3.66 \(\int \frac{\cot ^{-1}(x)}{\sqrt{a+a x^2}} \, dx\)

Optimal. Leaf size=155 \[ -\frac{i \sqrt{x^2+1} \text{PolyLog}\left (2,-\frac{i \sqrt{1+i x}}{\sqrt{1-i x}}\right )}{\sqrt{a x^2+a}}+\frac{i \sqrt{x^2+1} \text{PolyLog}\left (2,\frac{i \sqrt{1+i x}}{\sqrt{1-i x}}\right )}{\sqrt{a x^2+a}}-\frac{2 i \sqrt{x^2+1} \tan ^{-1}\left (\frac{\sqrt{1+i x}}{\sqrt{1-i x}}\right ) \cot ^{-1}(x)}{\sqrt{a x^2+a}} \]

[Out]

((-2*I)*Sqrt[1 + x^2]*ArcCot[x]*ArcTan[Sqrt[1 + I*x]/Sqrt[1 - I*x]])/Sqrt[a + a*x^2] - (I*Sqrt[1 + x^2]*PolyLo
g[2, ((-I)*Sqrt[1 + I*x])/Sqrt[1 - I*x]])/Sqrt[a + a*x^2] + (I*Sqrt[1 + x^2]*PolyLog[2, (I*Sqrt[1 + I*x])/Sqrt
[1 - I*x]])/Sqrt[a + a*x^2]

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Rubi [A]  time = 0.0447531, antiderivative size = 155, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {4891, 4887} \[ -\frac{i \sqrt{x^2+1} \text{PolyLog}\left (2,-\frac{i \sqrt{1+i x}}{\sqrt{1-i x}}\right )}{\sqrt{a x^2+a}}+\frac{i \sqrt{x^2+1} \text{PolyLog}\left (2,\frac{i \sqrt{1+i x}}{\sqrt{1-i x}}\right )}{\sqrt{a x^2+a}}-\frac{2 i \sqrt{x^2+1} \tan ^{-1}\left (\frac{\sqrt{1+i x}}{\sqrt{1-i x}}\right ) \cot ^{-1}(x)}{\sqrt{a x^2+a}} \]

Antiderivative was successfully verified.

[In]

Int[ArcCot[x]/Sqrt[a + a*x^2],x]

[Out]

((-2*I)*Sqrt[1 + x^2]*ArcCot[x]*ArcTan[Sqrt[1 + I*x]/Sqrt[1 - I*x]])/Sqrt[a + a*x^2] - (I*Sqrt[1 + x^2]*PolyLo
g[2, ((-I)*Sqrt[1 + I*x])/Sqrt[1 - I*x]])/Sqrt[a + a*x^2] + (I*Sqrt[1 + x^2]*PolyLog[2, (I*Sqrt[1 + I*x])/Sqrt
[1 - I*x]])/Sqrt[a + a*x^2]

Rule 4891

Int[((a_.) + ArcCot[(c_.)*(x_)]*(b_.))^(p_.)/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Dist[Sqrt[1 + c^2*x^2]/Sq
rt[d + e*x^2], Int[(a + b*ArcCot[c*x])^p/Sqrt[1 + c^2*x^2], x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*
d] && IGtQ[p, 0] &&  !GtQ[d, 0]

Rule 4887

Int[((a_.) + ArcCot[(c_.)*(x_)]*(b_.))/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[(-2*I*(a + b*ArcCot[c*x])*
ArcTan[Sqrt[1 + I*c*x]/Sqrt[1 - I*c*x]])/(c*Sqrt[d]), x] + (-Simp[(I*b*PolyLog[2, -((I*Sqrt[1 + I*c*x])/Sqrt[1
- I*c*x])])/(c*Sqrt[d]), x] + Simp[(I*b*PolyLog[2, (I*Sqrt[1 + I*c*x])/Sqrt[1 - I*c*x]])/(c*Sqrt[d]), x]) /;
FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && GtQ[d, 0]

Rubi steps

\begin{align*} \int \frac{\cot ^{-1}(x)}{\sqrt{a+a x^2}} \, dx &=\frac{\sqrt{1+x^2} \int \frac{\cot ^{-1}(x)}{\sqrt{1+x^2}} \, dx}{\sqrt{a+a x^2}}\\ &=-\frac{2 i \sqrt{1+x^2} \cot ^{-1}(x) \tan ^{-1}\left (\frac{\sqrt{1+i x}}{\sqrt{1-i x}}\right )}{\sqrt{a+a x^2}}-\frac{i \sqrt{1+x^2} \text{Li}_2\left (-\frac{i \sqrt{1+i x}}{\sqrt{1-i x}}\right )}{\sqrt{a+a x^2}}+\frac{i \sqrt{1+x^2} \text{Li}_2\left (\frac{i \sqrt{1+i x}}{\sqrt{1-i x}}\right )}{\sqrt{a+a x^2}}\\ \end{align*}

Mathematica [A]  time = 0.102603, size = 89, normalized size = 0.57 \[ -\frac{\sqrt{a \left (x^2+1\right )} \left (i \text{PolyLog}\left (2,-e^{i \cot ^{-1}(x)}\right )-i \text{PolyLog}\left (2,e^{i \cot ^{-1}(x)}\right )+\cot ^{-1}(x) \left (\log \left (1-e^{i \cot ^{-1}(x)}\right )-\log \left (1+e^{i \cot ^{-1}(x)}\right )\right )\right )}{a \sqrt{\frac{1}{x^2}+1} x} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[ArcCot[x]/Sqrt[a + a*x^2],x]

[Out]

-((Sqrt[a*(1 + x^2)]*(ArcCot[x]*(Log[1 - E^(I*ArcCot[x])] - Log[1 + E^(I*ArcCot[x])]) + I*PolyLog[2, -E^(I*Arc
Cot[x])] - I*PolyLog[2, E^(I*ArcCot[x])]))/(a*Sqrt[1 + x^(-2)]*x))

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Maple [A]  time = 0.57, size = 99, normalized size = 0.6 \begin{align*}{\frac{i}{a} \left ( i{\rm arccot} \left (x\right )\ln \left ( 1-{(x+i){\frac{1}{\sqrt{{x}^{2}+1}}}} \right ) -i{\rm arccot} \left (x\right )\ln \left ( 1+{(x+i){\frac{1}{\sqrt{{x}^{2}+1}}}} \right ) +{\it polylog} \left ( 2,{(x+i){\frac{1}{\sqrt{{x}^{2}+1}}}} \right ) -{\it polylog} \left ( 2,-{(x+i){\frac{1}{\sqrt{{x}^{2}+1}}}} \right ) \right ) \sqrt{a \left ( x+i \right ) \left ( x-i \right ) }{\frac{1}{\sqrt{{x}^{2}+1}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(arccot(x)/(a*x^2+a)^(1/2),x)

[Out]

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

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\operatorname{arccot}\left (x\right )}{\sqrt{a x^{2} + a}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(arccot(x)/sqrt(a*x^2 + a), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{acot}{\left (x \right )}}{\sqrt{a \left (x^{2} + 1\right )}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{arccot}\left (x\right )}{\sqrt{a x^{2} + a}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(arccot(x)/sqrt(a*x^2 + a), x)