Integrand size = 20, antiderivative size = 136 \[ \int \frac {\arctan (a x)}{x^4 \left (c+a^2 c x^2\right )^2} \, dx=-\frac {a}{6 c^2 x^2}+\frac {a^3}{4 c^2 \left (1+a^2 x^2\right )}-\frac {\arctan (a x)}{3 c^2 x^3}+\frac {2 a^2 \arctan (a x)}{c^2 x}+\frac {a^4 x \arctan (a x)}{2 c^2 \left (1+a^2 x^2\right )}+\frac {5 a^3 \arctan (a x)^2}{4 c^2}-\frac {7 a^3 \log (x)}{3 c^2}+\frac {7 a^3 \log \left (1+a^2 x^2\right )}{6 c^2} \]
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Time = 0.29 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.00, number of steps used = 23, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.550, Rules used = {5086, 5038, 4946, 272, 46, 36, 29, 31, 5004, 5012, 267} \[ \int \frac {\arctan (a x)}{x^4 \left (c+a^2 c x^2\right )^2} \, dx=\frac {5 a^3 \arctan (a x)^2}{4 c^2}-\frac {7 a^3 \log (x)}{3 c^2}+\frac {2 a^2 \arctan (a x)}{c^2 x}+\frac {a^4 x \arctan (a x)}{2 c^2 \left (a^2 x^2+1\right )}+\frac {a^3}{4 c^2 \left (a^2 x^2+1\right )}+\frac {7 a^3 \log \left (a^2 x^2+1\right )}{6 c^2}-\frac {\arctan (a x)}{3 c^2 x^3}-\frac {a}{6 c^2 x^2} \]
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Rule 29
Rule 31
Rule 36
Rule 46
Rule 267
Rule 272
Rule 4946
Rule 5004
Rule 5012
Rule 5038
Rule 5086
Rubi steps \begin{align*} \text {integral}& = -\left (a^2 \int \frac {\arctan (a x)}{x^2 \left (c+a^2 c x^2\right )^2} \, dx\right )+\frac {\int \frac {\arctan (a x)}{x^4 \left (c+a^2 c x^2\right )} \, dx}{c} \\ & = a^4 \int \frac {\arctan (a x)}{\left (c+a^2 c x^2\right )^2} \, dx+\frac {\int \frac {\arctan (a x)}{x^4} \, dx}{c^2}-2 \frac {a^2 \int \frac {\arctan (a x)}{x^2 \left (c+a^2 c x^2\right )} \, dx}{c} \\ & = -\frac {\arctan (a x)}{3 c^2 x^3}+\frac {a^4 x \arctan (a x)}{2 c^2 \left (1+a^2 x^2\right )}+\frac {a^3 \arctan (a x)^2}{4 c^2}-\frac {1}{2} a^5 \int \frac {x}{\left (c+a^2 c x^2\right )^2} \, dx+\frac {a \int \frac {1}{x^3 \left (1+a^2 x^2\right )} \, dx}{3 c^2}-2 \left (\frac {a^2 \int \frac {\arctan (a x)}{x^2} \, dx}{c^2}-\frac {a^4 \int \frac {\arctan (a x)}{c+a^2 c x^2} \, dx}{c}\right ) \\ & = \frac {a^3}{4 c^2 \left (1+a^2 x^2\right )}-\frac {\arctan (a x)}{3 c^2 x^3}+\frac {a^4 x \arctan (a x)}{2 c^2 \left (1+a^2 x^2\right )}+\frac {a^3 \arctan (a x)^2}{4 c^2}+\frac {a \text {Subst}\left (\int \frac {1}{x^2 \left (1+a^2 x\right )} \, dx,x,x^2\right )}{6 c^2}-2 \left (-\frac {a^2 \arctan (a x)}{c^2 x}-\frac {a^3 \arctan (a x)^2}{2 c^2}+\frac {a^3 \int \frac {1}{x \left (1+a^2 x^2\right )} \, dx}{c^2}\right ) \\ & = \frac {a^3}{4 c^2 \left (1+a^2 x^2\right )}-\frac {\arctan (a x)}{3 c^2 x^3}+\frac {a^4 x \arctan (a x)}{2 c^2 \left (1+a^2 x^2\right )}+\frac {a^3 \arctan (a x)^2}{4 c^2}+\frac {a \text {Subst}\left (\int \left (\frac {1}{x^2}-\frac {a^2}{x}+\frac {a^4}{1+a^2 x}\right ) \, dx,x,x^2\right )}{6 c^2}-2 \left (-\frac {a^2 \arctan (a x)}{c^2 x}-\frac {a^3 \arctan (a x)^2}{2 c^2}+\frac {a^3 \text {Subst}\left (\int \frac {1}{x \left (1+a^2 x\right )} \, dx,x,x^2\right )}{2 