\(\int \frac {a+b \log (c (4 d-d g x^2)^p)}{x \sqrt {4-g x^2}} \, dx\) [680]

Optimal result

Integrand size = 36, antiderivative size = 90 \[ \int \frac {a+b \log \left (c \left (4 d-d g x^2\right )^p\right )}{x \sqrt {4-g x^2}} \, dx=-\frac {1}{2} \text {arctanh}\left (\frac {1}{2} \sqrt {4-g x^2}\right ) \left (a+b \log \left (c \left (4 d-d g x^2\right )^p\right )\right )-\frac {1}{2} b p \operatorname {PolyLog}\left (2,-\frac {1}{2} \sqrt {4-g x^2}\right )+\frac {1}{2} b p \operatorname {PolyLog}\left (2,\frac {1}{2} \sqrt {4-g x^2}\right ) \] Output:

-1/2*arctanh(1/2*(-g*x^2+4)^(1/2))*(a+b*ln(c*(-d*g*x^2+4*d)^p))-1/2*b*p*po 
lylog(2,-1/2*(-g*x^2+4)^(1/2))+1/2*b*p*polylog(2,1/2*(-g*x^2+4)^(1/2))
 

Mathematica [A] (verified)

Time = 0.11 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.16 \[ \int \frac {a+b \log \left (c \left (4 d-d g x^2\right )^p\right )}{x \sqrt {4-g x^2}} \, dx=\frac {1}{4} \left (\left (a+b \log \left (c \left (-d \left (-4+g x^2\right )\right )^p\right )\right ) \left (\log \left (2-\sqrt {4-g x^2}\right )-\log \left (2+\sqrt {4-g x^2}\right )\right )-2 b p \operatorname {PolyLog}\left (2,-\frac {1}{2} \sqrt {4-g x^2}\right )+2 b p \operatorname {PolyLog}\left (2,\frac {1}{2} \sqrt {4-g x^2}\right )\right ) \] Input:

Integrate[(a + b*Log[c*(4*d - d*g*x^2)^p])/(x*Sqrt[4 - g*x^2]),x]
 

Output:

((a + b*Log[c*(-(d*(-4 + g*x^2)))^p])*(Log[2 - Sqrt[4 - g*x^2]] - Log[2 + 
Sqrt[4 - g*x^2]]) - 2*b*p*PolyLog[2, -1/2*Sqrt[4 - g*x^2]] + 2*b*p*PolyLog 
[2, Sqrt[4 - g*x^2]/2])/4
 

Rubi [F]

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.

\(\displaystyle \int \frac {a+b \log \left (c \left (4 d-d g x^2\right )^p\right )}{x \sqrt {4-g x^2}} \, dx\)

\(\Big \downarrow \) 2929

\(\displaystyle \int \frac {a+b \log \left (c \left (4 d-d g x^2\right )^p\right )}{x \sqrt {4-g x^2}}dx\)

Input:

Int[(a + b*Log[c*(4*d - d*g*x^2)^p])/(x*Sqrt[4 - g*x^2]),x]
 

Output:

$Aborted
 

Defintions of rubi rules used

rule 2929

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))^(q_.)*((h_.)* 
(x_))^(m_.)*((f_) + (g_.)*(x_)^(s_))^(r_.), x_Symbol] :> Unintegrable[(h*x) 
^m*(f + g*x^s)^r*(a + b*Log[c*(d + e*x^n)^p])^q, x] /; FreeQ[{a, b, c, d, e 
, f, g, h, m, n, p, q, r, s}, x]
 

Maple [F]

\[\int \frac {a +b \ln \left (c \left (-d g \,x^{2}+4 d \right )^{p}\right )}{x \sqrt {-g \,x^{2}+4}}d x\]

Input:

int((a+b*ln(c*(-d*g*x^2+4*d)^p))/x/(-g*x^2+4)^(1/2),x)
 

Output:

int((a+b*ln(c*(-d*g*x^2+4*d)^p))/x/(-g*x^2+4)^(1/2),x)
 

Fricas [F]

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

integrate((a+b*log(c*(-d*g*x^2+4*d)^p))/x/(-g*x^2+4)^(1/2),x, algorithm="f 
ricas")
 

Output:

integral(-(sqrt(-g*x^2 + 4)*b*log((-d*g*x^2 + 4*d)^p*c) + sqrt(-g*x^2 + 4) 
*a)/(g*x^3 - 4*x), x)
 

Sympy [F]

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

integrate((a+b*ln(c*(-d*g*x**2+4*d)**p))/x/(-g*x**2+4)**(1/2),x)
 

Output:

Integral((a + b*log(c*(-d*g*x**2 + 4*d)**p))/(x*sqrt(-g*x**2 + 4)), x)
 

Maxima [F]

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

integrate((a+b*log(c*(-d*g*x^2+4*d)^p))/x/(-g*x^2+4)^(1/2),x, algorithm="m 
axima")
 

Output:

b*integrate((p*log(d) + log((-g*x^2 + 4)^p) + log(c))/(sqrt(-g*x^2 + 4)*x) 
, x) - 1/2*a*log(4*sqrt(-g*x^2 + 4)/abs(x) + 8/abs(x))
 

Giac [F]

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

integrate((a+b*log(c*(-d*g*x^2+4*d)^p))/x/(-g*x^2+4)^(1/2),x, algorithm="g 
iac")
 

Output:

integrate((b*log((-d*g*x^2 + 4*d)^p*c) + a)/(sqrt(-g*x^2 + 4)*x), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {a+b \log \left (c \left (4 d-d g x^2\right )^p\right )}{x \sqrt {4-g x^2}} \, dx=\int \frac {a+b\,\ln \left (c\,{\left (4\,d-d\,g\,x^2\right )}^p\right )}{x\,\sqrt {4-g\,x^2}} \,d x \] Input:

int((a + b*log(c*(4*d - d*g*x^2)^p))/(x*(4 - g*x^2)^(1/2)),x)
 

Output:

int((a + b*log(c*(4*d - d*g*x^2)^p))/(x*(4 - g*x^2)^(1/2)), x)
 

Reduce [F]

\[ \int \frac {a+b \log \left (c \left (4 d-d g x^2\right )^p\right )}{x \sqrt {4-g x^2}} \, dx=\left (\int \frac {\mathrm {log}\left (\left (-d g \,x^{2}+4 d \right )^{p} c \right )}{\sqrt {-g \,x^{2}+4}\, x}d x \right ) b +\frac {\mathrm {log}\left (\tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {g}\, x}{2}\right )}{2}\right )\right ) a}{2} \] Input:

int((a+b*log(c*(-d*g*x^2+4*d)^p))/x/(-g*x^2+4)^(1/2),x)
 

Output:

(2*int(log(( - d*g*x**2 + 4*d)**p*c)/(sqrt( - g*x**2 + 4)*x),x)*b + log(ta 
n(asin((sqrt(g)*x)/2)/2))*a)/2