Optimal. Leaf size=225 \[ \frac {1}{7} a^4 c^2 x^7 \tan ^{-1}(a x)^2-\frac {8 i c^2 \text {Li}_2\left (1-\frac {2}{i a x+1}\right )}{105 a^3}-\frac {1}{21} a^3 c^2 x^6 \tan ^{-1}(a x)-\frac {8 i c^2 \tan ^{-1}(a x)^2}{105 a^3}+\frac {c^2 \tan ^{-1}(a x)}{210 a^3}-\frac {16 c^2 \log \left (\frac {2}{1+i a x}\right ) \tan ^{-1}(a x)}{105 a^3}+\frac {1}{105} a^2 c^2 x^5+\frac {2}{5} a^2 c^2 x^5 \tan ^{-1}(a x)^2-\frac {c^2 x}{210 a^2}-\frac {9}{70} a c^2 x^4 \tan ^{-1}(a x)+\frac {1}{3} c^2 x^3 \tan ^{-1}(a x)^2-\frac {8 c^2 x^2 \tan ^{-1}(a x)}{105 a}+\frac {17 c^2 x^3}{630} \]
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Rubi [A] time = 0.75, antiderivative size = 225, normalized size of antiderivative = 1.00, number of steps used = 44, number of rules used = 10, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.454, Rules used = {4948, 4852, 4916, 321, 203, 4920, 4854, 2402, 2315, 302} \[ -\frac {8 i c^2 \text {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )}{105 a^3}+\frac {1}{105} a^2 c^2 x^5+\frac {1}{7} a^4 c^2 x^7 \tan ^{-1}(a x)^2-\frac {1}{21} a^3 c^2 x^6 \tan ^{-1}(a x)+\frac {2}{5} a^2 c^2 x^5 \tan ^{-1}(a x)^2-\frac {c^2 x}{210 a^2}-\frac {8 i c^2 \tan ^{-1}(a x)^2}{105 a^3}+\frac {c^2 \tan ^{-1}(a x)}{210 a^3}-\frac {16 c^2 \log \left (\frac {2}{1+i a x}\right ) \tan ^{-1}(a x)}{105 a^3}-\frac {9}{70} a c^2 x^4 \tan ^{-1}(a x)+\frac {1}{3} c^2 x^3 \tan ^{-1}(a x)^2-\frac {8 c^2 x^2 \tan ^{-1}(a x)}{105 a}+\frac {17 c^2 x^3}{630} \]
Antiderivative was successfully verified.
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Rule 203
Rule 302
Rule 321
Rule 2315
Rule 2402
Rule 4852
Rule 4854
Rule 4916
Rule 4920
Rule 4948
Rubi steps
\begin {align*} \int x^2 \left (c+a^2 c x^2\right )^2 \tan ^{-1}(a x)^2 \, dx &=\int \left (c^2 x^2 \tan ^{-1}(a x)^2+2 a^2 c^2 x^4 \tan ^{-1}(a x)^2+a^4 c^2 x^6 \tan ^{-1}(a x)^2\right ) \, dx\\ &=c^2 \int x^2 \tan ^{-1}(a x)^2 \, dx+\left (2 a^2 c^2\right ) \int x^4 \tan ^{-1}(a x)^2 \, dx+\left (a^4 c^2\right ) \int x^6 \tan ^{-1}(a x)^2 \, dx\\ &=\frac {1}{3} c^2 x^3 \tan ^{-1}(a x)^2+\frac {2}{5} a^2 c^2 x^5 \tan ^{-1}(a x)^2+\frac {1}{7} a^4 c^2 x^7 \tan ^{-1}(a x)^2-\frac {1}{3} \left (2 a c^2\right ) \int \frac {x^3 \tan ^{-1}(a x)}{1+a^2 x^2} \, dx-\frac {1}{5} \left (4 a^3 c^2\right ) \int \frac {x^5 \tan ^{-1}(a x)}{1+a^2 x^2} \, dx-\frac {1}{7} \left (2 a^5 c^2\right ) \int \frac {x^7 \tan ^{-1}(a x)}{1+a^2 x^2} \, dx\\ &=\frac {1}{3} c^2 x^3 \tan ^{-1}(a x)^2+\frac {2}{5} a^2 c^2 x^5 \tan ^{-1}(a x)^2+\frac {1}{7} a^4 c^2 x^7 \tan ^{-1}(a x)^2-\frac {\left (2 c^2\right ) \int x \tan ^{-1}(a x) \, dx}{3 a}+\frac {\left (2 c^2\right ) \int \frac {x \tan ^{-1}(a x)}{1+a^2 x^2} \, dx}{3 a}-\frac {1}{5} \left (4 a c^2\right ) \int x^3 \tan ^{-1}(a x) \, dx+\frac {1}{5} \left (4 a c^2\right ) \int \frac {x^3 \tan ^{-1}(a x)}{1+a^2 x^2} \, dx-\frac {1}{7} \left (2 a^3 c^2\right ) \int x^5 \tan ^{-1}(a x) \, dx+\frac {1}{7} \left (2 a^3 c^2\right ) \int \frac {x^5 \tan ^{-1}(a x)}{1+a^2 x^2} \, dx\\ &=-\frac {c^2 x^2 \tan ^{-1}(a x)}{3 a}-\frac {1}{5} a c^2 x^4 \tan ^{-1}(a x)-\frac {1}{21} a^3 c^2 x^6 \tan ^{-1}(a x)-\frac {i c^2 \tan ^{-1}(a x)^2}{3 a^3}+\frac {1}{3} c^2 x^3 \tan ^{-1}(a x)^2+\frac {2}{5} a^2 c^2 x^5 \tan ^{-1}(a x)^2+\frac {1}{7} a^4 c^2 x^7 \tan ^{-1}(a x)^2+\frac {1}{3} c^2 \int \frac {x^2}{1+a^2 x^2} \, dx-\frac {\left (2 c^2\right ) \int \frac {\tan ^{-1}(a x)}{i-a x} \, dx}{3 a^2}+\frac {\left (4 c^2\right ) \int x \tan ^{-1}(a x) \, dx}{5 a}-\frac {\left (4 c^2\right ) \int \frac {x \tan ^{-1}(a x)}{1+a^2 x^2} \, dx}{5 a}+\frac {1}{7} \left (2 a c^2\right ) \int x^3 \tan ^{-1}(a x) \, dx-\frac {1}{7} \left (2 a c^2\right ) \int \frac {x^3 \tan ^{-1}(a x)}{1+a^2 x^2} \, dx+\frac {1}{5} \left (a^2 c^2\right ) \int \frac {x^4}{1+a^2 x^2} \, dx+\frac {1}{21} \left (a^4 c^2\right ) \int \frac {x^6}{1+a^2 x^2} \, dx\\ &=\frac {c^2 x}{3 a^2}+\frac {c^2 x^2 \tan ^{-1}(a x)}{15 a}-\frac {9}{70} a c^2 x^4 \tan ^{-1}(a x)-\frac {1}{21} a^3 c^2 x^6 \tan ^{-1}(a x)+\frac {i c^2 \tan ^{-1}(a x)^2}{15 a^3}+\frac {1}{3} c^2 x^3 \tan ^{-1}(a x)^2+\frac {2}{5} a^2 c^2 x^5 \tan ^{-1}(a x)^2+\frac {1}{7} a^4 c^2 x^7 \tan ^{-1}(a x)^2-\frac {2 c^2 \tan ^{-1}(a x) \log \left (\frac {2}{1+i a x}\right )}{3 a^3}-\frac {1}{5} \left (2 c^2\right ) \int \frac {x^2}{1+a^2 x^2} \, dx-\frac {c^2 \int \frac {1}{1+a^2 x^2} \, dx}{3 a^2}+\frac {\left (2 c^2\right ) \int \frac {\log \left (\frac {2}{1+i a x}\right )}{1+a^2 x^2} \, dx}{3 a^2}+\frac {\left (4 c^2\right ) \int \frac {\tan ^{-1}(a x)}{i-a x} \, dx}{5 a^2}-\frac {\left (2 c^2\right ) \int x \tan ^{-1}(a x) \, dx}{7 a}+\frac {\left (2 c^2\right ) \int \frac {x \tan ^{-1}(a x)}{1+a^2 x^2} \, dx}{7 