\(\int \cos ^3(a+\frac {1}{3} \sqrt {-\frac {1}{n^2}} \log (c x^n)) \, dx\) [109]

Optimal result

Integrand size = 24, antiderivative size = 128 \[ \int \cos ^3\left (a+\frac {1}{3} \sqrt {-\frac {1}{n^2}} \log \left (c x^n\right )\right ) \, dx=\frac {9}{16} e^{a \sqrt {-\frac {1}{n^2}} n} x \left (c x^n\right )^{\left .-\frac {1}{3}\right /n}+\frac {9}{32} e^{-a \sqrt {-\frac {1}{n^2}} n} x \left (c x^n\right )^{\left .\frac {1}{3}\right /n}+\frac {1}{16} e^{-3 a \sqrt {-\frac {1}{n^2}} n} x \left (c x^n\right )^{\frac {1}{n}}+\frac {1}{8} e^{3 a \sqrt {-\frac {1}{n^2}} n} x \left (c x^n\right )^{-1/n} \log (x) \] Output:

9/16*exp(a*(-1/n^2)^(1/2)*n)*x/((c*x^n)^(1/3/n))+9/32*x*(c*x^n)^(1/3/n)/ex 
p(a*(-1/n^2)^(1/2)*n)+1/16*x*(c*x^n)^(1/n)/exp(3*a*(-1/n^2)^(1/2)*n)+1/8*e 
xp(3*a*(-1/n^2)^(1/2)*n)*x*ln(x)/((c*x^n)^(1/n))
 

Mathematica [F]

\[ \int \cos ^3\left (a+\frac {1}{3} \sqrt {-\frac {1}{n^2}} \log \left (c x^n\right )\right ) \, dx=\int \cos ^3\left (a+\frac {1}{3} \sqrt {-\frac {1}{n^2}} \log \left (c x^n\right )\right ) \, dx \] Input:

Integrate[Cos[a + (Sqrt[-n^(-2)]*Log[c*x^n])/3]^3,x]
 

Output:

Integrate[Cos[a + (Sqrt[-n^(-2)]*Log[c*x^n])/3]^3, x]
 

Rubi [A] (warning: unable to verify)

Time = 0.35 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.08, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {4987, 4993, 2009}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \cos ^3\left (a+\frac {1}{3} \sqrt {-\frac {1}{n^2}} \log \left (c x^n\right )\right ) \, dx\)

\(\Big \downarrow \) 4987

\(\displaystyle \frac {x \left (c x^n\right )^{-1/n} \int \left (c x^n\right )^{\frac {1}{n}-1} \cos ^3\left (a+\frac {1}{3} \sqrt {-\frac {1}{n^2}} \log \left (c x^n\right )\right )d\left (c x^n\right )}{n}\)

\(\Big \downarrow \) 4993

\(\displaystyle \frac {x \left (c x^n\right )^{-1/n} \int \left (3 e^{a \sqrt {-\frac {1}{n^2}} n} \left (c x^n\right )^{\frac {2}{3 n}-1}+3 e^{-a \sqrt {-\frac {1}{n^2}} n} \left (c x^n\right )^{\frac {4}{3 n}-1}+e^{-3 a \sqrt {-\frac {1}{n^2}} n} \left (c x^n\right )^{\frac {2}{n}-1}+\frac {e^{3 a \sqrt {-\frac {1}{n^2}} n} x^{-n}}{c}\right )d\left (c x^n\right )}{8 n}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {x \left (c x^n\right )^{-1/n} \left (\frac {9}{2} n e^{a \sqrt {-\frac {1}{n^2}} n} \left (c x^n\right )^{\left .\frac {2}{3}\right /n}+\frac {9}{4} n e^{-a \sqrt {-\frac {1}{n^2}} n} \left (c x^n\right )^{\left .\frac {4}{3}\right /n}+\frac {1}{2} n e^{-3 a \sqrt {-\frac {1}{n^2}} n} \left (c x^n\right )^{2/n}+e^{3 a \sqrt {-\frac {1}{n^2}} n} \log \left (c x^n\right )\right )}{8 n}\)

