Optimal. Leaf size=237 \[ \frac {3}{128 a^3 c^3 \left (1+a^2 x^2\right )^2}-\frac {3}{128 a^3 c^3 \left (1+a^2 x^2\right )}+\frac {3 x \text {ArcTan}(a x)}{32 a^2 c^3 \left (1+a^2 x^2\right )^2}-\frac {3 x \text {ArcTan}(a x)}{64 a^2 c^3 \left (1+a^2 x^2\right )}-\frac {3 \text {ArcTan}(a x)^2}{128 a^3 c^3}-\frac {3 \text {ArcTan}(a x)^2}{16 a^3 c^3 \left (1+a^2 x^2\right )^2}+\frac {3 \text {ArcTan}(a x)^2}{16 a^3 c^3 \left (1+a^2 x^2\right )}-\frac {x \text {ArcTan}(a x)^3}{4 a^2 c^3 \left (1+a^2 x^2\right )^2}+\frac {x \text {ArcTan}(a x)^3}{8 a^2 c^3 \left (1+a^2 x^2\right )}+\frac {\text {ArcTan}(a x)^4}{32 a^3 c^3} \]
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Rubi [A]
time = 0.29, antiderivative size = 237, normalized size of antiderivative = 1.00, number of steps
used = 13, number of rules used = 6, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {5084, 5012,
5050, 267, 5020, 5016} \begin {gather*} \frac {\text {ArcTan}(a x)^4}{32 a^3 c^3}-\frac {3 \text {ArcTan}(a x)^2}{128 a^3 c^3}+\frac {x \text {ArcTan}(a x)^3}{8 a^2 c^3 \left (a^2 x^2+1\right )}-\frac {x \text {ArcTan}(a x)^3}{4 a^2 c^3 \left (a^2 x^2+1\right )^2}-\frac {3 x \text {ArcTan}(a x)}{64 a^2 c^3 \left (a^2 x^2+1\right )}+\frac {3 x \text {ArcTan}(a x)}{32 a^2 c^3 \left (a^2 x^2+1\right )^2}+\frac {3 \text {ArcTan}(a x)^2}{16 a^3 c^3 \left (a^2 x^2+1\right )}-\frac {3 \text {ArcTan}(a x)^2}{16 a^3 c^3 \left (a^2 x^2+1\right )^2}-\frac {3}{128 a^3 c^3 \left (a^2 x^2+1\right )}+\frac {3}{128 a^3 c^3 \left (a^2 x^2+1\right )^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 267
Rule 5012
Rule 5016
Rule 5020
Rule 5050
Rule 5084
Rubi steps
\begin {align*} \int \frac {x^2 \tan ^{-1}(a x)^3}{\left (c+a^2 c x^2\right )^3} \, dx &=-\frac {\int \frac {\tan ^{-1}(a x)^3}{\left (c+a^2 c x^2\right )^3} \, dx}{a^2}+\frac {\int \frac {\tan ^{-1}(a x)^3}{\left (c+a^2 c x^2\right )^2} \, dx}{a^2 c}\\ &=-\frac {3 \tan ^{-1}(a x)^2}{16 a^3 c^3 \left (1+a^2 x^2\right )^2}-\frac {x \tan ^{-1}(a x)^3}{4 a^2 c^3 \left (1+a^2 x^2\right )^2}+\frac {x \tan ^{-1}(a x)^3}{2 a^2 c^3 \left (1+a^2 x^2\right )}+\frac {\tan ^{-1}(a x)^4}{8 a^3 c^3}+\frac {3 \int \frac {\tan ^{-1}(a x)}{\left (c+a^2 c x^2\right )^3} \, dx}{8 a^2}-\frac {3 \int \frac {\tan ^{-1}(a x)^3}{\left (c+a^2 c x^2\right )^2} \, dx}{4 a^2 c}-\frac {3 \int \frac {x \tan ^{-1}(a x)^2}{\left (c+a^2 c x^2\right )^2} \, dx}{2 a c}\\ &=\frac {3}{128 a^3 c^3 \left (1+a^2 x^2\right )^2}+\frac {3 x \tan ^{-1}(a x)}{32 a^2 c^3 \left (1+a^2 x^2\right )^2}-\frac {3 \tan ^{-1}(a x)^2}{16 a^3 c^3 \left (1+a^2 x^2\right )^2}+\frac {3 \tan ^{-1}(a x)^2}{4 a^3 c^3 \left (1+a^2 x^2\right )}-\frac {x \tan ^{-1}(a x)^3}{4 a^2 c^3 \left (1+a^2 x^2\right )^2}+\frac {x \tan ^{-1}(a x)^3}{8 a^2 c^3 \left (1+a^2 x^2\right )}+\frac {\tan ^{-1}(a x)^4}{32 a^3 c^3}+\frac {9 \int \frac {\tan ^{-1}(a x)}{\left (c+a^2 c x^2\right )^2} \, dx}{32 