a}-\frac {1}{14} \left (a^2 c^2\right ) \int \frac {x^4}{1+a^2 x^2} \, dx+\frac {1}{5} \left (a^2 c^2\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 {1}{21} \left (a^4 c^2\right ) \int \left (\frac {1}{a^6}-\frac {x^2}{a^4}+\frac {x^4}{a^2}-\frac {1}{a^6 \left (1+a^2 x^2\right )}\right ) \, dx\\ &=-\frac {23 c^2 x}{105 a^2}+\frac {16 c^2 x^3}{315}+\frac {1}{105} a^2 c^2 x^5-\frac {c^2 \tan ^{-1}(a x)}{3 a^3}-\frac {8 c^2 x^2 \tan ^{-1}(a x)}{105 a}-\frac {9}{70} a c^2 x^4 \tan ^{-1}(a x)-\frac {1}{21} a^3 c^2 x^6 \tan ^{-1}(a x)-\frac {8 i c^2 \tan ^{-1}(a x)^2}{105 a^3}+\frac {1}{3} c^2 x^3 \tan ^{-1}(a x)^2+\frac {2}{5} a^2 c^2 x^5 \tan ^{-1}(a x)^2+\frac {1}{7} a^4 c^2 x^7 \tan ^{-1}(a x)^2+\frac {2 c^2 \tan ^{-1}(a x) \log \left (\frac {2}{1+i a x}\right )}{15 a^3}+\frac {1}{7} c^2 \int \frac {x^2}{1+a^2 x^2} \, dx-\frac {\left (2 i c^2\right ) \operatorname {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1+i a x}\right )}{3 a^3}-\frac {c^2 \int \frac {1}{1+a^2 x^2} \, dx}{21 a^2}+\frac {c^2 \int \frac {1}{1+a^2 x^2} \, dx}{5 a^2}-\frac {\left (2 c^2\right ) \int \frac {\tan ^{-1}(a x)}{i-a x} \, dx}{7 a^2}+\frac {\left (2 c^2\right ) \int \frac {1}{1+a^2 x^2} \, dx}{5 a^2}-\frac {\left (4 c^2\right ) \int \frac {\log \left (\frac {2}{1+i a x}\right )}{1+a^2 x^2} \, dx}{5 a^2}-\frac {1}{14} \left (a^2 c^2\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^2 x}{210 a^2}+\frac {17 c^2 x^3}{630}+\frac {1}{105} a^2 c^2 x^5+\frac {23 c^2 \tan ^{-1}(a x)}{105 a^3}-\frac {8 c^2 x^2 \tan ^{-1}(a x)}{105 a}-\frac {9}{70} a c^2 x^4 \tan ^{-1}(a x)-\frac {1}{21} a^3 c^2 x^6 \tan ^{-1}(a x)-\frac {8 i c^2 \tan ^{-1}(a x)^2}{105 a^3}+\frac {1}{3} c^2 x^3 \tan ^{-1}(a x)^2+\frac {2}{5} a^2 c^2 x^5 \tan ^{-1}(a x)^2+\frac {1}{7} a^4 c^2 x^7 \tan ^{-1}(a x)^2-\frac {16 c^2 \tan ^{-1}(a x) \log \left (\frac {2}{1+i a x}\right )}{105 a^3}-\frac {i c^2 \text {Li}_2\left (1-\frac {2}{1+i a x}\right )}{3 a^3}+\frac {\left (4 i c^2\right ) \operatorname {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1+i a x}\right )}{5 a^3}-\frac {c^2 \int \frac {1}{1+a^2 x^2} \, dx}{14 a^2}-\frac {c^2 \int \frac {1}{1+a^2 x^2} \, dx}{7 a^2}+\frac {\left (2 c^2\right ) \int \frac {\log \left (\frac {2}{1+i a x}\right )}{1+a^2 x^2} \, dx}{7 a^2}\\ &=-\frac {c^2 x}{210 a^2}+\frac {17 c^2 x^3}{630}+\frac {1}{105} a^2 