\(\int \frac {a+b \cot ^{-1}(\frac {\sqrt {1-c x}}{\sqrt {1+c x}})}{1-c^2 x^2} \, dx\) [6]

Optimal result

Integrand size = 38, antiderivative size = 98 \[ \int \frac {a+b \cot ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )}{1-c^2 x^2} \, dx=-\frac {a \log \left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )}{c}+\frac {i b \operatorname {PolyLog}\left (2,-\frac {i \sqrt {1+c x}}{\sqrt {1-c x}}\right )}{2 c}-\frac {i b \operatorname {PolyLog}\left (2,\frac {i \sqrt {1+c x}}{\sqrt {1-c x}}\right )}{2 c} \] Output:

-a*ln((-c*x+1)^(1/2)/(c*x+1)^(1/2))/c+1/2*I*b*polylog(2,-I*(c*x+1)^(1/2)/( 
-c*x+1)^(1/2))/c-1/2*I*b*polylog(2,I*(c*x+1)^(1/2)/(-c*x+1)^(1/2))/c
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.95 \[ \int \frac {a+b \cot ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )}{1-c^2 x^2} \, dx=-\frac {a \log \left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )-\frac {1}{2} i b \operatorname {PolyLog}\left (2,-\frac {i \sqrt {1+c x}}{\sqrt {1-c x}}\right )+\frac {1}{2} i b \operatorname {PolyLog}\left (2,\frac {i \sqrt {1+c x}}{\sqrt {1-c x}}\right )}{c} \] Input:

Integrate[(a + b*ArcCot[Sqrt[1 - c*x]/Sqrt[1 + c*x]])/(1 - c^2*x^2),x]
 

Output:

-((a*Log[Sqrt[1 - c*x]/Sqrt[1 + c*x]] - (I/2)*b*PolyLog[2, ((-I)*Sqrt[1 + 
c*x])/Sqrt[1 - c*x]] + (I/2)*b*PolyLog[2, (I*Sqrt[1 + c*x])/Sqrt[1 - c*x]] 
)/c)
 

Rubi [A] (verified)

Time = 0.33 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.95, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.079, Rules used = {7232, 5356, 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.

\(\displaystyle \int \frac {a+b \cot ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )}{1-c^2 x^2} \, dx\)

\(\Big \downarrow \) 7232

\(\displaystyle -\frac {\int \frac {\sqrt {c x+1} \left (a+b \cot ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )\right )}{\sqrt {1-c x}}d\frac {\sqrt {1-c x}}{\sqrt {c x+1}}}{c}\)

\(\Big \downarrow \) 5356

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

\(\Big \downarrow \) 2838

\(\displaystyle -\frac {a \log \left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )-\frac {1}{2} i b \operatorname {PolyLog}\left (2,-\frac {i \sqrt {c x+1}}{\sqrt {1-c x}}\right )+\frac {1}{2} i b \operatorname {PolyLog}\left (2,\frac {i \sqrt {c x+1}}{\sqrt {1-c x}}\right )}{c}\)

Input:

Int[(a + b*ArcCot[Sqrt[1 - c*x]/Sqrt[1 + c*x]])/(1 - c^2*x^2),x]
 

Output:

-((a*Log[Sqrt[1 - c*x]/Sqrt[1 + c*x]] - (I/2)*b*PolyLog[2, ((-I)*Sqrt[1 + 
c*x])/Sqrt[1 - c*x]] + (I/2)*b*PolyLog[2, (I*Sqrt[1 + c*x])/Sqrt[1 - c*x]] 
)/c)
 

Defintions of rubi rules used

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 5356

Int[((a_.) + ArcCot[(c_.)*(x_)]*(b_.))/(x_), x_Symbol] :> Simp[a*Log[x], x] 
 + (-Simp[I*(b/2)   Int[Log[1 + I/(c*x)]/x, x], x] + Simp[I*(b/2)   Int[Log 
[1 - I/(c*x)]/x, x], x]) /; FreeQ[{a, b, c}, x]
 

rule 7232

Int[((a_.) + (b_.)*(F_)[((c_.)*Sqrt[(d_.) + (e_.)*(x_)])/Sqrt[(f_.) + (g_.) 
*(x_)]])^(n_.)/((A_.) + (C_.)*(x_)^2), x_Symbol] :> Simp[2*e*(g/(C*(e*f - d 
*g)))   Subst[Int[(a + b*F[c*x])^n/x, x], x, Sqrt[d + e*x]/Sqrt[f + g*x]], 
x] /; FreeQ[{a, b, c, d, e, f, g, A, C, F}, x] && EqQ[C*d*f - A*e*g, 0] && 
EqQ[e*f + d*g, 0] && IGtQ[n, 0]
 

Maple [B] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 362 vs. \(2 (78 ) = 156\).

