\(\int \frac {a+b \log (c (d+e x)^n)}{\sqrt {2-g x} \sqrt {2+g x}} \, dx\) [277]

Optimal result

Integrand size = 34, antiderivative size = 278 \[ \int \frac {a+b \log \left (c (d+e x)^n\right )}{\sqrt {2-g x} \sqrt {2+g x}} \, dx=\frac {i b n \arcsin \left (\frac {g x}{2}\right )^2}{2 g}-\frac {b n \arcsin \left (\frac {g x}{2}\right ) \log \left (1+\frac {2 e e^{i \arcsin \left (\frac {g x}{2}\right )}}{i d g-\sqrt {4 e^2-d^2 g^2}}\right )}{g}-\frac {b n \arcsin \left (\frac {g x}{2}\right ) \log \left (1+\frac {2 e e^{i \arcsin \left (\frac {g x}{2}\right )}}{i d g+\sqrt {4 e^2-d^2 g^2}}\right )}{g}+\frac {\arcsin \left (\frac {g x}{2}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{g}+\frac {i b n \operatorname {PolyLog}\left (2,-\frac {2 e e^{i \arcsin \left (\frac {g x}{2}\right )}}{i d g-\sqrt {4 e^2-d^2 g^2}}\right )}{g}+\frac {i b n \operatorname {PolyLog}\left (2,-\frac {2 e e^{i \arcsin \left (\frac {g x}{2}\right )}}{i d g+\sqrt {4 e^2-d^2 g^2}}\right )}{g} \] Output:

1/2*I*b*n*arcsin(1/2*g*x)^2/g-b*n*arcsin(1/2*g*x)*ln(1+2*e*(1/2*I*g*x+1/2* 
(-g^2*x^2+4)^(1/2))/(I*d*g-(-d^2*g^2+4*e^2)^(1/2)))/g-b*n*arcsin(1/2*g*x)* 
ln(1+2*e*(1/2*I*g*x+1/2*(-g^2*x^2+4)^(1/2))/(I*d*g+(-d^2*g^2+4*e^2)^(1/2)) 
)/g+arcsin(1/2*g*x)*(a+b*ln(c*(e*x+d)^n))/g+I*b*n*polylog(2,-2*e*(1/2*I*g* 
x+1/2*(-g^2*x^2+4)^(1/2))/(I*d*g-(-d^2*g^2+4*e^2)^(1/2)))/g+I*b*n*polylog( 
2,-2*e*(1/2*I*g*x+1/2*(-g^2*x^2+4)^(1/2))/(I*d*g+(-d^2*g^2+4*e^2)^(1/2)))/ 
g
 

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 307, normalized size of antiderivative = 1.10 \[ \int \frac {a+b \log \left (c (d+e x)^n\right )}{\sqrt {2-g x} \sqrt {2+g x}} \, dx=\frac {a \arcsin \left (\frac {g x}{2}\right )}{g}+\frac {i b n \arcsin \left (\frac {g x}{2}\right )^2}{2 g}-\frac {b n \arcsin \left (\frac {g x}{2}\right ) \log \left (1+\frac {e e^{i \arcsin \left (\frac {g x}{2}\right )} g}{\frac {1}{2} i d g^2-\frac {1}{2} g \sqrt {4 e^2-d^2 g^2}}\right )}{g}-\frac {b n \arcsin \left (\frac {g x}{2}\right ) \log \left (1+\frac {e e^{i \arcsin \left (\frac {g x}{2}\right )} g}{\frac {1}{2} i d g^2+\frac {1}{2} g \sqrt {4 e^2-d^2 g^2}}\right )}{g}+\frac {b \arcsin \left (\frac {g x}{2}\right ) \log \left (c (d+e x)^n\right )}{g}+\frac {i b n \operatorname {PolyLog}\left (2,\frac {2 i e e^{i \arcsin \left (\frac {g x}{2}\right )}}{d g-i \sqrt {4 e^2-d^2 g^2}}\right )}{g}+\frac {i b n \operatorname {PolyLog}\left (2,\frac {2 i e e^{i \arcsin \left (\frac {g x}{2}\right )}}{d g+i \sqrt {4 e^2-d^2 g^2}}\right )}{g} \] Input:

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

Output:

