Integrand size = 14, antiderivative size = 244 \[ \int \left (c+d x^2\right )^4 \arctan (a x) \, dx=-\frac {d \left (420 a^6 c^3-378 a^4 c^2 d+180 a^2 c d^2-35 d^3\right ) x^2}{630 a^7}-\frac {d^2 \left (378 a^4 c^2-180 a^2 c d+35 d^2\right ) x^4}{1260 a^5}-\frac {\left (36 a^2 c-7 d\right ) d^3 x^6}{378 a^3}-\frac {d^4 x^8}{72 a}+c^4 x \arctan (a x)+\frac {4}{3} c^3 d x^3 \arctan (a x)+\frac {6}{5} c^2 d^2 x^5 \arctan (a x)+\frac {4}{7} c d^3 x^7 \arctan (a x)+\frac {1}{9} d^4 x^9 \arctan (a x)-\frac {\left (315 a^8 c^4-420 a^6 c^3 d+378 a^4 c^2 d^2-180 a^2 c d^3+35 d^4\right ) \log \left (1+a^2 x^2\right )}{630 a^9} \]
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Time = 0.12 (sec) , antiderivative size = 244, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {200, 5032, 1824, 266} \[ \int \left (c+d x^2\right )^4 \arctan (a x) \, dx=-\frac {d^3 x^6 \left (36 a^2 c-7 d\right )}{378 a^3}-\frac {d^2 x^4 \left (378 a^4 c^2-180 a^2 c d+35 d^2\right )}{1260 a^5}-\frac {d x^2 \left (420 a^6 c^3-378 a^4 c^2 d+180 a^2 c d^2-35 d^3\right )}{630 a^7}-\frac {\left (315 a^8 c^4-420 a^6 c^3 d+378 a^4 c^2 d^2-180 a^2 c d^3+35 d^4\right ) \log \left (a^2 x^2+1\right )}{630 a^9}+c^4 x \arctan (a x)+\frac {4}{3} c^3 d x^3 \arctan (a x)+\frac {6}{5} c^2 d^2 x^5 \arctan (a x)+\frac {4}{7} c d^3 x^7 \arctan (a x)+\frac {1}{9} d^4 x^9 \arctan (a x)-\frac {d^4 x^8}{72 a} \]
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Rule 200
Rule 266
Rule 1824
Rule 5032
Rubi steps \begin{align*} \text {integral}& = c^4 x \arctan (a x)+\frac {4}{3} c^3 d x^3 \arctan (a x)+\frac {6}{5} c^2 d^2 x^5 \arctan (a x)+\frac {4}{7} c d^3 x^7 \arctan (a x)+\frac {1}{9} d^4 x^9 \arctan (a x)-a \int \frac {c^4 x+\frac {4}{3} c^3 d x^3+\frac {6}{5} c^2 d^2 x^5+\frac {4}{7} c d^3 x^7+\frac {d^4 x^9}{9}}{1+a^2 x^2} \, dx \\ & = c^4 x \arctan (a x)+\frac {4}{3} c^3 d x^3 \arctan (a x)+\frac {6}{5} c^2 d^2 x^5 \arctan (a x)+\frac {4}{7} c d^3 x^7 \arctan (a x)+\frac {1}{9} d^4 x^9 \arctan (a x)-a \int \left (\frac {d \left (420 a^6 c^3-378 a^4 c^2 d+180 a^2 c d^2-35 d^3\right ) x}{315 a^8}+\frac {d^2 \left (378 a^4 c^2-180 a^2 c d+35 d^2\right ) x^3}{315 a^6}+\frac {\left (36 a^2 c-7 d\right ) d^3 x^5}{63 a^4}+\frac {d^4 x^7}{9 a^2}+\frac {\left (315 a^8 c^4-420 a^6 c^3 d+378 a^4 c^2 d^2-180 