Integrand size = 21, antiderivative size = 181 \[ \int \frac {e^{n \arctan (a x)}}{\left (c+a^2 c x^2\right )^4} \, dx=\frac {720 e^{n \arctan (a x)}}{a c^4 n \left (4+n^2\right ) \left (16+n^2\right ) \left (36+n^2\right )}+\frac {e^{n \arctan (a x)} (n+6 a x)}{a c^4 \left (36+n^2\right ) \left (1+a^2 x^2\right )^3}+\frac {30 e^{n \arctan (a x)} (n+4 a x)}{a c^4 \left (16+n^2\right ) \left (36+n^2\right ) \left (1+a^2 x^2\right )^2}+\frac {360 e^{n \arctan (a x)} (n+2 a x)}{a c^4 \left (4+n^2\right ) \left (16+n^2\right ) \left (36+n^2\right ) \left (1+a^2 x^2\right )} \]
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Time = 0.13 (sec) , antiderivative size = 181, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {5178, 5179} \[ \int \frac {e^{n \arctan (a x)}}{\left (c+a^2 c x^2\right )^4} \, dx=\frac {360 (2 a x+n) e^{n \arctan (a x)}}{a c^4 \left (n^2+4\right ) \left (n^2+16\right ) \left (n^2+36\right ) \left (a^2 x^2+1\right )}+\frac {30 (4 a x+n) e^{n \arctan (a x)}}{a c^4 \left (n^2+16\right ) \left (n^2+36\right ) \left (a^2 x^2+1\right )^2}+\frac {(6 a x+n) e^{n \arctan (a x)}}{a c^4 \left (n^2+36\right ) \left (a^2 x^2+1\right )^3}+\frac {720 e^{n \arctan (a x)}}{a c^4 n \left (n^2+4\right ) \left (n^2+16\right ) \left (n^2+36\right )} \]
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Rule 5178
Rule 5179
Rubi steps \begin{align*} \text {integral}& = \frac {e^{n \arctan (a x)} (n+6 a x)}{a c^4 \left (36+n^2\right ) \left (1+a^2 x^2\right )^3}+\frac {30 \int \frac {e^{n \arctan (a x)}}{\left (c+a^2 c x^2\right )^3} \, dx}{c \left (36+n^2\right )} \\ & = \frac {e^{n \arctan (a x)} (n+6 a x)}{a c^4 \left (36+n^2\right ) \left (1+a^2 x^2\right )^3}+\frac {30 e^{n \arctan (a x)} (n+4 a x)}{a c^4 \left (16+n^2\right ) \left (36+n^2\right ) \left (1+a^2 x^2\right )^2}+\frac {360 \int \frac {e^{n \arctan (a x)}}{\left (c+a^2 c x^2\right )^2} \, dx}{c^2 \left (16+n^2\right ) \left (36+n^2\right )} \\ & = \frac {e^{n \arctan (a x)} (n+6 a x)}{a c^4 \left (36+n^2\right ) \left (1+a^2 x^2\right )^3}+\frac {30 e^{n \arctan (a x)} (n+4 a x)}{a c^4 \left (16+n^2\right ) \left (36+n^2\right ) \left (1+a^2 x^2\right )^2}+\frac {360 e^{n \arctan (a x)} (n+2 a x)}{a c^4 \left (4+n^2\right ) \left (16+n^2\right ) \left (36+n^2\right ) \left (1+a^2 x^2\right )}+\frac {720 \int \frac {e^{n \arctan (a x)}}{c+a^2 c x^2} \, dx}{c^3 \left (4+n^2\right ) \left (16+n^2\right ) \left (36+n^2\right )} \\ & = \frac {720 e^{n \arctan (a x)}}{a c^4 n \left (4+n^2\right ) \left (16+n^2\right ) \left (36+n^2\right )}+\frac {e^{n \arctan (a x)} (n+6 a x)}{a c^4 \left (36+n^2\right ) \left (1+a^2 x^2\right )^3}+\frac {30 e^{n \arctan (a x)} (n+4 a x)}{a c^4 \left (16+n^2\right ) \left (36+n^2\right ) \left (1+a^2 x^2\right )^2}+\frac {360 e^{n \arctan (a x)} (n+2 a x)}{a c^4 \left (4+n^2\right ) \left (16+n^2\right ) \left (36+n^2\right ) \left (1+a^2 x^2\right )} \\ \end{align*}
Result contains complex when optimal does not.
