Optimal. Leaf size=105 \[ \frac {1}{4} i c d x^4 \left (a+b \tan ^{-1}(c x)\right )+\frac {1}{3} d x^3 \left (a+b \tan ^{-1}(c x)\right )-\frac {i b d \tan ^{-1}(c x)}{4 c^3}+\frac {i b d x}{4 c^2}+\frac {b d \log \left (c^2 x^2+1\right )}{6 c^3}-\frac {b d x^2}{6 c}-\frac {1}{12} i b d x^3 \]
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Rubi [A] time = 0.09, antiderivative size = 105, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {43, 4872, 12, 801, 635, 203, 260} \[ \frac {1}{4} i c d x^4 \left (a+b \tan ^{-1}(c x)\right )+\frac {1}{3} d x^3 \left (a+b \tan ^{-1}(c x)\right )+\frac {b d \log \left (c^2 x^2+1\right )}{6 c^3}+\frac {i b d x}{4 c^2}-\frac {i b d \tan ^{-1}(c x)}{4 c^3}-\frac {b d x^2}{6 c}-\frac {1}{12} i b d x^3 \]
Antiderivative was successfully verified.
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Rule 12
Rule 43
Rule 203
Rule 260
Rule 635
Rule 801
Rule 4872
Rubi steps
\begin {align*} \int x^2 (d+i c d x) \left (a+b \tan ^{-1}(c x)\right ) \, dx &=\frac {1}{3} d x^3 \left (a+b \tan ^{-1}(c x)\right )+\frac {1}{4} i c d x^4 \left (a+b \tan ^{-1}(c x)\right )-(b c) \int \frac {d x^3 (4+3 i c x)}{12 \left (1+c^2 x^2\right )} \, dx\\ &=\frac {1}{3} d x^3 \left (a+b \tan ^{-1}(c x)\right )+\frac {1}{4} i c d x^4 \left (a+b \tan ^{-1}(c x)\right )-\frac {1}{12} (b c d) \int \frac {x^3 (4+3 i c x)}{1+c^2 x^2} \, dx\\ &=\frac {1}{3} d x^3 \left (a+b \tan ^{-1}(c x)\right )+\frac {1}{4} i c d x^4 \left (a+b \tan ^{-1}(c x)\right )-\frac {1}{12} (b c d) \int \left (-\frac {3 i}{c^3}+\frac {4 x}{c^2}+\frac {3 i x^2}{c}+\frac {3 i-4 c x}{c^3 \left (1+c^2 x^2\right )}\right ) \, dx\\ &=\frac {i b d x}{4 c^2}-\frac {b d x^2}{6 c}-\frac {1}{12} i b d x^3+\frac {1}{3} d x^3 \left (a+b \tan ^{-1}(c x)\right )+\frac {1}{4} i c d x^4 \left (a+b \tan ^{-1}(c x)\right )-\frac {(b d) \int \frac {3 i-4 c x}{1+c^2 x^2} \, dx}{12 c^2}\\ &=\frac {i b d x}{4 c^2}-\frac {b d x^2}{6 c}-\frac {1}{12} i b d x^3+\frac {1}{3} d x^3 \left (a+b \tan ^{-1}(c x)\right )+\frac {1}{4} i c d x^4 \left (a+b \tan ^{-1}(c x)\right )-\frac {(i b d) \int \frac {1}{1+c^2 x^2} \, dx}{4 c^2}+\frac {(b d) \int \frac {x}{1+c^2 x^2} \, dx}{3 c}\\ &=\frac {i b d x}{4 c^2}-\frac {b d x^2}{6 c}-\frac {1}{12} i b d x^3-\frac {i b d \tan ^{-1}(c x)}{4 c^3}+\frac {1}{3} d x^3 \left (a+b \tan ^{-1}(c x)\right )+\frac {1}{4} i c d x^4 \left (a+b \tan ^{-1}(c x)\right )+\frac {b d \log \left (1+c^2 x^2\right )}{6 c^3}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 88, normalized size = 0.84 \[ \frac {d \left (a c^3 x^3 (4+3 i c x)+b c x \left (-i c^2 x^2-2 c x+3 i\right )+2 b \log \left (c^2 x^2+1\right )+b \left (3 i c^4 