Input:

Int[Cos[a + (Sqrt[-n^(-2)]*Log[c*x^n])/3]^3,x]
 

Output:

(x*((9*E^(a*Sqrt[-n^(-2)]*n)*n*(c*x^n)^(2/(3*n)))/2 + (9*n*(c*x^n)^(4/(3*n 
)))/(4*E^(a*Sqrt[-n^(-2)]*n)) + (n*(c*x^n)^(2/n))/(2*E^(3*a*Sqrt[-n^(-2)]* 
n)) + E^(3*a*Sqrt[-n^(-2)]*n)*Log[c*x^n]))/(8*n*(c*x^n)^n^(-1))
 

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 4987

Int[Cos[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*(d_.)]^(p_.), x_Symbol] :> Si 
mp[x/(n*(c*x^n)^(1/n))   Subst[Int[x^(1/n - 1)*Cos[d*(a + b*Log[x])]^p, x], 
 x, c*x^n], x] /; FreeQ[{a, b, c, d, n, p}, x] && (NeQ[c, 1] || NeQ[n, 1])
 

rule 4993

Int[Cos[((a_.) + Log[x_]*(b_.))*(d_.)]^(p_.)*((e_.)*(x_))^(m_.), x_Symbol] 
:> Simp[1/2^p   Int[ExpandIntegrand[(e*x)^m*(E^(a*b*d^2*(p/(m + 1)))/x^((m 
+ 1)/p) + x^((m + 1)/p)/E^(a*b*d^2*(p/(m + 1))))^p, x], x], x] /; FreeQ[{a, 
 b, d, e, m}, x] && IGtQ[p, 0] && EqQ[b^2*d^2*p^2 + (m + 1)^2, 0]
 

Maple [F]

\[\int {\cos \left (a +\frac {\sqrt {-\frac {1}{n^{2}}}\, \ln \left (c \,x^{n}\right )}{3}\right )}^{3}d x\]

Input:

int(cos(a+1/3*(-1/n^2)^(1/2)*ln(c*x^n))^3,x)
 

Output:

int(cos(a+1/3*(-1/n^2)^(1/2)*ln(c*x^n))^3,x)
 

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.07 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.66 \[ \int \cos ^3\left (a+\frac {1}{3} \sqrt {-\frac {1}{n^2}} \log \left (c x^n\right )\right ) \, dx=\frac {1}{32} \, {\left (9 \, x^{\frac {4}{3}} e^{\left (\frac {2 \, {\left (3 i \, a n - \log \left (c\right )\right )}}{3 \, n}\right )} + 2 \, x^{2} + 12 \, e^{\left (\frac {2 \, {\left (3 i \, a n - \log \left (c\right )\right )}}{n}\right )} \log \left (x^{\frac {1}{3}}\right ) + 18 \, x^{\frac {2}{3}} e^{\left (\frac {4 \, {\left (3 i \, a n - \log \left (c\right )\right )}}{3 \, n}\right )}\right )} e^{\left (-\frac {3 i \, a n - \log \left (c\right )}{n}\right )} \] Input:

integrate(cos(a+1/3*(-1/n^2)^(1/2)*log(c*x^n))^3,x, algorithm="fricas")
 

Output:

1/32*(9*x^(4/3)*e^(2/3*(3*I*a*n - log(c))/n) + 2*x^2 + 12*e^(2*(3*I*a*n - 
log(c))/n)*log(x^(1/3)) + 18*x^(2/3)*e^(4/3*(3*I*a*n - log(c))/n))*e^(-(3* 
I*a*n - log(c))/n)
 

Sympy [F]

\[ \int \cos ^3\left (a+\frac {1}{3} \sqrt {-\frac {1}{n^2}} \log \left (c x^n\right )\right ) \, dx=\int \cos ^{3}{\left (a + \frac {\sqrt {- \frac {1}{n^{2}}} \log {\left (c x^{n} \right )}}{3} \right )}\, dx \] Input:

integrate(cos(a+1/3*(-1/n**2)**(1/2)*ln(c*x**n))**3,x)
 

