Optimal. Leaf size=191 \[ -\frac {1}{6} i c^3 d^3 x^6 \left (a+b \tan ^{-1}(c x)\right )-\frac {3}{5} c^2 d^3 x^5 \left (a+b \tan ^{-1}(c x)\right )+\frac {3}{4} i c d^3 x^4 \left (a+b \tan ^{-1}(c x)\right )+\frac {1}{3} d^3 x^3 \left (a+b \tan ^{-1}(c x)\right )-\frac {11 i b d^3 \tan ^{-1}(c x)}{12 c^3}+\frac {1}{30} i b c^2 d^3 x^5+\frac {11 i b d^3 x}{12 c^2}+\frac {7 b d^3 \log \left (c^2 x^2+1\right )}{15 c^3}+\frac {3}{20} b c d^3 x^4-\frac {7 b d^3 x^2}{15 c}-\frac {11}{36} i b d^3 x^3 \]
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Rubi [A] time = 0.17, antiderivative size = 191, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.304, Rules used = {43, 4872, 12, 1802, 635, 203, 260} \[ -\frac {1}{6} i c^3 d^3 x^6 \left (a+b \tan ^{-1}(c x)\right )-\frac {3}{5} c^2 d^3 x^5 \left (a+b \tan ^{-1}(c x)\right )+\frac {3}{4} i c d^3 x^4 \left (a+b \tan ^{-1}(c x)\right )+\frac {1}{3} d^3 x^3 \left (a+b \tan ^{-1}(c x)\right )+\frac {1}{30} i b c^2 d^3 x^5+\frac {7 b d^3 \log \left (c^2 x^2+1\right )}{15 c^3}+\frac {11 i b d^3 x}{12 c^2}-\frac {11 i b d^3 \tan ^{-1}(c x)}{12 c^3}+\frac {3}{20} b c d^3 x^4-\frac {7 b d^3 x^2}{15 c}-\frac {11}{36} i b d^3 x^3 \]
Antiderivative was successfully verified.
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Rule 12
Rule 43
Rule 203
Rule 260
Rule 635
Rule 1802
Rule 4872
Rubi steps
\begin {align*} \int x^2 (d+i c d x)^3 \left (a+b \tan ^{-1}(c x)\right ) \, dx &=\frac {1}{3} d^3 x^3 \left (a+b \tan ^{-1}(c x)\right )+\frac {3}{4} i c d^3 x^4 \left (a+b \tan ^{-1}(c x)\right )-\frac {3}{5} c^2 d^3 x^5 \left (a+b \tan ^{-1}(c x)\right )-\frac {1}{6} i c^3 d^3 x^6 \left (a+b \tan ^{-1}(c x)\right )-(b c) \int \frac {d^3 x^3 \left (20+45 i c x-36 c^2 x^2-10 i c^3 x^3\right )}{60 \left (1+c^2 x^2\right )} \, dx\\ &=\frac {1}{3} d^3 x^3 \left (a+b \tan ^{-1}(c x)\right )+\frac {3}{4} i c d^3 x^4 \left (a+b \tan ^{-1}(c x)\right )-\frac {3}{5} c^2 d^3 x^5 \left (a+b \tan ^{-1}(c x)\right )-\frac {1}{6} i c^3 d^3 x^6 \left (a+b \tan ^{-1}(c x)\right )-\frac {1}{60} \left (b c d^3\right ) \int \frac {x^3 \left (20+45 i c x-36 c^2 x^2-10 i c^3 x^3\right )}{1+c^2 x^2} \, dx\\ &=\frac {1}{3} d^3 x^3 \left (a+b \tan ^{-1}(c x)\right )+\frac {3}{4} i c d^3 x^4 \left (a+b \tan ^{-1}(c x)\right )-\frac {3}{5} c^2 d^3 x^5 \left (a+b \tan ^{-1}(c x)\right )-\frac {1}{6} i c^3 d^3 x^6 \left (a+b \tan ^{-1}(c x)\right )-\frac {1}{60} \left (b c d^3\right ) \int \left (-\frac {55 i}{c^3}+\frac {56 x}{c^2}+\frac {55 i x^2}{c}-36 x^3-10 i c x^4+\frac {55 i-56 c x}{c^3 \left (1+c^2 x^2\right )}\right ) \, dx\\ &=\frac {11 i b d^3 x}{12 c^2}-\frac {7 b d^3 x^2}{15 c}-\frac {11}{36} i b d^3 x^3+\frac {3}{20} b c d^3 x^4+\frac {1}{30} i b c^2 d^3 x^5+\frac {1}{3} d^3 x^3 \left (a+b \tan ^{-1}(c x)\right )+\frac {3}{4} i c d^3 x^4 \left (a+b \tan ^{-1}(c x)\right )-\frac {3}{5} c^2 d^3 x^5 \left (a+b \tan ^{-1}(c x)\right )-\frac {1}{6} i c^3 d^3 x^6 \left (a+b \tan ^{-1}(c x)\right )-\frac {\left (b d^3\right ) \int \frac {55 i-56 c x}{1+c^2 