Optimal. Leaf size=212 \[ -\frac {3 x^3}{128 a c^3 \left (1+a^2 x^2\right )^2}-\frac {45 x}{256 a^3 c^3 \left (1+a^2 x^2\right )}-\frac {27 \text {ArcTan}(a x)}{256 a^4 c^3}-\frac {3 x^4 \text {ArcTan}(a x)}{32 c^3 \left (1+a^2 x^2\right )^2}+\frac {9 \text {ArcTan}(a x)}{32 a^4 c^3 \left (1+a^2 x^2\right )}+\frac {3 x^3 \text {ArcTan}(a x)^2}{16 a c^3 \left (1+a^2 x^2\right )^2}+\frac {9 x \text {ArcTan}(a x)^2}{32 a^3 c^3 \left (1+a^2 x^2\right )}-\frac {3 \text {ArcTan}(a x)^3}{32 a^4 c^3}+\frac {x^4 \text {ArcTan}(a x)^3}{4 c^3 \left (1+a^2 x^2\right )^2} \]
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Rubi [A]
time = 0.21, antiderivative size = 212, normalized size of antiderivative = 1.00, number of steps
used = 9, number of rules used = 7, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.318, Rules used = {5064, 5060,
5056, 5050, 205, 211, 294} \begin {gather*} -\frac {3 \text {ArcTan}(a x)^3}{32 a^4 c^3}-\frac {27 \text {ArcTan}(a x)}{256 a^4 c^3}+\frac {x^4 \text {ArcTan}(a x)^3}{4 c^3 \left (a^2 x^2+1\right )^2}-\frac {3 x^4 \text {ArcTan}(a x)}{32 c^3 \left (a^2 x^2+1\right )^2}+\frac {3 x^3 \text {ArcTan}(a x)^2}{16 a c^3 \left (a^2 x^2+1\right )^2}-\frac {3 x^3}{128 a c^3 \left (a^2 x^2+1\right )^2}+\frac {9 \text {ArcTan}(a x)}{32 a^4 c^3 \left (a^2 x^2+1\right )}+\frac {9 x \text {ArcTan}(a x)^2}{32 a^3 c^3 \left (a^2 x^2+1\right )}-\frac {45 x}{256 a^3 c^3 \left (a^2 x^2+1\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 205
Rule 211
Rule 294
Rule 5050
Rule 5056
Rule 5060
Rule 5064
Rubi steps
\begin {align*} \int \frac {x^3 \tan ^{-1}(a x)^3}{\left (c+a^2 c x^2\right )^3} \, dx &=\frac {x^4 \tan ^{-1}(a x)^3}{4 c^3 \left (1+a^2 x^2\right )^2}-\frac {1}{4} (3 a) \int \frac {x^4 \tan ^{-1}(a x)^2}{\left (c+a^2 c x^2\right )^3} \, dx\\ &=-\frac {3 x^4 \tan ^{-1}(a x)}{32 c^3 \left (1+a^2 x^2\right )^2}+\frac {3 x^3 \tan ^{-1}(a x)^2}{16 a c^3 \left (1+a^2 x^2\right )^2}+\frac {x^4 \tan ^{-1}(a x)^3}{4 c^3 \left (1+a^2 x^2\right )^2}+\frac {1}{32} (3 a) \int \frac {x^4}{\left (c+a^2 c x^2\right )^3} \, dx-\frac {9 \int \frac {x^2 \tan ^{-1}(a x)^2}{\left (c+a^2 c x^2\right )^2} \, dx}{16 a c}\\ &=-\frac {3 x^3}{128 a c^3 \left (1+a^2 x^2\right )^2}-\frac {3 x^4 \tan ^{-1}(a x)}{32 c^3 \left (1+a^2 x^2\right )^2}+\frac {3 x^3 \tan ^{-1}(a x)^2}{16 a c^3 \left (1+a^2 x^2\right )^2}+\frac {9 x \tan ^{-1}(a x)^2}{32 a^3 c^3 \left (1+a^2 x^2\right )}-\frac {3 \tan ^{-1}(a x)^3}{32 a^4 c^3}+\frac {x^4 \tan ^{-1}(a x)^3}{4 c^3 \left (1+a^2 x^2\right )^2}-\frac {9 \int \frac {x \tan ^{-1}(a