left (c\right )\right ) - \sin \left (b \log \left (c\right )\right )\right )} x \sin \left (b \log \left (x^{n}\right ) + a\right )}{2 \, {\left ({\left (b^{2} \cos \left (b \log \left (c\right )\right )^{2} + b^{2} \sin \left (b \log \left (c\right )\right )^{2}\right )} n^{2} + \cos \left (b \log \left (c\right )\right )^{2} + \sin \left (b \log \left (c\right )\right )^{2}\right )}} \]

[In]

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

[Out]

1/2*(((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*cos(b*log(x^n) + a) + ((b*cos(2*b*log(c))*c
os(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*sin(b*log(x^n) + a))/((b^2*cos(b*log(c))^2 + b^2*sin(b*log(c))^2)*n^2
+ 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. 878 vs. \(2 (51) = 102\).

Time = 0.30 (sec) , antiderivative size = 878, normalized size of antiderivative = 17.22 \[ \int \cos \left (a+b \log \left (c x^n\right )\right ) \, dx=\text {Too large to display} \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 26.94 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.76 \[ \int \cos \left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {x\,\left (\cos \left (a+b\,\ln \left (c\,x^n\right )\right )+b\,n\,\sin \left (a+b\,\ln \left (c\,x^n\right )\right )\right )}{b^2\,n^2+1} \]

[In]

int(cos(a + b*log(c*x^n)),x)

[Out]

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