Time = 0.28 (sec) , antiderivative size = 363, normalized size of antiderivative = 3.70

method	result	size
default	\(-\frac {a \ln \left (c x -1\right )}{2 c}+\frac {a \ln \left (c x +1\right )}{2 c}-b \left (-\frac {\operatorname {arccot}\left (\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}\right ) \ln \left (1-\frac {i+\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}}{\sqrt {\frac {-c x +1}{c x +1}+1}}\right )}{c}+\frac {i \operatorname {polylog}\left (2, \frac {i+\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}}{\sqrt {\frac {-c x +1}{c x +1}+1}}\right )}{c}+\frac {\operatorname {arccot}\left (\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}\right ) \ln \left (1+\frac {\left (i+\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}\right )^{2}}{\frac {-c x +1}{c x +1}+1}\right )}{c}-\frac {i \operatorname {polylog}\left (2, -\frac {\left (i+\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}\right )^{2}}{\frac {-c x +1}{c x +1}+1}\right )}{2 c}-\frac {\operatorname {arccot}\left (\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}\right ) \ln \left (1+\frac {i+\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}}{\sqrt {\frac {-c x +1}{c x +1}+1}}\right )}{c}+\frac {i \operatorname {polylog}\left (2, -\frac {i+\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}}{\sqrt {\frac {-c x +1}{c x +1}+1}}\right )}{c}\right )\)	\(363\)
parts	\(-\frac {a \ln \left (c x -1\right )}{2 c}+\frac {a \ln \left (c x +1\right )}{2 c}-b \left (-\frac {\operatorname {arccot}\left (\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}\right ) \ln \left (1-\frac {i+\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}}{\sqrt {\frac {-c x +1}{c x +1}+1}}\right )}{c}+\frac {i \operatorname {polylog}\left (2, \frac {i+\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}}{\sqrt {\frac {-c x +1}{c x +1}+1}}\right )}{c}+\frac {\operatorname {arccot}\left (\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}\right ) \ln \left (1+\frac {\left (i+\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}\right )^{2}}{\frac {-c x +1}{c x +1}+1}\right )}{c}-\frac {i \operatorname {polylog}\left (2, -\frac {\left (i+\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}\right )^{2}}{\frac {-c x +1}{c x +1}+1}\right )}{2 c}-\frac {\operatorname {arccot}\left (\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}\right ) \ln \left (1+\frac {i+\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}}{\sqrt {\frac {-c x +1}{c x +1}+1}}\right )}{c}+\frac {i \operatorname {polylog}\left (2, -\frac {i+\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}}{\sqrt {\frac {-c x +1}{c x +1}+1}}\right )}{c}\right )\)	\(363\)

Input:

int((a+b*arccot((-c*x+1)^(1/2)/(c*x+1)^(1/2)))/(-c^2*x^2+1),x,method=_RETU 
RNVERBOSE)
 

Output:

-1/2*a/c*ln(c*x-1)+1/2*a/c*ln(c*x+1)-b*(-1/c*arccot((-c*x+1)^(1/2)/(c*x+1) 
^(1/2))*ln(1-(I+(-c*x+1)^(1/2)/(c*x+1)^(1/2))/((-c*x+1)/(c*x+1)+1)^(1/2))+ 
I/c*polylog(2,(I+(-c*x+1)^(1/2)/(c*x+1)^(1/2))/((-c*x+1)/(c*x+1)+1)^(1/2)) 
+1/c*arccot((-c*x+1)^(1/2)/(c*x+1)^(1/2))*ln(1+(I+(-c*x+1)^(1/2)/(c*x+1)^( 
1/2))^2/((-c*x+1)/(c*x+1)+1))-1/2*I/c*polylog(2,-(I+(-c*x+1)^(1/2)/(c*x+1) 
^(1/2))^2/((-c*x+1)/(c*x+1)+1))-1/c*arccot((-c*x+1)^(1/2)/(c*x+1)^(1/2))*l 
n(1+(I+(-c*x+1)^(1/2)/(c*x+1)^(1/2))/((-c*x+1)/(c*x+1)+1)^(1/2))+I/c*polyl 
og(2,-(I+(-c*x+1)^(1/2)/(c*x+1)^(1/2))/((-c*x+1)/(c*x+1)+1)^(1/2)))
 

