Integrand size = 20, antiderivative size = 111 \[ \int x^3 \left (c+a^2 c x^2\right )^2 \arctan (a x) \, dx=\frac {c^2 x}{24 a^3}-\frac {c^2 x^3}{72 a}-\frac {1}{24} a c^2 x^5-\frac {1}{56} a^3 c^2 x^7-\frac {c^2 \arctan (a x)}{24 a^4}+\frac {1}{4} c^2 x^4 \arctan (a x)+\frac {1}{3} a^2 c^2 x^6 \arctan (a x)+\frac {1}{8} a^4 c^2 x^8 \arctan (a x) \]
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Time = 0.11 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {5068, 4946, 308, 209} \[ \int x^3 \left (c+a^2 c x^2\right )^2 \arctan (a x) \, dx=\frac {1}{8} a^4 c^2 x^8 \arctan (a x)-\frac {c^2 \arctan (a x)}{24 a^4}-\frac {1}{56} a^3 c^2 x^7+\frac {c^2 x}{24 a^3}+\frac {1}{3} a^2 c^2 x^6 \arctan (a x)+\frac {1}{4} c^2 x^4 \arctan (a x)-\frac {1}{24} a c^2 x^5-\frac {c^2 x^3}{72 a} \]
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Rule 209
Rule 308
Rule 4946
Rule 5068
Rubi steps \begin{align*} \text {integral}& = \int \left (c^2 x^3 \arctan (a x)+2 a^2 c^2 x^5 \arctan (a x)+a^4 c^2 x^7 \arctan (a x)\right ) \, dx \\ & = c^2 \int x^3 \arctan (a x) \, dx+\left (2 a^2 c^2\right ) \int x^5 \arctan (a x) \, dx+\left (a^4 c^2\right ) \int x^7 \arctan (a x) \, dx \\ & = \frac {1}{4} c^2 x^4 \arctan (a x)+\frac {1}{3} a^2 c^2 x^6 \arctan (a x)+\frac {1}{8} a^4 c^2 x^8 \arctan (a x)-\frac {1}{4} \left (a c^2\right ) \int \frac {x^4}{1+a^2 x^2} \, dx-\frac {1}{3} \left (a^3 c^2\right ) \int \frac {x^6}{1+a^2 x^2} \, dx-\frac {1}{8} \left (a^5 c^2\right ) \int \frac {x^8}{1+a^2 x^2} \, dx \\ & = \frac {1}{4} c^2 x^4 \arctan (a x)+\frac {1}{3} a^2 c^2 x^6 \arctan (a x)+\frac {1}{8} a^4 c^2 x^8 \arctan (a x)-\frac {1}{4} \left (a c^2\right ) \int \left (-\frac {1}{a^4}+\frac {x^2}{a^2}+\frac {1}{a^4 \left (1+a^2 x^2\right )}\right ) \, dx-\frac {1}{3} \left (a^3 c^2\right ) \int \left (\frac {1}{a^6}-\frac {x^2}{a^4}+\frac {x^4}{a^2}-\frac {1}{a^6 \left (1+a^2 x^2\right )}\right ) \, dx-\frac {1}{8} \left (a^5 c^2\right ) \int \left (-\frac {1}{a^8}+\frac {x^2}{a^6}-\frac {x^4}{a^4}+\frac {x^6}{a^2}+\frac {1}{a^8 \left (1+a^2 x^2\right )}\right ) \, dx \\ & = \frac {c^2 x}{24 a^3}-\frac {c^2 x^3}{72 a}-\frac {1}{24} a c^2 x^5-\frac {1}{56} a^3 c^2 x^7+\frac {1}{4} c^2 x^4 \arctan (a x)+\frac {1}{3} a^2 c^2 x^6 \arctan (a x)+\frac {1}{8} a^4 c^2 x^8 \arctan (a x)-\frac {c^2 \int \frac {1}{1+a^2 x^2} \, dx}{8 a^3}-\frac {c^2 \int \frac {1}{1+a^2 x^2} \, dx}{4 a^3}+\frac {c^2 \int \frac {1}{1+a^2 x^2} \, dx}{3 a^3} \\ & = \frac {c^2 x}{24 a^3}-\frac {c^2 x^3}{72 a}-\frac {1}{24} a c^2 x^5-\frac {1}{56} a^3 c^2 x^7-\frac {c^2 \arctan (a x)}{24 a^4}+\frac {1}{4} c^2 x^4 \arctan (a x)+\frac {1}{3} a^2 c^2 x^6 \arctan (a x)+\frac {1}{8} a^4 c^2 x^8 \arctan (a x) \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.00 \[ \int x^3 \left (c+a^2 c x^2\right )^2 \arctan (a x) \, dx=\frac {c^2 x}{24 a^3}-\frac {c^2 x^3}{72 a}-\frac {1}{24} a c^2 x^5-\frac {1}{56} a^3 c^2 x^7-\frac {c^2 \arctan (a x)}{24 a^4}+\frac {1}{4} c^2 x^4 \arctan (a x)+\frac {1}{3} a^2 c^2 x^6 \arctan (a x)+\frac {1}{8} a^4 c^2 x^8 \arctan (a x) \]
