Optimal. Leaf size=194 \[ \frac {2 b^2 \log \left (\frac {2}{1+i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )}{c^4}+\frac {b^2 x^2 \left (a+b \tan ^{-1}(c x)\right )}{4 c^2}-\frac {\left (a+b \tan ^{-1}(c x)\right )^3}{4 c^4}+\frac {i b \left (a+b \tan ^{-1}(c x)\right )^2}{c^4}+\frac {3 b x \left (a+b \tan ^{-1}(c x)\right )^2}{4 c^3}+\frac {1}{4} x^4 \left (a+b \tan ^{-1}(c x)\right )^3-\frac {b x^3 \left (a+b \tan ^{-1}(c x)\right )^2}{4 c}+\frac {i b^3 \text {Li}_2\left (1-\frac {2}{i c x+1}\right )}{c^4}+\frac {b^3 \tan ^{-1}(c x)}{4 c^4}-\frac {b^3 x}{4 c^3} \]
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Rubi [A] time = 0.55, antiderivative size = 194, normalized size of antiderivative = 1.00, number of steps used = 18, number of rules used = 10, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.714, Rules used = {4852, 4916, 321, 203, 4920, 4854, 2402, 2315, 4846, 4884} \[ \frac {i b^3 \text {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )}{c^4}+\frac {b^2 x^2 \left (a+b \tan ^{-1}(c x)\right )}{4 c^2}+\frac {2 b^2 \log \left (\frac {2}{1+i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )}{c^4}+\frac {3 b x \left (a+b \tan ^{-1}(c x)\right )^2}{4 c^3}-\frac {\left (a+b \tan ^{-1}(c x)\right )^3}{4 c^4}+\frac {i b \left (a+b \tan ^{-1}(c x)\right )^2}{c^4}+\frac {1}{4} x^4 \left (a+b \tan ^{-1}(c x)\right )^3-\frac {b x^3 \left (a+b \tan ^{-1}(c x)\right )^2}{4 c}-\frac {b^3 x}{4 c^3}+\frac {b^3 \tan ^{-1}(c x)}{4 c^4} \]
Antiderivative was successfully verified.
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Rule 203
Rule 321
Rule 2315
Rule 2402
Rule 4846
Rule 4852
Rule 4854
Rule 4884
Rule 4916
Rule 4920
Rubi steps
\begin {align*} \int x^3 \left (a+b \tan ^{-1}(c x)\right )^3 \, dx &=\frac {1}{4} x^4 \left (a+b \tan ^{-1}(c x)\right )^3-\frac {1}{4} (3 b c) \int \frac {x^4 \left (a+b \tan ^{-1}(c x)\right )^2}{1+c^2 x^2} \, dx\\ &=\frac {1}{4} x^4 \left (a+b \tan ^{-1}(c x)\right )^3-\frac {(3 b) \int x^2 \left (a+b \tan ^{-1}(c x)\right )^2 \, dx}{4 c}+\frac {(3 b) \int \frac {x^2 \left (a+b \tan ^{-1}(c x)\right )^2}{1+c^2 x^2} \, dx}{4 c}\\ &=-\frac {b x^3 \left (a+b \tan ^{-1}(c x)\right )^2}{4 c}+\frac {1}{4} x^4 \left (a+b \tan ^{-1}(c x)\right )^3+\frac {1}{2} b^2 \int \frac {x^3 \left (a+b \tan ^{-1}(c x)\right )}{1+c^2 x^2} \, dx+\frac {(3 b) \int \left (a+b \tan ^{-1}(c x)\right )^2 \, dx}{4 c^3}-\frac {(3 b) \int \frac {\left (a+b \tan ^{-1}(c x)\right )^2}{1+c^2 x^2} \, dx}{4 c^3}\\ &=\frac {3 b x \left (a+b \tan ^{-1}(c x)\right )^2}{4 c^3}-\frac {b x^3 \left (a+b \tan ^{-1}(c x)\right )^2}{4 c}-\frac {\left (a+b \tan ^{-1}(c x)\right )^3}{4 c^4}+\frac {1}{4} x^4 \left (a+b \tan ^{-1}(c x)\right )^3+\frac {b^2 \int x \left (a+b \tan ^{-1}(c x)\right ) \, dx}{2 c^2}-\frac {b^2 \int \frac {x \left (a+b \tan ^{-1}(c x)\right )}{1+c^2 