Integrand size = 19, antiderivative size = 149 \[ \int \frac {e^{\arctan (a x)}}{\left (c+a^2 c x^2\right )^5} \, dx=\frac {8064 e^{\arctan (a x)}}{40885 a c^5}+\frac {e^{\arctan (a x)} (1+8 a x)}{65 a c^5 \left (1+a^2 x^2\right )^4}+\frac {56 e^{\arctan (a x)} (1+6 a x)}{2405 a c^5 \left (1+a^2 x^2\right )^3}+\frac {336 e^{\arctan (a x)} (1+4 a x)}{8177 a c^5 \left (1+a^2 x^2\right )^2}+\frac {4032 e^{\arctan (a x)} (1+2 a x)}{40885 a c^5 \left (1+a^2 x^2\right )} \]
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Time = 0.11 (sec) , antiderivative size = 149, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {5178, 5179} \[ \int \frac {e^{\arctan (a x)}}{\left (c+a^2 c x^2\right )^5} \, dx=\frac {4032 (2 a x+1) e^{\arctan (a x)}}{40885 a c^5 \left (a^2 x^2+1\right )}+\frac {336 (4 a x+1) e^{\arctan (a x)}}{8177 a c^5 \left (a^2 x^2+1\right )^2}+\frac {56 (6 a x+1) e^{\arctan (a x)}}{2405 a c^5 \left (a^2 x^2+1\right )^3}+\frac {(8 a x+1) e^{\arctan (a x)}}{65 a c^5 \left (a^2 x^2+1\right )^4}+\frac {8064 e^{\arctan (a x)}}{40885 a c^5} \]
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Rule 5178
Rule 5179
Rubi steps \begin{align*} \text {integral}& = \frac {e^{\arctan (a x)} (1+8 a x)}{65 a c^5 \left (1+a^2 x^2\right )^4}+\frac {56 \int \frac {e^{\arctan (a x)}}{\left (c+a^2 c x^2\right )^4} \, dx}{65 c} \\ & = \frac {e^{\arctan (a x)} (1+8 a x)}{65 a c^5 \left (1+a^2 x^2\right )^4}+\frac {56 e^{\arctan (a x)} (1+6 a x)}{2405 a c^5 \left (1+a^2 x^2\right )^3}+\frac {336 \int \frac {e^{\arctan (a x)}}{\left (c+a^2 c x^2\right )^3} \, dx}{481 c^2} \\ & = \frac {e^{\arctan (a x)} (1+8 a x)}{65 a c^5 \left (1+a^2 x^2\right )^4}+\frac {56 e^{\arctan (a x)} (1+6 a x)}{2405 a c^5 \left (1+a^2 x^2\right )^3}+\frac {336 e^{\arctan (a x)} (1+4 a x)}{8177 a c^5 \left (1+a^2 x^2\right )^2}+\frac {4032 \int \frac {e^{\arctan (a x)}}{\left (c+a^2 c x^2\right )^2} \, dx}{8177 c^3} \\ & = \frac {e^{\arctan (a x)} (1+8 a x)}{65 a c^5 \left (1+a^2 x^2\right )^4}+\frac {56 e^{\arctan (a x)} (1+6 a x)}{2405 a c^5 \left (1+a^2 x^2\right )^3}+\frac {336 e^{\arctan (a x)} (1+4 a x)}{8177 a c^5 \left (1+a^2 x^2\right )^2}+\frac {4032 e^{\arctan (a x)} (1+2 a x)}{40885 a c^5 \left (1+a^2 x^2\right )}+\frac {8064 \int \frac {e^{\arctan (a x)}}{c+a^2 c x^2} \, dx}{40885 c^4} \\ & = \frac {8064 e^{\arctan (a x)}}{40885 a c^5}+\frac {e^{\arctan (a x)} (1+8 a x)}{65 a c^5 \left (1+a^2 x^2\right )^4}+\frac {56 e^{\arctan (a x)} (1+6 a x)}{2405 a c^5 \left (1+a^2 x^2\right )^3}+\frac {336 e^{\arctan (a x)} (1+4 a x)}{8177 a c^5 \left (1+a^2 x^2\right )^2}+\frac {4032 e^{\arctan (a x)} (1+2 a x)}{40885 a c^5 \left (1+a^2 x^2\right )} \\ \end{align*}
Result contains complex when optimal does not.
