\(\int x^2 \sin ^p(a+b \log (c x^n)) \, dx\) [80]

Optimal result

Integrand size = 17, antiderivative size = 114 \[ \int x^2 \sin ^p\left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {x^3 \left (1-e^{2 i a} \left (c x^n\right )^{2 i b}\right )^{-p} \operatorname {Hypergeometric2F1}\left (-p,-\frac {3 i+b n p}{2 b n},\frac {1}{2} \left (2-\frac {3 i}{b n}-p\right ),e^{2 i a} \left (c x^n\right )^{2 i b}\right ) \sin ^p\left (a+b \log \left (c x^n\right )\right )}{3-i b n p} \] Output:

x^3*hypergeom([-p, -1/2*(3*I+b*n*p)/b/n],[1-3/2*I/b/n-1/2*p],exp(2*I*a)*(c 
*x^n)^(2*I*b))*sin(a+b*ln(c*x^n))^p/(3-I*b*n*p)/((1-exp(2*I*a)*(c*x^n)^(2* 
I*b))^p)
 

Mathematica [A] (verified)

Time = 0.66 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.30 \[ \int x^2 \sin ^p\left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {i x^3 \left (2-2 e^{2 i a} \left (c x^n\right )^{2 i b}\right )^{-p} \left (-i e^{-i a} \left (c x^n\right )^{-i b} \left (-1+e^{2 i a} \left (c x^n\right )^{2 i b}\right )\right )^p \operatorname {Hypergeometric2F1}\left (-p,-\frac {3 i+b n p}{2 b n},1-\frac {3 i}{2 b n}-\frac {p}{2},e^{2 i a} \left (c x^n\right )^{2 i b}\right )}{3 i+b n p} \] Input:

Integrate[x^2*Sin[a + b*Log[c*x^n]]^p,x]
 

Output:

(I*x^3*(((-I)*(-1 + E^((2*I)*a)*(c*x^n)^((2*I)*b)))/(E^(I*a)*(c*x^n)^(I*b) 
))^p*Hypergeometric2F1[-p, -1/2*(3*I + b*n*p)/(b*n), 1 - ((3*I)/2)/(b*n) - 
 p/2, E^((2*I)*a)*(c*x^n)^((2*I)*b)])/((3*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, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {4996, 4994, 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.

\(\displaystyle \int x^2 \sin ^p\left (a+b \log \left (c x^n\right )\right ) \, dx\)

\(\Big \downarrow \) 4996

\(\displaystyle \frac {x^3 \left (c x^n\right )^{-3/n} \int \left (c x^n\right )^{\frac {3}{n}-1} \sin ^p\left (a+b \log \left (c x^n\right )\right )d\left (c x^n\right )}{n}\)

\(\Big \downarrow \) 4994

\(\displaystyle \frac {x^3 \left (c x^n\right )^{-\frac {3}{n}+i b p} \left (1-e^{2 i a} \left (c x^n\right )^{2 i b}\right )^{-p} \sin ^p\left (a+b \log \left (c x^n\right )\right ) \int \left (c x^n\right )^{-i b p+\frac {3}{n}-1} \left (1-e^{2 i a} \left (c x^n\right )^{2 i b}\right )^pd\left (c x^n\right )}{n}\)

\(\Big \downarrow \) 888

\(\displaystyle \frac {x^3 \left (1-e^{2 i a} \left (c x^n\right )^{2 i b}\right )^{-p} \operatorname {Hypergeometric2F1}\left (-p,-\frac {b n p+3 i}{2 b n},\frac {1}{2} \left (-p-\frac {3 i}{b n}+2\right ),e^{2 i a} \left (c x^n\right )^{2 i b}\right ) \sin ^p\left (a+b \log \left (c x^n\right )\right )}{3-i b n p}\)

Input:

Int[x^2*Sin[a + b*Log[c*x^n]]^p,x]
 

Output:

(x^3*Hypergeometric2F1[-p, -1/2*(3*I + b*n*p)/(b*n), (2 - (3*I)/(b*n) - p) 
/2, E^((2*I)*a)*(c*x^n)^((2*I)*b)]*Sin[a + b*Log[c*x^n]]^p)/((3 - 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 4994

Int[((e_.)*(x_))^(m_.)*Sin[((a_.) + Log[x_]*(b_.))*(d_.)]^(p_), x_Symbol] : 
> Simp[Sin[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 4996

Int[((e_.)*(x_))^(m_.)*Sin[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*(d_.)]^(p_ 
.), x_Symbol] :> Simp[(e*x)^(m + 1)/(e*n*(c*x^n)^((m + 1)/n))   Subst[Int[x 
^((m + 1)/n - 1)*Sin[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]

\[\int x^{2} {\sin \left (a +b \ln \left (c \,x^{n}\right )\right )}^{p}d x\]

Input:

int(x^2*sin(a+b*ln(c*x^n))^p,x)
 

Output:

int(x^2*sin(a+b*ln(c*x^n))^p,x)
 

Fricas [F]

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

integrate(x^2*sin(a+b*log(c*x^n))^p,x, algorithm="fricas")
 

Output:

integral(x^2*sin(b*log(c*x^n) + a)^p, x)
 

Sympy [F]

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

integrate(x**2*sin(a+b*ln(c*x**n))**p,x)
 

Output:

Integral(x**2*sin(a + b*log(c*x**n))**p, x)
 

Maxima [F]

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

integrate(x^2*sin(a+b*log(c*x^n))^p,x, algorithm="maxima")
 

Output:

integrate(x^2*sin(b*log(c*x^n) + a)^p, x)
 

Giac [F]

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

integrate(x^2*sin(a+b*log(c*x^n))^p,x, algorithm="giac")
 

Output:

integrate(x^2*sin(b*log(c*x^n) + a)^p, x)
 

Mupad [F(-1)]

Timed out. \[ \int x^2 \sin ^p\left (a+b \log \left (c x^n\right )\right ) \, dx=\int x^2\,{\sin \left (a+b\,\ln \left (c\,x^n\right )\right )}^p \,d x \] Input:

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

Output:

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

Reduce [F]

\[ \int x^2 \sin ^p\left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {{\sin \left (\mathrm {log}\left (x^{n} c \right ) b +a \right )}^{p} x^{3}}{3}-\frac {\left (\int \frac {{\sin \left (\mathrm {log}\left (x^{n} c \right ) b +a \right )}^{p} \cos \left (\mathrm {log}\left (x^{n} c \right ) b +a \right ) x^{2}}{\sin \left (\mathrm {log}\left (x^{n} c \right ) b +a \right )}d x \right ) b n p}{3} \] Input:

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

Output:

(sin(log(x**n*c)*b + a)**p*x**3 - int((sin(log(x**n*c)*b + a)**p*cos(log(x 
**n*c)*b + a)*x**2)/sin(log(x**n*c)*b + a),x)*b*n*p)/3