x^2} \, dx}{2 c^2}-\frac {\left (3 b^2\right ) \int \frac {x \left (a+b \tan ^{-1}(c x)\right )}{1+c^2 x^2} \, dx}{2 c^2}\\ &=\frac {b^2 x^2 \left (a+b \tan ^{-1}(c x)\right )}{4 c^2}+\frac {i b \left (a+b \tan ^{-1}(c x)\right )^2}{c^4}+\frac {3 b x \left (a+b \tan ^{-1}(c x)\right )^2}{4 c^3}-\frac {b x^3 \left (a+b \tan ^{-1}(c x)\right )^2}{4 c}-\frac {\left (a+b \tan ^{-1}(c x)\right )^3}{4 c^4}+\frac {1}{4} x^4 \left (a+b \tan ^{-1}(c x)\right )^3+\frac {b^2 \int \frac {a+b \tan ^{-1}(c x)}{i-c x} \, dx}{2 c^3}+\frac {\left (3 b^2\right ) \int \frac {a+b \tan ^{-1}(c x)}{i-c x} \, dx}{2 c^3}-\frac {b^3 \int \frac {x^2}{1+c^2 x^2} \, dx}{4 c}\\ &=-\frac {b^3 x}{4 c^3}+\frac {b^2 x^2 \left (a+b \tan ^{-1}(c x)\right )}{4 c^2}+\frac {i b \left (a+b \tan ^{-1}(c x)\right )^2}{c^4}+\frac {3 b x \left (a+b \tan ^{-1}(c x)\right )^2}{4 c^3}-\frac {b x^3 \left (a+b \tan ^{-1}(c x)\right )^2}{4 c}-\frac {\left (a+b \tan ^{-1}(c x)\right )^3}{4 c^4}+\frac {1}{4} x^4 \left (a+b \tan ^{-1}(c x)\right )^3+\frac {2 b^2 \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2}{1+i c x}\right )}{c^4}+\frac {b^3 \int \frac {1}{1+c^2 x^2} \, dx}{4 c^3}-\frac {b^3 \int \frac {\log \left (\frac {2}{1+i c x}\right )}{1+c^2 x^2} \, dx}{2 c^3}-\frac {\left (3 b^3\right ) \int \frac {\log \left (\frac {2}{1+i c x}\right )}{1+c^2 x^2} \, dx}{2 c^3}\\ &=-\frac {b^3 x}{4 c^3}+\frac {b^3 \tan ^{-1}(c x)}{4 c^4}+\frac {b^2 x^2 \left (a+b \tan ^{-1}(c x)\right )}{4 c^2}+\frac {i b \left (a+b \tan ^{-1}(c x)\right )^2}{c^4}+\frac {3 b x \left (a+b \tan ^{-1}(c x)\right )^2}{4 c^3}-\frac {b x^3 \left (a+b \tan ^{-1}(c x)\right )^2}{4 c}-\frac {\left (a+b \tan ^{-1}(c x)\right )^3}{4 c^4}+\frac {1}{4} x^4 \left (a+b \tan ^{-1}(c x)\right )^3+\frac {2 b^2 \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2}{1+i c x}\right )}{c^4}+\frac {\left (i b^3\right ) \operatorname {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1+i c x}\right )}{2 c^4}+\frac {\left (3 i b^3\right ) \operatorname {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1+i c x}\right )}{2 c^4}\\ &=-\frac {b^3 x}{4 c^3}+\frac {b^3 \tan ^{-1}(c x)}{4 c^4}+\frac {b^2 x^2 \left (a+b \tan ^{-1}(c x)\right )}{4 c^2}+\frac {i b \left (a+b \tan ^{-1}(c x)\right )^2}{c^4}+\frac {3 b x \left (a+b \tan ^{-1}(c x)\right )^2}{4 c^3}-\frac {b x^3 \left (a+b \tan ^{-1}(c x)\right )^2}{4 c}-\frac {\left (a+b \tan ^{-1}(c x)\right )^3}{4 c^4}+\frac {1}{4} x^4 \left (a+b \tan ^{-1}(c x)\right )^3+\frac {2 b^2 \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2}{1+i c x}\right )}{c^4}+\frac {i b^3 \text {Li}_2\left (1-\frac {2}{1+i c x}\right )}{c^4}\\ \end {align*}