x)}{\left (c+a^2 c x^2\right )^2} \, dx}{16 a^2 c}+\frac {9 \int \frac {x^2}{\left (c+a^2 c x^2\right )^2} \, dx}{128 a c}\\ &=-\frac {3 x^3}{128 a c^3 \left (1+a^2 x^2\right )^2}-\frac {9 x}{256 a^3 c^3 \left (1+a^2 x^2\right )}-\frac {3 x^4 \tan ^{-1}(a x)}{32 c^3 \left (1+a^2 x^2\right )^2}+\frac {9 \tan ^{-1}(a x)}{32 a^4 c^3 \left (1+a^2 x^2\right )}+\frac {3 x^3 \tan ^{-1}(a x)^2}{16 a c^3 \left (1+a^2 x^2\right )^2}+\frac {9 x \tan ^{-1}(a x)^2}{32 a^3 c^3 \left (1+a^2 x^2\right )}-\frac {3 \tan ^{-1}(a x)^3}{32 a^4 c^3}+\frac {x^4 \tan ^{-1}(a x)^3}{4 c^3 \left (1+a^2 x^2\right )^2}+\frac {9 \int \frac {1}{c+a^2 c x^2} \, dx}{256 a^3 c^2}-\frac {9 \int \frac {1}{\left (c+a^2 c x^2\right )^2} \, dx}{32 a^3 c}\\ &=-\frac {3 x^3}{128 a c^3 \left (1+a^2 x^2\right )^2}-\frac {45 x}{256 a^3 c^3 \left (1+a^2 x^2\right )}+\frac {9 \tan ^{-1}(a x)}{256 a^4 c^3}-\frac {3 x^4 \tan ^{-1}(a x)}{32 c^3 \left (1+a^2 x^2\right )^2}+\frac {9 \tan ^{-1}(a x)}{32 a^4 c^3 \left (1+a^2 x^2\right )}+\frac {3 x^3 \tan ^{-1}(a x)^2}{16 a c^3 \left (1+a^2 x^2\right )^2}+\frac {9 x \tan ^{-1}(a x)^2}{32 a^3 c^3 \left (1+a^2 x^2\right )}-\frac {3 \tan ^{-1}(a x)^3}{32 a^4 c^3}+\frac {x^4 \tan ^{-1}(a x)^3}{4 c^3 \left (1+a^2 x^2\right )^2}-\frac {9 \int \frac {1}{c+a^2 c x^2} \, dx}{64 a^3 c^2}\\ &=-\frac {3 x^3}{128 a c^3 \left (1+a^2 x^2\right )^2}-\frac {45 x}{256 a^3 c^3 \left (1+a^2 x^2\right )}-\frac {27 \tan ^{-1}(a x)}{256 a^4 c^3}-\frac {3 x^4 \tan ^{-1}(a x)}{32 c^3 \left (1+a^2 x^2\right )^2}+\frac {9 \tan ^{-1}(a x)}{32 a^4 c^3 \left (1+a^2 x^2\right )}+\frac {3 x^3 \tan ^{-1}(a x)^2}{16 a c^3 \left (1+a^2 x^2\right )^2}+\frac {9 x \tan ^{-1}(a x)^2}{32 a^3 c^3 \left (1+a^2 x^2\right )}-\frac {3 \tan ^{-1}(a x)^3}{32 a^4 c^3}+\frac {x^4 \tan ^{-1}(a x)^3}{4 c^3 \left (1+a^2 x^2\right )^2}\\ \end {align*}
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Mathematica [A]
time = 0.16, size = 105, normalized size = 0.50 \begin {gather*} \frac {-3 a x \left (15+17 a^2 x^2\right )+\left (45+18 a^2 x^2-51 a^4 x^4\right ) \text {ArcTan}(a x)+24 a x \left (3+5 a^2 x^2\right ) \text {ArcTan}(a x)^2+8 \left (-3-6 a^2 x^2+5 a^4 x^4\right ) \text {ArcTan}(a x)^3}{256 a^4 c^3 \left (1+a^2 x^2\right )^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 1.00, size = 176, normalized size = 0.83
method | result | size |
