\(\int x \cos ^p(a+b \log (c x^n)) \, dx\) [133]

Optimal result

Integrand size = 15, antiderivative size = 114 \[ \int x \cos ^p\left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {x^2 \left (1+e^{2 i a} \left (c x^n\right )^{2 i b}\right )^{-p} \cos ^p\left (a+b \log \left (c x^n\right )\right ) \operatorname {Hypergeometric2F1}\left (\frac {1}{2} \left (-\frac {2 i}{b n}-p\right ),-p,\frac {1}{2} \left (2-\frac {2 i}{b n}-p\right ),-e^{2 i a} \left (c x^n\right )^{2 i b}\right )}{2-i b n p} \] Output:

x^2*cos(a+b*ln(c*x^n))^p*hypergeom([-p, -I/b/n-1/2*p],[1-I/b/n-1/2*p],-exp 
(2*I*a)*(c*x^n)^(2*I*b))/(2-I*b*n*p)/((1+exp(2*I*a)*(c*x^n)^(2*I*b))^p)
 

Mathematica [A] (verified)

Time = 0.67 (sec) , antiderivative size = 141, normalized size of antiderivative = 1.24 \[ \int x \cos ^p\left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {i x^2 \left (e^{-i a} \left (c x^n\right )^{-i b}+e^{i a} \left (c x^n\right )^{i b}\right )^p \left (2+2 e^{2 i a} \left (c x^n\right )^{2 i b}\right )^{-p} \operatorname {Hypergeometric2F1}\left (-\frac {i}{b n}-\frac {p}{2},-p,1-\frac {i}{b n}-\frac {p}{2},-e^{2 i a} \left (c x^n\right )^{2 i b}\right )}{2 i+b n p} \] Input:

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

Output:

(I*x^2*(1/(E^(I*a)*(c*x^n)^(I*b)) + E^(I*a)*(c*x^n)^(I*b))^p*Hypergeometri 
c2F1[(-I)/(b*n) - p/2, -p, 1 - I/(b*n) - p/2, -(E^((2*I)*a)*(c*x^n)^((2*I) 
*b))])/((2*I + b*n*p)*(2 + 2*E^((2*I)*a)*(c*x^n)^((2*I)*b))^p)
 

Rubi [A] (verified)

Time = 0.33 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {4997, 4995, 888}

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.

\(\displaystyle \int x \cos ^p\left (a+b \log \left (c x^n\right )\right ) \, dx\)

\(\Big \downarrow \) 4997

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

\(\Big \downarrow \) 4995

\(\displaystyle \frac {x^2 \left (c x^n\right )^{-\frac {2}{n}+i b p} \left (1+e^{2 i a} \left (c x^n\right )^{2 i b}\right )^{-p} \cos ^p\left (a+b \log \left (c x^n\right )\right ) \int \left (c x^n\right )^{-i b p+\frac {2}{n}-1} \left (e^{2 i a} \left (c x^n\right )^{2 i b}+1\right )^pd\left (c x^n\right )}{n}\)

\(\Big \downarrow \) 888

\(\displaystyle \frac {x^2 \left (1+e^{2 i a} \left (c x^n\right )^{2 i b}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {1}{2} \left (-p-\frac {2 i}{b n}\right ),-p,\frac {1}{2} \left (-p-\frac {2 i}{b n}+2\right ),-e^{2 i a} \left (c x^n\right )^{2 i b}\right ) \cos ^p\left (a+b \log \left (c x^n\right )\right )}{2-i b n p}\)

Input:

Int[x*Cos[a + b*Log[c*x^n]]^p,x]
 

Output:

(x^2*Cos[a + b*Log[c*x^n]]^p*Hypergeometric2F1[((-2*I)/(b*n) - p)/2, -p, ( 
2 - (2*I)/(b*n) - p)/2, -(E^((2*I)*a)*(c*x^n)^((2*I)*b))])/((2 - I*b*n*p)* 
(1 + E^((2*I)*a)*(c*x^n)^((2*I)*b))^p)
 

Defintions of rubi rules used

rule 888

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^p 
*((c*x)^(m + 1)/(c*(m + 1)))*Hypergeometric2F1[-p, (m + 1)/n, (m + 1)/n + 1 
, (-b)*(x^n/a)], x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[p, 0] && (ILt 
Q[p, 0] || GtQ[a, 0])
 

rule 4995

Int[Cos[((a_.) + Log[x_]*(b_.))*(d_.)]^(p_)*((e_.)*(x_))^(m_.), x_Symbol] : 
> Simp[Cos[d*(a + b*Log[x])]^p*(x^(I*b*d*p)/(1 + E^(2*I*a*d)*x^(2*I*b*d))^p 
)   Int[(e*x)^m*((1 + E^(2*I*a*d)*x^(2*I*b*d))^p/x^(I*b*d*p)), x], x] /; Fr 
eeQ[{a, b, d, e, m, p}, x] &&  !IntegerQ[p]
 

rule 4997

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

Maple [F]

\[\int x {\cos \left (a +b \ln \left (c \,x^{n}\right )\right )}^{p}d x\]

Input:

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

Output:

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

Fricas [F]

\[ \int x \cos ^p\left (a+b \log \left (c x^n\right )\right ) \, dx=\int { x \cos \left (b \log \left (c x^{n}\right ) + a\right )^{p} \,d x } \] Input:

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

Output:

integral(x*cos(b*log(c*x^n) + a)^p, x)
 

Sympy [F]

\[ \int x \cos ^p\left (a+b \log \left (c x^n\right )\right ) \, dx=\int x \cos ^{p}{\left (a + b \log {\left (c x^{n} \right )} \right )}\, dx \] Input:

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

Output:

Integral(x*cos(a + b*log(c*x**n))**p, x)
 

Maxima [F]

\[ \int x \cos ^p\left (a+b \log \left (c x^n\right )\right ) \, dx=\int { x \cos \left (b \log \left (c x^{n}\right ) + a\right )^{p} \,d x } \] Input:

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

Output:

integrate(x*cos(b*log(c*x^n) + a)^p, x)
 

Giac [F]

\[ \int x \cos ^p\left (a+b \log \left (c x^n\right )\right ) \, dx=\int { x \cos \left (b \log \left (c x^{n}\right ) + a\right )^{p} \,d x } \] Input:

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

Output:

integrate(x*cos(b*log(c*x^n) + a)^p, x)
 

Mupad [F(-1)]

Timed out. \[ \int x \cos ^p\left (a+b \log \left (c x^n\right )\right ) \, dx=\int x\,{\cos \left (a+b\,\ln \left (c\,x^n\right )\right )}^p \,d x \] Input:

int(x*cos(a + b*log(c*x^n))^p,x)
 

Output:

int(x*cos(a + b*log(c*x^n))^p, x)
 

Reduce [F]

\[ \int x \cos ^p\left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {{\cos \left (\mathrm {log}\left (x^{n} c \right ) b +a \right )}^{p} x^{2}}{2}+\frac {\left (\int \frac {{\cos \left (\mathrm {log}\left (x^{n} c \right ) b +a \right )}^{p} \sin \left (\mathrm {log}\left (x^{n} c \right ) b +a \right ) x}{\cos \left (\mathrm {log}\left (x^{n} c \right ) b +a \right )}d x \right ) b n p}{2} \] Input:

int(x*cos(a+b*log(c*x^n))^p,x)
 

Output:

(cos(log(x**n*c)*b + a)**p*x**2 + int((cos(log(x**n*c)*b + a)**p*sin(log(x 
**n*c)*b + a)*x)/cos(log(x**n*c)*b + a),x)*b*n*p)/2