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

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

[Out]

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

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Rubi [A]  time = 0.0249069, antiderivative size = 16, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.048, Rules used = {4884} \[ -\frac{2}{a c \sqrt{\tan ^{-1}(a x)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

Mathematica [A]  time = 0.0049823, size = 16, normalized size = 1. \[ -\frac{2}{a c \sqrt{\tan ^{-1}(a x)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.087, size = 15, normalized size = 0.9 \begin{align*} -2\,{\frac{1}{ac\sqrt{\arctan \left ( ax \right ) }}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: RuntimeError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: RuntimeError

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Fricas [A]  time = 1.61517, size = 38, normalized size = 2.38 \begin{align*} -\frac{2}{a c \sqrt{\arctan \left (a x\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/(a*c*sqrt(arctan(a*x)))

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Sympy [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError

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Giac [A]  time = 1.11232, size = 19, normalized size = 1.19 \begin{align*} -\frac{2}{a c \sqrt{\arctan \left (a x\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/(a*c*sqrt(arctan(a*x)))