Integrand size = 19, antiderivative size = 225 \[ \int \frac {\arctan (a x)^3}{\left (c+a^2 c x^2\right )^3} \, dx=-\frac {3}{128 a c^3 \left (1+a^2 x^2\right )^2}-\frac {45}{128 a c^3 \left (1+a^2 x^2\right )}-\frac {3 x \arctan (a x)}{32 c^3 \left (1+a^2 x^2\right )^2}-\frac {45 x \arctan (a x)}{64 c^3 \left (1+a^2 x^2\right )}-\frac {45 \arctan (a x)^2}{128 a c^3}+\frac {3 \arctan (a x)^2}{16 a c^3 \left (1+a^2 x^2\right )^2}+\frac {9 \arctan (a x)^2}{16 a c^3 \left (1+a^2 x^2\right )}+\frac {x \arctan (a x)^3}{4 c^3 \left (1+a^2 x^2\right )^2}+\frac {3 x \arctan (a x)^3}{8 c^3 \left (1+a^2 x^2\right )}+\frac {3 \arctan (a x)^4}{32 a c^3} \]
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Time = 0.14 (sec) , antiderivative size = 225, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {5020, 5012, 5050, 267, 5016} \[ \int \frac {\arctan (a x)^3}{\left (c+a^2 c x^2\right )^3} \, dx=\frac {3 x \arctan (a x)^3}{8 c^3 \left (a^2 x^2+1\right )}+\frac {x \arctan (a x)^3}{4 c^3 \left (a^2 x^2+1\right )^2}+\frac {9 \arctan (a x)^2}{16 a c^3 \left (a^2 x^2+1\right )}+\frac {3 \arctan (a x)^2}{16 a c^3 \left (a^2 x^2+1\right )^2}-\frac {45 x \arctan (a x)}{64 c^3 \left (a^2 x^2+1\right )}-\frac {3 x \arctan (a x)}{32 c^3 \left (a^2 x^2+1\right )^2}-\frac {45}{128 a c^3 \left (a^2 x^2+1\right )}-\frac {3}{128 a c^3 \left (a^2 x^2+1\right )^2}+\frac {3 \arctan (a x)^4}{32 a c^3}-\frac {45 \arctan (a x)^2}{128 a c^3} \]
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Rule 267
Rule 5012
Rule 5016
Rule 5020
Rule 5050
Rubi steps \begin{align*} \text {integral}& = \frac {3 \arctan (a x)^2}{16 a c^3 \left (1+a^2 x^2\right )^2}+\frac {x \arctan (a x)^3}{4 c^3 \left (1+a^2 x^2\right )^2}-\frac {3}{8} \int \frac {\arctan (a x)}{\left (c+a^2 c x^2\right )^3} \, dx+\frac {3 \int \frac {\arctan (a x)^3}{\left (c+a^2 c x^2\right )^2} \, dx}{4 c} \\ & = -\frac {3}{128 a c^3 \left (1+a^2 x^2\right )^2}-\frac {3 x \arctan (a x)}{32 c^3 \left (1+a^2 x^2\right )^2}+\frac {3 \arctan (a x)^2}{16 a c^3 \left (1+a^2 x^2\right )^2}+\frac {x \arctan (a x)^3}{4 c^3 \left (1+a^2 x^2\right )^2}+\frac {3 x \arctan (a x)^3}{8 c^3 \left (1+a^2 x^2\right )}+\frac {3 \arctan (a x)^4}{32 a c^3}-\frac {9 \int \frac {\arctan (a x)}{\left (c+a^2 c x^2\right )^2} \, dx}{32 c}-\frac {(9 a) \int \frac {x \arctan (a x)^2}{\left (c+a^2 c x^2\right )^2} \, dx}{8 c} \\ & = -\frac {3}{128 a c^3 \left (1+a^2 x^2\right )^2}-\frac {3 x \arctan (a x)}{32 c^3 \left (1+a^2 x^2\right )^2}-\frac {9 x \arctan (a x)}{64 c^3 \left (1+a^2 x^2\right )}-\frac {9 \arctan (a x)^2}{128 a c^3}+\frac {3 \arctan (a x)^2}{16 a c^3 \left (1+a^2 x^2\right )^2}+\frac {9 \arctan (a x)^2}{16 a c^3 \left (1+a^2 x^2\right )}+\frac {x \arctan (a x)^3}{4 c^3 \left (1+a^2 x^2\right )^2}+\frac {3 x \arctan (a x)^3}{8 c^3 \left (1+a^2 x^2\right )}+\frac {3 \arctan (a x)^4}{32 a c^3}-\frac {9 \int \frac {\arctan (a x)}{\left (c+a^2 c x^2\right )^2} \, dx}{8 c}+\frac {(9 a) \int \frac {x}{\left (c+a^2 c x^2\right )^2} \, dx}{64 c} \\ & = -\frac {3}{128 a c^3 \left (1+a^2 x^2\right )^2}-\frac {9}{128 a c^3 \left (1+a^2 x^2\right )}-\frac {3 x \arctan (a x)}{32 c^3 \left (1+a^2 x^2\right )^2}-\frac {45 x \arctan (a x)}{64 c^3 \left (1+a^2 x^2\right )}-\frac {45 \arctan (a x)^2}{128 a c^3}+\frac {3 \arctan (a x)^2}{16 a c^3 \left (1+a^2 x^2\right )^2}+\frac {9 \arctan (a x)^2}{16 a c^3 \left (1+a^2 x^2\right )}+\frac {x \arctan (a x)^3}{4 c^3 \left (1+a^2 x^2\right )^2}+\frac {3 x \arctan (a x)^3}{8 c^3 \left (1+a^2 x^2\right )}+\frac {3 \arctan (a x)^4}{32 a c^3}+\frac {(9 a) \int \frac {x}{\left (c+a^2 c x^2\right )^2} \, dx}{16 c} \\ & = -\frac {3}{128 a c^3 \left (1+a^2 x^2\right )^2}-\frac {45}{128 a c^3 \left (1+a^2 x^2\right )}-\frac {3 x \arctan (a x)}{32 c^3 \left (1+a^2 x^2\right )^2}-\frac {45 x \arctan (a x)}{64 c^3 \left (1+a^2 x^2\right )}-\frac {45 \arctan (a x)^2}{128 a c^3}+\frac {3 \arctan (a x)^2}{16 a c^3 \left (1+a^2 x^2\right )^2}+\frac {9 \arctan (a x)^2}{16 a c^3 \left (1+a^2 x^2\right )}+\frac {x \arctan (a x)^3}{4 c^3 \left (1+a^2 x^2\right )^2}+\frac {3 x \arctan (a x)^3}{8 c^3 \left (1+a^2 x^2\right )}+\frac {3 \arctan (a x)^4}{32 a c^3} \\ \end{align*}
Time = 0.11 (sec) , antiderivative size = 114, normalized size of antiderivative = 0.51 \[ \int \frac {\arctan (a x)^3}{\left (c+a^2 c x^2\right )^3} \, dx=-\frac {48+45 a^2 x^2+6 a x \left (17+15 a^2 x^2\right ) \arctan (a x)+3 \left (-17+6 a^2 x^2+15 a^4 x^4\right ) \arctan (a x)^2-16 a x \left (5+3 a^2 x^2\right ) \arctan (a x)^3-12 \left (1+a^2 x^2\right )^2 \arctan (a x)^4}{128 a c^3 \left (1+a^2 x^2\right )^2} \]
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Time = 1.40 (sec) , antiderivative size = 153, normalized size of antiderivative = 0.68
method | result | size |
parallelrisch | \(\frac {12 a^{4} x^{4} \arctan \left (a x \right )^{4}-45 a^{4} \arctan \left (a x \right )^{2} x^{4}+48 \arctan \left (a x \right )^{3} a^{3} x^{3}+48 a^{4} x^{4}+24 \arctan \left (a x \right )^{4} x^{2} a^{2}-90 \arctan \left (a x \right ) x^{3} a^{3}-18 x^{2} \arctan \left (a x \right )^{2} a^{2}+80 \arctan \left (a x \right )^{3} a x +51 a^{2} x^{2}+12 \arctan \left (a x \right )^{4}-102 x \arctan \left (a x \right ) a +51 \arctan \left (a x \right )^{2}}{128 c^{3} \left (a^{2} x^{2}+1\right )^{2} a}\) | \(153\) |
