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

Optimal. Leaf size=12 \[ \frac {\log \left (\tan ^{-1}(a x)\right )}{a c} \]

[Out]

ln(arctan(a*x))/a/c

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Rubi [A]  time = 0.03, antiderivative size = 12, 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 = {4882} \[ \frac {\log \left (\tan ^{-1}(a x)\right )}{a c} \]

Antiderivative was successfully verified.

[In]

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

[Out]

Log[ArcTan[a*x]]/(a*c)

Rule 4882

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

Rubi steps

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

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Mathematica [A]  time = 0.02, size = 12, normalized size = 1.00 \[ \frac {\log \left (\tan ^{-1}(a x)\right )}{a c} \]

Antiderivative was successfully verified.

[In]

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

[Out]

Log[ArcTan[a*x]]/(a*c)

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fricas [A]  time = 0.40, size = 12, normalized size = 1.00 \[ \frac {\log \left (\arctan \left (a x\right )\right )}{a c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(arctan(a*x))/(a*c)

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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),x, algorithm="giac")

[Out]

sage0*x

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maple [A]  time = 0.08, size = 13, normalized size = 1.08 \[ \frac {\ln \left (\arctan \left (a x \right )\right )}{a c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

ln(arctan(a*x))/a/c

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maxima [A]  time = 0.32, size = 15, normalized size = 1.25 \[ \frac {\log \left (2 \, {\left | \arctan \left (a x\right ) \right |}\right )}{a c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(2*abs(arctan(a*x)))/(a*c)

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mupad [B]  time = 0.09, size = 12, normalized size = 1.00 \[ \frac {\ln \left (\mathrm {atan}\left (a\,x\right )\right )}{a\,c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(atan(a*x)*(c + a^2*c*x^2)),x)

[Out]

log(atan(a*x))/(a*c)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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