3.6.46 \(\int \frac {1}{(c+a^2 c x^2) \text {ArcTan}(a x)^2} \, dx\) [546]

Optimal. Leaf size=14 \[ -\frac {1}{a c \text {ArcTan}(a x)} \]

[Out]

-1/a/c/arctan(a*x)

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Rubi [A]
time = 0.02, antiderivative size = 14, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {5004} \begin {gather*} -\frac {1}{a c \text {ArcTan}(a x)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-(1/(a*c*ArcTan[a*x]))

Rule 5004

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)^2), x_Symbol] :> Simp[(a + b*ArcTan[c*x])^(p +
1)/(b*c*d*(p + 1)), x] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[e, c^2*d] && NeQ[p, -1]

Rubi steps

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

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Mathematica [A]
time = 0.00, size = 14, normalized size = 1.00 \begin {gather*} -\frac {1}{a c \text {ArcTan}(a x)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-(1/(a*c*ArcTan[a*x]))

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Maple [A]
time = 0.08, size = 15, normalized size = 1.07

method	result	size
derivativedivides	\(-\frac {1}{a c \arctan \left (a x \right )}\)	\(15\)
default	\(-\frac {1}{a c \arctan \left (a x \right )}\)	\(15\)
risch	\(\frac {2 i}{a c \left (\ln \left (-i a x +1\right )-\ln \left (i a x +1\right )\right )}\)	\(31\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/a/c/arctan(a*x)

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Maxima [A]
time = 0.28, size = 14, normalized size = 1.00 \begin {gather*} -\frac {1}{a c \arctan \left (a x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/(a*c*arctan(a*x))

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Fricas [A]
time = 0.75, size = 14, normalized size = 1.00 \begin {gather*} -\frac {1}{a c \arctan \left (a x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/(a*c*arctan(a*x))

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Sympy [A]
time = 0.37, size = 10, normalized size = 0.71 \begin {gather*} - \frac {1}{a c \operatorname {atan}{\left (a x \right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/(a*c*atan(a*x))

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Giac [F]
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/(a^2*c*x^2+c)/arctan(a*x)^2,x, algorithm="giac")

[Out]

sage0*x

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Mupad [B]
time = 0.34, size = 14, normalized size = 1.00 \begin {gather*} -\frac {1}{a\,c\,\mathrm {atan}\left (a\,x\right )} \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)),x)

[Out]

-1/(a*c*atan(a*x))

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