Integrand size = 21, antiderivative size = 123 \[ \int \frac {e^{2 \arctan (a x)}}{\left (c+a^2 c x^2\right )^4} \, dx=\frac {9 e^{2 \arctan (a x)}}{160 a c^4}+\frac {e^{2 \arctan (a x)} (1+3 a x)}{20 a c^4 \left (1+a^2 x^2\right )^3}+\frac {3 e^{2 \arctan (a x)} (1+2 a x)}{40 a c^4 \left (1+a^2 x^2\right )^2}+\frac {9 e^{2 \arctan (a x)} (1+a x)}{80 a c^4 \left (1+a^2 x^2\right )} \]
[Out]
Time = 0.09 (sec) , antiderivative size = 123, 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} \[ \int \frac {e^{2 \arctan (a x)}}{\left (c+a^2 c x^2\right )^4} \, dx=\frac {9 (a x+1) e^{2 \arctan (a x)}}{80 a c^4 \left (a^2 x^2+1\right )}+\frac {3 (2 a x+1) e^{2 \arctan (a x)}}{40 a c^4 \left (a^2 x^2+1\right )^2}+\frac {(3 a x+1) e^{2 \arctan (a x)}}{20 a c^4 \left (a^2 x^2+1\right )^3}+\frac {9 e^{2 \arctan (a x)}}{160 a c^4} \]
[In]
[Out]
Rule 5178
Rule 5179
Rubi steps \begin{align*} \text {integral}& = \frac {e^{2 \arctan (a x)} (1+3 a x)}{20 a c^4 \left (1+a^2 x^2\right )^3}+\frac {3 \int \frac {e^{2 \arctan (a x)}}{\left (c+a^2 c x^2\right )^3} \, dx}{4 c} \\ & = \frac {e^{2 \arctan (a x)} (1+3 a x)}{20 a c^4 \left (1+a^2 x^2\right )^3}+\frac {3 e^{2 \arctan (a x)} (1+2 a x)}{40 a c^4 \left (1+a^2 x^2\right )^2}+\frac {9 \int \frac {e^{2 \arctan (a x)}}{\left (c+a^2 c x^2\right )^2} \, dx}{20 c^2} \\ & = \frac {e^{2 \arctan (a x)} (1+3 a x)}{20 a c^4 \left (1+a^2 x^2\right )^3}+\frac {3 e^{2 \arctan (a x)} (1+2 a x)}{40 a c^4 \left (1+a^2 x^2\right )^2}+\frac {9 e^{2 \arctan (a x)} (1+a x)}{80 a c^4 \left (1+a^2 x^2\right )}+\frac {9 \int \frac {e^{2 \arctan (a x)}}{c+a^2 c x^2} \, dx}{80 c^3} \\ & = \frac {9 e^{2 \arctan (a x)}}{160 a c^4}+\frac {e^{2 \arctan (a x)} (1+3 a x)}{20 a c^4 \left (1+a^2 x^2\right )^3}+\frac {3 e^{2 \arctan (a x)} (1+2 a x)}{40 a c^4 \left (1+a^2 x^2\right )^2}+\frac {9 e^{2 \arctan (a x)} (1+a x)}{80 a c^4 \left (1+a^2 x^2\right )} \\ \end{align*}
Result contains complex when optimal does not.
Time = 0.20 (sec) , antiderivative size = 122, normalized size of antiderivative = 0.99 \[ \int \frac {e^{2 \arctan (a x)}}{\left (c+a^2 c x^2\right )^4} \, dx=\frac {8 c e^{2 \arctan (a x)} (1+3 a x)+3 \left (c+a^2 c x^2\right ) \left (4 e^{2 \arctan (a x)} (1+2 a x)+3 (1-i a x)^i (1+i a x)^{-i} (-i+a x) (i+a x) \left (3+2 a x+a^2 x^2\right )\right )}{160 a c^2 \left (c+a^2 c x^2\right )^3} \]
[In]
[Out]
Time = 31.91 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.59
method | result | size |
gosper | \(\frac {{\mathrm e}^{2 \arctan \left (a x \right )} \left (9 a^{6} x^{6}+18 a^{5} x^{5}+45 a^{4} x^{4}+60 a^{3} x^{3}+75 a^{2} x^{2}+66 a x +47\right )}{160 \left (a^{2} x^{2}+1\right )^{3} c^{4} a}\) | \(73\) |
parallelrisch | \(\frac {9 a^{6} {\mathrm e}^{2 \arctan \left (a x \right )} x^{6}+18 a^{5} {\mathrm e}^{2 \arctan \left (a x \right )} x^{5}+45 a^{4} {\mathrm e}^{2 \arctan \left (a x \right )} x^{4}+60 a^{3} x^{3} {\mathrm e}^{2 \arctan \left (a x \right )}+75 x^{2} {\mathrm e}^{2 \arctan \left (a x \right )} a^{2}+66 \,{\mathrm e}^{2 \arctan \left (a x \right )} a x +47 \,{\mathrm e}^{2 \arctan \left (a x \right )}}{160 c^{4} \left (a^{2} x^{2}+1\right )^{3} a}\) | \(116\) |
[In]
[Out]
none
Time = 0.29 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.77 \[ \int \frac {e^{2 \arctan (a x)}}{\left (c+a^2 c x^2\right )^4} \, dx=\frac {{\left (9 \, a^{6} x^{6} + 18 \, a^{5} x^{5} + 45 \, a^{4} x^{4} + 60 \, a^{3} x^{3} + 75 \, a^{2} x^{2} + 66 \, a x + 47\right )} e^{\left (2 \, \arctan \left (a x\right )\right )}}{160 \, {\left (a^{7} c^{4} x^{6} + 3 \, a^{5} c^{4} x^{4} + 3 \, a^{3} c^{4} x^{2} + a c^{4}\right )}} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 410 vs. \(2 (112) = 224\).
