Optimal. Leaf size=124 \[ -\frac {9 e^{-2 \text {ArcTan}(a x)}}{160 a c^4}-\frac {e^{-2 \text {ArcTan}(a x)} (1-3 a x)}{20 a c^4 \left (1+a^2 x^2\right )^3}-\frac {3 e^{-2 \text {ArcTan}(a x)} (1-2 a x)}{40 a c^4 \left (1+a^2 x^2\right )^2}-\frac {9 e^{-2 \text {ArcTan}(a x)} (1-a x)}{80 a c^4 \left (1+a^2 x^2\right )} \]
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Rubi [A]
time = 0.09, antiderivative size = 124, normalized size of antiderivative = 1.00, number of steps
used = 4, 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 {(1-3 a x) e^{-2 \text {ArcTan}(a x)}}{20 a c^4 \left (a^2 x^2+1\right )^3}-\frac {9 (1-a x) e^{-2 \text {ArcTan}(a x)}}{80 a c^4 \left (a^2 x^2+1\right )}-\frac {3 (1-2 a x) e^{-2 \text {ArcTan}(a x)}}{40 a c^4 \left (a^2 x^2+1\right )^2}-\frac {9 e^{-2 \text {ArcTan}(a x)}}{160 a c^4} \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 )^4} \, dx &=-\frac {e^{-2 \tan ^{-1}(a x)} (1-3 a x)}{20 a c^4 \left (1+a^2 x^2\right )^3}+\frac {3 \int \frac {e^{-2 \tan ^{-1}(a x)}}{\left (c+a^2 c x^2\right )^3} \, dx}{4 c}\\ &=-\frac {e^{-2 \tan ^{-1}(a x)} (1-3 a x)}{20 a c^4 \left (1+a^2 x^2\right )^3}-\frac {3 e^{-2 \tan ^{-1}(a x)} (1-2 a x)}{40 a c^4 \left (1+a^2 x^2\right )^2}+\frac {9 \int \frac {e^{-2 \tan ^{-1}(a x)}}{\left (c+a^2 c x^2\right )^2} \, dx}{20 c^2}\\ &=-\frac {e^{-2 \tan ^{-1}(a x)} (1-3 a x)}{20 a c^4 \left (1+a^2 x^2\right )^3}-\frac {3 e^{-2 \tan ^{-1}(a x)} (1-2 a x)}{40 a c^4 \left (1+a^2 x^2\right )^2}-\frac {9 e^{-2 \tan ^{-1}(a x)} (1-a x)}{80 a c^4 \left (1+a^2 x^2\right )}+\frac {9 \int \frac {e^{-2 \tan ^{-1}(a x)}}{c+a^2 c x^2} \, dx}{80 c^3}\\ &=-\frac {9 e^{-2 \tan ^{-1}(a x)}}{160 a c^4}-\frac {e^{-2 \tan ^{-1}(a x)} (1-3 a x)}{20 a c^4 \left (1+a^2 x^2\right )^3}-\frac {3 e^{-2 \tan ^{-1}(a x)} (1-2 a x)}{40 a c^4 \left (1+a^2 x^2\right )^2}-\frac {9 e^{-2 \tan ^{-1}(a x)} (1-a x)}{80 a c^4 \left (1+a^2 x^2\right )}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 0.25, size = 121, normalized size = 0.98 \begin {gather*} \frac {8 c e^{-2 \text {ArcTan}(a x)} (-1+3 a x)-3 \left (c+a^2 c x^2\right ) \left (e^{-2 \text {ArcTan}(a x)} (4-8 a x)+3 (1-i a x)^{-i} (1+i a x)^i (-i+a x) (i+a x) \left (3-2 a x+a^2 x^2\right )\right )}{160 a c^2 \left (c+a^2 c x^2\right )^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.08, size = 75, normalized size = 0.60
method | result | size |
gosper | \(-\frac {\left (9 a^{6} x^{6}-18 a^{5} x^{5}+45 a^{4} x^{4}-60 a^{3} x^{3}+75 a^{2} x^{2}-66 a x +47\right ) {\mathrm e}^{-2 \arctan \left (a x \right )}}{160 \left (a^{2} x^{2}+1\right )^{3} c^{4} a}\) | \(75\) |
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 = 6.90, size = 95, normalized size = 0.77 \begin {gather*} -\frac {{\left (9 \, a^{6} x^{6} - 18 \, a^{5} x^{5} + 45 \, a^{4} x^{4} - 60 \, a^{3} x^{3} + 75 \, a^{2} x^{2} - 66 \, a x + 47\right )} e^{\left (-2 \, \arctan \left (a x\right )\right )}}{160 \, {\left (a^{7} c^{4} x^{6} + 3 \, a^{5} c^{4} x^{4} + 3 \, a^{3} c^{4} x^{2} + a c^{4}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \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 = 0.67, size = 111, normalized size = 0.90 \begin {gather*} \frac {9\,{\mathrm {e}}^{-2\,\mathrm {atan}\left (a\,x\right )}\,\left (a\,x-1\right )}{80\,a\,c^4\,\left (a^2\,x^2+1\right )}-\frac {9\,{\mathrm {e}}^{-2\,\mathrm {atan}\left (a\,x\right )}}{160\,a\,c^4}+\frac {3\,{\mathrm {e}}^{-2\,\mathrm {atan}\left (a\,x\right )}\,\left (2\,a\,x-1\right )}{40\,a\,c^4\,{\left (a^2\,x^2+1\right )}^2}+\frac {{\mathrm {e}}^{-2\,\mathrm {atan}\left (a\,x\right )}\,\left (3\,a\,x-1\right )}{20\,a\,c^4\,{\left (a^2\,x^2+1\right )}^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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