Optimal. Leaf size=16 \[ \frac {\text {ArcTan}(a x)^2}{2 a c} \]
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Rubi [A]
time = 0.01, antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps
used = 1, number of rules used = 1, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {5004}
\begin {gather*} \frac {\text {ArcTan}(a x)^2}{2 a c} \end {gather*}
Antiderivative was successfully verified.
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Rule 5004
Rubi steps
\begin {align*} \int \frac {\tan ^{-1}(a x)}{c+a^2 c x^2} \, dx &=\frac {\tan ^{-1}(a x)^2}{2 a c}\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 16, normalized size = 1.00 \begin {gather*} \frac {\text {ArcTan}(a x)^2}{2 a c} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.06, size = 15, normalized size = 0.94
method | result | size |
derivativedivides | \(\frac {\arctan \left (a x \right )^{2}}{2 a c}\) | \(15\) |
default | \(\frac {\arctan \left (a x \right )^{2}}{2 a c}\) | \(15\) |
risch | \(-\frac {\ln \left (i a x +1\right )^{2}}{8 c a}+\frac {\ln \left (-i a x +1\right ) \ln \left (i a x +1\right )}{4 c a}-\frac {\ln \left (-i a x +1\right )^{2}}{8 c a}\) | \(62\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.47, size = 14, normalized size = 0.88 \begin {gather*} \frac {\arctan \left (a x\right )^{2}}{2 \, a c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 4.99, size = 14, normalized size = 0.88 \begin {gather*} \frac {\arctan \left (a x\right )^{2}}{2 \, a c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 36 vs.
\(2 (10) = 20\).
time = 1.45, size = 36, normalized size = 2.25 \begin {gather*} \begin {cases} 0 & \text {for}\: a = 0 \\\tilde {\infty } \left (\begin {cases} 0 & \text {for}\: a = 0 \\\frac {a x \operatorname {atan}{\left (a x \right )} - \frac {\log {\left (a^{2} x^{2} + 1 \right )}}{2}}{a} & \text {otherwise} \end {cases}\right ) & \text {for}\: c = 0 \\\frac {\operatorname {atan}^{2}{\left (a x \right )}}{2 a c} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 35 vs.
\(2 (14) = 28\).
time = 0.41, size = 35, normalized size = 2.19 \begin {gather*} -\frac {2 \, \pi \arctan \left (a x\right ) \left \lfloor \frac {\arctan \left (a x\right )}{\pi } + \frac {1}{2} \right \rfloor - \arctan \left (a x\right )^{2}}{2 \, a c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.38, size = 14, normalized size = 0.88 \begin {gather*} \frac {{\mathrm {atan}\left (a\,x\right )}^2}{2\,a\,c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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