a^2 c d^3+35 d^4\right ) x}{315 a^8 \left (1+a^2 x^2\right )}\right ) \, dx \\ & = -\frac {d \left (420 a^6 c^3-378 a^4 c^2 d+180 a^2 c d^2-35 d^3\right ) x^2}{630 a^7}-\frac {d^2 \left (378 a^4 c^2-180 a^2 c d+35 d^2\right ) x^4}{1260 a^5}-\frac {\left (36 a^2 c-7 d\right ) d^3 x^6}{378 a^3}-\frac {d^4 x^8}{72 a}+c^4 x \arctan (a x)+\frac {4}{3} c^3 d x^3 \arctan (a x)+\frac {6}{5} c^2 d^2 x^5 \arctan (a x)+\frac {4}{7} c d^3 x^7 \arctan (a x)+\frac {1}{9} d^4 x^9 \arctan (a x)-\frac {\left (315 a^8 c^4-420 a^6 c^3 d+378 a^4 c^2 d^2-180 a^2 c d^3+35 d^4\right ) \int \frac {x}{1+a^2 x^2} \, dx}{315 a^7} \\ & = -\frac {d \left (420 a^6 c^3-378 a^4 c^2 d+180 a^2 c d^2-35 d^3\right ) x^2}{630 a^7}-\frac {d^2 \left (378 a^4 c^2-180 a^2 c d+35 d^2\right ) x^4}{1260 a^5}-\frac {\left (36 a^2 c-7 d\right ) d^3 x^6}{378 a^3}-\frac {d^4 x^8}{72 a}+c^4 x \arctan (a x)+\frac {4}{3} c^3 d x^3 \arctan (a x)+\frac {6}{5} c^2 d^2 x^5 \arctan (a x)+\frac {4}{7} c d^3 x^7 \arctan (a x)+\frac {1}{9} d^4 x^9 \arctan (a x)-\frac {\left (315 a^8 c^4-420 a^6 c^3 d+378 a^4 c^2 d^2-180 a^2 c d^3+35 d^4\right ) \log \left (1+a^2 x^2\right )}{630 a^9} \\ \end{align*}
Time = 0.07 (sec) , antiderivative size = 212, normalized size of antiderivative = 0.87 \[ \int \left (c+d x^2\right )^4 \arctan (a x) \, dx=-\frac {a^2 d x^2 \left (-420 d^3+30 a^2 d^2 \left (72 c+7 d x^2\right )-4 a^4 d \left (1134 c^2+270 c d x^2+35 d^2 x^4\right )+3 a^6 \left (1680 c^3+756 c^2 d x^2+240 c d^2 x^4+35 d^3 x^6\right )\right )-24 a^9 x \left (315 c^4+420 c^3 d x^2+378 c^2 d^2 x^4+180 c d^3 x^6+35 d^4 x^8\right ) \arctan (a x)+12 \left (315 a^8 c^4-420 a^6 c^3 d+378 a^4 c^2 d^2-180 a^2 c d^3+35 d^4\right ) \log \left (1+a^2 x^2\right )}{7560 a^9} \]
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Time = 0.35 (sec) , antiderivative size = 245, normalized size of antiderivative = 1.00
method | result | size |
parts | \(\frac {d^{4} x^{9} \arctan \left (a x \right )}{9}+\frac {4 c \,d^{3} x^{7} \arctan \left (a x \right )}{7}+\frac {6 c^{2} d^{2} x^{5} \arctan \left (a x \right )}{5}+\frac {4 c^{3} d \,x^{3} \arctan \left (a x \right )}{3}+c^{4} x \arctan \left (a x \right )-\frac {a \left (\frac {d \left (\frac {35}{4} a^{6} d^{3} x^{8}+60 a^{6} c \,d^{2} x^{6}+189 a^{6} c^{2} d \,x^{4}+420 