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

Optimal. Leaf size=19 \[ \text {Int}\left (\frac {\cot ^{-1}(a x)}{\sqrt {c+d x^2}},x\right ) \]

[Out]

Unintegrable(arccot(a*x)/(d*x^2+c)^(1/2),x)

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Rubi [A]  time = 0.02, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {\cot ^{-1}(a x)}{\sqrt {c+d x^2}} \, dx \]

Verification is Not applicable to the result.

[In]

Int[ArcCot[a*x]/Sqrt[c + d*x^2],x]

[Out]

Defer[Int][ArcCot[a*x]/Sqrt[c + d*x^2], x]

Rubi steps

\begin {align*} \int \frac {\cot ^{-1}(a x)}{\sqrt {c+d x^2}} \, dx &=\int \frac {\cot ^{-1}(a x)}{\sqrt {c+d x^2}} \, dx\\ \end {align*}

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Mathematica [A]  time = 4.23, size = 0, normalized size = 0.00 \[ \int \frac {\cot ^{-1}(a x)}{\sqrt {c+d x^2}} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.96, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\operatorname {arccot}\left (a x\right )}{\sqrt {d x^{2} + c}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {arccot}\left (a x\right )}{\sqrt {d x^{2} + c}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 1.29, size = 0, normalized size = 0.00 \[ \int \frac {\mathrm {arccot}\left (a x \right )}{\sqrt {d \,x^{2}+c}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {arccot}\left (a x\right )}{\sqrt {d x^{2} + c}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [A]  time = 0.00, size = -1, normalized size = -0.05 \[ \int \frac {\mathrm {acot}\left (a\,x\right )}{\sqrt {d\,x^2+c}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(acot(a*x)/(c + d*x^2)^(1/2),x)

[Out]

int(acot(a*x)/(c + d*x^2)^(1/2), x)

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sympy [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {acot}{\left (a x \right )}}{\sqrt {c + d x^{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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