3.6.77 \(\int \frac {1}{\sqrt {c+a^2 c x^2} \text {ArcTan}(a x)^2} \, dx\) [577]

Optimal. Leaf size=24 \[ \text {Int}\left (\frac {1}{\sqrt {c+a^2 c x^2} \text {ArcTan}(a x)^2},x\right ) \]

[Out]

Unintegrable(1/arctan(a*x)^2/(a^2*c*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 = {} \begin {gather*} \int \frac {1}{\sqrt {c+a^2 c x^2} \text {ArcTan}(a x)^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[1/(Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2),x]

[Out]

Defer[Int][1/(Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2), x]

Rubi steps

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

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Mathematica [A]
time = 0.44, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt {c+a^2 c x^2} \text {ArcTan}(a x)^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[1/(Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2),x]

[Out]

Integrate[1/(Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2), x]

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Maple [A]
time = 0.19, size = 0, normalized size = 0.00 \[\int \frac {1}{\arctan \left (a x \right )^{2} \sqrt {a^{2} c \,x^{2}+c}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/(sqrt(a^2*c*x^2 + c)*arctan(a*x)^2), x)

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Fricas [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(1/(sqrt(a^2*c*x^2 + c)*arctan(a*x)^2), x)

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Sympy [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt {c \left (a^{2} x^{2} + 1\right )} \operatorname {atan}^{2}{\left (a x \right )}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sage0*x

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Mupad [A]
time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int \frac {1}{{\mathrm {atan}\left (a\,x\right )}^2\,\sqrt {c\,a^2\,x^2+c}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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