Time = 0.21 (sec) , antiderivative size = 153, normalized size of antiderivative = 1.03 \[ \int \frac {e^{\arctan (a x)}}{\left (c+a^2 c x^2\right )^5} \, dx=\frac {629 e^{\arctan (a x)} (1+8 a x)+\frac {56 \left (c+a^2 c x^2\right ) \left (17 c e^{\arctan (a x)} (1+6 a x)+6 \left (c+a^2 c x^2\right ) \left (5 e^{\arctan (a x)} (1+4 a x)+12 (1-i a x)^{\frac {i}{2}} (1+i a x)^{-\frac {i}{2}} (-i+a x) (i+a x) \left (3+2 a x+2 a^2 x^2\right )\right )\right )}{c^2}}{40885 a c \left (c+a^2 c x^2\right )^4} \]
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Time = 85.38 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.58
method | result | size |
gosper | \(\frac {{\mathrm e}^{\arctan \left (a x \right )} \left (8064 a^{8} x^{8}+8064 a^{7} x^{7}+36288 a^{6} x^{6}+30912 a^{5} x^{5}+62160 a^{4} x^{4}+43344 a^{3} x^{3}+48664 a^{2} x^{2}+25528 a x +15357\right )}{40885 \left (a^{2} x^{2}+1\right )^{4} c^{5} a}\) | \(87\) |
parallelrisch | \(\frac {8064 a^{8} {\mathrm e}^{\arctan \left (a x \right )} x^{8}+8064 a^{7} {\mathrm e}^{\arctan \left (a x \right )} x^{7}+36288 a^{6} {\mathrm e}^{\arctan \left (a x \right )} x^{6}+30912 a^{5} {\mathrm e}^{\arctan \left (a x \right )} x^{5}+62160 a^{4} {\mathrm e}^{\arctan \left (a x \right )} x^{4}+43344 a^{3} x^{3} {\mathrm e}^{\arctan \left (a x \right )}+48664 x^{2} {\mathrm e}^{\arctan \left (a x \right )} a^{2}+25528 \,{\mathrm e}^{\arctan \left (a x \right )} a x +15357 \,{\mathrm e}^{\arctan \left (a x \right )}}{40885 c^{5} \left (a^{2} x^{2}+1\right )^{4} a}\) | \(128\) |
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none
Time = 0.28 (sec) , antiderivative size = 120, normalized size of antiderivative = 0.81 \[ \int \frac {e^{\arctan (a x)}}{\left (c+a^2 c x^2\right )^5} \, dx=\frac {{\left (8064 \, a^{8} x^{8} + 8064 \, a^{7} x^{7} + 36288 \, a^{6} x^{6} + 30912 \, a^{5} x^{5} + 62160 \, a^{4} x^{4} + 43344 \, a^{3} x^{3} + 48664 \, a^{2} x^{2} + 25528 \, a x + 15357\right )} e^{\left (\arctan \left (a x\right )\right )}}{40885 \, {\left (a^{9} c^{5} x^{8} + 4 \, a^{7} c^{5} x^{6} + 6 \, a^{5} c^{5} x^{4} + 4 \, a^{3} c^{5} x^{2} + a c^{5}\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 620 vs. \(2 (139) = 278\).
