\(\int \frac {\cot ^{-1}(a x)}{\sqrt {c+d x^2}} \, dx\) [60]

Optimal result

Integrand size = 16, antiderivative size = 16 \[ \int \frac {\cot ^{-1}(a x)}{\sqrt {c+d x^2}} \, dx=\text {Int}\left (\frac {\cot ^{-1}(a x)}{\sqrt {c+d x^2}},x\right ) \]

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.02 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps used = 0, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {\cot ^{-1}(a x)}{\sqrt {c+d x^2}} \, dx=\int \frac {\cot ^{-1}(a x)}{\sqrt {c+d x^2}} \, dx \]

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

Mathematica [N/A]

Not integrable

Time = 2.88 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.12 \[ \int \frac {\cot ^{-1}(a x)}{\sqrt {c+d x^2}} \, dx=\int \frac {\cot ^{-1}(a x)}{\sqrt {c+d x^2}} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 0.71 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.88

\[\int \frac {\operatorname {arccot}\left (a x \right )}{\sqrt {d \,x^{2}+c}}d x\]

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.26 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00 \[ \int \frac {\cot ^{-1}(a x)}{\sqrt {c+d x^2}} \, dx=\int { \frac {\operatorname {arccot}\left (a x\right )}{\sqrt {d x^{2} + c}} \,d x } \]

[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)

Sympy [N/A]

Not integrable

Time = 1.44 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.94 \[ \int \frac {\cot ^{-1}(a x)}{\sqrt {c+d x^2}} \, dx=\int \frac {\operatorname {acot}{\left (a x \right )}}{\sqrt {c + d x^{2}}}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 0.92 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00 \[ \int \frac {\cot ^{-1}(a x)}{\sqrt {c+d x^2}} \, dx=\int { \frac {\operatorname {arccot}\left (a x\right )}{\sqrt {d x^{2} + c}} \,d x } \]

[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)

Giac [N/A]

Not integrable

Time = 0.29 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00 \[ \int \frac {\cot ^{-1}(a x)}{\sqrt {c+d x^2}} \, dx=\int { \frac {\operatorname {arccot}\left (a x\right )}{\sqrt {d x^{2} + c}} \,d x } \]

[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)

Mupad [N/A]

Not integrable

Time = 0.70 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00 \[ \int \frac {\cot ^{-1}(a x)}{\sqrt {c+d x^2}} \, dx=\int \frac {\mathrm {acot}\left (a\,x\right )}{\sqrt {d\,x^2+c}} \,d x \]

[In]

int(acot(a*x)/(c + d*x^2)^(1/2),x)

[Out]

int(acot(a*x)/(c + d*x^2)^(1/2), x)