\(\int \frac {a+b \log (c (d (e+f x)^p)^q)}{\sqrt {2-h x} \sqrt {2+h x}} \, dx\) [540]

Optimal result

Integrand size = 38, antiderivative size = 287 \[ \int \frac {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{\sqrt {2-h x} \sqrt {2+h x}} \, dx=\frac {i b p q \arcsin \left (\frac {h x}{2}\right )^2}{2 h}-\frac {b p q \arcsin \left (\frac {h x}{2}\right ) \log \left (1+\frac {2 e^{i \arcsin \left (\frac {h x}{2}\right )} f}{i e h-\sqrt {4 f^2-e^2 h^2}}\right )}{h}-\frac {b p q \arcsin \left (\frac {h x}{2}\right ) \log \left (1+\frac {2 e^{i \arcsin \left (\frac {h x}{2}\right )} f}{i e h+\sqrt {4 f^2-e^2 h^2}}\right )}{h}+\frac {\arcsin \left (\frac {h x}{2}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{h}+\frac {i b p q \operatorname {PolyLog}\left (2,-\frac {2 e^{i \arcsin \left (\frac {h x}{2}\right )} f}{i e h-\sqrt {4 f^2-e^2 h^2}}\right )}{h}+\frac {i b p q \operatorname {PolyLog}\left (2,-\frac {2 e^{i \arcsin \left (\frac {h x}{2}\right )} f}{i e h+\sqrt {4 f^2-e^2 h^2}}\right )}{h} \] Output:

1/2*I*b*p*q*arcsin(1/2*h*x)^2/h-b*p*q*arcsin(1/2*h*x)*ln(1+2*(1/2*I*h*x+1/ 
2*(-h^2*x^2+4)^(1/2))*f/(I*e*h-(-e^2*h^2+4*f^2)^(1/2)))/h-b*p*q*arcsin(1/2 
*h*x)*ln(1+2*(1/2*I*h*x+1/2*(-h^2*x^2+4)^(1/2))*f/(I*e*h+(-e^2*h^2+4*f^2)^ 
(1/2)))/h+arcsin(1/2*h*x)*(a+b*ln(c*(d*(f*x+e)^p)^q))/h+I*b*p*q*polylog(2, 
-2*(1/2*I*h*x+1/2*(-h^2*x^2+4)^(1/2))*f/(I*e*h-(-e^2*h^2+4*f^2)^(1/2)))/h+ 
I*b*p*q*polylog(2,-2*(1/2*I*h*x+1/2*(-h^2*x^2+4)^(1/2))*f/(I*e*h+(-e^2*h^2 
+4*f^2)^(1/2)))/h
 

Mathematica [A] (verified)

Time = 0.23 (sec) , antiderivative size = 257, normalized size of antiderivative = 0.90 \[ \int \frac {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{\sqrt {2-h x} \sqrt {2+h x}} \, dx=\frac {\arcsin \left (\frac {h x}{2}\right ) \left (2 a+i b p q \arcsin \left (\frac {h x}{2}\right )-2 b p q \log \left (1-\frac {2 e^{i \arcsin \left (\frac {h x}{2}\right )} f}{-i e h+\sqrt {4 f^2-e^2 h^2}}\right )-2 b p q \log \left (1+\frac {2 e^{i \arcsin \left (\frac {h x}{2}\right )} f}{i e h+\sqrt {4 f^2-e^2 h^2}}\right )+2 b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )+2 i b p q \operatorname {PolyLog}\left (2,\frac {2 e^{i \arcsin \left (\frac {h x}{2}\right )} f}{-i e h+\sqrt {4 f^2-e^2 h^2}}\right )+2 i b p q \operatorname {PolyLog}\left (2,-\frac {2 e^{i \arcsin \left (\frac {h x}{2}\right )} f}{i e h+\sqrt {4 f^2-e^2 h^2}}\right )}{2 h} \] Input:

Integrate[(a + b*Log[c*(d*(e + f*x)^p)^q])/(Sqrt[2 - h*x]*Sqrt[2 + h*x]),x 
]
 

Output:

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

Rubi [A] (verified)

Time = 2.57 (sec) , antiderivative size = 300, normalized size of antiderivative = 1.05, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.237, Rules used = {2895, 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.

