Integrand size = 21, antiderivative size = 89 \[ \int \frac {e^{-\arctan (a x)}}{\left (c+a^2 c x^2\right )^3} \, dx=-\frac {24 e^{-\arctan (a x)}}{85 a c^3}-\frac {e^{-\arctan (a x)} (1-4 a x)}{17 a c^3 \left (1+a^2 x^2\right )^2}-\frac {12 e^{-\arctan (a x)} (1-2 a x)}{85 a c^3 \left (1+a^2 x^2\right )} \]
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Time = 0.07 (sec) , antiderivative size = 89, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {5178, 5179} \[ \int \frac {e^{-\arctan (a x)}}{\left (c+a^2 c x^2\right )^3} \, dx=-\frac {(1-4 a x) e^{-\arctan (a x)}}{17 a c^3 \left (a^2 x^2+1\right )^2}-\frac {12 (1-2 a x) e^{-\arctan (a x)}}{85 a c^3 \left (a^2 x^2+1\right )}-\frac {24 e^{-\arctan (a x)}}{85 a c^3} \]
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Rule 5178
Rule 5179
Rubi steps \begin{align*} \text {integral}& = -\frac {e^{-\arctan (a x)} (1-4 a x)}{17 a c^3 \left (1+a^2 x^2\right )^2}+\frac {12 \int \frac {e^{-\arctan (a x)}}{\left (c+a^2 c x^2\right )^2} \, dx}{17 c} \\ & = -\frac {e^{-\arctan (a x)} (1-4 a x)}{17 a c^3 \left (1+a^2 x^2\right )^2}-\frac {12 e^{-\arctan (a x)} (1-2 a x)}{85 a c^3 \left (1+a^2 x^2\right )}+\frac {24 \int \frac {e^{-\arctan (a x)}}{c+a^2 c x^2} \, dx}{85 c^2} \\ & = -\frac {24 e^{-\arctan (a x)}}{85 a c^3}-\frac {e^{-\arctan (a x)} (1-4 a x)}{17 a c^3 \left (1+a^2 x^2\right )^2}-\frac {12 e^{-\arctan (a x)} (1-2 a x)}{85 a c^3 \left (1+a^2 x^2\right )} \\ \end{align*}
Result contains complex when optimal does not.
Time = 0.12 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.02 \[ \int \frac {e^{-\arctan (a x)}}{\left (c+a^2 c x^2\right )^3} \, dx=\frac {5 e^{-\arctan (a x)} (-1+4 a x)-12 (1-i a x)^{-\frac {i}{2}} (1+i a x)^{\frac {i}{2}} \left (1+a^2 x^2\right ) \left (3-2 a x+2 a^2 x^2\right )}{85 a c^3 \left (1+a^2 x^2\right )^2} \]
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Time = 12.62 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.64
method | result | size |
gosper | \(-\frac {\left (24 a^{4} x^{4}-24 a^{3} x^{3}+60 a^{2} x^{2}-44 a x +41\right ) {\mathrm e}^{-\arctan \left (a x \right )}}{85 \left (a^{2} x^{2}+1\right )^{2} c^{3} a}\) | \(57\) |
parallelrisch | \(\frac {\left (-24 a^{4} x^{4}+24 a^{3} x^{3}-60 a^{2} x^{2}+44 a x -41\right ) {\mathrm e}^{-\arctan \left (a x \right )}}{85 c^{3} \left (a^{2} x^{2}+1\right )^{2} a}\) | \(57\) |
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none
Time = 0.26 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.76 \[ \int \frac {e^{-\arctan (a x)}}{\left (c+a^2 c x^2\right )^3} \, dx=-\frac {{\left (24 \, a^{4} x^{4} - 24 \, a^{3} x^{3} + 60 \, a^{2} x^{2} - 44 \, a x + 41\right )} e^{\left (-\arctan \left (a x\right )\right )}}{85 \, {\left (a^{5} c^{3} x^{4} + 2 \, a^{3} c^{3} x^{2} + a c^{3}\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 291 vs. \(2 (76) = 152\).
Time = 87.60 (sec) , antiderivative size = 291, normalized size of antiderivative = 3.27 \[ \int \frac {e^{-\arctan (a x)}}{\left (c+a^2 c x^2\right )^3} \, dx=\begin {cases} - \frac {24 a^{4} x^{4}}{85 a^{5} c^{3} x^{4} e^{\operatorname {atan}{\left (a x \right )}} + 170 a^{3} c^{3} x^{2} e^{\operatorname {atan}{\left (a x \right )}} + 85 a c^{3} e^{\operatorname {atan}{\left (a x \right )}}} + \frac {24 a^{3} x^{3}}{85 a^{5} c^{3} x^{4} e^{\operatorname {atan}{\left (a x \right )}} + 170 a^{3} c^{3} x^{2} e^{\operatorname {atan}{\left (a x \right )}} + 85 a c^{3} e^{\operatorname {atan}{\left (a x \right )}}} - \frac {60 a^{2} x^{2}}{85 a^{5} c^{3} x^{4} e^{\operatorname {atan}{\left (a x \right )}} + 170 a^{3} c^{3} x^{2} e^{\operatorname {atan}{\left (a x \right )}} + 85 a c^{3} e^{\operatorname {atan}{\left (a x \right )}}} + \frac {44 a x}{85 a^{5} c^{3} x^{4} e^{\operatorname {atan}{\left (a x \right )}} + 170 a^{3} c^{3} x^{2} e^{\operatorname {atan}{\left (a x \right )}} + 85 a c^{3} e^{\operatorname {atan}{\left (a x \right )}}} - \frac {41}{85 a^{5} c^{3} x^{4} e^{\operatorname {atan}{\left (a x \right )}} + 170 a^{3} c^{3} x^{2} e^{\operatorname {atan}{\left (a x \right )}} + 85 a c^{3} e^{\operatorname {atan}{\left (a x \right )}}} & \text {for}\: a \neq 0 \\\frac {x}{c^{3}} & \text {otherwise} \end {cases} \]
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\[ \int \frac {e^{-\arctan (a x)}}{\left (c+a^2 c x^2\right )^3} \, dx=\int { \frac {e^{\left (-\arctan \left (a x\right )\right )}}{{\left (a^{2} c x^{2} + c\right )}^{3}} \,d x } \]
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\[ \int \frac {e^{-\arctan (a x)}}{\left (c+a^2 c x^2\right )^3} \, dx=\int { \frac {e^{\left (-\arctan \left (a x\right )\right )}}{{\left (a^{2} c x^{2} + c\right )}^{3}} \,d x } \]
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Time = 0.72 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.90 \[ \int \frac {e^{-\arctan (a x)}}{\left (c+a^2 c x^2\right )^3} \, dx=\frac {12\,{\mathrm {e}}^{-\mathrm {atan}\left (a\,x\right )}\,\left (2\,a\,x-1\right )}{85\,a\,c^3\,\left (a^2\,x^2+1\right )}-\frac {24\,{\mathrm {e}}^{-\mathrm {atan}\left (a\,x\right )}}{85\,a\,c^3}+\frac {{\mathrm {e}}^{-\mathrm {atan}\left (a\,x\right )}\,\left (4\,a\,x-1\right )}{17\,a\,c^3\,{\left (a^2\,x^2+1\right )}^2} \]
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