a^2 c}-\frac {3 \int \frac {\tan ^{-1}(a x)}{\left (c+a^2 c x^2\right )^2} \, dx}{2 a^2 c}+\frac {9 \int \frac {x \tan ^{-1}(a x)^2}{\left (c+a^2 c x^2\right )^2} \, dx}{8 a c}\\ &=\frac {3}{128 a^3 c^3 \left (1+a^2 x^2\right )^2}+\frac {3 x \tan ^{-1}(a x)}{32 a^2 c^3 \left (1+a^2 x^2\right )^2}-\frac {39 x \tan ^{-1}(a x)}{64 a^2 c^3 \left (1+a^2 x^2\right )}-\frac {39 \tan ^{-1}(a x)^2}{128 a^3 c^3}-\frac {3 \tan ^{-1}(a x)^2}{16 a^3 c^3 \left (1+a^2 x^2\right )^2}+\frac {3 \tan ^{-1}(a x)^2}{16 a^3 c^3 \left (1+a^2 x^2\right )}-\frac {x \tan ^{-1}(a x)^3}{4 a^2 c^3 \left (1+a^2 x^2\right )^2}+\frac {x \tan ^{-1}(a x)^3}{8 a^2 c^3 \left (1+a^2 x^2\right )}+\frac {\tan ^{-1}(a x)^4}{32 a^3 c^3}+\frac {9 \int \frac {\tan ^{-1}(a x)}{\left (c+a^2 c x^2\right )^2} \, dx}{8 a^2 c}-\frac {9 \int \frac {x}{\left (c+a^2 c x^2\right )^2} \, dx}{64 a c}+\frac {3 \int \frac {x}{\left (c+a^2 c x^2\right )^2} \, dx}{4 a c}\\ &=\frac {3}{128 a^3 c^3 \left (1+a^2 x^2\right )^2}-\frac {39}{128 a^3 c^3 \left (1+a^2 x^2\right )}+\frac {3 x \tan ^{-1}(a x)}{32 a^2 c^3 \left (1+a^2 x^2\right )^2}-\frac {3 x \tan ^{-1}(a x)}{64 a^2 c^3 \left (1+a^2 x^2\right )}-\frac {3 \tan ^{-1}(a x)^2}{128 a^3 c^3}-\frac {3 \tan ^{-1}(a x)^2}{16 a^3 c^3 \left (1+a^2 x^2\right )^2}+\frac {3 \tan ^{-1}(a x)^2}{16 a^3 c^3 \left (1+a^2 x^2\right )}-\frac {x \tan ^{-1}(a x)^3}{4 a^2 c^3 \left (1+a^2 x^2\right )^2}+\frac {x \tan ^{-1}(a x)^3}{8 a^2 c^3 \left (1+a^2 x^2\right )}+\frac {\tan ^{-1}(a x)^4}{32 a^3 c^3}-\frac {9 \int \frac {x}{\left (c+a^2 c x^2\right )^2} \, dx}{16 a c}\\ &=\frac {3}{128 a^3 c^3 \left (1+a^2 x^2\right )^2}-\frac {3}{128 a^3 c^3 \left (1+a^2 x^2\right )}+\frac {3 x \tan ^{-1}(a x)}{32 a^2 c^3 \left (1+a^2 x^2\right )^2}-\frac {3 x \tan ^{-1}(a x)}{64 a^2 c^3 \left (1+a^2 x^2\right )}-\frac {3 \tan ^{-1}(a x)^2}{128 a^3 c^3}-\frac {3 \tan ^{-1}(a x)^2}{16 a^3 c^3 \left (1+a^2 x^2\right )^2}+\frac {3 \tan ^{-1}(a x)^2}{16 a^3 c^3 \left (1+a^2 x^2\right )}-\frac {x \tan ^{-1}(a x)^3}{4 a^2 c^3 \left (1+a^2 x^2\right )^2}+\frac {x \tan ^{-1}(a x)^3}{8 a^2 c^3 \left (1+a^2 x^2\right )}+\frac {\tan ^{-1}(a x)^4}{32 a^3 c^3}\\ \end {align*}
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Mathematica [A]
time = 0.04, size = 111, normalized size = 0.47 \begin {gather*} \frac {-3 a^2 x^2+\left (6 a x-6 a^3 x^3\right ) \text {ArcTan}(a x)-3 \left (1-6 a^2 x^2+a^4 x^4\right ) \text {ArcTan}(a x)^2+16 a x \left (-1+a^2 x^2\right ) \text {ArcTan}(a x)^3+4 \left (1+a^2 x^2\right )^2 \text {ArcTan}(a x)^4}{128 a^3 c^3 \left (1+a^2 x^2\right )^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.99, size = 197, normalized size = 0.83
method | result | size |
