Optimal. Leaf size=156 \[ \frac {c x}{30 a^2}+\frac {c x^3}{30}-\frac {c \text {ArcTan}(a x)}{30 a^3}-\frac {2 c x^2 \text {ArcTan}(a x)}{15 a}-\frac {1}{10} a c x^4 \text {ArcTan}(a x)-\frac {2 i c \text {ArcTan}(a x)^2}{15 a^3}+\frac {1}{3} c x^3 \text {ArcTan}(a x)^2+\frac {1}{5} a^2 c x^5 \text {ArcTan}(a x)^2-\frac {4 c \text {ArcTan}(a x) \log \left (\frac {2}{1+i a x}\right )}{15 a^3}-\frac {2 i c \text {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )}{15 a^3} \]
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Rubi [A]
time = 0.30, antiderivative size = 156, normalized size of antiderivative = 1.00, number of steps
used = 24, number of rules used = 10, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5070, 4946,
5036, 327, 209, 5040, 4964, 2449, 2352, 308} \begin {gather*} -\frac {2 i c \text {ArcTan}(a x)^2}{15 a^3}-\frac {c \text {ArcTan}(a x)}{30 a^3}-\frac {4 c \text {ArcTan}(a x) \log \left (\frac {2}{1+i a x}\right )}{15 a^3}-\frac {2 i c \text {Li}_2\left (1-\frac {2}{i a x+1}\right )}{15 a^3}+\frac {1}{5} a^2 c x^5 \text {ArcTan}(a x)^2+\frac {c x}{30 a^2}-\frac {1}{10} a c x^4 \text {ArcTan}(a x)+\frac {1}{3} c x^3 \text {ArcTan}(a x)^2-\frac {2 c x^2 \text {ArcTan}(a x)}{15 a}+\frac {c x^3}{30} \end {gather*}
Antiderivative was successfully verified.
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Rule 209
Rule 308
Rule 327
Rule 2352
Rule 2449
Rule 4946
Rule 4964
Rule 5036
Rule 5040
Rule 5070
Rubi steps
\begin {align*} \int x^2 \left (c+a^2 c x^2\right ) \tan ^{-1}(a x)^2 \, dx &=c \int x^2 \tan ^{-1}(a x)^2 \, dx+\left (a^2 c\right ) \int x^4 \tan ^{-1}(a x)^2 \, dx\\ &=\frac {1}{3} c x^3 \tan ^{-1}(a x)^2+\frac {1}{5} a^2 c x^5 \tan ^{-1}(a x)^2-\frac {1}{3} (2 a c) \int \frac {x^3 \tan ^{-1}(a x)}{1+a^2 x^2} \, dx-\frac {1}{5} \left (2 a^3 c\right ) \int \frac {x^5 \tan ^{-1}(a x)}{1+a^2 x^2} \, dx\\ &=\frac {1}{3} c x^3 \tan ^{-1}(a x)^2+\frac {1}{5} a^2 c x^5 \tan ^{-1}(a x)^2-\frac {(2 c) \int x \tan ^{-1}(a x) \, dx}{3 a}+\frac {(2 c) \int \frac {x \tan ^{-1}(a x)}{1+a^2 x^2} \, dx}{3 a}-\frac {1}{5} (2 a c) \int x^3 \tan ^{-1}(a x) \, dx+\frac {1}{5} (2 a c) \int \frac {x^3 \tan ^{-1}(a x)}{1+a^2 x^2} \, dx\\ &=-\frac {c x^2 \tan ^{-1}(a x)}{3 a}-\frac {1}{10} a c x^4 \tan ^{-1}(a x)-\frac {i c \tan ^{-1}(a x)^2}{3 a^3}+\frac {1}{3} c x^3 \tan ^{-1}(a x)^2+\frac {1}{5} a^2 c x^5 \tan ^{-1}(a x)^2+\frac {1}{3} c \int \frac {x^2}{1+a^2 x^2} \, dx-\frac {(2 c) \int \frac {\tan ^{-1}(a x)}{i-a x} \, dx}{3 a^2}+\frac {(2 c) \int x \tan ^{-1}(a x) \, dx}{5 a}-\frac {(2 c) \int \frac {x \tan ^{-1}(a x)}{1+a^2 x^2} \, dx}{5 a}+\frac {1}{10} \left (a^2 c\right ) \int \frac {x^4}{1+a^2 x^2} \, dx\\ &=\frac {c x}{3 a^2}-\frac {2 c x^2 \tan ^{-1}(a x)}{15 a}-\frac {1}{10} a c x^4 \tan ^{-1}(a x)-\frac {2 i c \tan ^{-1}(a x)^2}{15 a^3}+\frac {1}{3} c x^3 \tan ^{-1}(a x)^2+\frac {1}{5} a^2 c x^5 \tan ^{-1}(a x)^2-\frac {2 c \tan ^{-1}(a x) \log \left (\frac {2}{1+i a x}\right )}{3 a^3}-\frac {1}{5} c \int \frac {x^2}{1+a^2 x^2} \, dx-\frac {c \int \frac {1}{1+a^2 x^2} \, dx}{3 a^2}+\frac {(2 c) \int \frac {\tan ^{-1}(a x)}{i-a x} \, dx}{5 a^2}+\frac {(2 c) \int \frac {\log \left (\frac {2}{1+i a x}\right )}{1+a^2 x^2} \, dx}{3 a^2}+\frac {1}{10} \left (a^2 c\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 {c x}{30 a^2}+\frac {c x^3}{30}-\frac {c \tan ^{-1}(a x)}{3 a^3}-\frac {2 c x^2 \tan ^{-1}(a x)}{15 a}-\frac {1}{10} a c x^4 \tan ^{-1}(a x)-\frac {2 i c \tan ^{-1}(a x)^2}{15 a^3}+\frac {1}{3} c x^3 \tan ^{-1}(a x)^2+\frac {1}{5} a^2 c x^5 \tan ^{-1}(a x)^2-\frac {4 c \tan ^{-1}(a x) \log \left (\frac {2}{1+i a x}\right )}{15 a^3}-\frac {(2 i c) \text {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1+i a x}\right )}{3 a^3}+\frac {c \int \frac {1}{1+a^2 x^2} \, dx}{10 a^2}+\frac {c \int \frac {1}{1+a^2 x^2} \, dx}{5 a^2}-\frac {(2 c) \int \frac {\log \left (\frac {2}{1+i a x}\right )}{1+a^2 x^2} \, dx}{5 a^2}\\ &=\frac {c x}{30 a^2}+\frac {c x^3}{30}-\frac {c \tan ^{-1}(a x)}{30 a^3}-\frac {2 c x^2 \tan ^{-1}(a x)}{15 a}-\frac {1}{10} a c x^4 \tan ^{-1}(a x)-\frac {2 i c \tan ^{-1}(a x)^2}{15 a^3}+\frac {1}{3} c x^3 \tan ^{-1}(a x)^2+\frac {1}{5} a^2 c x^5 \tan ^{-1}(a x)^2-\frac {4 c \tan ^{-1}(a x) \log \left (\frac {2}{1+i a x}\right )}{15 a^3}-\frac {i c \text {Li}_2\left (1-\frac {2}{1+i a x}\right )}{3 a^3}+\frac {(2 i c) \text {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1+i a x}\right )}{5 a^3}\\ &=\frac {c x}{30 a^2}+\frac {c x^3}{30}-\frac {c \tan ^{-1}(a x)}{30 a^3}-\frac {2 c x^2 \tan ^{-1}(a x)}{15 a}-\frac {1}{10} a c x^4 \tan ^{-1}(a x)-\frac {2 i c \tan ^{-1}(a x)^2}{15 a^3}+\frac {1}{3} c x^3 \tan ^{-1}(a x)^2+\frac {1}{5} a^2 c x^5 \tan ^{-1}(a x)^2-\frac {4 c \tan ^{-1}(a x) \log \left (\frac {2}{1+i a x}\right )}{15 a^3}-\frac {2 i c \text {Li}_2\left (1-\frac {2}{1+i a x}\right )}{15 a^3}\\ \end {align*}
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Mathematica [A]
time = 0.47, size = 104, normalized size = 0.67 \begin {gather*} \frac {c \left (a x+a^3 x^3+2 \left (2 i+5 a^3 x^3+3 a^5 x^5\right ) \text {ArcTan}(a x)^2-\text {ArcTan}(a x) \left (1+4 a^2 x^2+3 a^4 x^4+8 \log \left (1+e^{2 i \text {ArcTan}(a x)}\right )\right )+4 i \text {PolyLog}\left (2,-e^{2 i \text {ArcTan}(a x)}\right )\right )}{30 a^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.39, size = 227, normalized size = 1.46
method | result | size |
derivativedivides | \(\frac {\frac {c \arctan \left (a x \right )^{2} a^{5} x^{5}}{5}+\frac {c \arctan \left (a x \right )^{2} a^{3} x^{3}}{3}-\frac {2 c \left (\frac {3 \arctan \left (a x \right ) a^{4} x^{4}}{4}+\arctan \left (a x \right ) a^{2} x^{2}-\arctan \left (a x \right ) \ln \left (a^{2} x^{2}+1\right )-\frac {a^{3} x^{3}}{4}-\frac {a x}{4}+\frac {\arctan \left (a x \right )}{4}-\frac {i \ln \left (a x -i\right ) \ln \left (a^{2} x^{2}+1\right )}{2}+\frac {i \ln \left (a x -i\right )^{2}}{4}+\frac {i \ln \left (a x -i\right ) \ln \left (-\frac {i \left (a x +i\right )}{2}\right )}{2}+\frac {i \dilog \left (-\frac {i \left (a x +i\right )}{2}\right )}{2}+\frac {i \ln \left (a x +i\right ) \ln \left (a^{2} x^{2}+1\right )}{2}-\frac {i \ln \left (a x +i\right )^{2}}{4}-\frac {i \ln \left (a x +i\right ) \ln \left (\frac {i \left (a x -i\right )}{2}\right )}{2}-\frac {i \dilog \left (\frac {i \left (a x -i\right )}{2}\right )}{2}\right )}{15}}{a^{3}}\) | \(227\) |
