Optimal. Leaf size=54 \[ -\frac {e^{-2 \text {ArcTan}(a x)}}{8 a c^2}-\frac {e^{-2 \text {ArcTan}(a x)} (1-a x)}{4 a c^2 \left (1+a^2 x^2\right )} \]
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Rubi [A]
time = 0.04, antiderivative size = 54, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {5178, 5179}
\begin {gather*} -\frac {(1-a x) e^{-2 \text {ArcTan}(a x)}}{4 a c^2 \left (a^2 x^2+1\right )}-\frac {e^{-2 \text {ArcTan}(a x)}}{8 a c^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 5178
Rule 5179
Rubi steps
\begin {align*} \int \frac {e^{-2 \tan ^{-1}(a x)}}{\left (c+a^2 c x^2\right )^2} \, dx &=-\frac {e^{-2 \tan ^{-1}(a x)} (1-a x)}{4 a c^2 \left (1+a^2 x^2\right )}+\frac {\int \frac {e^{-2 \tan ^{-1}(a x)}}{c+a^2 c x^2} \, dx}{4 c}\\ &=-\frac {e^{-2 \tan ^{-1}(a x)}}{8 a c^2}-\frac {e^{-2 \tan ^{-1}(a x)} (1-a x)}{4 a c^2 \left (1+a^2 x^2\right )}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 0.02, size = 55, normalized size = 1.02 \begin {gather*} -\frac {(1-i a x)^{-i} (1+i a x)^i \left (3-2 a x+a^2 x^2\right )}{8 c^2 \left (a+a^3 x^2\right )} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.09, size = 42, normalized size = 0.78
method | result | size |
gosper | \(-\frac {\left (a^{2} x^{2}-2 a x +3\right ) {\mathrm e}^{-2 \arctan \left (a x \right )}}{8 \left (a^{2} x^{2}+1\right ) c^{2} a}\) | \(42\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.69, size = 40, normalized size = 0.74 \begin {gather*} -\frac {{\left (a^{2} x^{2} - 2 \, a x + 3\right )} e^{\left (-2 \, \arctan \left (a x\right )\right )}}{8 \, {\left (a^{3} c^{2} x^{2} + a c^{2}\right )}} \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. 124 vs.
\(2 (46) = 92\).
time = 44.95, size = 124, normalized size = 2.30 \begin {gather*} \begin {cases} - \frac {a^{2} x^{2}}{8 a^{3} c^{2} x^{2} e^{2 \operatorname {atan}{\left (a x \right )}} + 8 a c^{2} e^{2 \operatorname {atan}{\left (a x \right )}}} + \frac {2 a x}{8 a^{3} c^{2} x^{2} e^{2 \operatorname {atan}{\left (a x \right )}} + 8 a c^{2} e^{2 \operatorname {atan}{\left (a x \right )}}} - \frac {3}{8 a^{3} c^{2} x^{2} e^{2 \operatorname {atan}{\left (a x \right )}} + 8 a c^{2} e^{2 \operatorname {atan}{\left (a x \right )}}} & \text {for}\: a \neq 0 \\\frac {x}{c^{2}} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.57, size = 47, normalized size = 0.87 \begin {gather*} -\frac {{\mathrm {e}}^{-2\,\mathrm {atan}\left (a\,x\right )}\,\left (\frac {3}{8\,a^3\,c^2}-\frac {x}{4\,a^2\,c^2}+\frac {x^2}{8\,a\,c^2}\right )}{\frac {1}{a^2}+x^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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