derivativedivides | \(\frac {\frac {\arctan \left (a x \right )^{3} a x}{4 c^{3} \left (a^{2} x^{2}+1\right )^{2}}+\frac {3 \arctan \left (a x \right )^{3} a x}{8 c^{3} \left (a^{2} x^{2}+1\right )}+\frac {3 \arctan \left (a x \right )^{4}}{8 c^{3}}-\frac {3 \left (-\frac {3 \arctan \left (a x \right )^{2}}{2 \left (a^{2} x^{2}+1\right )}-\frac {\arctan \left (a x \right )^{2}}{2 \left (a^{2} x^{2}+1\right )^{2}}+\frac {15 \arctan \left (a x \right ) a^{3} x^{3}}{8 \left (a^{2} x^{2}+1\right )^{2}}+\frac {17 \arctan \left (a x \right ) a x}{8 \left (a^{2} x^{2}+1\right )^{2}}+\frac {15 \arctan \left (a x \right )^{2}}{16}+\frac {15}{16 \left (a^{2} x^{2}+1\right )}+\frac {1}{16 \left (a^{2} x^{2}+1\right )^{2}}+\frac {3 \arctan \left (a x \right )^{4}}{4}\right )}{8 c^{3}}}{a}\) | \(193\) |
default | \(\frac {\frac {\arctan \left (a x \right )^{3} a x}{4 c^{3} \left (a^{2} x^{2}+1\right )^{2}}+\frac {3 \arctan \left (a x \right )^{3} a x}{8 c^{3} \left (a^{2} x^{2}+1\right )}+\frac {3 \arctan \left (a x \right )^{4}}{8 c^{3}}-\frac {3 \left (-\frac {3 \arctan \left (a x \right )^{2}}{2 \left (a^{2} x^{2}+1\right )}-\frac {\arctan \left (a x \right )^{2}}{2 \left (a^{2} x^{2}+1\right )^{2}}+\frac {15 \arctan \left (a x \right ) a^{3} x^{3}}{8 \left (a^{2} x^{2}+1\right )^{2}}+\frac {17 \arctan \left (a x \right ) a x}{8 \left (a^{2} x^{2}+1\right )^{2}}+\frac {15 \arctan \left (a x \right )^{2}}{16}+\frac {15}{16 \left (a^{2} x^{2}+1\right )}+\frac {1}{16 \left (a^{2} x^{2}+1\right )^{2}}+\frac {3 \arctan \left (a x \right )^{4}}{4}\right )}{8 c^{3}}}{a}\) | \(193\) |
parts | \(\frac {x \arctan \left (a x \right )^{3}}{4 c^{3} \left (a^{2} x^{2}+1\right )^{2}}+\frac {3 x \arctan \left (a x \right )^{3}}{8 c^{3} \left (a^{2} x^{2}+1\right )}+\frac {3 \arctan \left (a x \right )^{4}}{8 a \,c^{3}}-\frac {3 \left (\frac {-\frac {3 \arctan \left (a x \right )^{2}}{2 \left (a^{2} x^{2}+1\right )}-\frac {\arctan \left (a x \right )^{2}}{2 \left (a^{2} x^{2}+1\right )^{2}}+\frac {15 \arctan \left (a x \right ) a^{3} x^{3}}{8 \left (a^{2} x^{2}+1\right )^{2}}+\frac {17 \arctan \left (a x \right ) a x}{8 \left (a^{2} x^{2}+1\right )^{2}}+\frac {15 \arctan \left (a x \right )^{2}}{16}+\frac {15}{16 \left (a^{2} x^{2}+1\right )}+\frac {1}{16 \left (a^{2} x^{2}+1\right )^{2}}}{a}+\frac {3 \arctan \left (a x \right )^{4}}{4 a}\right )}{8 c^{3}}\) | \(198\) |