c^2}\right ) \\ & = -\frac {a}{6 c^2 x^2}+\frac {a^3}{4 c^2 \left (1+a^2 x^2\right )}-\frac {\arctan (a x)}{3 c^2 x^3}+\frac {a^4 x \arctan (a x)}{2 c^2 \left (1+a^2 x^2\right )}+\frac {a^3 \arctan (a x)^2}{4 c^2}-\frac {a^3 \log (x)}{3 c^2}+\frac {a^3 \log \left (1+a^2 x^2\right )}{6 c^2}-2 \left (-\frac {a^2 \arctan (a x)}{c^2 x}-\frac {a^3 \arctan (a x)^2}{2 c^2}+\frac {a^3 \text {Subst}\left (\int \frac {1}{x} \, dx,x,x^2\right )}{2 c^2}-\frac {a^5 \text {Subst}\left (\int \frac {1}{1+a^2 x} \, dx,x,x^2\right )}{2 c^2}\right ) \\ & = -\frac {a}{6 c^2 x^2}+\frac {a^3}{4 c^2 \left (1+a^2 x^2\right )}-\frac {\arctan (a x)}{3 c^2 x^3}+\frac {a^4 x \arctan (a x)}{2 c^2 \left (1+a^2 x^2\right )}+\frac {a^3 \arctan (a x)^2}{4 c^2}-\frac {a^3 \log (x)}{3 c^2}+\frac {a^3 \log \left (1+a^2 x^2\right )}{6 c^2}-2 \left (-\frac {a^2 \arctan (a x)}{c^2 x}-\frac {a^3 \arctan (a x)^2}{2 c^2}+\frac {a^3 \log (x)}{c^2}-\frac {a^3 \log \left (1+a^2 x^2\right )}{2 c^2}\right ) \\ \end{align*}
Time = 0.11 (sec) , antiderivative size = 124, normalized size of antiderivative = 0.91 \[ \int \frac {\arctan (a x)}{x^4 \left (c+a^2 c x^2\right )^2} \, dx=-\frac {a}{6 c^2 x^2}+\frac {a^3}{4 c^2 \left (1+a^2 x^2\right )}+\frac {\left (-2+10 a^2 x^2+15 a^4 x^4\right ) \arctan (a x)}{6 c^2 x^3 \left (1+a^2 x^2\right )}+\frac {5 a^3 \arctan (a x)^2}{4 c^2}-\frac {7 a^3 \log (x)}{3 c^2}+\frac {7 a^3 \log \left (1+a^2 x^2\right )}{6 c^2} \]
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Time = 0.37 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.89
method | result | size |
derivativedivides | \(a^{3} \left (\frac {a x \arctan \left (a x \right )}{2 c^{2} \left (a^{2} x^{2}+1\right )}+\frac {5 \arctan \left (a x \right )^{2}}{2 c^{2}}-\frac {\arctan \left (a x \right )}{3 c^{2} a^{3} x^{3}}+\frac {2 \arctan \left (a x \right )}{c^{2} a x}-\frac {-7 \ln \left (a^{2} x^{2}+1\right )-\frac {3}{2 \left (a^{2} x^{2}+1\right )}+\frac {1}{a^{2} x^{2}}+14 \ln \left (a x \right )+\frac {15 \arctan \left (a x \right )^{2}}{2}}{6 c^{2}}\right )\) | \(121\) |
default | \(a^{3} \left (\frac {a x \arctan \left (a x \right )}{2 c^{2} \left (a^{2} x^{2}+1\right )}+\frac {5 \arctan \left (a x \right )^{2}}{2 c^{2}}-\frac {\arctan \left (a x \right )}{3 c^{2} a^{3} x^{3}}+\frac {2 \arctan \left (a x \right )}{c^{2} a x}-\frac {-7 \ln \left (a^{2} x^{2}+1\right )-\frac {3}{2 \left (a^{2} x^{2}+1\right )}+\frac {1}{a^{2} x^{2}}+14 \ln \left (a x \right )+\frac {15 \arctan \left (a x \right )^{2}}{2}}{6 c^{2}}\right )\) | \(121\) |
parts | \(\frac {a^{4} x \arctan \left (a x \right )}{2 c^{2} \left (a^{2} x^{2}+1\right )}+\frac {5 a^{3} \arctan \left (a x \right )^{2}}{2 c^{2}}-\frac {\arctan \left (a x \right )}{3 c^{2} x^{3}}+\frac {2 a^{2} \arctan \left (a x \right )}{c^{2} x}-\frac {\frac {15 a^{3} \arctan \left (a x \right )^{2}}{4}+\frac {a^{3} \left (-7 \ln \left (a^{2} x^{2}+1\right )-\frac {3}{2 \left (a^{2} x^{2}+1\right )}+\frac {1}{a^{2} x^{2}}+14 \ln \left (a x \right )\right )}{2}}{3 c^{2}}\) | \(128\) |