Output:

Integral(cos(a + sqrt(-1/n**2)*log(c*x**n)/3)**3, x)
                                                                                    
                                                                                    
 

Maxima [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.83 \[ \int \cos ^3\left (a+\frac {1}{3} \sqrt {-\frac {1}{n^2}} \log \left (c x^n\right )\right ) \, dx=\frac {9 \, c^{\frac {5}{3 \, n}} x {\left (x^{n}\right )}^{\frac {2}{3 \, n}} \cos \left (a\right ) + 4 \, c^{\frac {1}{3 \, n}} {\left (x^{n}\right )}^{\frac {1}{3 \, n}} \cos \left (3 \, a\right ) \log \left (x\right ) + 18 \, c^{\left (\frac {1}{n}\right )} x \cos \left (a\right ) + 2 \, c^{\frac {7}{3 \, n}} \cos \left (3 \, a\right ) e^{\left (\frac {\log \left (x^{n}\right )}{3 \, n} + 2 \, \log \left (x\right )\right )}}{32 \, c^{\frac {4}{3 \, n}} {\left (x^{n}\right )}^{\frac {1}{3 \, n}}} \] Input:

integrate(cos(a+1/3*(-1/n^2)^(1/2)*log(c*x^n))^3,x, algorithm="maxima")
 

Output:

1/32*(9*c^(5/3/n)*x*(x^n)^(2/3/n)*cos(a) + 4*c^(1/3/n)*(x^n)^(1/3/n)*cos(3 
*a)*log(x) + 18*c^(1/n)*x*cos(a) + 2*c^(7/3/n)*cos(3*a)*e^(1/3*log(x^n)/n 
+ 2*log(x)))/(c^(4/3/n)*(x^n)^(1/3/n))
 

Giac [F(-2)]

Exception generated. \[ \int \cos ^3\left (a+\frac {1}{3} \sqrt {-\frac {1}{n^2}} \log \left (c x^n\right )\right ) \, dx=\text {Exception raised: NotImplementedError} \] Input:

integrate(cos(a+1/3*(-1/n^2)^(1/2)*log(c*x^n))^3,x, algorithm="giac")
 

Output:

Exception raised: NotImplementedError >> unable to parse Giac output: (9*s 
ageVARn^4*sageVARx*exp((-3*i)*sageVARa)*exp((sageVARn*abs(sageVARn)*ln(sag 
eVARx)+abs(sageVARn)*ln(sageVARc))/sageVARn^2)+27*sageVARn^4*sageVARx*exp( 
(-i)*sageVARa)*exp(
 

Mupad [B] (verification not implemented)

Time = 20.29 (sec) , antiderivative size = 158, normalized size of antiderivative = 1.23 \[ \int \cos ^3\left (a+\frac {1}{3} \sqrt {-\frac {1}{n^2}} \log \left (c x^n\right )\right ) \, dx=x\,{\mathrm {e}}^{-a\,1{}\mathrm {i}}\,\frac {1}{{\left (c\,x^n\right )}^{\frac {\sqrt {-\frac {1}{n^2}}\,1{}\mathrm {i}}{3}}}\,\left (\frac {27}{64}+\frac {n\,\sqrt {-\frac {1}{n^2}}\,9{}\mathrm {i}}{64}\right )-x\,{\mathrm {e}}^{a\,1{}\mathrm {i}}\,{\left (c\,x^n\right )}^{\frac {\sqrt {-\frac {1}{n^2}}\,1{}\mathrm {i}}{3}}\,\left (-\frac {27}{64}+\frac {n\,\sqrt {-\frac {1}{n^2}}\,9{}\mathrm {i}}{64}\right )+\frac {x\,{\mathrm {e}}^{-a\,3{}\mathrm {i}}\,\frac {1}{{\left (c\,x^n\right )}^{\sqrt {-\frac {1}{n^2}}\,1{}\mathrm {i}}}\,1{}\mathrm {i}}{8\,n\,\sqrt {-\frac {1}{n^2}}+8{}\mathrm {i}}-\frac {x\,{\mathrm {e}}^{a\,3{}\mathrm {i}}\,{\left (c\,x^n\right )}^{\sqrt {-\frac {1}{n^2}}\,1{}\mathrm {i}}\,1{}\mathrm {i}}{8\,n\,\sqrt {-\frac {1}{n^2}}-8{}\mathrm {i}} \] Input:

int(cos(a + (log(c*x^n)*(-1/n^2)^(1/2))/3)^3,x)
 