risch | \(\frac {3 \ln \left (i a x +1\right )^{4}}{512 c^{3} a}-\frac {\left (3 x^{4} \ln \left (-i a x +1\right ) a^{4}+6 a^{2} x^{2} \ln \left (-i a x +1\right )-6 i a^{3} x^{3}+3 \ln \left (-i a x +1\right )-10 i a x \right ) \ln \left (i a x +1\right )^{3}}{128 c^{3} \left (a^{2} x^{2}+1\right )^{2} a}+\frac {3 \left (6 a^{4} x^{4} \ln \left (-i a x +1\right )^{2}-24 i x^{3} \ln \left (-i a x +1\right ) a^{3}+15 a^{4} x^{4}+12 a^{2} x^{2} \ln \left (-i a x +1\right )^{2}-40 i a x \ln \left (-i a x +1\right )+6 a^{2} x^{2}+6 \ln \left (-i a x +1\right )^{2}-17\right ) \ln \left (i a x +1\right )^{2}}{512 \left (a x +i\right )^{2} a \,c^{3} \left (a x -i\right )^{2}}-\frac {3 \left (2 a^{4} x^{4} \ln \left (-i a x +1\right )^{3}-12 i x^{3} \ln \left (-i a x +1\right )^{2} a^{3}+15 x^{4} \ln \left (-i a x +1\right ) a^{4}-30 i a^{3} x^{3}+4 a^{2} x^{2} \ln \left (-i a x +1\right )^{3}-20 i a x \ln \left (-i a x +1\right )^{2}+6 a^{2} x^{2} \ln \left (-i a x +1\right )-34 i a x +2 \ln \left (-i a x +1\right )^{3}-17 \ln \left (-i a x +1\right )\right ) \ln \left (i a x +1\right )}{256 \left (a x +i\right )^{2} a \,c^{3} \left (a x -i\right )^{2}}+\frac {3 a^{4} x^{4} \ln \left (-i a x +1\right )^{4}+45 a^{4} x^{4} \ln \left (-i a x +1\right )^{2}+6 a^{2} x^{2} \ln \left (-i a x +1\right )^{4}-24 i a^{3} x^{3} \ln \left (-i a x +1\right )^{3}+18 a^{2} x^{2} \ln \left (-i a x +1\right )^{2}-180 i x^{3} \ln \left (-i a x +1\right ) a^{3}-180 a^{2} x^{2}+3 \ln \left (-i a x +1\right )^{4}-40 i a x \ln \left (-i a x +1\right )^{3}-51 \ln \left (-i a x +1\right )^{2}-204 i a x \ln \left (-i a x +1\right )-192}{512 \left (a x +i\right )^{2} a \,c^{3} \left (a x -i\right )^{2}}\) | \(601\) |
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Time = 0.25 (sec) , antiderivative size = 132, normalized size of antiderivative = 0.59 \[ \int \frac {\arctan (a x)^3}{\left (c+a^2 c x^2\right )^3} \, dx=\frac {12 \, {\left (a^{4} x^{4} + 2 \, a^{2} x^{2} + 1\right )} \arctan \left (a x\right )^{4} - 45 \, a^{2} x^{2} + 16 \, {\left (3 \, a^{3} x^{3} + 5 \, a x\right )} \arctan \left (a x\right )^{3} - 3 \, {\left (15 \, a^{4} x^{4} + 6 \, a^{2} x^{2} - 17\right )} \arctan \left (a x\right )^{2} - 6 \, {\left (15 \, a^{3} x^{3} + 17 \, a x\right )} \arctan \left (a x\right ) - 48}{128 \, {\left (a^{5} c^{3} x^{4} + 2 \, a^{3} c^{3} x^{2} + a c^{3}\right )}} \]