parallelrisch | \(-\frac {-15 a^{5} \arctan \left (a x \right )^{2} x^{5}+28 \ln \left (x \right ) x^{5} a^{5}-14 \ln \left (a^{2} x^{2}+1\right ) x^{5} a^{5}-3 a^{5} x^{5}-30 \arctan \left (a x \right ) a^{4} x^{4}-15 a^{3} \arctan \left (a x \right )^{2} x^{3}+28 \ln \left (x \right ) a^{3} x^{3}-14 a^{3} \ln \left (a^{2} x^{2}+1\right ) x^{3}-4 a^{3} x^{3}-20 a^{2} \arctan \left (a x \right ) x^{2}+2 a x +4 \arctan \left (a x \right )}{12 x^{3} c^{2} \left (a^{2} x^{2}+1\right )}\) | \(155\) |
risch | \(\frac {i a^{2} \ln \left (-i a x +1\right )}{c^{2} x}-\frac {i a^{2} \ln \left (i a x +1\right )}{c^{2} x}+\frac {a^{3} \ln \left (-i a x +1\right )}{8 c^{2} \left (-i a x +1\right )}+\frac {a^{3} \ln \left (-i a x +1\right )}{16 c^{2} \left (-i a x -1\right )}+\frac {i \ln \left (i a x +1\right )}{6 c^{2} x^{3}}+\frac {5 a^{3} \ln \left (\frac {1}{2}-\frac {i a x}{2}\right ) \ln \left (i a x +1\right )}{8 c^{2}}+\frac {a^{3} \ln \left (i a x +1\right )}{8 c^{2} \left (i a x +1\right )}+\frac {a^{3} \ln \left (i a x +1\right )}{16 c^{2} \left (i a x -1\right )}-\frac {5 a^{3} \ln \left (\frac {1}{2}+\frac {i a x}{2}\right ) \ln \left (\frac {1}{2}-\frac {i a x}{2}\right )}{4 c^{2}}+\frac {5 a^{3} \ln \left (\frac {1}{2}+\frac {i a x}{2}\right ) \ln \left (-i a x +1\right )}{8 c^{2}}-\frac {i \ln \left (-i a x +1\right )}{6 c^{2} x^{3}}-\frac {a}{6 c^{2} x^{2}}-\frac {a^{3} \ln \left (a^{2} x^{2}+1\right )}{16 c^{2}}+\frac {a^{3}}{8 c^{2} \left (i a x +1\right )}-\frac {7 a^{3} \ln \left (i a x \right )}{6 c^{2}}+\frac {7 a^{3} \ln \left (i a x +1\right )}{6 c^{2}}-\frac {5 a^{3} \ln \left (i a x +1\right )^{2}}{16 c^{2}}-\frac {5 a^{3} \operatorname {dilog}\left (\frac {1}{2}+\frac {i a x}{2}\right )}{8 c^{2}}+\frac {a^{3}}{8 c^{2} \left (-i a x +1\right )}-\frac {7 a^{3} \ln \left (-i a x \right )}{6 c^{2}}+\frac {7 a^{3} \ln \left (-i a x +1\right )}{6 c^{2}}-\frac {5 a^{3} \ln \left (-i a x +1\right )^{2}}{16 c^{2}}-\frac {5 a^{3} \operatorname {dilog}\left (\frac {1}{2}-\frac {i a x}{2}\right )}{8 c^{2}}-\frac {i a^{4} \ln \left (-i a x +1\right ) x}{16 c^{2} \left (-i a x -1\right )}+\frac {i a^{4} \ln \left (i a x +1\right ) x}{16 c^{2} \left (i a x -1\right )}\) | \(491\) |
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Time = 0.24 (sec) , antiderivative size = 127, normalized size of antiderivative = 0.93 \[ \int \frac {\arctan (a x)}{x^4 \left (c+a^2 c x^2\right )^2} \, dx=\frac {a^{3} x^{3} + 15 \, {\left (a^{5} x^{5} + a^{3} x^{3}\right )} \arctan \left (a x\right )^{2} - 2 \, a x + 2 \, {\left (15 \, a^{4} x^{4} + 10 \, a^{2} x^{2} - 2\right )} \arctan \left (a x\right ) + 14 \, {\left (a^{5} x^{5} + a^{3} x^{3}\right )} \log \left (a^{2} x^{2} + 1\right ) - 28 \, {\left (a^{5} x^{5} + a^{3} x^{3}\right )} \log \left (x\right )}{12 \, {\left (a^{2} c^{2} x^{5} + c^{2} x^{3}\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 362 vs. \(2 (129) = 258\).