Output:

x*exp(-a*1i)/(c*x^n)^(((-1/n^2)^(1/2)*1i)/3)*((n*(-1/n^2)^(1/2)*9i)/64 + 2 
7/64) - x*exp(a*1i)*(c*x^n)^(((-1/n^2)^(1/2)*1i)/3)*((n*(-1/n^2)^(1/2)*9i) 
/64 - 27/64) + (x*exp(-a*3i)/(c*x^n)^((-1/n^2)^(1/2)*1i)*1i)/(8*n*(-1/n^2) 
^(1/2) + 8i) - (x*exp(a*3i)*(c*x^n)^((-1/n^2)^(1/2)*1i)*1i)/(8*n*(-1/n^2)^ 
(1/2) - 8i)
 

Reduce [B] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 229, normalized size of antiderivative = 1.79 \[ \int \cos ^3\left (a+\frac {1}{3} \sqrt {-\frac {1}{n^2}} \log \left (c x^n\right )\right ) \, dx=\frac {x \left (-4 \cos \left (\frac {\mathrm {log}\left (x^{n} c \right ) i +3 a n}{3 n}\right ) \mathrm {log}\left (x^{n} c \right ) {\sin \left (\frac {\mathrm {log}\left (x^{n} c \right ) i +3 a n}{3 n}\right )}^{2}+\cos \left (\frac {\mathrm {log}\left (x^{n} c \right ) i +3 a n}{3 n}\right ) \mathrm {log}\left (x^{n} c \right )-\cos \left (\frac {\mathrm {log}\left (x^{n} c \right ) i +3 a n}{3 n}\right ) {\sin \left (\frac {\mathrm {log}\left (x^{n} c \right ) i +3 a n}{3 n}\right )}^{2} n +7 \cos \left (\frac {\mathrm {log}\left (x^{n} c \right ) i +3 a n}{3 n}\right ) n -4 \,\mathrm {log}\left (x^{n} c \right ) {\sin \left (\frac {\mathrm {log}\left (x^{n} c \right ) i +3 a n}{3 n}\right )}^{3} i +3 \,\mathrm {log}\left (x^{n} c \right ) \sin \left (\frac {\mathrm {log}\left (x^{n} c \right ) i +3 a n}{3 n}\right ) i +3 {\sin \left (\frac {\mathrm {log}\left (x^{n} c \right ) i +3 a n}{3 n}\right )}^{3} i n \right )}{8 n} \] Input:

int(cos(a+1/3*(-1/n^2)^(1/2)*log(c*x^n))^3,x)
 

Output:

(x*( - 4*cos((log(x**n*c)*i + 3*a*n)/(3*n))*log(x**n*c)*sin((log(x**n*c)*i 
 + 3*a*n)/(3*n))**2 + cos((log(x**n*c)*i + 3*a*n)/(3*n))*log(x**n*c) - cos 
((log(x**n*c)*i + 3*a*n)/(3*n))*sin((log(x**n*c)*i + 3*a*n)/(3*n))**2*n + 
7*cos((log(x**n*c)*i + 3*a*n)/(3*n))*n - 4*log(x**n*c)*sin((log(x**n*c)*i 
+ 3*a*n)/(3*n))**3*i + 3*log(x**n*c)*sin((log(x**n*c)*i + 3*a*n)/(3*n))*i 
+ 3*sin((log(x**n*c)*i + 3*a*n)/(3*n))**3*i*n))/(8*n)