x^2} \, dx}{60 c^2}\\ &=\frac {11 i b d^3 x}{12 c^2}-\frac {7 b d^3 x^2}{15 c}-\frac {11}{36} i b d^3 x^3+\frac {3}{20} b c d^3 x^4+\frac {1}{30} i b c^2 d^3 x^5+\frac {1}{3} d^3 x^3 \left (a+b \tan ^{-1}(c x)\right )+\frac {3}{4} i c d^3 x^4 \left (a+b \tan ^{-1}(c x)\right )-\frac {3}{5} c^2 d^3 x^5 \left (a+b \tan ^{-1}(c x)\right )-\frac {1}{6} i c^3 d^3 x^6 \left (a+b \tan ^{-1}(c x)\right )-\frac {\left (11 i b d^3\right ) \int \frac {1}{1+c^2 x^2} \, dx}{12 c^2}+\frac {\left (14 b d^3\right ) \int \frac {x}{1+c^2 x^2} \, dx}{15 c}\\ &=\frac {11 i b d^3 x}{12 c^2}-\frac {7 b d^3 x^2}{15 c}-\frac {11}{36} i b d^3 x^3+\frac {3}{20} b c d^3 x^4+\frac {1}{30} i b c^2 d^3 x^5-\frac {11 i b d^3 \tan ^{-1}(c x)}{12 c^3}+\frac {1}{3} d^3 x^3 \left (a+b \tan ^{-1}(c x)\right )+\frac {3}{4} i c d^3 x^4 \left (a+b \tan ^{-1}(c x)\right )-\frac {3}{5} c^2 d^3 x^5 \left (a+b \tan ^{-1}(c x)\right )-\frac {1}{6} i c^3 d^3 x^6 \left (a+b \tan ^{-1}(c x)\right )+\frac {7 b d^3 \log \left (1+c^2 x^2\right )}{15 c^3}\\ \end {align*}
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Mathematica [A] time = 0.09, size = 234, normalized size = 1.23 \[ -\frac {1}{6} i a c^3 d^3 x^6-\frac {3}{5} a c^2 d^3 x^5+\frac {3}{4} i a c d^3 x^4+\frac {1}{3} a d^3 x^3-\frac {1}{6} i b c^3 d^3 x^6 \tan ^{-1}(c x)-\frac {11 i b d^3 \tan ^{-1}(c x)}{12 c^3}+\frac {1}{30} i b c^2 d^3 x^5-\frac {3}{5} b c^2 d^3 x^5 \tan ^{-1}(c x)+\frac {11 i b d^3 x}{12 c^2}+\frac {7 b d^3 \log \left (c^2 x^2+1\right )}{15 c^3}+\frac {3}{20} b c d^3 x^4+\frac {3}{4} i b c d^3 x^4 \tan ^{-1}(c x)+\frac {1}{3} b d^3 x^3 \tan ^{-1}(c x)-\frac {7 b d^3 x^2}{15 c}-\frac {11}{36} i b d^3 x^3 \]
Antiderivative was successfully verified.
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fricas [A] time = 0.44, size = 188, normalized size = 0.98 \[ \frac {-60 i \, a c^{6} d^{3} x^{6} - 12 \, {\left (18 \, a - i \, b\right )} c^{5} d^{3} x^{5} + {\left (270 i \, a + 54 \, b\right )} c^{4} d^{3} x^{4} + 10 \, {\left (12 \, a - 11 i \, b\right )} c^{3} d^{3} x^{3} - 168 \, b c^{2} d^{3} x^{2} + 330 i \, b c d^{3} x + 333 \, b d^{3} \log \left (\frac {c x + i}{c}\right ) + 3 \, b d^{3} \log \left (\frac {c x - i}{c}\right ) + {\left (30 \, b c^{6} d^{3} x^{6} - 108 i \, b c^{5} d^{3} x^{5} - 135 \, b c^{4} d^{3} x^{4} + 60 i \, b c^{3} d^{3} x^{3}\right )} \log \left (-\frac {c x + i}{c x - i}\right )}{360 \, c^{3}} \]
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 = 197, normalized size = 1.03 \[ -\frac {i c^{3} d^{3} a \,x^{6}}{6}-\frac {3 c^{2} d^{3} a \,x^{5}}{5}+\frac {3 i c \,d^{3} a \,x^{4}}{4}+\frac {d^{3} a \,x^{3}}{3}-\frac {i c^{3} d^{3} b \arctan \left (c x \right ) x^{6}}{6}-\frac {3 c^{2} d^{3} b \arctan \left (c x \right ) x^{5}}{5}+\frac {3 i c \,d^{3} b \arctan \left (c x \right ) x^{4}}{4}+\frac {d^{3} b \arctan \left (c x \right ) x^{3}}{3}+\frac {11 i b \,d^{3} x}{12 c^{2}}+\frac {i b \,c^{2} d^{3} x^{5}}{30}+\frac {3 b c \,d^{3} x^{4}}{20}-\frac {11 i b \,d^{3} x^{3}}{36}-\frac {7 b \,d^{3} x^{2}}{15 c}+\frac {7 b \,d^{3} \ln \left (c^{2} x^{2}+1\right )}{15 c^{3}}-\frac {11 i b \,d^{3} \arctan \left (c x \right )}{12 c^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.42, size = 242, normalized size = 1.27 \[ -\frac {1}{6} i \, a c^{3} d^{3} x^{6} - \frac {3}{5} \, a c^{2} d^{3} x^{5} + \frac {3}{4} i \, a c d^{3} x^{4} - \frac {1}{90} i \, {\left (15 \, x^{6} \arctan \left (c x\right ) - c {\left (\frac {3 \, c^{4} x^{5} - 5 \, c^{2} x^{3} + 15 \, x}{c^{6}} - \frac {15 \, \arctan \left (c x\right )}{c^{7}}\right )}\right )} b c^{3} d^{3} - \frac {3}{20} \, {\left (4 \, x^{5} \arctan \left (c x\right ) - c {\left (\frac {c^{2} x^{4} - 2 \, x^{2}}{c^{4}} + \frac {2 \, \log \left (c^{2} x^{2} + 1\right )}{c^{6}}\right )}\right )} b c^{2} d^{3} + \frac {1}{3} \, a d^{3} x^{3} + \frac {1}{4} i \, {\left (3 \, x^{4} \arctan \left (c x\right ) - c {\left (\frac {c^{2} x^{3} - 3 \, x}{c^{4}} + \frac {3 \, \arctan \left (c x\right )}{c^{5}}\right )}\right )} b c d^{3} + \frac {1}{6} \, {\left (2 \, x^{3} \arctan \left (c x\right ) - c {\left (\frac {x^{2}}{c^{2}} - \frac {\log \left (c^{2} x^{2} + 1\right )}{c^{4}}\right )}\right )} b d^{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.83, size = 174, normalized size = 0.91 \[ -\frac {\frac {d^3\,\left (-84\,b\,\ln \left (c^2\,x^2+1\right )+b\,\mathrm {atan}\left (c\,x\right )\,165{}\mathrm {i}\right )}{180}+\frac {7\,b\,c^2\,d^3\,x^2}{15}-\frac {b\,c\,d^3\,x\,11{}\mathrm {i}}{12}}{c^3}+\frac {d^3\,\left (60\,a\,x^3+60\,b\,x^3\,\mathrm {atan}\left (c\,x\right )-b\,x^3\,55{}\mathrm {i}\right )}{180}-\frac {c^3\,d^3\,\left (a\,x^6\,30{}\mathrm {i}+b\,x^6\,\mathrm {atan}\left (c\,x\right )\,30{}\mathrm {i}\right )}{180}+\frac {c\,d^3\,\left (a\,x^4\,135{}\mathrm {i}+27\,b\,x^4+b\,x^4\,\mathrm {atan}\left (c\,x\right )\,135{}\mathrm {i}\right )}{180}-\frac {c^2\,d^3\,\left (108\,a\,x^5+108\,b\,x^5\,\mathrm {atan}\left (c\,x\right )-b\,x^5\,6{}\mathrm {i}\right )}{180} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 4.90, size = 316, normalized size = 1.65 \[ - \frac {i a c^{3} d^{3} x^{6}}{6} - \frac {7 b d^{3} x^{2}}{15 c} + \frac {11 i b d^{3} x}{12 c^{2}} - \frac {b d^{3} \left (- \frac {\log {\left (310 b c d^{3} x - 310 i b d^{3} \right )}}{120} - \frac {209 \log {\left (310 b c d^{3} x + 310 i b d^{3} \right )}}{280}\right )}{c^{3}} - x^{5} \left (\frac {3 a c^{2} d^{3}}{5} - \frac {i b c^{2} d^{3}}{30}\right ) - x^{4} \left (- \frac {3 i a c d^{3}}{4} - \frac {3 b c d^{3}}{20}\right ) - x^{3} \left (- \frac {a d^{3}}{3} + \frac {11 i b d^{3}}{36}\right ) + \left (- \frac {b c^{3} d^{3} x^{6}}{12} + \frac {3 i b c^{2} d^{3} x^{5}}{10} + \frac {3 b c d^{3} x^{4}}{8} - \frac {i b d^{3} x^{3}}{6}\right ) \log {\left (i c x + 1 \right )} - \frac {\left (- 70 b c^{6} d^{3} x^{6} + 252 i b c^{5} d^{3} x^{5} + 315 b c^{4} d^{3} x^{4} - 140 i b c^{3} d^{3} x^{3} - 150 b d^{3}\right ) \log {\left (- i c x + 1 \right )}}{840 c^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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