Optimal. Leaf size=89 \[ -\frac {(1-2 a x) e^{-2 \tan ^{-1}(a x)}}{10 a c^3 \left (a^2 x^2+1\right )^2}-\frac {3 (1-a x) e^{-2 \tan ^{-1}(a x)}}{20 a c^3 \left (a^2 x^2+1\right )}-\frac {3 e^{-2 \tan ^{-1}(a x)}}{40 a c^3} \]
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Rubi [A] time = 0.09, antiderivative size = 89, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {5070, 5071} \[ -\frac {(1-2 a x) e^{-2 \tan ^{-1}(a x)}}{10 a c^3 \left (a^2 x^2+1\right )^2}-\frac {3 (1-a x) e^{-2 \tan ^{-1}(a x)}}{20 a c^3 \left (a^2 x^2+1\right )}-\frac {3 e^{-2 \tan ^{-1}(a x)}}{40 a c^3} \]
Antiderivative was successfully verified.
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Rule 5070
Rule 5071
Rubi steps
\begin {align*} \int \frac {e^{-2 \tan ^{-1}(a x)}}{\left (c+a^2 c x^2\right )^3} \, dx &=-\frac {e^{-2 \tan ^{-1}(a x)} (1-2 a x)}{10 a c^3 \left (1+a^2 x^2\right )^2}+\frac {3 \int \frac {e^{-2 \tan ^{-1}(a x)}}{\left (c+a^2 c x^2\right )^2} \, dx}{5 c}\\ &=-\frac {e^{-2 \tan ^{-1}(a x)} (1-2 a x)}{10 a c^3 \left (1+a^2 x^2\right )^2}-\frac {3 e^{-2 \tan ^{-1}(a x)} (1-a x)}{20 a c^3 \left (1+a^2 x^2\right )}+\frac {3 \int \frac {e^{-2 \tan ^{-1}(a x)}}{c+a^2 c x^2} \, dx}{20 c^2}\\ &=-\frac {3 e^{-2 \tan ^{-1}(a x)}}{40 a c^3}-\frac {e^{-2 \tan ^{-1}(a x)} (1-2 a x)}{10 a c^3 \left (1+a^2 x^2\right )^2}-\frac {3 e^{-2 \tan ^{-1}(a x)} (1-a x)}{20 a c^3 \left (1+a^2 x^2\right )}\\ \end {align*}
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Mathematica [C] time = 0.17, size = 85, normalized size = 0.96 \[ \frac {(8 a x-4) e^{-2 \tan ^{-1}(a x)}-3 (1-i a x)^{-i} (1+i a x)^i \left (a^2 x^2+1\right ) \left (a^2 x^2-2 a x+3\right )}{40 a c^3 \left (a^2 x^2+1\right )^2} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.44, size = 68, normalized size = 0.76 \[ -\frac {{\left (3 \, a^{4} x^{4} - 6 \, a^{3} x^{3} + 12 \, a^{2} x^{2} - 14 \, a x + 13\right )} e^{\left (-2 \, \arctan \left (a x\right )\right )}}{40 \, {\left (a^{5} c^{3} x^{4} + 2 \, a^{3} c^{3} x^{2} + a c^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 59, normalized size = 0.66 \[ -\frac {\left (3 a^{4} x^{4}-6 a^{3} x^{3}+12 a^{2} x^{2}-14 a x +13\right ) {\mathrm e}^{-2 \arctan \left (a x \right )}}{40 \left (a^{2} x^{2}+1\right )^{2} c^{3} a} \]
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 \[ \int \frac {e^{\left (-2 \, \arctan \left (a x\right )\right )}}{{\left (a^{2} c x^{2} + c\right )}^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.62, size = 79, normalized size = 0.89 \[ \frac {3\,{\mathrm {e}}^{-2\,\mathrm {atan}\left (a\,x\right )}\,\left (a\,x-1\right )}{20\,a\,c^3\,\left (a^2\,x^2+1\right )}-\frac {3\,{\mathrm {e}}^{-2\,\mathrm {atan}\left (a\,x\right )}}{40\,a\,c^3}+\frac {{\mathrm {e}}^{-2\,\mathrm {atan}\left (a\,x\right )}\,\left (2\,a\,x-1\right )}{10\,a\,c^3\,{\left (a^2\,x^2+1\right )}^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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