(a*ArcSin[(g*x)/2])/g + ((I/2)*b*n*ArcSin[(g*x)/2]^2)/g - (b*n*ArcSin[(g*x 
)/2]*Log[1 + (e*E^(I*ArcSin[(g*x)/2])*g)/((I/2)*d*g^2 - (g*Sqrt[4*e^2 - d^ 
2*g^2])/2)])/g - (b*n*ArcSin[(g*x)/2]*Log[1 + (e*E^(I*ArcSin[(g*x)/2])*g)/ 
((I/2)*d*g^2 + (g*Sqrt[4*e^2 - d^2*g^2])/2)])/g + (b*ArcSin[(g*x)/2]*Log[c 
*(d + e*x)^n])/g + (I*b*n*PolyLog[2, ((2*I)*e*E^(I*ArcSin[(g*x)/2]))/(d*g 
- I*Sqrt[4*e^2 - d^2*g^2])])/g + (I*b*n*PolyLog[2, ((2*I)*e*E^(I*ArcSin[(g 
*x)/2]))/(d*g + I*Sqrt[4*e^2 - d^2*g^2])])/g
 

Rubi [A] (verified)

Time = 1.54 (sec) , antiderivative size = 295, normalized size of antiderivative = 1.06, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {2852, 27, 5240, 5032, 27, 2620, 2715, 2838}

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.

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

\(\Big \downarrow \) 2852

\(\displaystyle \frac {\arcsin \left (\frac {g x}{2}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{g}-b e n \int \frac {\arcsin \left (\frac {g x}{2}\right )}{g (d+e x)}dx\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\arcsin \left (\frac {g x}{2}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{g}-\frac {b e n \int \frac {\arcsin \left (\frac {g x}{2}\right )}{d+e x}dx}{g}\)

\(\Big \downarrow \) 5240

\(\displaystyle \frac {\arcsin \left (\frac {g x}{2}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{g}-\frac {b e n \int \frac {\sqrt {1-\frac {g^2 x^2}{4}} \arcsin \left (\frac {g x}{2}\right )}{\frac {d g}{2}+\frac {e x g}{2}}d\arcsin \left (\frac {g x}{2}\right )}{g}\)

\(\Big \downarrow \) 5032

\(\displaystyle \frac {\arcsin \left (\frac {g x}{2}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{g}-\frac {b e n \left (i \int \frac {2 e^{i \arcsin \left (\frac {g x}{2}\right )} \arcsin \left (\frac {g x}{2}\right )}{2 e^{i \arcsin \left (\frac {g x}{2}\right )} e+i d g-\sqrt {4 e^2-d^2 g^2}}d\arcsin \left (\frac {g x}{2}\right )+i \int \frac {2 e^{i \arcsin \left (\frac {g x}{2}\right )} \arcsin \left (\frac {g x}{2}\right )}{2 e^{i \arcsin \left (\frac {g x}{2}\right )} e+i d g+\sqrt {4 e^2-d^2 g^2}}d\arcsin \left (\frac {g x}{2}\right )-\frac {i \arcsin \left (\frac {g x}{2}\right )^2}{2 e}\right )}{g}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\arcsin \left (\frac {g x}{2}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{g}-\frac {b e n \left (2 i \int \frac {e^{i \arcsin \left (\frac {g x}{2}\right )} \arcsin \left (\frac {g x}{2}\right )}{2 e^{i \arcsin \left (\frac {g x}{2}\right )} e+i d g-\sqrt {4 e^2-d^2 g^2}}d\arcsin \left (\frac {g x}{2}\right )+2 i \int \frac {e^{i \arcsin \left (\frac {g x}{2}\right )} \arcsin \left (\frac {g x}{2}\right )}{2 e^{i \arcsin \left (\frac {g x}{2}\right )} e+i d g+\sqrt {4 e^2-d^2 g^2}}d\arcsin \left (\frac {g x}{2}\right )-\frac {i \arcsin \left (\frac {g x}{2}\right )^2}{2 e}\right )}{g}\)