c^2 x^5+\frac {c^2 \tan ^{-1}(a x)}{210 a^3}-\frac {8 c^2 x^2 \tan ^{-1}(a x)}{105 a}-\frac {9}{70} a c^2 x^4 \tan ^{-1}(a x)-\frac {1}{21} a^3 c^2 x^6 \tan ^{-1}(a x)-\frac {8 i c^2 \tan ^{-1}(a x)^2}{105 a^3}+\frac {1}{3} c^2 x^3 \tan ^{-1}(a x)^2+\frac {2}{5} a^2 c^2 x^5 \tan ^{-1}(a x)^2+\frac {1}{7} a^4 c^2 x^7 \tan ^{-1}(a x)^2-\frac {16 c^2 \tan ^{-1}(a x) \log \left (\frac {2}{1+i a x}\right )}{105 a^3}+\frac {i c^2 \text {Li}_2\left (1-\frac {2}{1+i a x}\right )}{15 a^3}-\frac {\left (2 i c^2\right ) \operatorname {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1+i a x}\right )}{7 a^3}\\ &=-\frac {c^2 x}{210 a^2}+\frac {17 c^2 x^3}{630}+\frac {1}{105} a^2 c^2 x^5+\frac {c^2 \tan ^{-1}(a x)}{210 a^3}-\frac {8 c^2 x^2 \tan ^{-1}(a x)}{105 a}-\frac {9}{70} a c^2 x^4 \tan ^{-1}(a x)-\frac {1}{21} a^3 c^2 x^6 \tan ^{-1}(a x)-\frac {8 i c^2 \tan ^{-1}(a x)^2}{105 a^3}+\frac {1}{3} c^2 x^3 \tan ^{-1}(a x)^2+\frac {2}{5} a^2 c^2 x^5 \tan ^{-1}(a x)^2+\frac {1}{7} a^4 c^2 x^7 \tan ^{-1}(a x)^2-\frac {16 c^2 \tan ^{-1}(a x) \log \left (\frac {2}{1+i a x}\right )}{105 a^3}-\frac {8 i c^2 \text {Li}_2\left (1-\frac {2}{1+i a x}\right )}{105 a^3}\\ \end {align*}
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Mathematica [A] time = 1.30, size = 133, normalized size = 0.59 \[ \frac {c^2 \left (a x \left (6 a^4 x^4+17 a^2 x^2-3\right )+6 \left (15 a^7 x^7+42 a^5 x^5+35 a^3 x^3+8 i\right ) \tan ^{-1}(a x)^2-3 \tan ^{-1}(a x) \left (10 a^6 x^6+27 a^4 x^4+16 a^2 x^2+32 \log \left (1+e^{2 i \tan ^{-1}(a x)}\right )-1\right )+48 i \text {Li}_2\left (-e^{2 i \tan ^{-1}(a x)}\right )\right )}{630 a^3} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.61, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (a^{4} c^{2} x^{6} + 2 \, a^{2} c^{2} x^{4} + c^{2} x^{2}\right )} \arctan \left (a x\right )^{2}, x\right ) \]
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 = 0.10, size = 333, normalized size = 1.48 \[ \frac {a^{4} c^{2} x^{7} \arctan \left (a x \right )^{2}}{7}+\frac {2 a^{2} c^{2} x^{5} \arctan \left (a x \right )^{2}}{5}+\frac {c^{2} x^{3} \arctan \left (a x \right )^{2}}{3}-\frac {a^{3} c^{2} x^{6} \arctan \left (a x \right )}{21}-\frac {9 a \,c^{2} x^{4} \arctan \left (a x \right )}{70}-\frac {8 c^{2} x^{2} \arctan \left (a x \right )}{105 a}+\frac {8 c^{2} \arctan \left (a x \right ) \ln \left (a^{2} x^{2}+1\right )}{105 a^{3}}+\frac {a^{2} c^{2} x^{5}}{105}+\frac {17 c^{2} x^{3}}{630}-\frac {c^{2} x}{210 a^{2}}+\frac {c^{2} \arctan \left (a x \right )}{210 a^{3}}+\frac {4 i c^{2} \ln \left (a x -i\right ) \ln \left (a^{2} x^{2}+1\right )}{105 a^{3}}+\frac {2 i c^{2} \ln \left (a x +i\right )^{2}}{105 a^{3}}-\frac {4 i c^{2} \ln \left (a x -i\right ) \ln \left (-\frac {i \left (a x +i\right )}{2}\right )}{105 a^{3}}+\frac {4 i c^{2} \dilog \left (\frac {i \left (a x -i\right )}{2}\right )}{105 a^{3}}-\frac {4 i c^{2} \dilog \left (-\frac {i \left (a x +i\right )}{2}\right )}{105 a^{3}}+\frac {4 i c^{2} \ln \left (a x +i\right ) \ln \left (\frac {i \left (a x -i\right )}{2}\right )}{105 a^{3}}-\frac {4 i c^{2} \ln \left (a x +i\right ) \ln \left (a^{2} x^{2}+1\right )}{105 a^{3}}-\frac {2 i c^{2} \ln \left (a x -i\right )^{2}}{105 a^{3}} \]
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 \[ \frac {1}{420} \, {\left (15 \, a^{4} c^{2} x^{7} + 42 \, a^{2} c^{2} x^{5} + 35 \, c^{2} x^{3}\right )} \arctan \left (a x\right )^{2} - \frac {1}{1680} \, {\left (15 \, a^{4} c^{2} x^{7} + 42 \, a^{2} c^{2} x^{5} + 35 \, c^{2} x^{3}\right )} \log \left (a^{2} x^{2} + 1\right )^{2} + \int \frac {1260 \, {\left (a^{6} c^{2} x^{8} + 3 \, a^{4} c^{2} x^{6} + 3 \, a^{2} c^{2} x^{4} + c^{2} x^{2}\right )} \arctan \left (a x\right )^{2} + 105 \, {\left (a^{6} c^{2} x^{8} + 3 \, a^{4} c^{2} x^{6} + 3 \, a^{2} c^{2} x^{4} + c^{2} x^{2}\right )} \log \left (a^{2} x^{2} + 1\right )^{2} - 8 \, {\left (15 \, a^{5} c^{2} x^{7} + 42 \, a^{3} c^{2} x^{5} + 35 \, a c^{2} x^{3}\right )} \arctan \left (a x\right ) + 4 \, {\left (15 \, a^{6} c^{2} x^{8} + 42 \, a^{4} c^{2} x^{6} + 35 \, a^{2} c^{2} x^{4}\right )} \log \left (a^{2} x^{2} + 1\right )}{1680 \, {\left (a^{2} x^{2} + 1\right )}}\,{d x} \]
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 \[ \int x^2\,{\mathrm {atan}\left (a\,x\right )}^2\,{\left (c\,a^2\,x^2+c\right )}^2 \,d x \]
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 \[ c^{2} \left (\int x^{2} \operatorname {atan}^{2}{\left (a x \right )}\, dx + \int 2 a^{2} x^{4} \operatorname {atan}^{2}{\left (a x \right )}\, dx + \int a^{4} x^{6} \operatorname {atan}^{2}{\left (a x \right )}\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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