3.4.11 \(\int \frac {a+b \log (c x^n)}{x \sqrt {d-e x} \sqrt {d+e x}} \, dx\) [311]

3.4.11.1 Optimal result

Integrand size = 33, antiderivative size = 301 \[ \int \frac {a+b \log \left (c x^n\right )}{x \sqrt {d-e x} \sqrt {d+e x}} \, dx=\frac {b n \sqrt {1-\frac {e^2 x^2}{d^2}} \text {arctanh}\left (\sqrt {1-\frac {e^2 x^2}{d^2}}\right )^2}{2 \sqrt {d-e x} \sqrt {d+e x}}-\frac {\sqrt {1-\frac {e^2 x^2}{d^2}} \text {arctanh}\left (\sqrt {1-\frac {e^2 x^2}{d^2}}\right ) \left (a+b \log \left (c x^n\right )\right )}{\sqrt {d-e x} \sqrt {d+e x}}-\frac {b n \sqrt {1-\frac {e^2 x^2}{d^2}} \text {arctanh}\left (\sqrt {1-\frac {e^2 x^2}{d^2}}\right ) \log \left (\frac {2}{1-\sqrt {1-\frac {e^2 x^2}{d^2}}}\right )}{\sqrt {d-e x} \sqrt {d+e x}}-\frac {b n \sqrt {1-\frac {e^2 x^2}{d^2}} \operatorname {PolyLog}\left (2,-\frac {1+\sqrt {1-\frac {e^2 x^2}{d^2}}}{1-\sqrt {1-\frac {e^2 x^2}{d^2}}}\right )}{2 \sqrt {d-e x} \sqrt {d+e x}} \]

output

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

3.4.11.2 Mathematica [A] (verified)

Time = 1.20 (sec) , antiderivative size = 310, normalized size of antiderivative = 1.03 \[ \int \frac {a+b \log \left (c x^n\right )}{x \sqrt {d-e x} \sqrt {d+e x}} \, dx=\frac {\log (x) \left (a-b n \log (x)+b \log \left (c x^n\right )\right )}{d}-\frac {\left (a-b n \log (x)+b \log \left (c x^n\right )\right ) \log \left (d+\sqrt {d-e x} \sqrt {d+e x}\right )}{d}+\frac {b n \sqrt {-d^2+e^2 x^2} \left (-\frac {4 \text {arctanh}\left (\frac {\sqrt {-d^2+e^2 x^2}}{\sqrt {-d^2}}\right ) \left (2 \log (x)-\log \left (\frac {e^2 x^2}{d^2}\right )\right )}{\sqrt {-d^2}}+\frac {\sqrt {1-\frac {e^2 x^2}{d^2}} \left (\log ^2\left (\frac {e^2 x^2}{d^2}\right )-4 \log \left (\frac {e^2 x^2}{d^2}\right ) \log \left (\frac {1}{2} \left (1+\sqrt {1-\frac {e^2 x^2}{d^2}}\right )\right )+2 \log ^2\left (\frac {1}{2} \left (1+\sqrt {1-\frac {e^2 x^2}{d^2}}\right )\right )-4 \operatorname {PolyLog}\left (2,\frac {1}{2}-\frac {1}{2} \sqrt {1-\frac {e^2 x^2}{d^2}}\right )\right )}{\sqrt {-d^2+e^2 x^2}}\right )}{8 \sqrt {d-e x} \sqrt {d+e x}} \]

input

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

output

(Log[x]*(a - b*n*Log[x] + b*Log[c*x^n]))/d - ((a - b*n*Log[x] + b*Log[c*x^ 
n])*Log[d + Sqrt[d - e*x]*Sqrt[d + e*x]])/d + (b*n*Sqrt[-d^2 + e^2*x^2]*(( 
-4*ArcTanh[Sqrt[-d^2 + e^2*x^2]/Sqrt[-d^2]]*(2*Log[x] - Log[(e^2*x^2)/d^2] 
))/Sqrt[-d^2] + (Sqrt[1 - (e^2*x^2)/d^2]*(Log[(e^2*x^2)/d^2]^2 - 4*Log[(e^ 
2*x^2)/d^2]*Log[(1 + Sqrt[1 - (e^2*x^2)/d^2])/2] + 2*Log[(1 + Sqrt[1 - (e^ 
2*x^2)/d^2])/2]^2 - 4*PolyLog[2, 1/2 - Sqrt[1 - (e^2*x^2)/d^2]/2]))/Sqrt[- 
d^2 + e^2*x^2]))/(8*Sqrt[d - e*x]*Sqrt[d + e*x])
 

3.4.11.3 Rubi [A] (verified)

Time = 1.10 (sec) , antiderivative size = 175, normalized size of antiderivative = 0.58, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.303, Rules used = {2787, 2790, 25, 7282, 7267, 25, 6546, 6470, 2849, 2752}

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.