Fricas [F]

\[ \int \frac {a+b \cot ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )}{1-c^2 x^2} \, dx=\int { -\frac {b \operatorname {arccot}\left (\frac {\sqrt {-c x + 1}}{\sqrt {c x + 1}}\right ) + a}{c^{2} x^{2} - 1} \,d x } \] Input:

integrate((a+b*arccot((-c*x+1)^(1/2)/(c*x+1)^(1/2)))/(-c^2*x^2+1),x, algor 
ithm="fricas")
 

Output:

integral(-(b*arccot(sqrt(-c*x + 1)/sqrt(c*x + 1)) + a)/(c^2*x^2 - 1), x)
 

Sympy [F]

\[ \int \frac {a+b \cot ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )}{1-c^2 x^2} \, dx=- \int \frac {a}{c^{2} x^{2} - 1}\, dx - \int \frac {b \operatorname {acot}{\left (\frac {\sqrt {- c x + 1}}{\sqrt {c x + 1}} \right )}}{c^{2} x^{2} - 1}\, dx \] Input:

integrate((a+b*acot((-c*x+1)**(1/2)/(c*x+1)**(1/2)))/(-c**2*x**2+1),x)
 

Output:

-Integral(a/(c**2*x**2 - 1), x) - Integral(b*acot(sqrt(-c*x + 1)/sqrt(c*x 
+ 1))/(c**2*x**2 - 1), x)
 

Maxima [F]

\[ \int \frac {a+b \cot ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )}{1-c^2 x^2} \, dx=\int { -\frac {b \operatorname {arccot}\left (\frac {\sqrt {-c x + 1}}{\sqrt {c x + 1}}\right ) + a}{c^{2} x^{2} - 1} \,d x } \] Input:

integrate((a+b*arccot((-c*x+1)^(1/2)/(c*x+1)^(1/2)))/(-c^2*x^2+1),x, algor 
ithm="maxima")
 

Output:

1/2*a*(log(c*x + 1)/c - log(c*x - 1)/c) + 1/2*((log(c*x + 1) - log(-c*x + 
1))*arctan2(sqrt(c*x + 1), sqrt(-c*x + 1)) + 2*c*integrate(1/2*(e^(1/2*log 
(c*x + 1) + 1/2*log(-c*x + 1))*log(c*x + 1) - e^(1/2*log(c*x + 1) + 1/2*lo 
g(-c*x + 1))*log(-c*x + 1))/((c^2*x^2 - 1)*(c*x + 1) - (c^2*x^2 - 1)*(c*x 
- 1)), x))*b/c
 

Giac [F]

\[ \int \frac {a+b \cot ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )}{1-c^2 x^2} \, dx=\int { -\frac {b \operatorname {arccot}\left (\frac {\sqrt {-c x + 1}}{\sqrt {c x + 1}}\right ) + a}{c^{2} x^{2} - 1} \,d x } \] Input:

integrate((a+b*arccot((-c*x+1)^(1/2)/(c*x+1)^(1/2)))/(-c^2*x^2+1),x, algor 
ithm="giac")
 

Output:

integrate(-(b*arccot(sqrt(-c*x + 1)/sqrt(c*x + 1)) + a)/(c^2*x^2 - 1), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {a+b \cot ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )}{1-c^2 x^2} \, dx=\int -\frac {a+b\,\mathrm {acot}\left (\frac {\sqrt {1-c\,x}}{\sqrt {c\,x+1}}\right )}{c^2\,x^2-1} \,d x \] Input:

int(-(a + b*acot((1 - c*x)^(1/2)/(c*x + 1)^(1/2)))/(c^2*x^2 - 1),x)
 

Output:

int(-(a + b*acot((1 - c*x)^(1/2)/(c*x + 1)^(1/2)))/(c^2*x^2 - 1), x)
 

Reduce [F]

\[ \int \frac {a+b \cot ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )}{1-c^2 x^2} \, dx=\frac {-2 \left (\int \frac {\mathit {acot} \left (\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}\right )}{c^{2} x^{2}-1}d x \right ) b c -\mathrm {log}\left (c^{2} x -c \right ) a +\mathrm {log}\left (c^{2} x +c \right ) a}{2 c} \] Input:

int((a+b*acot((-c*x+1)^(1/2)/(c*x+1)^(1/2)))/(-c^2*x^2+1),x)
 

Output:

( - 2*int(acot(sqrt( - c*x + 1)/sqrt(c*x + 1))/(c**2*x**2 - 1),x)*b*c - lo 
g(c**2*x - c)*a + log(c**2*x + c)*a)/(2*c)