\(\int (e x)^m \cos ^p(d (a+b \log (c x^n))) \, dx\) [132]

Optimal result

Integrand size = 21, antiderivative size = 144 \[ \int (e x)^m \cos ^p\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\frac {(e x)^{1+m} \left (1+e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )^{-p} \cos ^p\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \operatorname {Hypergeometric2F1}\left (-p,-\frac {i+i m+b d n p}{2 b d n},\frac {1}{2} \left (2-\frac {i (1+m)}{b d n}-p\right ),-e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )}{e (1+m-i b d n p)} \] Output:

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

Mathematica [A] (warning: unable to verify)

Time = 1.38 (sec) , antiderivative size = 170, normalized size of antiderivative = 1.18 \[ \int (e x)^m \cos ^p\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\frac {x (e x)^m \left (e^{-i a d} \left (c x^n\right )^{-i b d}+e^{i a d} \left (c x^n\right )^{i b d}\right )^p \left (2+2 e^{-2 i a d} \left (c x^n\right )^{-2 i b d}\right )^{-p} \operatorname {Hypergeometric2F1}\left (-p,\frac {i (1+m+i b d n p)}{2 b d n},1+\frac {i (1+m)}{2 b d n}-\frac {p}{2},-e^{-2 i a d} \left (c x^n\right )^{-2 i b d}\right )}{1+m+i b d n p} \] Input:

Integrate[(e*x)^m*Cos[d*(a + b*Log[c*x^n])]^p,x]
 

Output:

(x*(e*x)^m*(1/(E^(I*a*d)*(c*x^n)^(I*b*d)) + E^(I*a*d)*(c*x^n)^(I*b*d))^p*H 
ypergeometric2F1[-p, ((I/2)*(1 + m + I*b*d*n*p))/(b*d*n), 1 + ((I/2)*(1 + 
m))/(b*d*n) - p/2, -(1/(E^((2*I)*a*d)*(c*x^n)^((2*I)*b*d)))])/((1 + m + I* 
b*d*n*p)*(2 + 2/(E^((2*I)*a*d)*(c*x^n)^((2*I)*b*d)))^p)
 

Rubi [A] (verified)

Time = 0.38 (sec) , antiderivative size = 181, normalized size of antiderivative = 1.26, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, 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.

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

\(\Big \downarrow \) 4997

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

\(\Big \downarrow \) 4995

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

\(\Big \downarrow \) 888

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

Input:

Int[(e*x)^m*Cos[d*(a + b*Log[c*x^n])]^p,x]
 

Output:

((e*x)^(1 + m)*(c*x^n)^(-((1 + m)/n) + I*b*d*p + (1 + m - I*b*d*n*p)/n)*Co 
s[d*(a + b*Log[c*x^n])]^p*Hypergeometric2F1[-p, -1/2*(I + I*m + b*d*n*p)/( 
b*d*n), (2 - (I*(1 + m))/(b*d*n) - p)/2, -(E^((2*I)*a*d)*(c*x^n)^((2*I)*b* 
d))])/(e*(1 + m - I*b*d*n*p)*(1 + E^((2*I)*a*d)*(c*x^n)^((2*I)*b*d))^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]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int (e x)^m \cos ^p\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\frac {e^{m} \left (x^{m} {\cos \left (\mathrm {log}\left (x^{n} c \right ) b d +a d \right )}^{p} x +\left (\int \frac {x^{m} {\cos \left (\mathrm {log}\left (x^{n} c \right ) b d +a d \right )}^{p} \sin \left (\mathrm {log}\left (x^{n} c \right ) b d +a d \right )}{\cos \left (\mathrm {log}\left (x^{n} c \right ) b d +a d \right )}d x \right ) b d n p \right )}{m +1} \] Input:

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

Output:

(e**m*(x**m*cos(log(x**n*c)*b*d + a*d)**p*x + int((x**m*cos(log(x**n*c)*b* 
d + a*d)**p*sin(log(x**n*c)*b*d + a*d))/cos(log(x**n*c)*b*d + a*d),x)*b*d* 
n*p))/(m + 1)