Time = 21.90 (sec) , antiderivative size = 620, normalized size of antiderivative = 4.16 \[ \int \frac {e^{\arctan (a x)}}{\left (c+a^2 c x^2\right )^5} \, dx=\begin {cases} \frac {8064 a^{8} x^{8} e^{\operatorname {atan}{\left (a x \right )}}}{40885 a^{9} c^{5} x^{8} + 163540 a^{7} c^{5} x^{6} + 245310 a^{5} c^{5} x^{4} + 163540 a^{3} c^{5} x^{2} + 40885 a c^{5}} + \frac {8064 a^{7} x^{7} e^{\operatorname {atan}{\left (a x \right )}}}{40885 a^{9} c^{5} x^{8} + 163540 a^{7} c^{5} x^{6} + 245310 a^{5} c^{5} x^{4} + 163540 a^{3} c^{5} x^{2} + 40885 a c^{5}} + \frac {36288 a^{6} x^{6} e^{\operatorname {atan}{\left (a x \right )}}}{40885 a^{9} c^{5} x^{8} + 163540 a^{7} c^{5} x^{6} + 245310 a^{5} c^{5} x^{4} + 163540 a^{3} c^{5} x^{2} + 40885 a c^{5}} + \frac {30912 a^{5} x^{5} e^{\operatorname {atan}{\left (a x \right )}}}{40885 a^{9} c^{5} x^{8} + 163540 a^{7} c^{5} x^{6} + 245310 a^{5} c^{5} x^{4} + 163540 a^{3} c^{5} x^{2} + 40885 a c^{5}} + \frac {62160 a^{4} x^{4} e^{\operatorname {atan}{\left (a x \right )}}}{40885 a^{9} c^{5} x^{8} + 163540 a^{7} c^{5} x^{6} + 245310 a^{5} c^{5} x^{4} + 163540 a^{3} c^{5} x^{2} + 40885 a c^{5}} + \frac {43344 a^{3} x^{3} e^{\operatorname {atan}{\left (a x \right )}}}{40885 a^{9} c^{5} x^{8} + 163540 a^{7} c^{5} x^{6} + 245310 a^{5} c^{5} x^{4} + 163540 a^{3} c^{5} x^{2} + 40885 a c^{5}} + \frac {48664 a^{2} x^{2} e^{\operatorname {atan}{\left (a x \right )}}}{40885 a^{9} c^{5} x^{8} + 163540 a^{7} c^{5} x^{6} + 245310 a^{5} c^{5} x^{4} + 163540 a^{3} c^{5} x^{2} + 40885 a c^{5}} + \frac {25528 a x e^{\operatorname {atan}{\left (a x \right )}}}{40885 a^{9} c^{5} x^{8} + 163540 a^{7} c^{5} x^{6} + 245310 a^{5} c^{5} x^{4} + 163540 a^{3} c^{5} x^{2} + 40885 a c^{5}} + \frac {15357 e^{\operatorname {atan}{\left (a x \right )}}}{40885 a^{9} c^{5} x^{8} + 163540 a^{7} c^{5} x^{6} + 245310 a^{5} c^{5} x^{4} + 163540 a^{3} c^{5} x^{2} + 40885 a c^{5}} & \text {for}\: a \neq 0 \\\frac {x}{c^{5}} & \text {otherwise} \end {cases} \]
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\[ \int \frac {e^{\arctan (a x)}}{\left (c+a^2 c x^2\right )^5} \, dx=\int { \frac {e^{\left (\arctan \left (a x\right )\right )}}{{\left (a^{2} c x^{2} + c\right )}^{5}} \,d x } \]
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\[ \int \frac {e^{\arctan (a x)}}{\left (c+a^2 c x^2\right )^5} \, dx=\int { \frac {e^{\left (\arctan \left (a x\right )\right )}}{{\left (a^{2} c x^{2} + c\right )}^{5}} \,d x } \]
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Time = 0.86 (sec) , antiderivative size = 134, normalized size of antiderivative = 0.90 \[ \int \frac {e^{\arctan (a x)}}{\left (c+a^2 c x^2\right )^5} \, dx=\frac {{\mathrm {e}}^{\mathrm {atan}\left (a\,x\right )}\,\left (\frac {15357}{40885\,a^9\,c^5}+\frac {25528\,x}{40885\,a^8\,c^5}+\frac {8064\,x^8}{40885\,a\,c^5}+\frac {8064\,x^7}{40885\,a^2\,c^5}+\frac {36288\,x^6}{40885\,a^3\,c^5}+\frac {30912\,x^5}{40885\,a^4\,c^5}+\frac {336\,x^4}{221\,a^5\,c^5}+\frac {43344\,x^3}{40885\,a^6\,c^5}+\frac {48664\,x^2}{40885\,a^7\,c^5}\right )}{\frac {1}{a^8}+x^8+\frac {4\,x^6}{a^2}+\frac {6\,x^4}{a^4}+\frac {4\,x^2}{a^6}} \]
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