\(\int \sin ^{\frac {3}{2}}(a+b \log (c x^n)) \, dx\) [59]

Optimal result

Integrand size = 15, antiderivative size = 109 \[ \int \sin ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {2 x \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},\frac {1}{4} \left (-3-\frac {2 i}{b n}\right ),\frac {1}{4} \left (1-\frac {2 i}{b n}\right ),e^{2 i a} \left (c x^n\right )^{2 i b}\right ) \sin ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )}{(2-3 i b n) \left (1-e^{2 i a} \left (c x^n\right )^{2 i b}\right )^{3/2}} \] Output:

2*x*hypergeom([-3/2, -3/4-1/2*I/b/n],[1/4-1/2*I/b/n],exp(2*I*a)*(c*x^n)^(2 
*I*b))*sin(a+b*ln(c*x^n))^(3/2)/(2-3*I*b*n)/(1-exp(2*I*a)*(c*x^n)^(2*I*b)) 
^(3/2)
 

Mathematica [A] (verified)

Time = 0.90 (sec) , antiderivative size = 218, normalized size of antiderivative = 2.00 \[ \int \sin ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right ) \, dx=-\frac {6 i b^2 \sqrt {2-2 e^{2 i \left (a+b \log \left (c x^n\right )\right )}} n^2 x \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{4}-\frac {i}{2 b n},\frac {5}{4}-\frac {i}{2 b n},e^{2 i \left (a+b \log \left (c x^n\right )\right )}\right )}{\sqrt {-i e^{-i \left (a+b \log \left (c x^n\right )\right )} \left (-1+e^{2 i \left (a+b \log \left (c x^n\right )\right )}\right )} (-2 i+b n) (-2 i+3 b n) (2 i+3 b n)}+\frac {2 x \sqrt {\sin \left (a+b \log \left (c x^n\right )\right )} \left (-3 b n \cos \left (a+b \log \left (c x^n\right )\right )+2 \sin \left (a+b \log \left (c x^n\right )\right )\right )}{4+9 b^2 n^2} \] Input:

Integrate[Sin[a + b*Log[c*x^n]]^(3/2),x]
 

Output:

((-6*I)*b^2*Sqrt[2 - 2*E^((2*I)*(a + b*Log[c*x^n]))]*n^2*x*Hypergeometric2 
F1[1/2, 1/4 - (I/2)/(b*n), 5/4 - (I/2)/(b*n), E^((2*I)*(a + b*Log[c*x^n])) 
])/(Sqrt[((-I)*(-1 + E^((2*I)*(a + b*Log[c*x^n]))))/E^(I*(a + b*Log[c*x^n] 
))]*(-2*I + b*n)*(-2*I + 3*b*n)*(2*I + 3*b*n)) + (2*x*Sqrt[Sin[a + b*Log[c 
*x^n]]]*(-3*b*n*Cos[a + b*Log[c*x^n]] + 2*Sin[a + b*Log[c*x^n]]))/(4 + 9*b 
^2*n^2)
 

Rubi [A] (verified)

Time = 0.30 (sec) , antiderivative size = 109, 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 = {4986, 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.

\(\displaystyle \int \sin ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right ) \, dx\)

\(\Big \downarrow \) 4986

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

\(\Big \downarrow \) 4994

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

\(\Big \downarrow \) 888

\(\displaystyle \frac {2 x \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},\frac {1}{4} \left (-3-\frac {2 i}{b n}\right ),\frac {1}{4} \left (1-\frac {2 i}{b n}\right ),e^{2 i a} \left (c x^n\right )^{2 i b}\right ) \sin ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )}{(2-3 i b n) \left (1-e^{2 i a} \left (c x^n\right )^{2 i b}\right )^{3/2}}\)

Input:

Int[Sin[a + b*Log[c*x^n]]^(3/2),x]
 

Output:

(2*x*Hypergeometric2F1[-3/2, (-3 - (2*I)/(b*n))/4, (1 - (2*I)/(b*n))/4, E^ 
((2*I)*a)*(c*x^n)^((2*I)*b)]*Sin[a + b*Log[c*x^n]]^(3/2))/((2 - (3*I)*b*n) 
*(1 - E^((2*I)*a)*(c*x^n)^((2*I)*b))^(3/2))
 

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 4986

Int[Sin[((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)*Sin[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 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]
 

Maple [F]

\[\int {\sin \left (a +b \ln \left (c \,x^{n}\right )\right )}^{\frac {3}{2}}d x\]

Input:

int(sin(a+b*ln(c*x^n))^(3/2),x)
 

Output:

int(sin(a+b*ln(c*x^n))^(3/2),x)
 

Fricas [F(-2)]

Exception generated. \[ \int \sin ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right ) \, dx=\text {Exception raised: TypeError} \] Input:

integrate(sin(a+b*log(c*x^n))^(3/2),x, algorithm="fricas")
 

Output:

Exception raised: TypeError >>  Error detected within library code:   inte 
grate: implementation incomplete (has polynomial part)
 

Sympy [F]

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

integrate(sin(a+b*ln(c*x**n))**(3/2),x)
 

Output:

Integral(sin(a + b*log(c*x**n))**(3/2), x)
 

Maxima [F]

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

integrate(sin(a+b*log(c*x^n))^(3/2),x, algorithm="maxima")
 

Output:

integrate(sin(b*log(c*x^n) + a)^(3/2), x)
 

Giac [F]

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

integrate(sin(a+b*log(c*x^n))^(3/2),x, algorithm="giac")
 

Output:

integrate(sin(b*log(c*x^n) + a)^(3/2), x)
 

Mupad [F(-1)]

Timed out. \[ \int \sin ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right ) \, dx=\int {\sin \left (a+b\,\ln \left (c\,x^n\right )\right )}^{3/2} \,d x \] Input:

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

Output:

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

Reduce [F]

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

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

Output:

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