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

\(\Big \downarrow \) 2787

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

\(\Big \downarrow \) 2790

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 7282

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

\(\Big \downarrow \) 7267

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 6546

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

\(\Big \downarrow \) 6470

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

\(\Big \downarrow \) 2849

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

\(\Big \downarrow \) 2752

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

input

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

output

(Sqrt[1 - (e^2*x^2)/d^2]*(-(ArcTanh[Sqrt[1 - (e^2*x^2)/d^2]]*(a + b*Log[c* 
x^n])) + b*n*(ArcTanh[Sqrt[1 - (e^2*x^2)/d^2]]^2/2 - ArcTanh[Sqrt[1 - (e^2 
*x^2)/d^2]]*Log[2/(1 - Sqrt[1 - (e^2*x^2)/d^2])] - PolyLog[2, 1 - 2/(1 - S 
qrt[1 - (e^2*x^2)/d^2])]/2)))/(Sqrt[d - e*x]*Sqrt[d + e*x])
 

3.4.11.3.1 Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 2752

Int[Log[(c_.)*(x_)]/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[(-e^(-1))*PolyLo 
g[2, 1 - c*x], x] /; FreeQ[{c, d, e}, x] && EqQ[e + c*d, 0]
 

rule 2787

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*(x_)^(m_.)*((d1_) + (e1_.)*(x_))^ 
(q_)*((d2_) + (e2_.)*(x_))^(q_), x_Symbol] :> Simp[(d1 + e1*x)^q*((d2 + e2* 
x)^q/(1 + e1*(e2/(d1*d2))*x^2)^q)   Int[x^m*(1 + e1*(e2/(d1*d2))*x^2)^q*(a 
+ b*Log[c*x^n]), x], x] /; FreeQ[{a, b, c, d1, e1, d2, e2, n}, x] && EqQ[d2 
*e1 + d1*e2, 0] && IntegerQ[m] && IntegerQ[q - 1/2]
 

rule 2790

Int[(((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((d_) + (e_.)*(x_)^(r_.))^(q_.)) 
/(x_), x_Symbol] :> With[{u = IntHide[(d + e*x^r)^q/x, x]}, Simp[u*(a + b*L 
og[c*x^n]), x] - Simp[b*n   Int[1/x   u, x], x]] /; FreeQ[{a, b, c, d, e, n 
, r}, x] && IntegerQ[q - 1/2]
 

rule 2849

Int[Log[(c_.)/((d_) + (e_.)*(x_))]/((f_) + (g_.)*(x_)^2), x_Symbol] :> Simp 
[-e/g   Subst[Int[Log[2*d*x]/(1 - 2*d*x), x], x, 1/(d + e*x)], x] /; FreeQ[ 
{c, d, e, f, g}, x] && EqQ[c, 2*d] && EqQ[e^2*f + d^2*g, 0]
 

rule 6470

Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)), x_Symbol 
] :> Simp[(-(a + b*ArcTanh[c*x])^p)*(Log[2/(1 + e*(x/d))]/e), x] + Simp[b*c 
*(p/e)   Int[(a + b*ArcTanh[c*x])^(p - 1)*(Log[2/(1 + e*(x/d))]/(1 - c^2*x^ 
2)), x], x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 0] && EqQ[c^2*d^2 - e^2 
, 0]
 

rule 6546

Int[(((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)*(x_))/((d_) + (e_.)*(x_)^2), 
 x_Symbol] :> Simp[(a + b*ArcTanh[c*x])^(p + 1)/(b*e*(p + 1)), x] + Simp[1/ 
(c*d)   Int[(a + b*ArcTanh[c*x])^p/(1 - c*x), x], x] /; FreeQ[{a, b, c, d, 
e}, x] && EqQ[c^2*d + e, 0] && IGtQ[p, 0]
 

rule 7267

Int[u_, x_Symbol] :> With[{lst = SubstForFractionalPowerOfLinear[u, x]}, Si 
mp[lst[[2]]*lst[[4]]   Subst[Int[lst[[1]], x], x, lst[[3]]^(1/lst[[2]])], x 
] /;  !FalseQ[lst] && SubstForFractionalPowerQ[u, lst[[3]], x]]
 

rule 7282

Int[(u_)/(x_), x_Symbol] :> With[{lst = PowerVariableExpn[u, 0, x]}, Simp[1 
/lst[[2]]   Subst[Int[NormalizeIntegrand[Simplify[lst[[1]]/x], x], x], x, ( 
lst[[3]]*x)^lst[[2]]], x] /;  !FalseQ[lst] && NeQ[lst[[2]], 0]] /; NonsumQ[ 
u] &&  !RationalFunctionQ[u, x]
 

3.4.11.4 Maple [F]

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

input

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

output

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

3.4.11.5 Fricas [F]

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

input

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

output

integral(-(sqrt(e*x + d)*sqrt(-e*x + d)*b*log(c*x^n) + sqrt(e*x + d)*sqrt( 
-e*x + d)*a)/(e^2*x^3 - d^2*x), x)
 

3.4.11.6 Sympy [F]

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

input

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

output

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

3.4.11.7 Maxima [F]

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

input

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

output

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

3.4.11.8 Giac [F]

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

input

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

output

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

3.4.11.9 Mupad [F(-1)]

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

input

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

output

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