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

\(\Big \downarrow \) 2895

\(\displaystyle \int \frac {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{\sqrt {2-h x} \sqrt {h x+2}}dx\)

\(\Big \downarrow \) 2852

\(\displaystyle \frac {\arcsin \left (\frac {h x}{2}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{h}-b f p q \int \frac {\arcsin \left (\frac {h x}{2}\right )}{h (e+f x)}dx\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\arcsin \left (\frac {h x}{2}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{h}-\frac {b f p q \int \frac {\arcsin \left (\frac {h x}{2}\right )}{e+f x}dx}{h}\)

\(\Big \downarrow \) 5240

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

\(\Big \downarrow \) 5032

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 2620

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

\(\Big \downarrow \) 2715

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

\(\Big \downarrow \) 2838

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

Input:

Int[(a + b*Log[c*(d*(e + f*x)^p)^q])/(Sqrt[2 - h*x]*Sqrt[2 + h*x]),x]
 

Output:

(ArcSin[(h*x)/2]*(a + b*Log[c*(d*(e + f*x)^p)^q]))/h - (b*f*p*q*(((-1/2*I) 
*ArcSin[(h*x)/2]^2)/f + (2*I)*(((-1/2*I)*ArcSin[(h*x)/2]*Log[1 + (2*E^(I*A 
rcSin[(h*x)/2])*f)/(I*e*h - Sqrt[4*f^2 - e^2*h^2])])/f - PolyLog[2, (-2*E^ 
(I*ArcSin[(h*x)/2])*f)/(I*e*h - Sqrt[4*f^2 - e^2*h^2])]/(2*f)) + (2*I)*((( 
-1/2*I)*ArcSin[(h*x)/2]*Log[1 + (2*E^(I*ArcSin[(h*x)/2])*f)/(I*e*h + Sqrt[ 
4*f^2 - e^2*h^2])])/f - PolyLog[2, (-2*E^(I*ArcSin[(h*x)/2])*f)/(I*e*h + S 
qrt[4*f^2 - e^2*h^2])]/(2*f))))/h
 

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 2895

Int[((a_.) + Log[(c_.)*((d_.)*((e_.) + (f_.)*(x_))^(m_.))^(n_)]*(b_.))^(p_. 
)*(u_.), x_Symbol] :> Subst[Int[u*(a + b*Log[c*d^n*(e + f*x)^(m*n)])^p, x], 
 c*d^n*(e + f*x)^(m*n), c*(d*(e + f*x)^m)^n] /; FreeQ[{a, b, c, d, e, f, m, 
 n, p}, x] &&  !IntegerQ[n] &&  !(EqQ[d, 1] && EqQ[m, 1]) && IntegralFreeQ[ 
IntHide[u*(a + b*Log[c*d^n*(e + f*x)^(m*n)])^p, x]]
 

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]

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

Input:

int((a+b*ln(c*(d*(f*x+e)^p)^q))/(-h*x+2)^(1/2)/(h*x+2)^(1/2),x)
 

Output:

int((a+b*ln(c*(d*(f*x+e)^p)^q))/(-h*x+2)^(1/2)/(h*x+2)^(1/2),x)
 

Fricas [F]

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

integrate((a+b*log(c*(d*(f*x+e)^p)^q))/(-h*x+2)^(1/2)/(h*x+2)^(1/2),x, alg 
orithm="fricas")
 

Output:

integral(-(sqrt(h*x + 2)*sqrt(-h*x + 2)*b*log(((f*x + e)^p*d)^q*c) + sqrt( 
h*x + 2)*sqrt(-h*x + 2)*a)/(h^2*x^2 - 4), x)
 

Sympy [F]

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

integrate((a+b*ln(c*(d*(f*x+e)**p)**q))/(-h*x+2)**(1/2)/(h*x+2)**(1/2),x)
 

Output:

Integral((a + b*log(c*(d*(e + f*x)**p)**q))/(sqrt(-h*x + 2)*sqrt(h*x + 2)) 
, x)
 

Maxima [F]

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

integrate((a+b*log(c*(d*(f*x+e)^p)^q))/(-h*x+2)^(1/2)/(h*x+2)^(1/2),x, alg 
orithm="maxima")
 

Output:

b*integrate((q*log(d) + log(((f*x + e)^p)^q) + log(c))/(sqrt(h*x + 2)*sqrt 
(-h*x + 2)), x) + a*arcsin(1/2*h*x)/h
                                                                                    
                                                                                    
 

Giac [F]

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

integrate((a+b*log(c*(d*(f*x+e)^p)^q))/(-h*x+2)^(1/2)/(h*x+2)^(1/2),x, alg 
orithm="giac")
 

Output:

integrate((b*log(((f*x + e)^p*d)^q*c) + a)/(sqrt(h*x + 2)*sqrt(-h*x + 2)), 
 x)
 

Mupad [F(-1)]

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

int((a + b*log(c*(d*(e + f*x)^p)^q))/((2 - h*x)^(1/2)*(h*x + 2)^(1/2)),x)
 

Output:

int((a + b*log(c*(d*(e + f*x)^p)^q))/((2 - h*x)^(1/2)*(h*x + 2)^(1/2)), x)
 

Reduce [F]

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

int((a+b*log(c*(d*(f*x+e)^p)^q))/(-h*x+2)^(1/2)/(h*x+2)^(1/2),x)
 

Output:

( - 2*asin(sqrt( - h*x + 2)/2)*a + int(log(d**q*(e + f*x)**(p*q)*c)/(sqrt( 
h*x + 2)*sqrt( - h*x + 2)),x)*b*h)/h