Time = 0.93 (sec) , antiderivative size = 362, normalized size of antiderivative = 2.66 \[ \int \frac {\arctan (a x)}{x^4 \left (c+a^2 c x^2\right )^2} \, dx=\begin {cases} - \frac {28 a^{5} x^{5} \log {\left (x \right )}}{12 a^{2} c^{2} x^{5} + 12 c^{2} x^{3}} + \frac {14 a^{5} x^{5} \log {\left (x^{2} + \frac {1}{a^{2}} \right )}}{12 a^{2} c^{2} x^{5} + 12 c^{2} x^{3}} + \frac {15 a^{5} x^{5} \operatorname {atan}^{2}{\left (a x \right )}}{12 a^{2} c^{2} x^{5} + 12 c^{2} x^{3}} + \frac {30 a^{4} x^{4} \operatorname {atan}{\left (a x \right )}}{12 a^{2} c^{2} x^{5} + 12 c^{2} x^{3}} - \frac {28 a^{3} x^{3} \log {\left (x \right )}}{12 a^{2} c^{2} x^{5} + 12 c^{2} x^{3}} + \frac {14 a^{3} x^{3} \log {\left (x^{2} + \frac {1}{a^{2}} \right )}}{12 a^{2} c^{2} x^{5} + 12 c^{2} x^{3}} + \frac {15 a^{3} x^{3} \operatorname {atan}^{2}{\left (a x \right )}}{12 a^{2} c^{2} x^{5} + 12 c^{2} x^{3}} + \frac {a^{3} x^{3}}{12 a^{2} c^{2} x^{5} + 12 c^{2} x^{3}} + \frac {20 a^{2} x^{2} \operatorname {atan}{\left (a x \right )}}{12 a^{2} c^{2} x^{5} + 12 c^{2} x^{3}} - \frac {2 a x}{12 a^{2} c^{2} x^{5} + 12 c^{2} x^{3}} - \frac {4 \operatorname {atan}{\left (a x \right )}}{12 a^{2} c^{2} x^{5} + 12 c^{2} x^{3}} & \text {for}\: a \neq 0 \\0 & \text {otherwise} \end {cases} \]
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Time = 0.30 (sec) , antiderivative size = 160, normalized size of antiderivative = 1.18 \[ \int \frac {\arctan (a x)}{x^4 \left (c+a^2 c x^2\right )^2} \, dx=\frac {1}{6} \, {\left (\frac {15 \, a^{3} \arctan \left (a x\right )}{c^{2}} + \frac {15 \, a^{4} x^{4} + 10 \, a^{2} x^{2} - 2}{a^{2} c^{2} x^{5} + c^{2} x^{3}}\right )} \arctan \left (a x\right ) + \frac {{\left (a^{2} x^{2} - 15 \, {\left (a^{4} x^{4} + a^{2} x^{2}\right )} \arctan \left (a x\right )^{2} + 14 \, {\left (a^{4} x^{4} + a^{2} x^{2}\right )} \log \left (a^{2} x^{2} + 1\right ) - 28 \, {\left (a^{4} x^{4} + a^{2} x^{2}\right )} \log \left (x\right ) - 2\right )} a}{12 \, {\left (a^{2} c^{2} x^{4} + c^{2} x^{2}\right )}} \]
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\[ \int \frac {\arctan (a x)}{x^4 \left (c+a^2 c x^2\right )^2} \, dx=\int { \frac {\arctan \left (a x\right )}{{\left (a^{2} c x^{2} + c\right )}^{2} x^{4}} \,d x } \]
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Time = 0.61 (sec) , antiderivative size = 123, normalized size of antiderivative = 0.90 \[ \int \frac {\arctan (a x)}{x^4 \left (c+a^2 c x^2\right )^2} \, dx=\frac {\mathrm {atan}\left (a\,x\right )\,\left (\frac {5\,x^2}{3\,c^2}-\frac {1}{3\,a^2\,c^2}+\frac {5\,a^2\,x^4}{2\,c^2}\right )}{x^5+\frac {x^3}{a^2}}-\frac {a-\frac {a^3\,x^2}{2}}{6\,a^2\,c^2\,x^4+6\,c^2\,x^2}+\frac {7\,a^3\,\ln \left (a^2\,x^2+1\right )}{6\,c^2}-\frac {7\,a^3\,\ln \left (x\right )}{3\,c^2}+\frac {5\,a^3\,{\mathrm {atan}\left (a\,x\right )}^2}{4\,c^2} \]
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