Optimal. Leaf size=268 \[ \frac {38 c^3 x}{105}+\frac {19}{315} a^2 c^3 x^3+\frac {1}{105} a^4 c^3 x^5-\frac {8 c^3 \left (1+a^2 x^2\right ) \text {ArcTan}(a x)}{35 a}-\frac {3 c^3 \left (1+a^2 x^2\right )^2 \text {ArcTan}(a x)}{35 a}-\frac {c^3 \left (1+a^2 x^2\right )^3 \text {ArcTan}(a x)}{21 a}+\frac {16 i c^3 \text {ArcTan}(a x)^2}{35 a}+\frac {16}{35} c^3 x \text {ArcTan}(a x)^2+\frac {8}{35} c^3 x \left (1+a^2 x^2\right ) \text {ArcTan}(a x)^2+\frac {6}{35} c^3 x \left (1+a^2 x^2\right )^2 \text {ArcTan}(a x)^2+\frac {1}{7} c^3 x \left (1+a^2 x^2\right )^3 \text {ArcTan}(a x)^2+\frac {32 c^3 \text {ArcTan}(a x) \log \left (\frac {2}{1+i a x}\right )}{35 a}+\frac {16 i c^3 \text {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )}{35 a} \]
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Rubi [A]
time = 0.14, antiderivative size = 268, normalized size of antiderivative = 1.00, number of steps
used = 12, number of rules used = 8, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.421, Rules used = {5000, 4930,
5040, 4964, 2449, 2352, 8, 200} \begin {gather*} \frac {1}{105} a^4 c^3 x^5+\frac {1}{7} c^3 x \left (a^2 x^2+1\right )^3 \text {ArcTan}(a x)^2+\frac {6}{35} c^3 x \left (a^2 x^2+1\right )^2 \text {ArcTan}(a x)^2+\frac {8}{35} c^3 x \left (a^2 x^2+1\right ) \text {ArcTan}(a x)^2-\frac {c^3 \left (a^2 x^2+1\right )^3 \text {ArcTan}(a x)}{21 a}-\frac {3 c^3 \left (a^2 x^2+1\right )^2 \text {ArcTan}(a x)}{35 a}-\frac {8 c^3 \left (a^2 x^2+1\right ) \text {ArcTan}(a x)}{35 a}+\frac {19}{315} a^2 c^3 x^3+\frac {16}{35} c^3 x \text {ArcTan}(a x)^2+\frac {16 i c^3 \text {ArcTan}(a x)^2}{35 a}+\frac {32 c^3 \text {ArcTan}(a x) \log \left (\frac {2}{1+i a x}\right )}{35 a}+\frac {16 i c^3 \text {Li}_2\left (1-\frac {2}{i a x+1}\right )}{35 a}+\frac {38 c^3 x}{105} \end {gather*}
Antiderivative was successfully verified.
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Rule 8
Rule 200
Rule 2352
Rule 2449
Rule 4930
Rule 4964
Rule 5000
Rule 5040
Rubi steps
\begin {align*} \int \left (c+a^2 c x^2\right )^3 \tan ^{-1}(a x)^2 \, dx &=-\frac {c^3 \left (1+a^2 x^2\right )^3 \tan ^{-1}(a x)}{21 a}+\frac {1}{7} c^3 x \left (1+a^2 x^2\right )^3 \tan ^{-1}(a x)^2+\frac {1}{21} c \int \left (c+a^2 c x^2\right )^2 \, dx+\frac {1}{7} (6 c) \int \left (c+a^2 c x^2\right )^2 \tan ^{-1}(a x)^2 \, dx\\ &=-\frac {3 c^3 \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)}{35 a}-\frac {c^3 \left (1+a^2 x^2\right )^3 \tan ^{-1}(a x)}{21 a}+\frac {6}{35} c^3 x \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)^2+\frac {1}{7} c^3 x \left (1+a^2 x^2\right )^3 \tan ^{-1}(a x)^2+\frac {1}{21} c \int \left (c^2+2 a^2 c^2 x^2+a^4 c^2 x^4\right ) \, dx+\frac {1}{35} \left (3 c^2\right ) \int \left (c+a^2 c x^2\right ) \, dx+\frac {1}{35} \left (24 c^2\right ) \int \left (c+a^2 c x^2\right ) \tan ^{-1}(a x)^2 \, dx\\ &=\frac {2 c^3 x}{15}+\frac {19}{315} a^2 c^3 x^3+\frac {1}{105} a^4 c^3 x^5-\frac {8 c^3 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)}{35 a}-\frac {3 c^3 \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)}{35 