Optimal. Leaf size=308 \[ -\frac {1}{280} a^3 c^3 x^5-\frac {6 i c^3 \text {Li}_2\left (1-\frac {2}{i a x+1}\right )}{35 a^2}-\frac {3 c^3 x \left (a^2 x^2+1\right )^3 \tan ^{-1}(a x)^2}{56 a}-\frac {9 c^3 x \left (a^2 x^2+1\right )^2 \tan ^{-1}(a x)^2}{140 a}-\frac {3 c^3 x \left (a^2 x^2+1\right ) \tan ^{-1}(a x)^2}{35 a}+\frac {c^3 \left (a^2 x^2+1\right )^4 \tan ^{-1}(a x)^3}{8 a^2}+\frac {c^3 \left (a^2 x^2+1\right )^3 \tan ^{-1}(a x)}{56 a^2}+\frac {9 c^3 \left (a^2 x^2+1\right )^2 \tan ^{-1}(a x)}{280 a^2}+\frac {3 c^3 \left (a^2 x^2+1\right ) \tan ^{-1}(a x)}{35 a^2}-\frac {6 i c^3 \tan ^{-1}(a x)^2}{35 a^2}-\frac {12 c^3 \log \left (\frac {2}{1+i a x}\right ) \tan ^{-1}(a x)}{35 a^2}-\frac {19}{840} a c^3 x^3-\frac {19 c^3 x}{140 a}-\frac {6 c^3 x \tan ^{-1}(a x)^2}{35 a} \]
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Rubi [A] time = 0.25, antiderivative size = 308, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 9, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.450, Rules used = {4930, 4880, 4846, 4920, 4854, 2402, 2315, 8, 194} \[ -\frac {6 i c^3 \text {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )}{35 a^2}-\frac {1}{280} a^3 c^3 x^5-\frac {3 c^3 x \left (a^2 x^2+1\right )^3 \tan ^{-1}(a x)^2}{56 a}-\frac {9 c^3 x \left (a^2 x^2+1\right )^2 \tan ^{-1}(a x)^2}{140 a}-\frac {3 c^3 x \left (a^2 x^2+1\right ) \tan ^{-1}(a x)^2}{35 a}+\frac {c^3 \left (a^2 x^2+1\right )^4 \tan ^{-1}(a x)^3}{8 a^2}+\frac {c^3 \left (a^2 x^2+1\right )^3 \tan ^{-1}(a x)}{56 a^2}+\frac {9 c^3 \left (a^2 x^2+1\right )^2 \tan ^{-1}(a x)}{280 a^2}+\frac {3 c^3 \left (a^2 x^2+1\right ) \tan ^{-1}(a x)}{35 a^2}-\frac {6 i c^3 \tan ^{-1}(a x)^2}{35 a^2}-\frac {12 c^3 \log \left (\frac {2}{1+i a x}\right ) \tan ^{-1}(a x)}{35 a^2}-\frac {19}{840} a c^3 x^3-\frac {19 c^3 x}{140 a}-\frac {6 c^3 x \tan ^{-1}(a x)^2}{35 a} \]
Antiderivative was successfully verified.
