\(\int \frac {\sqrt {\sin (a+b \log (c x^n))}}{x^3} \, dx\) [57]

Optimal result

Integrand size = 19, antiderivative size = 111 \[ \int \frac {\sqrt {\sin \left (a+b \log \left (c x^n\right )\right )}}{x^3} \, dx=-\frac {2 \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},\frac {1}{4} \left (-1+\frac {4 i}{b n}\right ),\frac {1}{4} \left (3+\frac {4 i}{b n}\right ),e^{2 i a} \left (c x^n\right )^{2 i b}\right ) \sqrt {\sin \left (a+b \log \left (c x^n\right )\right )}}{(4+i b n) x^2 \sqrt {1-e^{2 i a} \left (c x^n\right )^{2 i b}}} \] Output:

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

Mathematica [A] (verified)

Time = 10.47 (sec) , antiderivative size = 145, normalized size of antiderivative = 1.31 \[ \int \frac {\sqrt {\sin \left (a+b \log \left (c x^n\right )\right )}}{x^3} \, dx=\frac {i \sqrt {2} \sqrt {-i e^{-i a} \left (c x^n\right )^{-i b} \left (-1+e^{2 i a} \left (c x^n\right )^{2 i b}\right )} \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},-\frac {1}{4}+\frac {i}{b n},\frac {3}{4}+\frac {i}{b n},e^{2 i a} \left (c x^n\right )^{2 i b}\right )}{(-4 i+b n) x^2 \sqrt {1-e^{2 i a} \left (c x^n\right )^{2 i b}}} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.32 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, 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.

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

\(\Big \downarrow \) 4996

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

\(\Big \downarrow \) 4994

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

\(\Big \downarrow \) 888

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

Input:

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

Output:

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

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]

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

Input:

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

Output:

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

Fricas [F(-2)]

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

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

Output:

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

Sympy [F]

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

integrate(sin(a+b*ln(c*x**n))**(1/2)/x**3,x)
 

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {\sqrt {\sin \left (a+b \log \left (c x^n\right )\right )}}{x^3} \, dx=\frac {-2 \sqrt {\sin \left (\mathrm {log}\left (x^{n} c \right ) b +a \right )}+\left (\int \frac {\sqrt {\sin \left (\mathrm {log}\left (x^{n} c \right ) b +a \right )}\, \cos \left (\mathrm {log}\left (x^{n} c \right ) b +a \right )}{\sin \left (\mathrm {log}\left (x^{n} c \right ) b +a \right ) x^{3}}d x \right ) b n \,x^{2}}{4 x^{2}} \] Input:

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

Output:

( - 2*sqrt(sin(log(x**n*c)*b + a)) + int((sqrt(sin(log(x**n*c)*b + a))*cos 
(log(x**n*c)*b + a))/(sin(log(x**n*c)*b + a)*x**3),x)*b*n*x**2)/(4*x**2)