Time = 0.35 (sec) , antiderivative size = 165, normalized size of antiderivative = 0.91 \[ \int \frac {e^{n \arctan (a x)}}{\left (c+a^2 c x^2\right )^4} \, dx=\frac {e^{n \arctan (a x)} (n+6 a x)+\frac {30 \left (c+a^2 c x^2\right ) \left (e^{n \arctan (a x)} n (-2 i+n) (2 i+n) (n+4 a x)+12 (1-i a x)^{\frac {i n}{2}} (1+i a x)^{-\frac {i n}{2}} (-i+a x) (i+a x) \left (2+n^2+2 a n x+2 a^2 x^2\right )\right )}{c n \left (64+20 n^2+n^4\right )}}{a c \left (36+n^2\right ) \left (c+a^2 c x^2\right )^3} \]
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Time = 35.41 (sec) , antiderivative size = 166, normalized size of antiderivative = 0.92
method | result | size |
gosper | \(\frac {\left (720 a^{6} x^{6}+720 a^{5} n \,x^{5}+360 a^{4} n^{2} x^{4}+120 a^{3} n^{3} x^{3}+2160 a^{4} x^{4}+30 a^{2} n^{4} x^{2}+1920 a^{3} n \,x^{3}+6 a \,n^{5} x +840 a^{2} n^{2} x^{2}+n^{6}+240 a \,n^{3} x +2160 a^{2} x^{2}+50 n^{4}+1584 n a x +544 n^{2}+720\right ) {\mathrm e}^{n \arctan \left (a x \right )}}{\left (a^{2} x^{2}+1\right )^{3} c^{4} a n \left (n^{6}+56 n^{4}+784 n^{2}+2304\right )}\) | \(166\) |
parallelrisch | \(\frac {720 a^{6} {\mathrm e}^{n \arctan \left (a x \right )} x^{6}+720 \,{\mathrm e}^{n \arctan \left (a x \right )}+2160 a^{2} {\mathrm e}^{n \arctan \left (a x \right )} x^{2}+360 x^{4} {\mathrm e}^{n \arctan \left (a x \right )} a^{4} n^{2}+120 x^{3} {\mathrm e}^{n \arctan \left (a x \right )} a^{3} n^{3}+30 x^{2} {\mathrm e}^{n \arctan \left (a x \right )} a^{2} n^{4}+1584 \,{\mathrm e}^{n \arctan \left (a x \right )} x n a +1920 a^{3} x^{3} {\mathrm e}^{n \arctan \left (a x \right )} n +720 a^{5} {\mathrm e}^{n \arctan \left (a x \right )} x^{5} n +6 x \,{\mathrm e}^{n \arctan \left (a x \right )} a \,n^{5}+840 x^{2} {\mathrm e}^{n \arctan \left (a x \right )} a^{2} n^{2}+240 x \,{\mathrm e}^{n \arctan \left (a x \right )} a \,n^{3}+2160 a^{4} {\mathrm e}^{n \arctan \left (a x \right )} x^{4}+{\mathrm e}^{n \arctan \left (a x \right )} n^{6}+50 \,{\mathrm e}^{n \arctan \left (a x \right )} n^{4}+544 \,{\mathrm e}^{n \arctan \left (a x \right )} n^{2}}{c^{4} \left (a^{2} x^{2}+1\right )^{3} \left (n^{4}+40 n^{2}+144\right ) n a \left (n^{2}+16\right )}\) | \(275\) |