derivativedivides | \(\frac {\frac {\arctan \left (a x \right )^{3}}{4 c^{3} \left (a^{2} x^{2}+1\right )^{2}}-\frac {\arctan \left (a x \right )^{3}}{2 c^{3} \left (a^{2} x^{2}+1\right )}-\frac {3 \left (-\frac {5 \arctan \left (a x \right )^{2} a^{3} x^{3}}{8 \left (a^{2} x^{2}+1\right )^{2}}-\frac {3 \arctan \left (a x \right )^{2} a x}{8 \left (a^{2} x^{2}+1\right )^{2}}-\frac {5 \arctan \left (a x \right )^{3}}{24}+\frac {\arctan \left (a x \right )}{8 \left (a^{2} x^{2}+1\right )^{2}}-\frac {5 \arctan \left (a x \right )}{8 \left (a^{2} x^{2}+1\right )}+\frac {\frac {17}{8} a^{3} x^{3}+\frac {15}{8} a x}{8 \left (a^{2} x^{2}+1\right )^{2}}+\frac {17 \arctan \left (a x \right )}{64}\right )}{4 c^{3}}}{a^{4}}\) | \(176\) |
default | \(\frac {\frac {\arctan \left (a x \right )^{3}}{4 c^{3} \left (a^{2} x^{2}+1\right )^{2}}-\frac {\arctan \left (a x \right )^{3}}{2 c^{3} \left (a^{2} x^{2}+1\right )}-\frac {3 \left (-\frac {5 \arctan \left (a x \right )^{2} a^{3} x^{3}}{8 \left (a^{2} x^{2}+1\right )^{2}}-\frac {3 \arctan \left (a x \right )^{2} a x}{8 \left (a^{2} x^{2}+1\right )^{2}}-\frac {5 \arctan \left (a x \right )^{3}}{24}+\frac {\arctan \left (a x \right )}{8 \left (a^{2} x^{2}+1\right )^{2}}-\frac {5 \arctan \left (a x \right )}{8 \left (a^{2} x^{2}+1\right )}+\frac {\frac {17}{8} a^{3} x^{3}+\frac {15}{8} a x}{8 \left (a^{2} x^{2}+1\right )^{2}}+\frac {17 \arctan \left (a x \right )}{64}\right )}{4 c^{3}}}{a^{4}}\) | \(176\) |
risch | \(\frac {i \left (5 a^{4} x^{4}-6 a^{2} x^{2}-3\right ) \ln \left (i a x +1\right )^{3}}{256 a^{4} c^{3} \left (a^{2} x^{2}+1\right )^{2}}-\frac {3 i \left (-6 a^{2} x^{2} \ln \left (-i a x +1\right )-3 \ln \left (-i a x +1\right )+5 x^{4} \ln \left (-i a x +1\right ) a^{4}-10 i a^{3} x^{3}-6 i a x \right ) \ln \left (i a x +1\right )^{2}}{256 a^{4} \left (a x +i\right )^{2} \left (a x -i\right )^{2} c^{3}}+\frac {3 i \left (5 a^{4} x^{4} \ln \left (-i a x +1\right )^{2}-6 a^{2} x^{2} \ln \left (-i a x +1\right )^{2}-3 \ln \left (-i a x +1\right )^{2}-20 i a^{3} x^{3} \ln \left (-i a x +1\right )-12 i a x \ln \left (-i a x +1\right )-20 a^{2} x^{2}-16\right ) \ln \left (i a x +1\right )}{256 a^{4} \left (a x +i\right )^{2} \left (a x -i\right )^{2} c^{3}}-\frac {i \left (-120 a^{2} x^{2} \ln \left (-i a x +1\right )-96 \ln \left (-i a x +1\right )+10 a^{4} x^{4} \ln \left (-i a x +1\right )^{3}-12 a^{2} x^{2} \ln \left (-i a x +1\right )^{3}-6 \ln \left (-i a x +1\right )^{3}-60 i a^{3} x^{3} \ln \left (-i a x +1\right )^{2}-36 i a x \ln \left (-i a x +1\right )^{2}+51 \ln \left (a x +i\right ) a^{4} x^{4}+102 \ln \left (a x +i\right ) a^{2} x^{2}+51 \ln \left (a x +i\right )-51 \ln \left (-a x +i\right ) a^{4} x^{4}-102 \ln \left (-a x +i\right ) a^{2} x^{2}-51 \ln \left (-a x +i\right )-102 i a^{3} x^{3}-90 i a x \right )}{512 a^{4} \left (a x +i\right )^{2} \left (a x -i\right )^{2} c^{3}}\) | \(492\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.51, size = 289, normalized size = 1.36 \begin {gather*} \frac {3}{32} \, a {\left (\frac {5 \, a^{2} x^{3} + 3 \, x}{a^{8} c^{3} x^{4} + 2 \, a^{6} c^{3} x^{2} + a^{4} c^{3}} + \frac {5 \, \arctan \left (a x\right )}{a^{5} c^{3}}\right )} \arctan \left (a x\right )^{2} - \frac {{\left (2 \, a^{2} x^{2} + 1\right )} \arctan \left (a x\right )^{3}}{4 \, {\left (a^{8} c^{3} x^{4} + 2 \, a^{6} c^{3} x^{2} + a^{4} c^{3}\right )}} - \frac {1}{256} \, {\left (\frac {{\left (51 \, a^{3} x^{3} - 40 \, {\left (a^{4} x^{4} + 2 \, a^{2} x^{2} + 1\right )} \arctan \left (a x\right )^{3} + 45 \, a x + 51 \, {\left (a^{4} x^{4} + 2 \, a^{2} x^{2} + 1\right )} \arctan \left (a x\right )\right )} a^{2}}{a^{11} c^{3} x^{4} + 2 \, a^{9} c^{3} x^{2} + a^{7} c^{3}} - \frac {24 \, {\left (5 \, a^{2} x^{2} - 5 \, {\left (a^{4} x^{4} + 2 \, a^{2} x^{2} + 1\right )} \arctan \left (a x\right )^{2} + 4\right )} a \arctan \left (a x\right )}{a^{10} c^{3} x^{4} + 2 \, a^{8} c^{3} x^{2} + a^{6} c^{3}}\right )} a \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 4.82, size = 117, normalized size = 0.55 \begin {gather*} -\frac {51 \, a^{3} x^{3} - 8 \, {\left (5 \, a^{4} x^{4} - 6 \, a^{2} x^{2} - 3\right )} \arctan \left (a x\right )^{3} - 24 \, {\left (5 \, a^{3} x^{3} + 3 \, a x\right )} \arctan \left (a x\right )^{2} + 45 \, a x + 3 \, {\left (17 \, a^{4} x^{4} - 6 \, a^{2} x^{2} - 15\right )} \arctan \left (a x\right )}{256 \, {\left (a^{8} c^{3} x^{4} + 2 \, a^{6} c^{3} x^{2} + a^{4} 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^{3} \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 = 0.59, size = 205, normalized size = 0.97 \begin {gather*} \frac {{\mathrm {atan}\left (a\,x\right )}^2\,\left (\frac {9\,x}{32\,a^5\,c^3}+\frac {15\,x^3}{32\,a^3\,c^3}\right )}{\frac {1}{a^2}+2\,x^2+a^2\,x^4}-{\mathrm {atan}\left (a\,x\right )}^3\,\left (\frac {\frac {1}{4\,a^6\,c^3}+\frac {x^2}{2\,a^4\,c^3}}{\frac {1}{a^2}+2\,x^2+a^2\,x^4}-\frac {5}{32\,a^4\,c^3}\right )-\frac {\frac {51\,a^2\,x^3}{8}+\frac {45\,x}{8}}{32\,a^7\,c^3\,x^4+64\,a^5\,c^3\,x^2+32\,a^3\,c^3}-\frac {51\,\mathrm {atan}\left (a\,x\right )}{256\,a^4\,c^3}+\frac {\mathrm {atan}\left (a\,x\right )\,\left (\frac {3}{8\,a^6\,c^3}+\frac {15\,x^2}{32\,a^4\,c^3}\right )}{\frac {1}{a^2}+2\,x^2+a^2\,x^4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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