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Time = 0.36 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.79
method | result | size |
derivativedivides | \(\frac {\frac {c^{2} \arctan \left (a x \right ) a^{8} x^{8}}{8}+\frac {c^{2} \arctan \left (a x \right ) a^{6} x^{6}}{3}+\frac {a^{4} c^{2} x^{4} \arctan \left (a x \right )}{4}-\frac {c^{2} \left (\frac {3 a^{7} x^{7}}{7}+a^{5} x^{5}+\frac {a^{3} x^{3}}{3}-a x +\arctan \left (a x \right )\right )}{24}}{a^{4}}\) | \(88\) |
default | \(\frac {\frac {c^{2} \arctan \left (a x \right ) a^{8} x^{8}}{8}+\frac {c^{2} \arctan \left (a x \right ) a^{6} x^{6}}{3}+\frac {a^{4} c^{2} x^{4} \arctan \left (a x \right )}{4}-\frac {c^{2} \left (\frac {3 a^{7} x^{7}}{7}+a^{5} x^{5}+\frac {a^{3} x^{3}}{3}-a x +\arctan \left (a x \right )\right )}{24}}{a^{4}}\) | \(88\) |
parts | \(\frac {a^{4} c^{2} x^{8} \arctan \left (a x \right )}{8}+\frac {a^{2} c^{2} x^{6} \arctan \left (a x \right )}{3}+\frac {c^{2} x^{4} \arctan \left (a x \right )}{4}-\frac {c^{2} a \left (\frac {\frac {3}{7} a^{6} x^{7}+a^{4} x^{5}+\frac {1}{3} a^{2} x^{3}-x}{a^{4}}+\frac {\arctan \left (a x \right )}{a^{5}}\right )}{24}\) | \(90\) |
parallelrisch | \(\frac {63 c^{2} \arctan \left (a x \right ) a^{8} x^{8}-9 a^{7} c^{2} x^{7}+168 c^{2} \arctan \left (a x \right ) a^{6} x^{6}-21 a^{5} c^{2} x^{5}+126 a^{4} c^{2} x^{4} \arctan \left (a x \right )-7 a^{3} c^{2} x^{3}+21 a \,c^{2} x -21 c^{2} \arctan \left (a x \right )}{504 a^{4}}\) | \(101\) |
risch | \(-\frac {i c^{2} x^{4} \left (3 a^{4} x^{4}+8 a^{2} x^{2}+6\right ) \ln \left (i a x +1\right )}{48}+\frac {i c^{2} a^{4} x^{8} \ln \left (-i a x +1\right )}{16}-\frac {a^{3} c^{2} x^{7}}{56}+\frac {i c^{2} a^{2} x^{6} \ln \left (-i a x +1\right )}{6}-\frac {a \,c^{2} x^{5}}{24}+\frac {i c^{2} x^{4} \ln \left (-i a x +1\right )}{8}-\frac {c^{2} x^{3}}{72 a}+\frac {c^{2} x}{24 a^{3}}-\frac {c^{2} \arctan \left (a x \right )}{24 a^{4}}\) | \(146\) |