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Time = 0.28 (sec) , antiderivative size = 298, normalized size of antiderivative = 1.65 \[ \int \frac {e^{n \arctan (a x)}}{\left (c+a^2 c x^2\right )^4} \, dx=\frac {{\left (720 \, a^{6} x^{6} + 720 \, a^{5} n x^{5} + n^{6} + 360 \, {\left (a^{4} n^{2} + 6 \, a^{4}\right )} x^{4} + 50 \, n^{4} + 120 \, {\left (a^{3} n^{3} + 16 \, a^{3} n\right )} x^{3} + 30 \, {\left (a^{2} n^{4} + 28 \, a^{2} n^{2} + 72 \, a^{2}\right )} x^{2} + 544 \, n^{2} + 6 \, {\left (a n^{5} + 40 \, a n^{3} + 264 \, a n\right )} x + 720\right )} e^{\left (n \arctan \left (a x\right )\right )}}{a c^{4} n^{7} + 56 \, a c^{4} n^{5} + 784 \, a c^{4} n^{3} + {\left (a^{7} c^{4} n^{7} + 56 \, a^{7} c^{4} n^{5} + 784 \, a^{7} c^{4} n^{3} + 2304 \, a^{7} c^{4} n\right )} x^{6} + 2304 \, a c^{4} n + 3 \, {\left (a^{5} c^{4} n^{7} + 56 \, a^{5} c^{4} n^{5} + 784 \, a^{5} c^{4} n^{3} + 2304 \, a^{5} c^{4} n\right )} x^{4} + 3 \, {\left (a^{3} c^{4} n^{7} + 56 \, a^{3} c^{4} n^{5} + 784 \, a^{3} c^{4} n^{3} + 2304 \, a^{3} c^{4} n\right )} x^{2}} \]
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Timed out. \[ \int \frac {e^{n \arctan (a x)}}{\left (c+a^2 c x^2\right )^4} \, dx=\text {Timed out} \]
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\[ \int \frac {e^{n \arctan (a x)}}{\left (c+a^2 c x^2\right )^4} \, dx=\int { \frac {e^{\left (n \arctan \left (a x\right )\right )}}{{\left (a^{2} c x^{2} + c\right )}^{4}} \,d x } \]
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\[ \int \frac {e^{n \arctan (a x)}}{\left (c+a^2 c x^2\right )^4} \, dx=\int { \frac {e^{\left (n \arctan \left (a x\right )\right )}}{{\left (a^{2} c x^{2} + c\right )}^{4}} \,d x } \]
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Time = 0.98 (sec) , antiderivative size = 281, normalized size of antiderivative = 1.55 \[ \int \frac {e^{n \arctan (a x)}}{\left (c+a^2 c x^2\right )^4} \, dx=\frac {{\mathrm {e}}^{n\,\mathrm {atan}\left (a\,x\right )}\,\left (\frac {720\,x^5}{a^2\,c^4\,\left (n^6+56\,n^4+784\,n^2+2304\right )}+\frac {n^6+50\,n^4+544\,n^2+720}{a^7\,c^4\,n\,\left (n^6+56\,n^4+784\,n^2+2304\right )}+\frac {720\,x^6}{a\,c^4\,n\,\left (n^6+56\,n^4+784\,n^2+2304\right )}+\frac {6\,x\,\left (n^4+40\,n^2+264\right )}{a^6\,c^4\,\left (n^6+56\,n^4+784\,n^2+2304\right )}+\frac {120\,x^3\,\left (n^2+16\right )}{a^4\,c^4\,\left (n^6+56\,n^4+784\,n^2+2304\right )}+\frac {360\,x^4\,\left (n^2+6\right )}{a^3\,c^4\,n\,\left (n^6+56\,n^4+784\,n^2+2304\right )}+\frac {30\,x^2\,\left (n^4+28\,n^2+72\right )}{a^5\,c^4\,n\,\left (n^6+56\,n^4+784\,n^2+2304\right )}\right )}{\frac {1}{a^6}+x^6+\frac {3\,x^4}{a^2}+\frac {3\,x^2}{a^4}} \]
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