a}-\frac {c^3 \left (1+a^2 x^2\right )^3 \tan ^{-1}(a x)}{21 a}+\frac {8}{35} c^3 x \left (1+a^2 x^2\right ) \tan ^{-1}(a x)^2+\frac {6}{35} c^3 x \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)^2+\frac {1}{7} c^3 x \left (1+a^2 x^2\right )^3 \tan ^{-1}(a x)^2+\frac {1}{35} \left (8 c^3\right ) \int 1 \, dx+\frac {1}{35} \left (16 c^3\right ) \int \tan ^{-1}(a x)^2 \, dx\\ &=\frac {38 c^3 x}{105}+\frac {19}{315} a^2 c^3 x^3+\frac {1}{105} a^4 c^3 x^5-\frac {8 c^3 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)}{35 a}-\frac {3 c^3 \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)}{35 a}-\frac {c^3 \left (1+a^2 x^2\right )^3 \tan ^{-1}(a x)}{21 a}+\frac {16}{35} c^3 x \tan ^{-1}(a x)^2+\frac {8}{35} c^3 x \left (1+a^2 x^2\right ) \tan ^{-1}(a x)^2+\frac {6}{35} c^3 x \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)^2+\frac {1}{7} c^3 x \left (1+a^2 x^2\right )^3 \tan ^{-1}(a x)^2-\frac {1}{35} \left (32 a c^3\right ) \int \frac {x \tan ^{-1}(a x)}{1+a^2 x^2} \, dx\\ &=\frac {38 c^3 x}{105}+\frac {19}{315} a^2 c^3 x^3+\frac {1}{105} a^4 c^3 x^5-\frac {8 c^3 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)}{35 a}-\frac {3 c^3 \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)}{35 a}-\frac {c^3 \left (1+a^2 x^2\right )^3 \tan ^{-1}(a x)}{21 a}+\frac {16 i c^3 \tan ^{-1}(a x)^2}{35 a}+\frac {16}{35} c^3 x \tan ^{-1}(a x)^2+\frac {8}{35} c^3 x \left (1+a^2 x^2\right ) \tan ^{-1}(a x)^2+\frac {6}{35} c^3 x \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)^2+\frac {1}{7} c^3 x \left (1+a^2 x^2\right )^3 \tan ^{-1}(a x)^2+\frac {1}{35} \left (32 c^3\right ) \int \frac {\tan ^{-1}(a x)}{i-a x} \, dx\\ &=\frac {38 c^3 x}{105}+\frac {19}{315} a^2 c^3 x^3+\frac {1}{105} a^4 c^3 x^5-\frac {8 c^3 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)}{35 a}-\frac {3 c^3 \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)}{35 a}-\frac {c^3 \left (1+a^2 x^2\right )^3 \tan ^{-1}(a x)}{21 a}+\frac {16 i c^3 \tan ^{-1}(a x)^2}{35 a}+\frac {16}{35} c^3 x \tan ^{-1}(a x)^2+\frac {8}{35} c^3 x \left (1+a^2 x^2\right ) \tan ^{-1}(a x)^2+\frac {6}{35} c^3 x \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)^2+\frac {1}{7} c^3 x \left (1+a^2 x^2\right )^3 \tan ^{-1}(a x)^2+\frac {32 c^3 \tan ^{-1}(a x) \log \left (\frac {2}{1+i a x}\right )}{35 a}-\frac {1}{35} \left (32 c^3\right ) \int \frac {\log \left (\frac {2}{1+i a x}\right )}{1+a^2 x^2} \, dx\\ &=\frac {38 c^3 x}{105}+\frac {19}{315} a^2 c^3 x^3+\frac {1}{105} a^4 c^3 x^5-\frac {8 c^3 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)}{35 a}-\frac {3 c^3 \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)}{35 a}-\frac {c^3 \left (1+a^2 x^2\right )^3 \tan ^{-1}(a x)}{21 a}+\frac {16 i c^3 \tan ^{-1}(a x)^2}{35 a}+\frac {16}{35} c^3 x \tan ^{-1}(a x)^2+\frac {8}{35} c^3 x \left (1+a^2 x^2\right ) \tan ^{-1}(a x)^2+\frac {6}{35} c^3 x \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)^2+\frac {1}{7} c^3 x \left (1+a^2 x^2\right )^3 \tan ^{-1}(a x)^2+\frac {32 c^3 \tan ^{-1}(a x) \log \left (\frac {2}{1+i a x}\right )}{35 a}+\frac {\left (32 i c^3\right ) \text {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1+i a x}\right )}{35 a}\\ &=\frac {38 c^3 x}{105}+\frac {19}{315} a^2 c^3 x^3+\frac {1}{105} a^4 c^3 x^5-\frac {8 c^3 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)}{35 a}-\frac {3 c^3 \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)}{35 a}-\frac {c^3 \left (1+a^2 x^2\right )^3 \tan ^{-1}(a x)}{21 a}+\frac {16 i c^3 \tan ^{-1}(a x)^2}{35 a}+\frac {16}{35} c^3 x \tan ^{-1}(a x)^2+\frac {8}{35} c^3 x \left (1+a^2 x^2\right ) \tan ^{-1}(a x)^2+\frac {6}{35} c^3 x \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)^2+\frac {1}{7} c^3 x \left (1+a^2 x^2\right )^3 \tan ^{-1}(a x)^2+\frac {32 c^3 \tan ^{-1}(a x) \log \left (\frac {2}{1+i a x}\right )}{35 a}+\frac {16 i c^3 \text {Li}_2\left (1-\frac {2}{1+i a x}\right )}{35 a}\\ \end {align*}
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Mathematica [A]
time = 0.87, size = 137, normalized size = 0.51 \begin {gather*} \frac {c^3 \left (a x \left (114+19 a^2 x^2+3 a^4 x^4\right )+9 \left (-16 i+35 a x+35 a^3 x^3+21 a^5 x^5+5 a^7 x^7\right ) \text {ArcTan}(a x)^2-3 \text {ArcTan}(a x) \left (38+57 a^2 x^2+24 a^4 x^4+5 a^6 x^6-96 \log \left (1+e^{2 i \text {ArcTan}(a x)}\right )\right )-144 i \text {PolyLog}\left (2,-e^{2 i \text {ArcTan}(a x)}\right )\right )}{315 a} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.21, size = 282, normalized size = 1.05
method | result | size |
derivativedivides | \(\frac {\frac {c^{3} \arctan \left (a x \right )^{2} a^{7} x^{7}}{7}+\frac {3 a^{5} c^{3} x^{5} \arctan \left (a x \right )^{2}}{5}+a^{3} c^{3} x^{3} \arctan \left (a x \right )^{2}+a \,c^{3} x \arctan \left (a x \right )^{2}-\frac {2 c^{3} \left (\frac {5 \arctan \left (a x \right ) a^{6} x^{6}}{6}+4 \arctan \left (a x \right ) a^{4} x^{4}+\frac {19 \arctan \left (a x \right ) a^{2} x^{2}}{2}+8 \arctan \left (a x \right ) \ln \left (a^{2} x^{2}+1\right )-\frac {a^{5} x^{5}}{6}-\frac {19 a^{3} x^{3}}{18}-\frac {19 a x}{3}+\frac {19 \arctan \left (a x \right )}{3}-4 i \ln \left (a x -i\right ) \ln \left (-\frac {i \left (a x +i\right )}{2}\right )-2 i \ln \left (a x -i\right )^{2}-4 i \dilog \left (-\frac {i \left (a x +i\right )}{2}\right )+4 i \dilog \left (\frac {i \left (a x -i\right )}{2}\right )+2 i \ln \left (a x +i\right )^{2}+4 i \ln \left (a x -i\right ) \ln \left (a^{2} x^{2}+1\right )+4 i \ln \left (a x +i\right ) \ln \left (\frac {i \left (a x -i\right )}{2}\right )-4 i \ln \left (a x +i\right ) \ln \left (a^{2} x^{2}+1\right )\right )}{35}}{a}\) | \(282\) |
default | \(\frac {\frac {c^{3} \arctan \left (a x \right )^{2} a^{7} x^{7}}{7}+\frac {3 a^{5} c^{3} x^{5} \arctan \left (a x \right )^{2}}{5}+a^{3} c^{3} x^{3} \arctan \left (a x \right )^{2}+a \,c^{3} x \arctan \left (a x \right )^{2}-\frac {2 c^{3} \left (\frac {5 \arctan \left (a x \right ) a^{6} x^{6}}{6}+4 \arctan \left (a x \right ) a^{4} x^{4}+\frac {19 \arctan \left (a x \right ) a^{2} x^{2}}{2}+8 \arctan \left (a x \right ) \ln \left (a^{2} x^{2}+1\right )-\frac {a^{5} x^{5}}{6}-\frac {19 a^{3} x^{3}}{18}-\frac {19 a x}{3}+\frac {19 \arctan \left (a x \right )}{3}-4 i \ln \left (a x -i\right ) \ln \left (-\frac {i \left (a x +i\right )}{2}\right )-2 i \ln \left (a x -i\right )^{2}-4 i \dilog \left (-\frac {i \left (a x +i\right )}{2}\right )+4 i \dilog \left (\frac {i \left (a x -i\right )}{2}\right )+2 i \ln \left (a x +i\right )^{2}+4 i \ln \left (a x -i\right ) \ln \left (a^{2} x^{2}+1\right )+4 i \ln \left (a x +i\right ) \ln \left (\frac {i \left (a x -i\right )}{2}\right )-4 i \ln \left (a x +i\right ) \ln \left (a^{2} x^{2}+1\right )\right )}{35}}{a}\) | \(282\) |
risch | \(\frac {c^{3} \ln \left (i a x +1\right ) \ln \left (-i a x +1\right ) x}{2}+\frac {38 c^{3} x}{105}+\frac {19 i c^{3} a \ln \left (i a x +1\right ) x^{2}}{70}+\frac {20469 i c^{3}}{42875 a}-\frac {c^{3} \ln \left (-i a x +1\right )^{2} x}{4}-\frac {c^{3} \ln \left (i a x +1\right )^{2} x}{4}+\frac {19 a^{2} c^{3} x^{3}}{315}+\frac {a^{4} c^{3} x^{5}}{105}-\frac {38 c^{3} \arctan \left (a x \right )}{105 a}+\frac {c^{3} a^{2} \ln \left (i a x +1\right ) \ln \left (-i a x +1\right ) x^{3}}{2}+\frac {c^{3} a^{6} \ln \left (i a x +1\right ) \ln \left (-i a x +1\right ) x^{7}}{14}+\frac {3 c^{3} a^{4} \ln \left (i a x +1\right ) \ln \left (-i a x +1\right ) x^{5}}{10}+\frac {8 i c^{3} \ln \left (i a x +1\right ) \ln \left (-i a x +1\right )}{35 a}-\frac {16 i c^{3} \ln \left (\frac {1}{2}+\frac {i a x}{2}\right ) \ln \left (-i a x +1\right )}{35 a}+\frac {16 i c^{3} \ln \left (\frac {1}{2}+\frac {i a x}{2}\right ) \ln \left (\frac {1}{2}-\frac {i a x}{2}\right )}{35 a}-\frac {i c^{3} a^{5} \ln \left (-i a x +1\right ) x^{6}}{42}+\frac {i c^{3} a^{5} \ln \left (i a x +1\right ) x^{6}}{42}-\frac {19 i c^{3} a \ln \left (-i a x +1\right ) x^{2}}{70}-\frac {4 i c^{3} a^{3} \ln \left (-i a x +1\right ) x^{4}}{35}+\frac {4 i c^{3} a^{3} \ln \left (i a x +1\right ) x^{4}}{35}+\frac {16 i c^{3} \dilog \left (\frac {1}{2}-\frac {i a x}{2}\right )}{35 a}-\frac {4 i c^{3} \ln \left (-i a x +1\right )^{2}}{35 a}+\frac {4 i c^{3} \ln \left (i a x +1\right )^{2}}{35 a}-\frac {c^{3} a^{6} \ln \left (-i a x +1\right )^{2} x^{7}}{28}-\frac {c^{3} a^{2} \ln \left (-i a x +1\right )^{2} x^{3}}{4}-\frac {3 c^{3} a^{4} \ln \left (-i a x +1\right )^{2} x^{5}}{20}-\frac {3 c^{3} a^{4} \ln \left (i a x +1\right )^{2} x^{5}}{20}-\frac {c^{3} a^{2} \ln \left (i a x +1\right )^{2} x^{3}}{4}-\frac {c^{3} a^{6} \ln \left (i a x +1\right )^{2} x^{7}}{28}\) | \(558\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} c^{3} \left (\int 3 a^{2} x^{2} \operatorname {atan}^{2}{\left (a x \right )}\, dx + \int 3 a^{4} x^{4} \operatorname {atan}^{2}{\left (a x \right )}\, dx + \int a^{6} x^{6} \operatorname {atan}^{2}{\left (a x \right )}\, dx + \int \operatorname {atan}^{2}{\left (a x \right )}\, dx\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int {\mathrm {atan}\left (a\,x\right )}^2\,{\left (c\,a^2\,x^2+c\right )}^3 \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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