a^{6} c^{3} x^{2}-\frac {35}{3} a^{4} d^{3} x^{6}-90 a^{4} c \,d^{2} x^{4}-378 a^{4} c^{2} d \,x^{2}+\frac {35}{2} a^{2} d^{3} x^{4}+180 a^{2} c \,d^{2} x^{2}-35 d^{3} x^{2}\right )}{2 a^{8}}+\frac {\left (315 a^{8} c^{4}-420 a^{6} c^{3} d +378 a^{4} c^{2} d^{2}-180 a^{2} c \,d^{3}+35 d^{4}\right ) \ln \left (a^{2} x^{2}+1\right )}{2 a^{10}}\right )}{315}\) | \(245\) |
derivativedivides | \(\frac {\arctan \left (a x \right ) a x \,c^{4}+\frac {4 \arctan \left (a x \right ) a \,c^{3} d \,x^{3}}{3}+\frac {6 \arctan \left (a x \right ) a \,c^{2} d^{2} x^{5}}{5}+\frac {4 \arctan \left (a x \right ) a c \,d^{3} x^{7}}{7}+\frac {\arctan \left (a x \right ) a \,d^{4} x^{9}}{9}-\frac {210 c^{3} a^{8} d \,x^{2}+\frac {189 c^{2} a^{8} d^{2} x^{4}}{2}-189 c^{2} a^{6} d^{2} x^{2}+30 c \,a^{8} d^{3} x^{6}-45 a^{6} c \,d^{3} x^{4}+\frac {35 d^{4} a^{8} x^{8}}{8}+90 a^{4} c \,d^{3} x^{2}-\frac {35 d^{4} a^{6} x^{6}}{6}+\frac {35 d^{4} a^{4} x^{4}}{4}-\frac {35 d^{4} a^{2} x^{2}}{2}+\frac {\left (315 a^{8} c^{4}-420 a^{6} c^{3} d +378 a^{4} c^{2} d^{2}-180 a^{2} c \,d^{3}+35 d^{4}\right ) \ln \left (a^{2} x^{2}+1\right )}{2}}{315 a^{8}}}{a}\) | \(254\) |
default | \(\frac {\arctan \left (a x \right ) a x \,c^{4}+\frac {4 \arctan \left (a x \right ) a \,c^{3} d \,x^{3}}{3}+\frac {6 \arctan \left (a x \right ) a \,c^{2} d^{2} x^{5}}{5}+\frac {4 \arctan \left (a x \right ) a c \,d^{3} x^{7}}{7}+\frac {\arctan \left (a x \right ) a \,d^{4} x^{9}}{9}-\frac {210 c^{3} a^{8} d \,x^{2}+\frac {189 c^{2} a^{8} d^{2} x^{4}}{2}-189 c^{2} a^{6} d^{2} x^{2}+30 c \,a^{8} d^{3} x^{6}-45 a^{6} c \,d^{3} x^{4}+\frac {35 d^{4} a^{8} x^{8}}{8}+90 a^{4} c \,d^{3} x^{2}-\frac {35 d^{4} a^{6} x^{6}}{6}+\frac {35 d^{4} a^{4} x^{4}}{4}-\frac {35 d^{4} a^{2} x^{2}}{2}+\frac {\left (315 a^{8} c^{4}-420 a^{6} c^{3} d +378 a^{4} c^{2} d^{2}-180 a^{2} c \,d^{3}+35 d^{4}\right ) \ln \left (a^{2} x^{2}+1\right )}{2}}{315 a^{8}}}{a}\) | \(254\) |
parallelrisch | \(-\frac {-840 x^{9} \arctan \left (a x \right ) a^{9} d^{4}-4320 x^{7} \arctan \left (a x \right ) a^{9} c \,d^{3}+105 d^{4} a^{8} x^{8}-9072 x^{5} \arctan \left (a x \right ) a^{9} c^{2} d^{2}+720 c \,a^{8} d^{3} x^{6}-10080 x^{3} \arctan \left (a x \right ) a^{9} c^{3} d -140 d^{4} a^{6} x^{6}+2268 c^{2} a^{8} d^{2} x^{4}-7560 x \arctan \left (a x \right ) a^{9} c^{4}-1080 a^{6} c \,d^{3} x^{4}+5040 c^{3} a^{8} d \,x^{2}+3780 \ln \left (a^{2} x^{2}+1\right ) a^{8} c^{4}+210 d^{4} a^{4} x^{4}-4536 c^{2} a^{6} d^{2} x^{2}-5040 \ln \left (a^{2} x^{2}+1\right ) a^{6} c^{3} d +2160 a^{4} c \,d^{3} x^{2}+4536 \ln \left (a^{2} x^{2}+1\right ) a^{4} c^{2} d^{2}-420 d^{4} a^{2} x^{2}-2160 \ln \left (a^{2} x^{2}+1\right ) a^{2} c \,d^{3}+420 \ln \left (a^{2} x^{2}+1\right ) d^{4}}{7560 a^{9}}\) | \(297\) |
meijerg | \(\frac {d^{4} \left (\frac {x^{2} a^{2} \left (-15 x^{6} a^{6}+20 a^{4} x^{4}-30 a^{2} x^{2}+60\right )}{270}+\frac {4 x^{10} a^{10} \arctan \left (\sqrt {a^{2} x^{2}}\right )}{9 \sqrt {a^{2} x^{2}}}-\frac {2 \ln \left (a^{2} x^{2}+1\right )}{9}\right )}{4 a^{9}}+\frac {d^{3} c \left (-\frac {x^{2} a^{2} \left (4 a^{4} x^{4}-6 a^{2} x^{2}+12\right )}{42}+\frac {4 x^{8} a^{8} \arctan \left (\sqrt {a^{2} x^{2}}\right )}{7 \sqrt {a^{2} x^{2}}}+\frac {2 \ln \left (a^{2} x^{2}+1\right )}{7}\right )}{a^{7}}+\frac {3 c^{2} d^{2} \left (\frac {x^{2} a^{2} \left (-3 a^{2} x^{2}+6\right )}{15}+\frac {4 x^{6} a^{6} \arctan \left (\sqrt {a^{2} x^{2}}\right )}{5 \sqrt {a^{2} x^{2}}}-\frac {2 \ln \left (a^{2} x^{2}+1\right )}{5}\right )}{2 a^{5}}+\frac {d \,c^{3} \left (-\frac {2 a^{2} x^{2}}{3}+\frac {4 a^{4} x^{4} \arctan \left (\sqrt {a^{2} x^{2}}\right )}{3 \sqrt {a^{2} x^{2}}}+\frac {2 \ln \left (a^{2} x^{2}+1\right )}{3}\right )}{a^{3}}+\frac {c^{4} \left (\frac {4 a^{2} x^{2} \arctan \left (\sqrt {a^{2} x^{2}}\right )}{\sqrt {a^{2} x^{2}}}-2 \ln \left (a^{2} x^{2}+1\right )\right )}{4 a}\) | \(331\) |
risch | \(\frac {i \ln \left (-i a x +1\right ) d^{4} x^{9}}{18}+\frac {i \ln \left (-i a x +1\right ) x \,c^{4}}{2}+\frac {2 i \ln \left (-i a x +1\right ) d^{3} c \,x^{7}}{7}-\frac {d^{4} x^{8}}{72 a}+\frac {3 i \ln \left (-i a x +1\right ) c^{2} d^{2} x^{5}}{5}-\frac {2 c \,d^{3} x^{6}}{21 a}+\frac {i \left (-\frac {1}{9} d^{4} x^{9}-\frac {4}{7} d^{3} c \,x^{7}-\frac {6}{5} c^{2} d^{2} x^{5}-\frac {4}{3} c^{3} d \,x^{3}-x \,c^{4}\right ) \ln \left (i a x +1\right )}{2}-\frac {3 c^{2} d^{2} x^{4}}{10 a}+\frac {d^{4} x^{6}}{54 a^{3}}+\frac {2 i \ln \left (-i a x +1\right ) c^{3} d \,x^{3}}{3}-\frac {2 c^{3} d \,x^{2}}{3 a}+\frac {c \,d^{3} x^{4}}{7 a^{3}}-\frac {\ln \left (-a^{2} x^{2}-1\right ) c^{4}}{2 