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Mathematica [A] time = 0.56, size = 225, normalized size = 1.16 \[ \frac {a^3 c^4 x^4+b \tan ^{-1}(c x) \left (3 a^2 \left (c^4 x^4-1\right )-2 a b c x \left (c^2 x^2-3\right )+b^2 \left (c^2 x^2+1\right )+8 b^2 \log \left (1+e^{2 i \tan ^{-1}(c x)}\right )\right )-a^2 b c^3 x^3+3 a^2 b c x+a b^2 c^2 x^2-4 a b^2 \log \left (c^2 x^2+1\right )-b^2 \tan ^{-1}(c x)^2 \left (a \left (3-3 c^4 x^4\right )+b \left (c^3 x^3-3 c x+4 i\right )\right )+a b^2+b^3 \left (c^4 x^4-1\right ) \tan ^{-1}(c x)^3-4 i b^3 \text {Li}_2\left (-e^{2 i \tan ^{-1}(c x)}\right )-b^3 c x}{4 c^4} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.47, size = 0, normalized size = 0.00 \[ {\rm integral}\left (b^{3} x^{3} \arctan \left (c x\right )^{3} + 3 \, a b^{2} x^{3} \arctan \left (c x\right )^{2} + 3 \, a^{2} b x^{3} \arctan \left (c x\right ) + a^{3} x^{3}, x\right ) \]
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 [B] time = 0.02, size = 445, normalized size = 2.29 \[ \frac {i b^{3} \ln \left (c x -i\right )^{2}}{4 c^{4}}-\frac {i b^{3} \dilog \left (\frac {i \left (c x -i\right )}{2}\right )}{2 c^{4}}+\frac {i b^{3} \dilog \left (-\frac {i \left (c x +i\right )}{2}\right )}{2 c^{4}}-\frac {i b^{3} \ln \left (c x +i\right )^{2}}{4 c^{4}}-\frac {b^{3} \arctan \left (c x \right ) \ln \left (c^{2} x^{2}+1\right )}{c^{4}}-\frac {b^{3} \arctan \left (c x \right )^{2} x^{3}}{4 c}+\frac {3 b^{3} \arctan \left (c x \right )^{2} x}{4 c^{3}}+\frac {b^{3} \arctan \left (c x \right ) x^{2}}{4 c^{2}}-\frac {a^{2} b \,x^{3}}{4 c}+\frac {x^{2} a \,b^{2}}{4 c^{2}}+\frac {3 x \,a^{2} b}{4 c^{3}}-\frac {a \,b^{2} \ln \left (c^{2} x^{2}+1\right )}{c^{4}}-\frac {3 a^{2} b \arctan \left (c x \right )}{4 c^{4}}+\frac {3 x^{4} a^{2} b \arctan \left (c x \right )}{4}+\frac {3 a \,b^{2} x^{4} \arctan \left (c x \right )^{2}}{4}-\frac {3 a \,b^{2} \arctan \left (c x \right )^{2}}{4 c^{4}}-\frac {b^{3} x}{4 c^{3}}+\frac {b^{3} \arctan \left (c x \right )}{4 c^{4}}+\frac {x^{4} a^{3}}{4}+\frac {b^{3} x^{4} \arctan \left (c x \right )^{3}}{4}-\frac {b^{3} \arctan \left (c x \right )^{3}}{4 c^{4}}-\frac {i b^{3} \ln \left (c x -i\right ) \ln \left (c^{2} x^{2}+1\right )}{2 c^{4}}+\frac {i b^{3} \ln \left (c x +i\right ) \ln \left (c^{2} x^{2}+1\right )}{2 c^{4}}-\frac {i b^{3} \ln \left (c x +i\right ) \ln \left (\frac {i \left (c x -i\right )}{2}\right )}{2 c^{4}}+\frac {i b^{3} \ln \left (c x -i\right ) \ln \left (-\frac {i \left (c x +i\right )}{2}\right )}{2 c^{4}}-\frac {a \,b^{2} x^{3} \arctan \left (c x \right )}{2 c}+\frac {3 a \,b^{2} x \arctan \left (c x \right )}{2 c^{3}} \]
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 = -0.01 \[ \int x^3\,{\left (a+b\,\mathrm {atan}\left (c\,x\right )\right )}^3 \,d x \]
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 \[ \int x^{3} \left (a + b \operatorname {atan}{\left (c x \right )}\right )^{3}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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