3.6.36 \(\int \frac {c+a^2 c x^2}{x \text {ArcTan}(a x)^2} \, dx\) [536]

Optimal. Leaf size=23 \[ \text {Int}\left (\frac {c+a^2 c x^2}{x \text {ArcTan}(a x)^2},x\right ) \]

[Out]

Unintegrable((a^2*c*x^2+c)/x/arctan(a*x)^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 {c+a^2 c x^2}{x \text {ArcTan}(a x)^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]
time = 0.68, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {c+a^2 c x^2}{x \text {ArcTan}(a x)^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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Maple [A]
time = 1.24, size = 0, normalized size = 0.00 \[\int \frac {a^{2} c \,x^{2}+c}{x \arctan \left (a x \right )^{2}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

-(a^4*c*x^4 + 2*a^2*c*x^2 - x*arctan(a*x)*integrate((3*a^4*c*x^4 + 2*a^2*c*x^2 - c)/(x^2*arctan(a*x)), x) + c)
/(a*x*arctan(a*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((a^2*c*x^2+c)/x/arctan(a*x)^2,x, algorithm="fricas")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

c*(Integral(1/(x*atan(a*x)**2), x) + Integral(a**2*x/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((a^2*c*x^2+c)/x/arctan(a*x)^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 {c\,a^2\,x^2+c}{x\,{\mathrm {atan}\left (a\,x\right )}^2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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