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\[ \int \frac {\arctan (a x)^3}{\left (c+a^2 c x^2\right )^3} \, dx=\frac {\int \frac {\operatorname {atan}^{3}{\left (a x \right )}}{a^{6} x^{6} + 3 a^{4} x^{4} + 3 a^{2} x^{2} + 1}\, dx}{c^{3}} \]
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Time = 0.35 (sec) , antiderivative size = 335, normalized size of antiderivative = 1.49 \[ \int \frac {\arctan (a x)^3}{\left (c+a^2 c x^2\right )^3} \, dx=\frac {1}{8} \, {\left (\frac {3 \, a^{2} x^{3} + 5 \, x}{a^{4} c^{3} x^{4} + 2 \, a^{2} c^{3} x^{2} + c^{3}} + \frac {3 \, \arctan \left (a x\right )}{a c^{3}}\right )} \arctan \left (a x\right )^{3} + \frac {3 \, {\left (3 \, a^{2} x^{2} - 3 \, {\left (a^{4} x^{4} + 2 \, a^{2} x^{2} + 1\right )} \arctan \left (a x\right )^{2} + 4\right )} a \arctan \left (a x\right )^{2}}{16 \, {\left (a^{6} c^{3} x^{4} + 2 \, a^{4} c^{3} x^{2} + a^{2} c^{3}\right )}} - \frac {3}{128} \, {\left (\frac {{\left (4 \, {\left (a^{4} x^{4} + 2 \, a^{2} x^{2} + 1\right )} \arctan \left (a x\right )^{4} + 15 \, a^{2} x^{2} - 15 \, {\left (a^{4} x^{4} + 2 \, a^{2} x^{2} + 1\right )} \arctan \left (a x\right )^{2} + 16\right )} a^{2}}{a^{8} c^{3} x^{4} + 2 \, a^{6} c^{3} x^{2} + a^{4} c^{3}} + \frac {2 \, {\left (15 \, a^{3} x^{3} - 8 \, {\left (a^{4} x^{4} + 2 \, a^{2} x^{2} + 1\right )} \arctan \left (a x\right )^{3} + 17 \, a x + 15 \, {\left (a^{4} x^{4} + 2 \, a^{2} x^{2} + 1\right )} \arctan \left (a x\right )\right )} a \arctan \left (a x\right )}{a^{7} c^{3} x^{4} + 2 \, a^{5} c^{3} x^{2} + a^{3} c^{3}}\right )} a \]
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\[ \int \frac {\arctan (a x)^3}{\left (c+a^2 c x^2\right )^3} \, dx=\int { \frac {\arctan \left (a x\right )^{3}}{{\left (a^{2} c x^{2} + c\right )}^{3}} \,d x } \]
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Time = 0.60 (sec) , antiderivative size = 199, normalized size of antiderivative = 0.88 \[ \int \frac {\arctan (a x)^3}{\left (c+a^2 c x^2\right )^3} \, dx={\mathrm {atan}\left (a\,x\right )}^2\,\left (\frac {\frac {3}{4\,a^3\,c^3}+\frac {9\,x^2}{16\,a\,c^3}}{\frac {1}{a^2}+2\,x^2+a^2\,x^4}-\frac {45}{128\,a\,c^3}\right )-\frac {\frac {45\,a\,x^2}{2}+\frac {24}{a}}{64\,a^4\,c^3\,x^4+128\,a^2\,c^3\,x^2+64\,c^3}-\frac {\mathrm {atan}\left (a\,x\right )\,\left (\frac {45\,x^3}{64\,c^3}+\frac {51\,x}{64\,a^2\,c^3}\right )}{\frac {1}{a^2}+2\,x^2+a^2\,x^4}+\frac {{\mathrm {atan}\left (a\,x\right )}^3\,\left (\frac {3\,x^3}{8\,c^3}+\frac {5\,x}{8\,a^2\,c^3}\right )}{\frac {1}{a^2}+2\,x^2+a^2\,x^4}+\frac {3\,{\mathrm {atan}\left (a\,x\right )}^4}{32\,a\,c^3} \]
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