\(\Big \downarrow \) 2620

\(\displaystyle \frac {\arcsin \left (\frac {g x}{2}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{g}-\frac {b e n \left (2 i \left (\frac {i \int \log \left (\frac {2 e^{i \arcsin \left (\frac {g x}{2}\right )} e}{i d g-\sqrt {4 e^2-d^2 g^2}}+1\right )d\arcsin \left (\frac {g x}{2}\right )}{2 e}-\frac {i \arcsin \left (\frac {g x}{2}\right ) \log \left (1+\frac {2 e e^{i \arcsin \left (\frac {g x}{2}\right )}}{-\sqrt {4 e^2-d^2 g^2}+i d g}\right )}{2 e}\right )+2 i \left (\frac {i \int \log \left (\frac {2 e^{i \arcsin \left (\frac {g x}{2}\right )} e}{i d g+\sqrt {4 e^2-d^2 g^2}}+1\right )d\arcsin \left (\frac {g x}{2}\right )}{2 e}-\frac {i \arcsin \left (\frac {g x}{2}\right ) \log \left (1+\frac {2 e e^{i \arcsin \left (\frac {g x}{2}\right )}}{\sqrt {4 e^2-d^2 g^2}+i d g}\right )}{2 e}\right )-\frac {i \arcsin \left (\frac {g x}{2}\right )^2}{2 e}\right )}{g}\)

\(\Big \downarrow \) 2715

\(\displaystyle \frac {\arcsin \left (\frac {g x}{2}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{g}-\frac {b e n \left (2 i \left (\frac {\int e^{-i \arcsin \left (\frac {g x}{2}\right )} \log \left (\frac {2 e^{i \arcsin \left (\frac {g x}{2}\right )} e}{i d g-\sqrt {4 e^2-d^2 g^2}}+1\right )de^{i \arcsin \left (\frac {g x}{2}\right )}}{2 e}-\frac {i \arcsin \left (\frac {g x}{2}\right ) \log \left (1+\frac {2 e e^{i \arcsin \left (\frac {g x}{2}\right )}}{-\sqrt {4 e^2-d^2 g^2}+i d g}\right )}{2 e}\right )+2 i \left (\frac {\int e^{-i \arcsin \left (\frac {g x}{2}\right )} \log \left (\frac {2 e^{i \arcsin \left (\frac {g x}{2}\right )} e}{i d g+\sqrt {4 e^2-d^2 g^2}}+1\right )de^{i \arcsin \left (\frac {g x}{2}\right )}}{2 e}-\frac {i \arcsin \left (\frac {g x}{2}\right ) \log \left (1+\frac {2 e e^{i \arcsin \left (\frac {g x}{2}\right )}}{\sqrt {4 e^2-d^2 g^2}+i d g}\right )}{2 e}\right )-\frac {i \arcsin \left (\frac {g x}{2}\right )^2}{2 e}\right )}{g}\)

\(\Big \downarrow \) 2838

\(\displaystyle \frac {\arcsin \left (\frac {g x}{2}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{g}-\frac {b e n \left (2 i \left (-\frac {\operatorname {PolyLog}\left (2,-\frac {2 e e^{i \arcsin \left (\frac {g x}{2}\right )}}{i d g-\sqrt {4 e^2-d^2 g^2}}\right )}{2 e}-\frac {i \arcsin \left (\frac {g x}{2}\right ) \log \left (1+\frac {2 e e^{i \arcsin \left (\frac {g x}{2}\right )}}{-\sqrt {4 e^2-d^2 g^2}+i d g}\right )}{2 e}\right )+2 i \left (-\frac {\operatorname {PolyLog}\left (2,-\frac {2 e e^{i \arcsin \left (\frac {g x}{2}\right )}}{i d g+\sqrt {4 e^2-d^2 g^2}}\right )}{2 e}-\frac {i \arcsin \left (\frac {g x}{2}\right ) \log \left (1+\frac {2 e e^{i \arcsin \left (\frac {g x}{2}\right )}}{\sqrt {4 e^2-d^2 g^2}+i d g}\right )}{2 e}\right )-\frac {i \arcsin \left (\frac {g x}{2}\right )^2}{2 e}\right )}{g}\)

Input:

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

Output:

(ArcSin[(g*x)/2]*(a + b*Log[c*(d + e*x)^n]))/g - (b*e*n*(((-1/2*I)*ArcSin[ 
(g*x)/2]^2)/e + (2*I)*(((-1/2*I)*ArcSin[(g*x)/2]*Log[1 + (2*e*E^(I*ArcSin[ 
(g*x)/2]))/(I*d*g - Sqrt[4*e^2 - d^2*g^2])])/e - PolyLog[2, (-2*e*E^(I*Arc 
Sin[(g*x)/2]))/(I*d*g - Sqrt[4*e^2 - d^2*g^2])]/(2*e)) + (2*I)*(((-1/2*I)* 
ArcSin[(g*x)/2]*Log[1 + (2*e*E^(I*ArcSin[(g*x)/2]))/(I*d*g + Sqrt[4*e^2 - 
d^2*g^2])])/e - PolyLog[2, (-2*e*E^(I*ArcSin[(g*x)/2]))/(I*d*g + Sqrt[4*e^ 
2 - d^2*g^2])]/(2*e))))/g
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 2620

Int[(((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.))/ 
((a_) + (b_.)*((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)), x_Symbol] :> Simp 
[((c + d*x)^m/(b*f*g*n*Log[F]))*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x] - Si 
mp[d*(m/(b*f*g*n*Log[F]))   Int[(c + d*x)^(m - 1)*Log[1 + b*((F^(g*(e + f*x 
)))^n/a)], x], x] /; FreeQ[{F, a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]
 

rule 2715

Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] 
:> Simp[1/(d*e*n*Log[F])   Subst[Int[Log[a + b*x]/x, x], x, (F^(e*(c + d*x) 
))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]
 

rule 2838

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2 
, (-c)*e*x^n]/n, x] /; FreeQ[{c, d, e, n}, x] && EqQ[c*d, 1]
 

rule 2852

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))/(Sqrt[(f1_) + (g1_ 
.)*(x_)]*Sqrt[(f2_) + (g2_.)*(x_)]), x_Symbol] :> With[{u = IntHide[1/Sqrt[ 
f1*f2 + g1*g2*x^2], x]}, Simp[u*(a + b*Log[c*(d + e*x)^n]), x] - Simp[b*e*n 
   Int[SimplifyIntegrand[u/(d + e*x), x], x], x]] /; FreeQ[{a, b, c, d, e, 
f1, g1, f2, g2, n}, x] && EqQ[f2*g1 + f1*g2, 0] && GtQ[f1, 0] && GtQ[f2, 0]
 

rule 5032

Int[(Cos[(c_.) + (d_.)*(x_)]*((e_.) + (f_.)*(x_))^(m_.))/((a_) + (b_.)*Sin[ 
(c_.) + (d_.)*(x_)]), x_Symbol] :> Simp[(-I)*((e + f*x)^(m + 1)/(b*f*(m + 1 
))), x] + (Simp[I   Int[(e + f*x)^m*(E^(I*(c + d*x))/(I*a - Rt[-a^2 + b^2, 
2] + b*E^(I*(c + d*x)))), x], x] + Simp[I   Int[(e + f*x)^m*(E^(I*(c + d*x) 
)/(I*a + Rt[-a^2 + b^2, 2] + b*E^(I*(c + d*x)))), x], x]) /; FreeQ[{a, b, c 
, d, e, f}, x] && IGtQ[m, 0] && NegQ[a^2 - b^2]
 

rule 5240

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)/((d_) + (e_.)*(x_)), x_Symbol] 
 :> Subst[Int[(a + b*x)^n*(Cos[x]/(c*d + e*Sin[x])), x], x, ArcSin[c*x]] /; 
 FreeQ[{a, b, c, d, e}, x] && IGtQ[n, 0]
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

integrate((a+b*log(c*(e*x+d)^n))/(-g*x+2)^(1/2)/(g*x+2)^(1/2),x, algorithm 
="fricas")
 

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

integrate((a+b*log(c*(e*x+d)^n))/(-g*x+2)^(1/2)/(g*x+2)^(1/2),x, algorithm 
="maxima")
 

Output:

b*integrate((log((e*x + d)^n) + log(c))/(sqrt(g*x + 2)*sqrt(-g*x + 2)), x) 
 + a*arcsin(1/2*g*x)/g
 

Giac [F]

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

integrate((a+b*log(c*(e*x+d)^n))/(-g*x+2)^(1/2)/(g*x+2)^(1/2),x, algorithm 
="giac")
 

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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