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

Optimal. Leaf size=195 \[ -\frac{i a \sqrt{x^2+1} \text{PolyLog}\left (2,-\frac{i \sqrt{1+i x}}{\sqrt{1-i x}}\right )}{2 \sqrt{a x^2+a}}+\frac{i a \sqrt{x^2+1} \text{PolyLog}\left (2,\frac{i \sqrt{1+i x}}{\sqrt{1-i x}}\right )}{2 \sqrt{a x^2+a}}+\frac{1}{2} \sqrt{a x^2+a}+\frac{1}{2} x \sqrt{a x^2+a} \cot ^{-1}(x)-\frac{i a \sqrt{x^2+1} \tan ^{-1}\left (\frac{\sqrt{1+i x}}{\sqrt{1-i x}}\right ) \cot ^{-1}(x)}{\sqrt{a x^2+a}} \]

[Out]

Sqrt[a + a*x^2]/2 + (x*Sqrt[a + a*x^2]*ArcCot[x])/2 - (I*a*Sqrt[1 + x^2]*ArcCot[x]*ArcTan[Sqrt[1 + I*x]/Sqrt[1
- I*x]])/Sqrt[a + a*x^2] - ((I/2)*a*Sqrt[1 + x^2]*PolyLog[2, ((-I)*Sqrt[1 + I*x])/Sqrt[1 - I*x]])/Sqrt[a + a*
x^2] + ((I/2)*a*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 = 0.0718277, antiderivative size = 195, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214, Rules used = {4879, 4891, 4887} \[ -\frac{i a \sqrt{x^2+1} \text{PolyLog}\left (2,-\frac{i \sqrt{1+i x}}{\sqrt{1-i x}}\right )}{2 \sqrt{a x^2+a}}+\frac{i a \sqrt{x^2+1} \text{PolyLog}\left (2,\frac{i \sqrt{1+i x}}{\sqrt{1-i x}}\right )}{2 \sqrt{a x^2+a}}+\frac{1}{2} \sqrt{a x^2+a}+\frac{1}{2} x \sqrt{a x^2+a} \cot ^{-1}(x)-\frac{i a \sqrt{x^2+1} \tan ^{-1}\left (\frac{\sqrt{1+i x}}{\sqrt{1-i x}}\right ) \cot ^{-1}(x)}{\sqrt{a x^2+a}} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[a + a*x^2]*ArcCot[x],x]

[Out]

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

Rule 4879

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

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

Mathematica [A]  time = 1.11405, size = 136, normalized size = 0.7 \[ -\frac{\left (a \left (x^2+1\right )\right )^{3/2} \left (4 i \text{PolyLog}\left (2,-e^{i \cot ^{-1}(x)}\right )-4 i \text{PolyLog}\left (2,e^{i \cot ^{-1}(x)}\right )-2 \cot \left (\frac{1}{2} \cot ^{-1}(x)\right )+4 \cot ^{-1}(x) \log \left (1-e^{i \cot ^{-1}(x)}\right )-4 \cot ^{-1}(x) \log \left (1+e^{i \cot ^{-1}(x)}\right )-2 \tan \left (\frac{1}{2} \cot ^{-1}(x)\right )-\cot ^{-1}(x) \csc ^2\left (\frac{1}{2} \cot ^{-1}(x)\right )+\cot ^{-1}(x) \sec ^2\left (\frac{1}{2} \cot ^{-1}(x)\right )\right )}{8 a \left (\frac{1}{x^2}+1\right )^{3/2} x^3} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

-((a*(1 + x^2))^(3/2)*(-2*Cot[ArcCot[x]/2] - ArcCot[x]*Csc[ArcCot[x]/2]^2 + 4*ArcCot[x]*Log[1 - E^(I*ArcCot[x]
)] - 4*ArcCot[x]*Log[1 + E^(I*ArcCot[x])] + (4*I)*PolyLog[2, -E^(I*ArcCot[x])] - (4*I)*PolyLog[2, E^(I*ArcCot[
x])] + ArcCot[x]*Sec[ArcCot[x]/2]^2 - 2*Tan[ArcCot[x]/2]))/(8*a*(1 + x^(-2))^(3/2)*x^3)

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Maple [A]  time = 0.7, size = 117, normalized size = 0.6 \begin{align*}{\frac{x{\rm arccot} \left (x\right )+1}{2}\sqrt{a \left ( x+i \right ) \left ( x-i \right ) }}-{{\frac{i}{2}}\sqrt{a \left ( x+i \right ) \left ( x-i \right ) } \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 ){\frac{1}{\sqrt{{x}^{2}+1}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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((a*x^2+a)^(1/2)*arccot(x),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 (\sqrt{a x^{2} + a} \operatorname{arccot}\left (x\right ), x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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