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Rule 8
Rule 194
Rule 2315
Rule 2402
Rule 4846
Rule 4854
Rule 4880
Rule 4920
Rule 4930
Rubi steps
\begin {align*} \int x \left (c+a^2 c x^2\right )^3 \tan ^{-1}(a x)^3 \, dx &=\frac {c^3 \left (1+a^2 x^2\right )^4 \tan ^{-1}(a x)^3}{8 a^2}-\frac {3 \int \left (c+a^2 c x^2\right )^3 \tan ^{-1}(a x)^2 \, dx}{8 a}\\ &=\frac {c^3 \left (1+a^2 x^2\right )^3 \tan ^{-1}(a x)}{56 a^2}-\frac {3 c^3 x \left (1+a^2 x^2\right )^3 \tan ^{-1}(a x)^2}{56 a}+\frac {c^3 \left (1+a^2 x^2\right )^4 \tan ^{-1}(a x)^3}{8 a^2}-\frac {c \int \left (c+a^2 c x^2\right )^2 \, dx}{56 a}-\frac {(9 c) \int \left (c+a^2 c x^2\right )^2 \tan ^{-1}(a x)^2 \, dx}{28 a}\\ &=\frac {9 c^3 \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)}{280 a^2}+\frac {c^3 \left (1+a^2 x^2\right )^3 \tan ^{-1}(a x)}{56 a^2}-\frac {9 c^3 x \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)^2}{140 a}-\frac {3 c^3 x \left (1+a^2 x^2\right )^3 \tan ^{-1}(a x)^2}{56 a}+\frac {c^3 \left (1+a^2 x^2\right )^4 \tan ^{-1}(a x)^3}{8 a^2}-\frac {c \int \left (c^2+2 a^2 c^2 x^2+a^4 c^2 x^4\right ) \, dx}{56 a}-\frac {\left (9 c^2\right ) \int \left (c+a^2 c x^2\right ) \, dx}{280 a}-\frac {\left (9 c^2\right ) \int \left (c+a^2 c x^2\right ) \tan ^{-1}(a x)^2 \, dx}{35 a}\\ &=-\frac {c^3 x}{20 a}-\frac {19}{840} a c^3 x^3-\frac {1}{280} a^3 c^3 x^5+\frac {3 c^3 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)}{35 a^2}+\frac {9 c^3 \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)}{280 a^2}+\frac {c^3 \left (1+a^2 x^2\right )^3 \tan ^{-1}(a x)}{56 a^2}-\frac {3 c^3 x \left (1+a^2 x^2\right ) \tan ^{-1}(a x)^2}{35 a}-\frac {9 c^3 x \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)^2}{140 a}-\frac {3 c^3 x \left (1+a^2 x^2\right )^3 \tan ^{-1}(a x)^2}{56 a}+\frac {c^3 \left (1+a^2 x^2\right )^4 \tan ^{-1}(a x)^3}{8 a^2}-\frac {\left (3 c^3\right ) \int 1 \, dx}{35 a}-\frac {\left (6 c^3\right ) \int \tan ^{-1}(a x)^2 \, dx}{35 a}\\ &=-\frac {19 c^3 x}{140 a}-\frac {19}{840} a c^3 x^3-\frac {1}{280} a^3 c^3 x^5+\frac {3 c^3 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)}{35 a^2}+\frac {9 c^3 \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)}{280 a^2}+\frac {c^3 \left (1+a^2 x^2\right )^3 \tan ^{-1}(a x)}{56 a^2}-\frac {6 c^3 x \tan ^{-1}(a x)^2}{35 a}-\frac {3 c^3 x \left (1+a^2 x^2\right ) \tan ^{-1}(a x)^2}{35 a}-\frac {9 c^3 x \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)^2}{140 a}-\frac {3 c^3 x \left (1+a^2 x^2\right )^3 \tan ^{-1}(a x)^2}{56 a}+\frac {c^3 \left (1+a^2 x^2\right )^4 \tan ^{-1}(a x)^3}{8 a^2}+\frac {1}{35} \left (12 c^3\right ) \int \frac {x \tan ^{-1}(a x)}{1+a^2 x^2} \, dx\\ &=-\frac {19 c^3 x}{140 a}-\frac {19}{840} a c^3 x^3-\frac {1}{280} a^3 c^3 x^5+\frac {3 c^3 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)}{35 a^2}+\frac {9 c^3 \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)}{280 a^2}+\frac {c^3 \left (1+a^2 x^2\right )^3 \tan ^{-1}(a x)}{56 a^2}-\frac {6 i c^3 \tan ^{-1}(a x)^2}{35 a^2}-\frac {6 c^3 x \tan ^{-1}(a x)^2}{35 a}-\frac {3 c^3 x \left (1+a^2 x^2\right ) \tan ^{-1}(a x)^2}{35 a}-\frac {9 c^3 x \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)^2}{140 a}-\frac {3 c^3 x \left (1+a^2 x^2\right )^3 \tan ^{-1}(a x)^2}{56 a}+\frac {c^3 \left (1+a^2 x^2\right )^4 \tan ^{-1}(a x)^3}{8 a^2}-\frac {\left (12 c^3\right ) \int \frac {\tan ^{-1}(a x)}{i-a x} \, dx}{35 a}\\ &=-\frac {19 c^3 x}{140 a}-\frac {19}{840} a c^3 x^3-\frac {1}{280} a^3 c^3 x^5+\frac {3 c^3 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)}{35 a^2}+\frac {9 c^3 \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)}{280 a^2}+\frac {c^3 \left (1+a^2 x^2\right )^3 \tan ^{-1}(a x)}{56 a^2}-\frac {6 i c^3 \tan ^{-1}(a x)^2}{35 a^2}-\frac {6 c^3 x \tan ^{-1}(a x)^2}{35 a}-\frac {3 c^3 x \left (1+a^2 x^2\right ) \tan ^{-1}(a x)^2}{35 a}-\frac {9 c^3 x \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)^2}{140 a}-\frac {3 c^3 x \left (1+a^2 x^2\right )^3 \tan ^{-1}(a x)^2}{56 a}+\frac {c^3 \left (1+a^2 x^2\right )^4 \tan ^{-1}(a x)^3}{8 a^2}-\frac {12 c^3 \tan ^{-1}(a x) \log \left (\frac {2}{1+i a x}\right )}{35 a^2}+\frac {\left (12 c^3\right ) \int \frac {\log \left (\frac {2}{1+i a x}\right )}{1+a^2 x^2} \, dx}{35 a}\\ &=-\frac {19 c^3 x}{140 a}-\frac {19}{840} a c^3 x^3-\frac {1}{280} a^3 c^3 x^5+\frac {3 c^3 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)}{35 a^2}+\frac {9 c^3 \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)}{280 a^2}+\frac {c^3 \left (1+a^2 x^2\right )^3 \tan ^{-1}(a x)}{56 a^2}-\frac {6 i c^3 \tan ^{-1}(a x)^2}{35 a^2}-\frac {6 c^3 x \tan ^{-1}(a x)^2}{35 a}-\frac {3 c^3 x \left (1+a^2 x^2\right ) \tan ^{-1}(a x)^2}{35 a}-\frac {9 c^3 x \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)^2}{140 a}-\frac {3 c^3 x \left (1+a^2 x^2\right )^3 \tan ^{-1}(a x)^2}{56 a}+\frac {c^3 \left (1+a^2 x^2\right )^4 \tan ^{-1}(a x)^3}{8 a^2}-\frac {12 c^3 \tan ^{-1}(a x) \log \left (\frac {2}{1+i a x}\right )}{35 a^2}-\frac {\left (12 i c^3\right ) \operatorname {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1+i a x}\right )}{35 a^2}\\ &=-\frac {19 c^3 x}{140 a}-\frac {19}{840} a c^3 x^3-\frac {1}{280} a^3 c^3 x^5+\frac {3 c^3 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)}{35 a^2}+\frac {9 c^3 \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)}{280 a^2}+\frac {c^3 \left (1+a^2 x^2\right )^3 \tan ^{-1}(a x)}{56 a^2}-\frac {6 i c^3 \tan ^{-1}(a x)^2}{35 a^2}-\frac {6 c^3 x \tan ^{-1}(a x)^2}{35 a}-\frac {3 c^3 x \left (1+a^2 x^2\right ) \tan ^{-1}(a x)^2}{35 a}-\frac {9 c^3 x \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)^2}{140 a}-\frac {3 c^3 x \left (1+a^2 x^2\right )^3 \tan ^{-1}(a x)^2}{56 a}+\frac {c^3 \left (1+a^2 x^2\right )^4 \tan ^{-1}(a x)^3}{8 a^2}-\frac {12 c^3 \tan ^{-1}(a x) \log \left (\frac {2}{1+i a x}\right )}{35 a^2}-\frac {6 i c^3 \text {Li}_2\left (1-\frac {2}{1+i a x}\right )}{35 a^2}\\ \end {align*}