a}+\frac {3 c^{2} d^{2} x^{2}}{5 a^{3}}-\frac {d^{4} x^{4}}{36 a^{5}}+\frac {2 \ln \left (-a^{2} x^{2}-1\right ) c^{3} d}{3 a^{3}}-\frac {2 c \,d^{3} x^{2}}{7 a^{5}}-\frac {3 \ln \left (-a^{2} x^{2}-1\right ) c^{2} d^{2}}{5 a^{5}}+\frac {d^{4} x^{2}}{18 a^{7}}+\frac {2 \ln \left (-a^{2} x^{2}-1\right ) c \,d^{3}}{7 a^{7}}-\frac {\ln \left (-a^{2} x^{2}-1\right ) d^{4}}{18 a^{9}}\) | \(365\) |
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Time = 0.26 (sec) , antiderivative size = 237, normalized size of antiderivative = 0.97 \[ \int \left (c+d x^2\right )^4 \arctan (a x) \, dx=-\frac {105 \, a^{8} d^{4} x^{8} + 20 \, {\left (36 \, a^{8} c d^{3} - 7 \, a^{6} d^{4}\right )} x^{6} + 6 \, {\left (378 \, a^{8} c^{2} d^{2} - 180 \, a^{6} c d^{3} + 35 \, a^{4} d^{4}\right )} x^{4} + 12 \, {\left (420 \, a^{8} c^{3} d - 378 \, a^{6} c^{2} d^{2} + 180 \, a^{4} c d^{3} - 35 \, a^{2} d^{4}\right )} x^{2} - 24 \, {\left (35 \, a^{9} d^{4} x^{9} + 180 \, a^{9} c d^{3} x^{7} + 378 \, a^{9} c^{2} d^{2} x^{5} + 420 \, a^{9} c^{3} d x^{3} + 315 \, a^{9} c^{4} x\right )} \arctan \left (a x\right ) + 12 \, {\left (315 \, a^{8} c^{4} - 420 \, a^{6} c^{3} d + 378 \, a^{4} c^{2} d^{2} - 180 \, a^{2} c d^{3} + 35 \, d^{4}\right )} \log \left (a^{2} x^{2} + 1\right )}{7560 \, a^{9}} \]
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Time = 0.66 (sec) , antiderivative size = 314, normalized size of antiderivative = 1.29 \[ \int \left (c+d x^2\right )^4 \arctan (a x) \, dx=\begin {cases} c^{4} x \operatorname {atan}{\left (a x \right )} + \frac {4 c^{3} d x^{3} \operatorname {atan}{\left (a x \right )}}{3} + \frac {6 c^{2} d^{2} x^{5} \operatorname {atan}{\left (a x \right )}}{5} + \frac {4 c d^{3} x^{7} \operatorname {atan}{\left (a x \right )}}{7} + \frac {d^{4} x^{9} \operatorname {atan}{\left (a x \right )}}{9} - \frac {c^{4} \log {\left (x^{2} + \frac {1}{a^{2}} \right )}}{2 a} - \frac {2 c^{3} d x^{2}}{3 a} - \frac {3 c^{2} d^{2} x^{4}}{10 a} - \frac {2 c d^{3} x^{6}}{21 a} - \frac {d^{4} x^{8}}{72 a} + \frac {2 c^{3} d \log {\left (x^{2} + \frac {1}{a^{2}} \right )}}{3 a^{3}} + \frac {3 c^{2} d^{2} x^{2}}{5 a^{3}} + \frac {c d^{3} x^{4}}{7 a^{3}} + \frac {d^{4} x^{6}}{54 a^{3}} - \frac {3 c^{2} d^{2} \log {\left (x^{2} + \frac {1}{a^{2}} \right )}}{5 a^{5}} - \frac {2 c d^{3} x^{2}}{7 a^{5}} - \frac {d^{4} x^{4}}{36 a^{5}} + \frac {2 c d^{3} \log {\left (x^{2} + \frac {1}{a^{2}} \right )}}{7 a^{7}} + \frac {d^{4} x^{2}}{18 a^{7}} - \frac {d^{4} \log {\left (x^{2} + \frac {1}{a^{2}} \right )}}{18 a^{9}} & \text {for}\: a \neq 0 \\0 & \text {otherwise} \end {cases} \]