meijerg | \(\frac {c^{2} \left (\frac {x a \left (-45 a^{6} x^{6}+63 a^{4} x^{4}-105 a^{2} x^{2}+315\right )}{630}-\frac {x a \left (-9 a^{8} x^{8}+9\right ) \arctan \left (\sqrt {a^{2} x^{2}}\right )}{18 \sqrt {a^{2} x^{2}}}\right )}{4 a^{4}}+\frac {c^{2} \left (-\frac {2 a x \left (21 a^{4} x^{4}-35 a^{2} x^{2}+105\right )}{315}+\frac {2 a x \left (7 a^{6} x^{6}+7\right ) \arctan \left (\sqrt {a^{2} x^{2}}\right )}{21 \sqrt {a^{2} x^{2}}}\right )}{2 a^{4}}+\frac {c^{2} \left (\frac {a x \left (-5 a^{2} x^{2}+15\right )}{15}-\frac {a x \left (-5 a^{4} x^{4}+5\right ) \arctan \left (\sqrt {a^{2} x^{2}}\right )}{5 \sqrt {a^{2} x^{2}}}\right )}{4 a^{4}}\) | \(194\) |
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Time = 0.24 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.82 \[ \int x^3 \left (c+a^2 c x^2\right )^2 \arctan (a x) \, dx=-\frac {9 \, a^{7} c^{2} x^{7} + 21 \, a^{5} c^{2} x^{5} + 7 \, a^{3} c^{2} x^{3} - 21 \, a c^{2} x - 21 \, {\left (3 \, a^{8} c^{2} x^{8} + 8 \, a^{6} c^{2} x^{6} + 6 \, a^{4} c^{2} x^{4} - c^{2}\right )} \arctan \left (a x\right )}{504 \, a^{4}} \]
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Time = 0.43 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.94 \[ \int x^3 \left (c+a^2 c x^2\right )^2 \arctan (a x) \, dx=\begin {cases} \frac {a^{4} c^{2} x^{8} \operatorname {atan}{\left (a x \right )}}{8} - \frac {a^{3} c^{2} x^{7}}{56} + \frac {a^{2} c^{2} x^{6} \operatorname {atan}{\left (a x \right )}}{3} - \frac {a c^{2} x^{5}}{24} + \frac {c^{2} x^{4} \operatorname {atan}{\left (a x \right )}}{4} - \frac {c^{2} x^{3}}{72 a} + \frac {c^{2} x}{24 a^{3}} - \frac {c^{2} \operatorname {atan}{\left (a x \right )}}{24 a^{4}} & \text {for}\: a \neq 0 \\0 & \text {otherwise} \end {cases} \]
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Time = 0.28 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.88 \[ \int x^3 \left (c+a^2 c x^2\right )^2 \arctan (a x) \, dx=-\frac {1}{504} \, a {\left (\frac {21 \, c^{2} \arctan \left (a x\right )}{a^{5}} + \frac {9 \, a^{6} c^{2} x^{7} + 21 \, a^{4} c^{2} x^{5} + 7 \, a^{2} c^{2} x^{3} - 21 \, c^{2} x}{a^{4}}\right )} + \frac {1}{24} \, {\left (3 \, a^{4} c^{2} x^{8} + 8 \, a^{2} c^{2} x^{6} + 6 \, c^{2} x^{4}\right )} \arctan \left (a x\right ) \]
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\[ \int x^3 \left (c+a^2 c x^2\right )^2 \arctan (a x) \, dx=\int { {\left (a^{2} c x^{2} + c\right )}^{2} x^{3} \arctan \left (a x\right ) \,d x } \]
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Time = 0.49 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.80 \[ \int x^3 \left (c+a^2 c x^2\right )^2 \arctan (a x) \, dx=\mathrm {atan}\left (a\,x\right )\,\left (\frac {a^4\,c^2\,x^8}{8}+\frac {a^2\,c^2\,x^6}{3}+\frac {c^2\,x^4}{4}\right )+\frac {c^2\,x}{24\,a^3}-\frac {a\,c^2\,x^5}{24}-\frac {c^2\,\mathrm {atan}\left (a\,x\right )}{24\,a^4}-\frac {c^2\,x^3}{72\,a}-\frac {a^3\,c^2\,x^7}{56} \]
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