Integrand size = 14, antiderivative size = 155 \[ \int \frac {\cot ^{-1}(x)}{\sqrt {a+a x^2}} \, dx=-\frac {2 i \sqrt {1+x^2} \cot ^{-1}(x) \arctan \left (\frac {\sqrt {1+i x}}{\sqrt {1-i x}}\right )}{\sqrt {a+a x^2}}-\frac {i \sqrt {1+x^2} \operatorname {PolyLog}\left (2,-\frac {i \sqrt {1+i x}}{\sqrt {1-i x}}\right )}{\sqrt {a+a x^2}}+\frac {i \sqrt {1+x^2} \operatorname {PolyLog}\left (2,\frac {i \sqrt {1+i x}}{\sqrt {1-i x}}\right )}{\sqrt {a+a x^2}} \] Output:
-2*I*(x^2+1)^(1/2)*arccot(x)*arctan((1+I*x)^(1/2)/(1-I*x)^(1/2))/(a*x^2+a) ^(1/2)-I*(x^2+1)^(1/2)*polylog(2,-I*(1+I*x)^(1/2)/(1-I*x)^(1/2))/(a*x^2+a) ^(1/2)+I*(x^2+1)^(1/2)*polylog(2,I*(1+I*x)^(1/2)/(1-I*x)^(1/2))/(a*x^2+a)^ (1/2)
Time = 0.13 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.57 \[ \int \frac {\cot ^{-1}(x)}{\sqrt {a+a x^2}} \, dx=-\frac {\sqrt {a \left (1+x^2\right )} \left (\cot ^{-1}(x) \left (\log \left (1-e^{i \cot ^{-1}(x)}\right )-\log \left (1+e^{i \cot ^{-1}(x)}\right )\right )+i \operatorname {PolyLog}\left (2,-e^{i \cot ^{-1}(x)}\right )-i \operatorname {PolyLog}\left (2,e^{i \cot ^{-1}(x)}\right )\right )}{a \sqrt {1+\frac {1}{x^2}} x} \] Input:
Integrate[ArcCot[x]/Sqrt[a + a*x^2],x]
Output:
-((Sqrt[a*(1 + x^2)]*(ArcCot[x]*(Log[1 - E^(I*ArcCot[x])] - Log[1 + E^(I*A rcCot[x])]) + I*PolyLog[2, -E^(I*ArcCot[x])] - I*PolyLog[2, E^(I*ArcCot[x] )]))/(a*Sqrt[1 + x^(-2)]*x))
Time = 0.27 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.75, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {5426, 5422}
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.
\(\displaystyle \int \frac {\cot ^{-1}(x)}{\sqrt {a x^2+a}} \, dx\) |
\(\Big \downarrow \) 5426 |
\(\displaystyle \frac {\sqrt {x^2+1} \int \frac {\cot ^{-1}(x)}{\sqrt {x^2+1}}dx}{\sqrt {a x^2+a}}\) |
\(\Big \downarrow \) 5422 |
\(\displaystyle \frac {\sqrt {x^2+1} \left (-2 i \arctan \left (\frac {\sqrt {1+i x}}{\sqrt {1-i x}}\right ) \cot ^{-1}(x)-i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i x+1}}{\sqrt {1-i x}}\right )+i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i x+1}}{\sqrt {1-i x}}\right )\right )}{\sqrt {a x^2+a}}\) |
Input:
Int[ArcCot[x]/Sqrt[a + a*x^2],x]
Output:
(Sqrt[1 + x^2]*((-2*I)*ArcCot[x]*ArcTan[Sqrt[1 + I*x]/Sqrt[1 - I*x]] - I*P olyLog[2, ((-I)*Sqrt[1 + I*x])/Sqrt[1 - I*x]] + I*PolyLog[2, (I*Sqrt[1 + I *x])/Sqrt[1 - I*x]]))/Sqrt[a + a*x^2]
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]
