Optimal. Leaf size=89 \[ -\frac {24 e^{-\text {ArcTan}(a x)}}{85 a c^3}-\frac {e^{-\text {ArcTan}(a x)} (1-4 a x)}{17 a c^3 \left (1+a^2 x^2\right )^2}-\frac {12 e^{-\text {ArcTan}(a x)} (1-2 a x)}{85 a c^3 \left (1+a^2 x^2\right )} \]
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Rubi [A]
time = 0.06, 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 = {5178, 5179}
\begin {gather*} -\frac {(1-4 a x) e^{-\text {ArcTan}(a x)}}{17 a c^3 \left (a^2 x^2+1\right )^2}-\frac {12 (1-2 a x) e^{-\text {ArcTan}(a x)}}{85 a c^3 \left (a^2 x^2+1\right )}-\frac {24 e^{-\text {ArcTan}(a x)}}{85 a c^3} \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 )^3} \, dx &=-\frac {e^{-\tan ^{-1}(a x)} (1-4 a x)}{17 a c^3 \left (1+a^2 x^2\right )^2}+\frac {12 \int \frac {e^{-\tan ^{-1}(a x)}}{\left (c+a^2 c x^2\right )^2} \, dx}{17 c}\\ &=-\frac {e^{-\tan ^{-1}(a x)} (1-4 a x)}{17 a c^3 \left (1+a^2 x^2\right )^2}-\frac {12 e^{-\tan ^{-1}(a x)} (1-2 a x)}{85 a c^3 \left (1+a^2 x^2\right )}+\frac {24 \int \frac {e^{-\tan ^{-1}(a x)}}{c+a^2 c x^2} \, dx}{85 c^2}\\ &=-\frac {24 e^{-\tan ^{-1}(a x)}}{85 a c^3}-\frac {e^{-\tan ^{-1}(a x)} (1-4 a x)}{17 a c^3 \left (1+a^2 x^2\right )^2}-\frac {12 e^{-\tan ^{-1}(a x)} (1-2 a x)}{85 a c^3 \left (1+a^2 x^2\right )}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 0.12, size = 91, normalized size = 1.02 \begin {gather*} \frac {5 e^{-\text {ArcTan}(a x)} (-1+4 a x)-12 (1-i a x)^{-\frac {i}{2}} (1+i a x)^{\frac {i}{2}} \left (1+a^2 x^2\right ) \left (3-2 a x+2 a^2 x^2\right )}{85 a c^3 \left (1+a^2 x^2\right )^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.08, size = 57, normalized size = 0.64
method | result | size |
gosper | \(-\frac {\left (24 a^{4} x^{4}-24 a^{3} x^{3}+60 a^{2} x^{2}-44 a x +41\right ) {\mathrm e}^{-\arctan \left (a x \right )}}{85 \left (a^{2} x^{2}+1\right )^{2} c^{3} a}\) | \(57\) |
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 = 1.48, size = 68, normalized size = 0.76 \begin {gather*} -\frac {{\left (24 \, a^{4} x^{4} - 24 \, a^{3} x^{3} + 60 \, a^{2} x^{2} - 44 \, a x + 41\right )} e^{\left (-\arctan \left (a x\right )\right )}}{85 \, {\left (a^{5} c^{3} x^{4} + 2 \, a^{3} c^{3} x^{2} + a c^{3}\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. 291 vs.
\(2 (76) = 152\).
time = 103.57, size = 291, normalized size = 3.27 \begin {gather*} \begin {cases} - \frac {24 a^{4} x^{4}}{85 a^{5} c^{3} x^{4} e^{\operatorname {atan}{\left (a x \right )}} + 170 a^{3} c^{3} x^{2} e^{\operatorname {atan}{\left (a x \right )}} + 85 a c^{3} e^{\operatorname {atan}{\left (a x \right )}}} + \frac {24 a^{3} x^{3}}{85 a^{5} c^{3} x^{4} e^{\operatorname {atan}{\left (a x \right )}} + 170 a^{3} c^{3} x^{2} e^{\operatorname {atan}{\left (a x \right )}} + 85 a c^{3} e^{\operatorname {atan}{\left (a x \right )}}} - \frac {60 a^{2} x^{2}}{85 a^{5} c^{3} x^{4} e^{\operatorname {atan}{\left (a x \right )}} + 170 a^{3} c^{3} x^{2} e^{\operatorname {atan}{\left (a x \right )}} + 85 a c^{3} e^{\operatorname {atan}{\left (a x \right )}}} + \frac {44 a x}{85 a^{5} c^{3} x^{4} e^{\operatorname {atan}{\left (a x \right )}} + 170 a^{3} c^{3} x^{2} e^{\operatorname {atan}{\left (a x \right )}} + 85 a c^{3} e^{\operatorname {atan}{\left (a x \right )}}} - \frac {41}{85 a^{5} c^{3} x^{4} e^{\operatorname {atan}{\left (a x \right )}} + 170 a^{3} c^{3} x^{2} e^{\operatorname {atan}{\left (a x \right )}} + 85 a c^{3} e^{\operatorname {atan}{\left (a x \right )}}} & \text {for}\: a \neq 0 \\\frac {x}{c^{3}} & \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 = 0.62, size = 80, normalized size = 0.90 \begin {gather*} \frac {12\,{\mathrm {e}}^{-\mathrm {atan}\left (a\,x\right )}\,\left (2\,a\,x-1\right )}{85\,a\,c^3\,\left (a^2\,x^2+1\right )}-\frac {24\,{\mathrm {e}}^{-\mathrm {atan}\left (a\,x\right )}}{85\,a\,c^3}+\frac {{\mathrm {e}}^{-\mathrm {atan}\left (a\,x\right )}\,\left (4\,a\,x-1\right )}{17\,a\,c^3\,{\left (a^2\,x^2+1\right )}^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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