x^4+4 c^3 x^3-3 i\right ) \tan ^{-1}(c x)\right )}{12 c^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.87, size = 113, normalized size = 1.08 \[ \frac {6 i \, a c^{4} d x^{4} + 2 \, {\left (4 \, a - i \, b\right )} c^{3} d x^{3} - 4 \, b c^{2} d x^{2} + 6 i \, b c d x + 7 \, b d \log \left (\frac {c x + i}{c}\right ) + b d \log \left (\frac {c x - i}{c}\right ) - {\left (3 \, b c^{4} d x^{4} - 4 i \, b c^{3} d x^{3}\right )} \log \left (-\frac {c x + i}{c x - i}\right )}{24 \, c^{3}} \]
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 = 0.03, size = 98, normalized size = 0.93 \[ \frac {i c d a \,x^{4}}{4}+\frac {d a \,x^{3}}{3}+\frac {i c d b \arctan \left (c x \right ) x^{4}}{4}+\frac {d b \arctan \left (c x \right ) x^{3}}{3}+\frac {i b d x}{4 c^{2}}-\frac {i b d \,x^{3}}{12}-\frac {b d \,x^{2}}{6 c}+\frac {b d \ln \left (c^{2} x^{2}+1\right )}{6 c^{3}}-\frac {i b d \arctan \left (c x \right )}{4 c^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.41, size = 99, normalized size = 0.94 \[ \frac {1}{4} i \, a c d x^{4} + \frac {1}{3} \, a d x^{3} + \frac {1}{12} i \, {\left (3 \, x^{4} \arctan \left (c x\right ) - c {\left (\frac {c^{2} x^{3} - 3 \, x}{c^{4}} + \frac {3 \, \arctan \left (c x\right )}{c^{5}}\right )}\right )} b c d + \frac {1}{6} \, {\left (2 \, x^{3} \arctan \left (c x\right ) - c {\left (\frac {x^{2}}{c^{2}} - \frac {\log \left (c^{2} x^{2} + 1\right )}{c^{4}}\right )}\right )} b d \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.71, size = 99, normalized size = 0.94 \[ -\frac {\frac {d\,\left (-2\,b\,\ln \left (c^2\,x^2+1\right )+b\,\mathrm {atan}\left (c\,x\right )\,3{}\mathrm {i}\right )}{12}+\frac {b\,c^2\,d\,x^2}{6}-\frac {b\,c\,d\,x\,1{}\mathrm {i}}{4}}{c^3}+\frac {d\,\left (4\,a\,x^3+4\,b\,x^3\,\mathrm {atan}\left (c\,x\right )-b\,x^3\,1{}\mathrm {i}\right )}{12}+\frac {c\,d\,\left (a\,x^4\,3{}\mathrm {i}+b\,x^4\,\mathrm {atan}\left (c\,x\right )\,3{}\mathrm {i}\right )}{12} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 3.08, size = 167, normalized size = 1.59 \[ \frac {i a c d x^{4}}{4} - \frac {b d x^{2}}{6 c} + \frac {i b d x}{4 c^{2}} + \frac {b d \left (\frac {\log {\left (11 b c d x - 11 i b d \right )}}{24} + \frac {9 \log {\left (11 b c d x + 11 i b d \right )}}{40}\right )}{c^{3}} + x^{3} \left (\frac {a d}{3} - \frac {i b d}{12}\right ) + \left (\frac {b c d x^{4}}{8} - \frac {i b d x^{3}}{6}\right ) \log {\left (i c x + 1 \right )} - \frac {\left (15 b c^{4} d x^{4} - 20 i b c^{3} d x^{3} - 8 b d\right ) \log {\left (- i c x + 1 \right )}}{120 c^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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