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Mathematica [A] time = 1.44, size = 157, normalized size = 0.51 \[ \frac {c^3 \left (105 \left (a^2 x^2+1\right )^4 \tan ^{-1}(a x)^3-a x \left (3 a^4 x^4+19 a^2 x^2+114\right )-9 \left (5 a^7 x^7+21 a^5 x^5+35 a^3 x^3+35 a x-16 i\right ) \tan ^{-1}(a x)^2+3 \tan ^{-1}(a x) \left (5 a^6 x^6+24 a^4 x^4+57 a^2 x^2-96 \log \left (1+e^{2 i \tan ^{-1}(a x)}\right )+38\right )+144 i \text {Li}_2\left (-e^{2 i \tan ^{-1}(a x)}\right )\right )}{840 a^2} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.57, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (a^{6} c^{3} x^{7} + 3 \, a^{4} c^{3} x^{5} + 3 \, a^{2} c^{3} x^{3} + c^{3} x\right )} \arctan \left (a x\right )^{3}, x\right ) \]
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 \[ \mathit {sage}_{0} x \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.12, size = 428, normalized size = 1.39 \[ \frac {3 i c^{3} \dilog \left (\frac {i \left (a x -i\right )}{2}\right )}{35 a^{2}}-\frac {3 i c^{3} \dilog \left (-\frac {i \left (a x +i\right )}{2}\right )}{35 a^{2}}+\frac {3 i c^{3} \ln \left (a x +i\right )^{2}}{70 a^{2}}-\frac {3 i c^{3} \ln \left (a x -i\right )^{2}}{70 a^{2}}-\frac {9 a^{3} c^{3} \arctan \left (a x \right )^{2} x^{5}}{40}-\frac {3 a \,c^{3} \arctan \left (a x \right )^{2} x^{3}}{8}+\frac {a^{4} c^{3} \arctan \left (a x \right ) x^{6}}{56}+\frac {3 a^{2} c^{3} \arctan \left (a x \right ) x^{4}}{35}+\frac {6 c^{3} \arctan \left (a x \right ) \ln \left (a^{2} x^{2}+1\right )}{35 a^{2}}+\frac {3 a^{2} c^{3} \arctan \left (a x \right )^{3} x^{4}}{4}+\frac {a^{6} c^{3} \arctan \left (a x \right )^{3} x^{8}}{8}+\frac {a^{4} c^{3} \arctan \left (a x \right )^{3} x^{6}}{2}-\frac {3 a^{5} c^{3} \arctan \left (a x \right )^{2} x^{7}}{56}-\frac {a^{3} c^{3} x^{5}}{280}-\frac {19 c^{3} x}{140 a}-\frac {19 a \,c^{3} x^{3}}{840}-\frac {3 c^{3} x \arctan \left (a x \right )^{2}}{8 a}-\frac {3 i c^{3} \ln \left (a x -i\right ) \ln \left (-\frac {i \left (a x +i\right )}{2}\right )}{35 a^{2}}+\frac {3 i c^{3} \ln \left (a x -i\right ) \ln \left (a^{2} x^{2}+1\right )}{35 a^{2}}+\frac {3 i c^{3} \ln \left (a x +i\right ) \ln \left (\frac {i \left (a x -i\right )}{2}\right )}{35 a^{2}}-\frac {3 i c^{3} \ln \left (a x +i\right ) \ln \left (a^{2} x^{2}+1\right )}{35 a^{2}}+\frac {57 c^{3} \arctan \left (a x \right ) x^{2}}{280}+\frac {c^{3} \arctan \left (a x \right )^{3} x^{2}}{2}+\frac {c^{3} \arctan \left (a x \right )^{3}}{8 a^{2}}+\frac {19 c^{3} \arctan \left (a x \right )}{140 a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
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 \[ \int x\,{\mathrm {atan}\left (a\,x\right )}^3\,{\left (c\,a^2\,x^2+c\right )}^3 \,d x \]
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 \[ c^{3} \left (\int x \operatorname {atan}^{3}{\left (a x \right )}\, dx + \int 3 a^{2} x^{3} \operatorname {atan}^{3}{\left (a x \right )}\, dx + \int 3 a^{4} x^{5} \operatorname {atan}^{3}{\left (a x \right )}\, dx + \int a^{6} x^{7} \operatorname {atan}^{3}{\left (a x \right )}\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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