derivativedivides | \(\frac {\frac {\arctan \left (a x \right )^{3} a^{3} x^{3}}{8 c^{3} \left (a^{2} x^{2}+1\right )^{2}}-\frac {\arctan \left (a x \right )^{3} a x}{8 c^{3} \left (a^{2} x^{2}+1\right )^{2}}+\frac {\arctan \left (a x \right )^{4}}{8 c^{3}}-\frac {3 \left (\frac {\arctan \left (a x \right )^{4}}{4}+\frac {\arctan \left (a x \right )^{2}}{2 \left (a^{2} x^{2}+1\right )^{2}}-\frac {\arctan \left (a x \right )^{2}}{2 \left (a^{2} x^{2}+1\right )}+\frac {\arctan \left (a x \right ) a^{3} x^{3}}{8 \left (a^{2} x^{2}+1\right )^{2}}-\frac {a x \arctan \left (a x \right )}{8 \left (a^{2} x^{2}+1\right )^{2}}+\frac {\arctan \left (a x \right )^{2}}{16}-\frac {1}{16 \left (a^{2} x^{2}+1\right )^{2}}+\frac {1}{16 a^{2} x^{2}+16}\right )}{8 c^{3}}}{a^{3}}\) | \(197\) |
default | \(\frac {\frac {\arctan \left (a x \right )^{3} a^{3} x^{3}}{8 c^{3} \left (a^{2} x^{2}+1\right )^{2}}-\frac {\arctan \left (a x \right )^{3} a x}{8 c^{3} \left (a^{2} x^{2}+1\right )^{2}}+\frac {\arctan \left (a x \right )^{4}}{8 c^{3}}-\frac {3 \left (\frac {\arctan \left (a x \right )^{4}}{4}+\frac {\arctan \left (a x \right )^{2}}{2 \left (a^{2} x^{2}+1\right )^{2}}-\frac {\arctan \left (a x \right )^{2}}{2 \left (a^{2} x^{2}+1\right )}+\frac {\arctan \left (a x \right ) a^{3} x^{3}}{8 \left (a^{2} x^{2}+1\right )^{2}}-\frac {a x \arctan \left (a x \right )}{8 \left (a^{2} x^{2}+1\right )^{2}}+\frac {\arctan \left (a x \right )^{2}}{16}-\frac {1}{16 \left (a^{2} x^{2}+1\right )^{2}}+\frac {1}{16 a^{2} x^{2}+16}\right )}{8 c^{3}}}{a^{3}}\) | \(197\) |
risch | \(\frac {\ln \left (i a x +1\right )^{4}}{512 c^{3} a^{3}}-\frac {\left (x^{4} \ln \left (-i a x +1\right ) a^{4}+2 a^{2} x^{2} \ln \left (-i a x +1\right )-2 i a^{3} x^{3}+\ln \left (-i a x +1\right )+2 i a x \right ) \ln \left (i a x +1\right )^{3}}{128 a^{3} c^{3} \left (a^{2} x^{2}+1\right )^{2}}+\frac {3 \left (2 a^{4} x^{4} \ln \left (-i a x +1\right )^{2}-8 i a^{3} x^{3} \ln \left (-i a x +1\right )+a^{4} x^{4}+4 a^{2} x^{2} \ln \left (-i a x +1\right )^{2}+8 i a x \ln \left (-i a x +1\right )-6 a^{2} x^{2}+2 \ln \left (-i a x +1\right )^{2}+1\right ) \ln \left (i a x +1\right )^{2}}{512 \left (a x +i\right )^{2} c^{3} \left (a x -i\right )^{2} a^{3}}-\frac {\left (2 a^{4} x^{4} \ln \left (-i a x +1\right )^{3}-12 i a^{3} x^{3} \ln \left (-i a x +1\right )^{2}+3 x^{4} \ln \left (-i a x +1\right ) a^{4}-6 i a^{3} x^{3}+4 a^{2} x^{2} \ln \left (-i a x +1\right )^{3}+12 i a x \ln \left (-i a x +1\right )^{2}-18 a^{2} x^{2} \ln \left (-i a x +1\right )+6 i a x +2 \ln \left (-i a x +1\right )^{3}+3 \ln \left (-i a x +1\right )\right ) \ln \left (i a x +1\right )}{256 \left (a x +i\right )^{2} c^{3} \left (a x -i\right )^{2} a^{3}}+\frac {a^{4} x^{4} \ln \left (-i a x +1\right )^{4}+3 a^{4} x^{4} \ln \left (-i a x +1\right )^{2}+2 a^{2} x^{2} \ln \left (-i a x +1\right )^{4}-8 i a^{3} x^{3} \ln \left (-i a x +1\right )^{3}-18 a^{2} x^{2} \ln \left (-i a x +1\right )^{2}-12 i a^{3} x^{3} \ln \left (-i a x +1\right )-12 a^{2} x^{2}+\ln \left (-i a x +1\right )^{4}+8 i a x \ln \left (-i a x +1\right )^{3}+3 \ln \left (-i a x +1\right )^{2}+12 i a x \ln \left (-i a x +1\right )}{512 \left (a x +i\right )^{2} c^{3} \left (a x -i\right )^{2} a^{3}}\) | \(593\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.54, size = 334, normalized size = 1.41 \begin {gather*} \frac {1}{8} \, {\left (\frac {a^{2} x^{3} - x}{a^{6} c^{3} x^{4} + 2 \, a^{4} c^{3} x^{2} + a^{2} c^{3}} + \frac {\arctan \left (a x\right )}{a^{3} c^{3}}\right )} \arctan \left (a x\right )^{3} + \frac {3 \, {\left (a^{2} x^{2} - {\left (a^{4} x^{4} + 2 \, a^{2} x^{2} + 1\right )} \arctan \left (a x\right )^{2}\right )} a \arctan \left (a x\right )^{2}}{16 \, {\left (a^{8} c^{3} x^{4} + 2 \, a^{6} c^{3} x^{2} + a^{4} c^{3}\right )}} - \frac {1}{128} \, {\left (\frac {{\left (4 \, {\left (a^{4} x^{4} + 2 \, a^{2} x^{2} + 1\right )} \arctan \left (a x\right )^{4} + 3 \, a^{2} x^{2} - 3 \, {\left (a^{4} x^{4} + 2 \, a^{2} x^{2} + 1\right )} \arctan \left (a x\right )^{2}\right )} a^{2}}{a^{10} c^{3} x^{4} + 2 \, a^{8} c^{3} x^{2} + a^{6} c^{3}} + \frac {2 \, {\left (3 \, a^{3} x^{3} - 8 \, {\left (a^{4} x^{4} + 2 \, a^{2} x^{2} + 1\right )} \arctan \left (a x\right )^{3} - 3 \, a x + 3 \, {\left (a^{4} x^{4} + 2 \, a^{2} x^{2} + 1\right )} \arctan \left (a x\right )\right )} a \arctan \left (a x\right )}{a^{9} c^{3} x^{4} + 2 \, a^{7} c^{3} x^{2} + a^{5} c^{3}}\right )} a \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 4.76, size = 130, normalized size = 0.55 \begin {gather*} \frac {4 \, {\left (a^{4} x^{4} + 2 \, a^{2} x^{2} + 1\right )} \arctan \left (a x\right )^{4} - 3 \, a^{2} x^{2} + 16 \, {\left (a^{3} x^{3} - a x\right )} \arctan \left (a x\right )^{3} - 3 \, {\left (a^{4} x^{4} - 6 \, a^{2} x^{2} + 1\right )} \arctan \left (a x\right )^{2} - 6 \, {\left (a^{3} x^{3} - a x\right )} \arctan \left (a x\right )}{128 \, {\left (a^{7} c^{3} x^{4} + 2 \, a^{5} c^{3} x^{2} + a^{3} c^{3}\right )}} \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*} \frac {\int \frac {x^{2} \operatorname {atan}^{3}{\left (a x \right )}}{a^{6} x^{6} + 3 a^{4} x^{4} + 3 a^{2} x^{2} + 1}\, dx}{c^{3}} \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 [B]
time = 0.54, size = 188, normalized size = 0.79 \begin {gather*} \frac {\mathrm {atan}\left (a\,x\right )\,\left (\frac {3\,x}{64\,a^4\,c^3}-\frac {3\,x^3}{64\,a^2\,c^3}\right )}{\frac {1}{a^2}+2\,x^2+a^2\,x^4}-\frac {3\,x^2}{2\,\left (64\,a^5\,c^3\,x^4+128\,a^3\,c^3\,x^2+64\,a\,c^3\right )}-{\mathrm {atan}\left (a\,x\right )}^2\,\left (\frac {3}{128\,a^3\,c^3}-\frac {3\,x^2}{16\,a^3\,c^3\,\left (\frac {1}{a^2}+2\,x^2+a^2\,x^4\right )}\right )-\frac {{\mathrm {atan}\left (a\,x\right )}^3\,\left (\frac {x}{8\,a^4\,c^3}-\frac {x^3}{8\,a^2\,c^3}\right )}{\frac {1}{a^2}+2\,x^2+a^2\,x^4}+\frac {{\mathrm {atan}\left (a\,x\right )}^4}{32\,a^3\,c^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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