3.622 \(\int \frac {1}{(c+a^2 c x^2) \tan ^{-1}(a x)^3} \, dx\)

Optimal. Leaf size=16 \[ -\frac {1}{2 a c \tan ^{-1}(a x)^2} \]

[Out]

-1/2/a/c/arctan(a*x)^2

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Rubi [A]  time = 0.02, antiderivative size = 16, 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 = {4884} \[ -\frac {1}{2 a c \tan ^{-1}(a x)^2} \]

Antiderivative was successfully verified.

[In]

Int[1/((c + a^2*c*x^2)*ArcTan[a*x]^3),x]

[Out]

-1/(2*a*c*ArcTan[a*x]^2)

Rule 4884

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)^3} \, dx &=-\frac {1}{2 a c \tan ^{-1}(a x)^2}\\ \end {align*}

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Mathematica [A]  time = 0.00, size = 16, normalized size = 1.00 \[ -\frac {1}{2 a c \tan ^{-1}(a x)^2} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((c + a^2*c*x^2)*ArcTan[a*x]^3),x]

[Out]

-1/2*1/(a*c*ArcTan[a*x]^2)

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fricas [A]  time = 0.38, size = 14, normalized size = 0.88 \[ -\frac {1}{2 \, a c \arctan \left (a x\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a^2*c*x^2+c)/arctan(a*x)^3,x, algorithm="fricas")

[Out]

-1/2/(a*c*arctan(a*x)^2)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \mathit {sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a^2*c*x^2+c)/arctan(a*x)^3,x, algorithm="giac")

[Out]

sage0*x

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maple [A]  time = 0.06, size = 15, normalized size = 0.94 \[ -\frac {1}{2 a c \arctan \left (a x \right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a^2*c*x^2+c)/arctan(a*x)^3,x)

[Out]

-1/2/a/c/arctan(a*x)^2

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maxima [A]  time = 0.43, size = 14, normalized size = 0.88 \[ -\frac {1}{2 \, a c \arctan \left (a x\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a^2*c*x^2+c)/arctan(a*x)^3,x, algorithm="maxima")

[Out]

-1/2/(a*c*arctan(a*x)^2)

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mupad [B]  time = 0.35, size = 14, normalized size = 0.88 \[ -\frac {1}{2\,a\,c\,{\mathrm {atan}\left (a\,x\right )}^2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(atan(a*x)^3*(c + a^2*c*x^2)),x)

[Out]

-1/(2*a*c*atan(a*x)^2)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \begin {cases} - \frac {1}{2 a c \operatorname {atan}^{2}{\left (a x \right )}} & \text {for}\: c \neq 0 \\\tilde {\infty } \int \frac {1}{\operatorname {atan}^{3}{\left (a x \right )}}\, dx & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a**2*c*x**2+c)/atan(a*x)**3,x)

[Out]

Piecewise((-1/(2*a*c*atan(a*x)**2), Ne(c, 0)), (zoo*Integral(atan(a*x)**(-3), x), True))

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