Time = 7.64 (sec) , antiderivative size = 410, normalized size of antiderivative = 3.33 \[ \int \frac {e^{2 \arctan (a x)}}{\left (c+a^2 c x^2\right )^4} \, dx=\begin {cases} \frac {9 a^{6} x^{6} e^{2 \operatorname {atan}{\left (a x \right )}}}{160 a^{7} c^{4} x^{6} + 480 a^{5} c^{4} x^{4} + 480 a^{3} c^{4} x^{2} + 160 a c^{4}} + \frac {18 a^{5} x^{5} e^{2 \operatorname {atan}{\left (a x \right )}}}{160 a^{7} c^{4} x^{6} + 480 a^{5} c^{4} x^{4} + 480 a^{3} c^{4} x^{2} + 160 a c^{4}} + \frac {45 a^{4} x^{4} e^{2 \operatorname {atan}{\left (a x \right )}}}{160 a^{7} c^{4} x^{6} + 480 a^{5} c^{4} x^{4} + 480 a^{3} c^{4} x^{2} + 160 a c^{4}} + \frac {60 a^{3} x^{3} e^{2 \operatorname {atan}{\left (a x \right )}}}{160 a^{7} c^{4} x^{6} + 480 a^{5} c^{4} x^{4} + 480 a^{3} c^{4} x^{2} + 160 a c^{4}} + \frac {75 a^{2} x^{2} e^{2 \operatorname {atan}{\left (a x \right )}}}{160 a^{7} c^{4} x^{6} + 480 a^{5} c^{4} x^{4} + 480 a^{3} c^{4} x^{2} + 160 a c^{4}} + \frac {66 a x e^{2 \operatorname {atan}{\left (a x \right )}}}{160 a^{7} c^{4} x^{6} + 480 a^{5} c^{4} x^{4} + 480 a^{3} c^{4} x^{2} + 160 a c^{4}} + \frac {47 e^{2 \operatorname {atan}{\left (a x \right )}}}{160 a^{7} c^{4} x^{6} + 480 a^{5} c^{4} x^{4} + 480 a^{3} c^{4} x^{2} + 160 a c^{4}} & \text {for}\: a \neq 0 \\\frac {x}{c^{4}} & \text {otherwise} \end {cases} \]
[In]
[Out]
\[ \int \frac {e^{2 \arctan (a x)}}{\left (c+a^2 c x^2\right )^4} \, dx=\int { \frac {e^{\left (2 \, \arctan \left (a x\right )\right )}}{{\left (a^{2} c x^{2} + c\right )}^{4}} \,d x } \]
[In]
[Out]
\[ \int \frac {e^{2 \arctan (a x)}}{\left (c+a^2 c x^2\right )^4} \, dx=\int { \frac {e^{\left (2 \, \arctan \left (a x\right )\right )}}{{\left (a^{2} c x^{2} + c\right )}^{4}} \,d x } \]
[In]
[Out]
Time = 0.78 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.90 \[ \int \frac {e^{2 \arctan (a x)}}{\left (c+a^2 c x^2\right )^4} \, dx=\frac {9\,{\mathrm {e}}^{2\,\mathrm {atan}\left (a\,x\right )}}{160\,a\,c^4}+\frac {9\,{\mathrm {e}}^{2\,\mathrm {atan}\left (a\,x\right )}\,\left (a\,x+1\right )}{80\,a\,c^4\,\left (a^2\,x^2+1\right )}+\frac {3\,{\mathrm {e}}^{2\,\mathrm {atan}\left (a\,x\right )}\,\left (2\,a\,x+1\right )}{40\,a\,c^4\,{\left (a^2\,x^2+1\right )}^2}+\frac {{\mathrm {e}}^{2\,\mathrm {atan}\left (a\,x\right )}\,\left (3\,a\,x+1\right )}{20\,a\,c^4\,{\left (a^2\,x^2+1\right )}^3} \]
[In]
[Out]