Int[((a_.) + ArcCot[(c_.)*(x_)]*(b_.))^(p_.)/Sqrt[(d_) + (e_.)*(x_)^2], x_S ymbol] :> Simp[Sqrt[1 + c^2*x^2]/Sqrt[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]
Time = 1.08 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.64
method | result | size |
default | \(-\frac {i \left (i \operatorname {arccot}\left (x \right ) \ln \left (\frac {i+x}{\sqrt {x^{2}+1}}+1\right )-i \operatorname {arccot}\left (x \right ) \ln \left (1-\frac {i+x}{\sqrt {x^{2}+1}}\right )+\operatorname {polylog}\left (2, -\frac {i+x}{\sqrt {x^{2}+1}}\right )-\operatorname {polylog}\left (2, \frac {i+x}{\sqrt {x^{2}+1}}\right )\right ) \sqrt {a \left (i+x \right ) \left (x -i\right )}}{\sqrt {x^{2}+1}\, a}\) | \(99\) |
Input:
int(arccot(x)/(a*x^2+a)^(1/2),x,method=_RETURNVERBOSE)
Output:
-I*(I*arccot(x)*ln((I+x)/(x^2+1)^(1/2)+1)-I*arccot(x)*ln(1-(I+x)/(x^2+1)^( 1/2))+polylog(2,-(I+x)/(x^2+1)^(1/2))-polylog(2,(I+x)/(x^2+1)^(1/2)))*(a*( I+x)*(x-I))^(1/2)/(x^2+1)^(1/2)/a
\[ \int \frac {\cot ^{-1}(x)}{\sqrt {a+a x^2}} \, dx=\int { \frac {\operatorname {arccot}\left (x\right )}{\sqrt {a x^{2} + a}} \,d x } \] Input:
integrate(arccot(x)/(a*x^2+a)^(1/2),x, algorithm="fricas")
Output:
integral(arccot(x)/sqrt(a*x^2 + a), x)
\[ \int \frac {\cot ^{-1}(x)}{\sqrt {a+a x^2}} \, dx=\int \frac {\operatorname {acot}{\left (x \right )}}{\sqrt {a \left (x^{2} + 1\right )}}\, dx \] Input:
integrate(acot(x)/(a*x**2+a)**(1/2),x)
Output:
Integral(acot(x)/sqrt(a*(x**2 + 1)), x)
\[ \int \frac {\cot ^{-1}(x)}{\sqrt {a+a x^2}} \, dx=\int { \frac {\operatorname {arccot}\left (x\right )}{\sqrt {a x^{2} + a}} \,d x } \] Input:
integrate(arccot(x)/(a*x^2+a)^(1/2),x, algorithm="maxima")
Output:
integrate(arccot(x)/sqrt(a*x^2 + a), x)
\[ \int \frac {\cot ^{-1}(x)}{\sqrt {a+a x^2}} \, dx=\int { \frac {\operatorname {arccot}\left (x\right )}{\sqrt {a x^{2} + a}} \,d x } \] Input:
integrate(arccot(x)/(a*x^2+a)^(1/2),x, algorithm="giac")
Output:
integrate(arccot(x)/sqrt(a*x^2 + a), x)
Timed out. \[ \int \frac {\cot ^{-1}(x)}{\sqrt {a+a x^2}} \, dx=\int \frac {\mathrm {acot}\left (x\right )}{\sqrt {a\,x^2+a}} \,d x \] Input:
int(acot(x)/(a + a*x^2)^(1/2),x)
Output:
int(acot(x)/(a + a*x^2)^(1/2), x)
\[ \int \frac {\cot ^{-1}(x)}{\sqrt {a+a x^2}} \, dx=\frac {\int \frac {\mathit {acot} \left (x \right )}{\sqrt {x^{2}+1}}d x}{\sqrt {a}} \] Input:
int(acot(x)/(a*x^2+a)^(1/2),x)
Output:
int(acot(x)/sqrt(x**2 + 1),x)/sqrt(a)