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

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

[Out]

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

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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 = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.048, Rules used = {4884} \[ \frac {2 \sqrt {\tan ^{-1}(a x)}}{a c} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

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

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.14, size = 14, normalized size = 0.88 \[ \frac {2 \, \sqrt {\arctan \left (a x\right )}}{a c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.12, size = 15, normalized size = 0.94 \[ \frac {2 \sqrt {\arctan \left (a x \right )}}{a c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: RuntimeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: RuntimeError >> ECL says: Error executing code in Maxima: expt: undefined: 0 to a negative e
xponent.

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mupad [B]  time = 0.34, size = 14, normalized size = 0.88 \[ \frac {2\,\sqrt {\mathrm {atan}\left (a\,x\right )}}{a\,c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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