Optimal. Leaf size=50 \[ \frac {2 e^{\text {ArcTan}(a x)}}{5 a c^2}+\frac {e^{\text {ArcTan}(a x)} (1+2 a x)}{5 a c^2 \left (1+a^2 x^2\right )} \]
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Rubi [A]
time = 0.04, antiderivative size = 50, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {5178, 5179}
\begin {gather*} \frac {(2 a x+1) e^{\text {ArcTan}(a x)}}{5 a c^2 \left (a^2 x^2+1\right )}+\frac {2 e^{\text {ArcTan}(a x)}}{5 a c^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 5178
Rule 5179
Rubi steps
\begin {align*} \int \frac {e^{\tan ^{-1}(a x)}}{\left (c+a^2 c x^2\right )^2} \, dx &=\frac {e^{\tan ^{-1}(a x)} (1+2 a x)}{5 a c^2 \left (1+a^2 x^2\right )}+\frac {2 \int \frac {e^{\tan ^{-1}(a x)}}{c+a^2 c x^2} \, dx}{5 c}\\ &=\frac {2 e^{\tan ^{-1}(a x)}}{5 a c^2}+\frac {e^{\tan ^{-1}(a x)} (1+2 a x)}{5 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 = 60, normalized size = 1.20 \begin {gather*} \frac {(1-i a x)^{\frac {i}{2}} (1+i a x)^{-\frac {i}{2}} \left (3+2 a x+2 a^2 x^2\right )}{5 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 = 39, normalized size = 0.78
method | result | size |
gosper | \(\frac {{\mathrm e}^{\arctan \left (a x \right )} \left (2 a^{2} x^{2}+2 a x +3\right )}{5 \left (a^{2} x^{2}+1\right ) a \,c^{2}}\) | \(39\) |
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 = 3.35, size = 39, normalized size = 0.78 \begin {gather*} \frac {{\left (2 \, a^{2} x^{2} + 2 \, a x + 3\right )} e^{\left (\arctan \left (a x\right )\right )}}{5 \, {\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. 95 vs.
\(2 (42) = 84\).
time = 1.43, size = 95, normalized size = 1.90 \begin {gather*} \begin {cases} \frac {2 a^{2} x^{2} e^{\operatorname {atan}{\left (a x \right )}}}{5 a^{3} c^{2} x^{2} + 5 a c^{2}} + \frac {2 a x e^{\operatorname {atan}{\left (a x \right )}}}{5 a^{3} c^{2} x^{2} + 5 a c^{2}} + \frac {3 e^{\operatorname {atan}{\left (a x \right )}}}{5 a^{3} c^{2} x^{2} + 5 a c^{2}} & \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 = 0.56, size = 44, normalized size = 0.88 \begin {gather*} \frac {{\mathrm {e}}^{\mathrm {atan}\left (a\,x\right )}\,\left (\frac {3}{5\,a^3\,c^2}+\frac {2\,x}{5\,a^2\,c^2}+\frac {2\,x^2}{5\,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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