default | \(\frac {\frac {c \arctan \left (a x \right )^{2} a^{5} x^{5}}{5}+\frac {c \arctan \left (a x \right )^{2} a^{3} x^{3}}{3}-\frac {2 c \left (\frac {3 \arctan \left (a x \right ) a^{4} x^{4}}{4}+\arctan \left (a x \right ) a^{2} x^{2}-\arctan \left (a x \right ) \ln \left (a^{2} x^{2}+1\right )-\frac {a^{3} x^{3}}{4}-\frac {a x}{4}+\frac {\arctan \left (a x \right )}{4}-\frac {i \ln \left (a x -i\right ) \ln \left (a^{2} x^{2}+1\right )}{2}+\frac {i \ln \left (a x -i\right )^{2}}{4}+\frac {i \ln \left (a x -i\right ) \ln \left (-\frac {i \left (a x +i\right )}{2}\right )}{2}+\frac {i \dilog \left (-\frac {i \left (a x +i\right )}{2}\right )}{2}+\frac {i \ln \left (a x +i\right ) \ln \left (a^{2} x^{2}+1\right )}{2}-\frac {i \ln \left (a x +i\right )^{2}}{4}-\frac {i \ln \left (a x +i\right ) \ln \left (\frac {i \left (a x -i\right )}{2}\right )}{2}-\frac {i \dilog \left (\frac {i \left (a x -i\right )}{2}\right )}{2}\right )}{15}}{a^{3}}\) | \(227\) |
risch | \(\frac {c x}{30 a^{2}}-\frac {c \arctan \left (a x \right )}{30 a^{3}}-\frac {c \ln \left (i a x +1\right )^{2} x^{3}}{12}-\frac {c \ln \left (-i a x +1\right )^{2} x^{3}}{12}+\frac {c \,a^{2} \ln \left (i a x +1\right ) \ln \left (-i a x +1\right ) x^{5}}{10}+\frac {i c a \ln \left (i a x +1\right ) x^{4}}{20}+\frac {i c \ln \left (i a x +1\right ) x^{2}}{15 a}-\frac {443 i c}{3375 a^{3}}+\frac {i c \ln \left (-i a x +1\right )^{2}}{30 a^{3}}-\frac {2 i c \dilog \left (\frac {1}{2}-\frac {i a x}{2}\right )}{15 a^{3}}+\frac {2 i c \ln \left (\frac {1}{2}+\frac {i a x}{2}\right ) \ln \left (-i a x +1\right )}{15 a^{3}}-\frac {i c \ln \left (i a x +1\right )^{2}}{30 a^{3}}+\frac {c \,x^{3}}{30}-\frac {i c \ln \left (i a x +1\right ) \ln \left (-i a x +1\right )}{15 a^{3}}+\frac {c \ln \left (i a x +1\right ) \ln \left (-i a x +1\right ) x^{3}}{6}-\frac {c \,a^{2} \ln \left (-i a x +1\right )^{2} x^{5}}{20}-\frac {c \,a^{2} \ln \left (i a x +1\right )^{2} x^{5}}{20}-\frac {i c \ln \left (-i a x +1\right ) x^{2}}{15 a}-\frac {2 i c \ln \left (\frac {1}{2}+\frac {i a x}{2}\right ) \ln \left (\frac {1}{2}-\frac {i a x}{2}\right )}{15 a^{3}}-\frac {i c a \ln \left (-i a x +1\right ) x^{4}}{20}\) | \(335\) |
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 \left (\int x^{2} \operatorname {atan}^{2}{\left (a x \right )}\, dx + \int a^{2} x^{4} \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 = -0.01 \begin {gather*} \int x^2\,{\mathrm {atan}\left (a\,x\right )}^2\,\left (c\,a^2\,x^2+c\right ) \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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