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Time = 0.20 (sec) , antiderivative size = 226, normalized size of antiderivative = 0.93 \[ \int \left (c+d x^2\right )^4 \arctan (a x) \, dx=-\frac {1}{7560} \, a {\left (\frac {105 \, a^{6} d^{4} x^{8} + 20 \, {\left (36 \, a^{6} c d^{3} - 7 \, a^{4} d^{4}\right )} x^{6} + 6 \, {\left (378 \, a^{6} c^{2} d^{2} - 180 \, a^{4} c d^{3} + 35 \, a^{2} d^{4}\right )} x^{4} + 12 \, {\left (420 \, a^{6} c^{3} d - 378 \, a^{4} c^{2} d^{2} + 180 \, a^{2} c d^{3} - 35 \, d^{4}\right )} x^{2}}{a^{8}} + \frac {12 \, {\left (315 \, a^{8} c^{4} - 420 \, a^{6} c^{3} d + 378 \, a^{4} c^{2} d^{2} - 180 \, a^{2} c d^{3} + 35 \, d^{4}\right )} \log \left (a^{2} x^{2} + 1\right )}{a^{10}}\right )} + \frac {1}{315} \, {\left (35 \, d^{4} x^{9} + 180 \, c d^{3} x^{7} + 378 \, c^{2} d^{2} x^{5} + 420 \, c^{3} d x^{3} + 315 \, c^{4} x\right )} \arctan \left (a x\right ) \]
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\[ \int \left (c+d x^2\right )^4 \arctan (a x) \, dx=\int { {\left (d x^{2} + c\right )}^{4} \arctan \left (a x\right ) \,d x } \]
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Time = 0.23 (sec) , antiderivative size = 233, normalized size of antiderivative = 0.95 \[ \int \left (c+d x^2\right )^4 \arctan (a x) \, dx=\mathrm {atan}\left (a\,x\right )\,\left (c^4\,x+\frac {4\,c^3\,d\,x^3}{3}+\frac {6\,c^2\,d^2\,x^5}{5}+\frac {4\,c\,d^3\,x^7}{7}+\frac {d^4\,x^9}{9}\right )+x^2\,\left (\frac {\frac {\frac {d^4}{9\,a^3}-\frac {4\,c\,d^3}{7\,a}}{a^2}+\frac {6\,c^2\,d^2}{5\,a}}{2\,a^2}-\frac {2\,c^3\,d}{3\,a}\right )+x^6\,\left (\frac {d^4}{54\,a^3}-\frac {2\,c\,d^3}{21\,a}\right )-x^4\,\left (\frac {\frac {d^4}{9\,a^3}-\frac {4\,c\,d^3}{7\,a}}{4\,a^2}+\frac {3\,c^2\,d^2}{10\,a}\right )-\frac {\ln \left (a^2\,x^2+1\right )\,\left (315\,a^8\,c^4-420\,a^6\,c^3\,d+378\,a^4\,c^2\,d^2-180\,a^2\,c\,d^3+35\,d^4\right )}{630\,a^9}-\frac {d^4\,x^8}{72\,a} \]
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