\(\int \frac {a+b \cot ^{-1}(\frac {\sqrt {1-c x}}{\sqrt {1+c x}})}{1-c^2 x^2} \, dx\) [154]

Optimal result

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

[Out]

-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

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {212, 6813, 4941, 2438} \[ \int \frac {a+b \cot ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )}{1-c^2 x^2} \, dx=-\frac {a \log \left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )}{c}+\frac {i b \operatorname {PolyLog}\left (2,-\frac {i \sqrt {c x+1}}{\sqrt {1-c x}}\right )}{2 c}-\frac {i b \operatorname {PolyLog}\left (2,\frac {i \sqrt {c x+1}}{\sqrt {1-c x}}\right )}{2 c} \]

[In]

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

[Out]

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

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 2438

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 4941

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

Rule 6813

Int[((a_.) + (b_.)*(F_)[((c_.)*Sqrt[(d_.) + (e_.)*(x_)])/Sqrt[(f_.) + (g_.)*(x_)]])^(n_.)/((A_.) + (C_.)*(x_)^
2), x_Symbol] :> Dist[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]

Rubi steps \begin{align*} \text {integral}& = -\frac {\text {Subst}\left (\int \frac {a+b \cot ^{-1}(x)}{x} \, dx,x,\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )}{c} \\ & = -\frac {a \log \left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )}{c}-\frac {(i b) \text {Subst}\left (\int \frac {\log \left (1-\frac {i}{x}\right )}{x} \, dx,x,\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )}{2 c}+\frac {(i b) \text {Subst}\left (\int \frac {\log \left (1+\frac {i}{x}\right )}{x} \, dx,x,\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )}{2 c} \\ & = -\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} \\ \end{align*}

Mathematica [A] (verified)

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

[In]

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

[Out]

-((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*PolyL
og[2, (I*Sqrt[1 + c*x])/Sqrt[1 - c*x]])/c)

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 = 0.43 (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	\(-\frac {a \ln \left (c x -1\right )}{2 c}+\frac {a \ln \left (c x +1\right )}{2 c}-b \left (-\frac {\operatorname {arccot}\left (\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}\right ) \ln \left (1-\frac {i+\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}}{\sqrt {\frac {-c x +1}{c x +1}+1}}\right )}{c}+\frac {i \operatorname {polylog}\left (2, \frac {i+\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}}{\sqrt {\frac {-c x +1}{c x +1}+1}}\right )}{c}+\frac {\operatorname {arccot}\left (\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}\right ) \ln \left (1+\frac {\left (i+\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}\right )^{2}}{\frac {-c x +1}{c x +1}+1}\right )}{c}-\frac {i \operatorname {polylog}\left (2, -\frac {\left (i+\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}\right )^{2}}{\frac {-c x +1}{c x +1}+1}\right )}{2 c}-\frac {\operatorname {arccot}\left (\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}\right ) \ln \left (1+\frac {i+\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}}{\sqrt {\frac {-c x +1}{c x +1}+1}}\right )}{c}+\frac {i \operatorname {polylog}\left (2, -\frac {i+\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}}{\sqrt {\frac {-c x +1}{c x +1}+1}}\right )}{c}\right )\)	\(363\)

[In]

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

[Out]

-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
))*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)))

Fricas [F]

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

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

-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]

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

[In]

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

[Out]

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*
log(-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]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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