Optimal. Leaf size=238 \[ \frac {1}{8} c^4 d^4 x^8 \left (a+b \tan ^{-1}(c x)\right )-\frac {4}{7} i c^3 d^4 x^7 \left (a+b \tan ^{-1}(c x)\right )-c^2 d^4 x^6 \left (a+b \tan ^{-1}(c x)\right )+\frac {4}{5} i c d^4 x^5 \left (a+b \tan ^{-1}(c x)\right )+\frac {1}{4} d^4 x^4 \left (a+b \tan ^{-1}(c x)\right )-\frac {11 b d^4 \tan ^{-1}(c x)}{8 c^4}-\frac {1}{56} b c^3 d^4 x^7+\frac {11 b d^4 x}{8 c^3}+\frac {2}{21} i b c^2 d^4 x^6+\frac {24 i b d^4 x^2}{35 c^2}-\frac {24 i b d^4 \log \left (c^2 x^2+1\right )}{35 c^4}+\frac {9}{40} b c d^4 x^5-\frac {11 b d^4 x^3}{24 c}-\frac {12}{35} i b d^4 x^4 \]
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Rubi [A] time = 0.21, antiderivative size = 238, 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} \[ \frac {1}{8} c^4 d^4 x^8 \left (a+b \tan ^{-1}(c x)\right )-\frac {4}{7} i c^3 d^4 x^7 \left (a+b \tan ^{-1}(c x)\right )-c^2 d^4 x^6 \left (a+b \tan ^{-1}(c x)\right )+\frac {4}{5} i c d^4 x^5 \left (a+b \tan ^{-1}(c x)\right )+\frac {1}{4} d^4 x^4 \left (a+b \tan ^{-1}(c x)\right )-\frac {1}{56} b c^3 d^4 x^7+\frac {2}{21} i b c^2 d^4 x^6+\frac {24 i b d^4 x^2}{35 c^2}-\frac {24 i b d^4 \log \left (c^2 x^2+1\right )}{35 c^4}+\frac {11 b d^4 x}{8 c^3}-\frac {11 b d^4 \tan ^{-1}(c x)}{8 c^4}+\frac {9}{40} b c d^4 x^5-\frac {11 b d^4 x^3}{24 c}-\frac {12}{35} i b d^4 x^4 \]
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^3 (d+i c d x)^4 \left (a+b \tan ^{-1}(c x)\right ) \, dx &=\frac {1}{4} d^4 x^4 \left (a+b \tan ^{-1}(c x)\right )+\frac {4}{5} i c d^4 x^5 \left (a+b \tan ^{-1}(c x)\right )-c^2 d^4 x^6 \left (a+b \tan ^{-1}(c x)\right )-\frac {4}{7} i c^3 d^4 x^7 \left (a+b \tan ^{-1}(c x)\right )+\frac {1}{8} c^4 d^4 x^8 \left (a+b \tan ^{-1}(c x)\right )-(b c) \int \frac {d^4 x^4 \left (70+224 i c x-280 c^2 x^2-160 i c^3 x^3+35 c^4 x^4\right )}{280 \left (1+c^2 x^2\right )} \, dx\\ &=\frac {1}{4} d^4 x^4 \left (a+b \tan ^{-1}(c x)\right )+\frac {4}{5} i c d^4 x^5 \left (a+b \tan ^{-1}(c x)\right )-c^2 d^4 x^6 \left (a+b \tan ^{-1}(c x)\right )-\frac {4}{7} i c^3 d^4 x^7 \left (a+b \tan ^{-1}(c x)\right )+\frac {1}{8} c^4 d^4 x^8 \left (a+b \tan ^{-1}(c x)\right )-\frac {1}{280} \left (b c d^4\right ) \int \frac {x^4 \left (70+224 i c x-280 c^2 x^2-160 i c^3 x^3+35 c^4 x^4\right )}{1+c^2 x^2} \, dx\\ &=\frac {1}{4} d^4 x^4 \left (a+b \tan ^{-1}(c x)\right )+\frac {4}{5} i c d^4 x^5 \left (a+b \tan ^{-1}(c x)\right )-c^2 d^4 x^6 \left (a+b \tan ^{-1}(c x)\right )-\frac {4}{7} i c^3 d^4 x^7 \left (a+b \tan ^{-1}(c x)\right )+\frac {1}{8} c^4 d^4 x^8 \left (a+b \tan ^{-1}(c x)\right )-\frac {1}{280} \left (b c d^4\right ) \int \left (-\frac {385}{c^4}-\frac {384 i x}{c^3}+\frac {385 x^2}{c^2}+\frac {384 i x^3}{c}-315 x^4-160 i c x^5+35 c^2 x^6+\frac {385+384 i c x}{c^4 \left (1+c^2 x^2\right )}\right ) \, dx\\ &=\frac {11 b d^4 x}{8 c^3}+\frac {24 i b d^4 x^2}{35 c^2}-\frac {11 b d^4 x^3}{24 c}-\frac {12}{35} i b d^4 x^4+\frac {9}{40} b c d^4 x^5+\frac {2}{21} i b c^2 d^4 x^6-\frac {1}{56} b c^3 d^4 x^7+\frac {1}{4} d^4 x^4 \left (a+b \tan ^{-1}(c x)\right )+\frac {4}{5} i c d^4 x^5 \left (a+b \tan ^{-1}(c x)\right )-c^2 d^4 x^6 \left (a+b \tan ^{-1}(c x)\right )-\frac {4}{7} i c^3 d^4 x^7 \left (a+b \tan ^{-1}(c x)\right )+\frac {1}{8} c^4 d^4 x^8 \left (a+b \tan ^{-1}(c x)\right )-\frac {\left (b d^4\right ) \int \frac {385+384 i c x}{1+c^2 x^2} \, dx}{280 c^3}\\ &=\frac {11 b d^4 x}{8 c^3}+\frac {24 i b d^4 x^2}{35 c^2}-\frac {11 b d^4 x^3}{24 c}-\frac {12}{35} i b d^4 x^4+\frac {9}{40} b c d^4 x^5+\frac {2}{21} i b c^2 d^4 x^6-\frac {1}{56} b c^3 d^4 x^7+\frac {1}{4} d^4 x^4 \left (a+b \tan ^{-1}(c x)\right )+\frac {4}{5} i c d^4 x^5 \left (a+b \tan ^{-1}(c x)\right )-c^2 d^4 x^6 \left (a+b \tan ^{-1}(c x)\right )-\frac {4}{7} i c^3 d^4 x^7 \left (a+b \tan ^{-1}(c x)\right )+\frac {1}{8} c^4 d^4 x^8 \left (a+b \tan ^{-1}(c x)\right )-\frac {\left (11 b d^4\right ) \int \frac {1}{1+c^2 x^2} \, dx}{8 c^3}-\frac {\left (48 i b d^4\right ) \int \frac {x}{1+c^2 x^2} \, dx}{35 c^2}\\ &=\frac {11 b d^4 x}{8 c^3}+\frac {24 i b d^4 x^2}{35 c^2}-\frac {11 b d^4 x^3}{24 c}-\frac {12}{35} i b d^4 x^4+\frac {9}{40} b c d^4 x^5+\frac {2}{21} i b c^2 d^4 x^6-\frac {1}{56} b c^3 d^4 x^7-\frac {11 b d^4 \tan ^{-1}(c x)}{8 c^4}+\frac {1}{4} d^4 x^4 \left (a+b \tan ^{-1}(c x)\right )+\frac {4}{5} i c d^4 x^5 \left (a+b \tan ^{-1}(c x)\right )-c^2 d^4 x^6 \left (a+b \tan ^{-1}(c x)\right )-\frac {4}{7} i c^3 d^4 x^7 \left (a+b \tan ^{-1}(c x)\right )+\frac {1}{8} c^4 d^4 x^8 \left (a+b \tan ^{-1}(c x)\right )-\frac {24 i b d^4 \log \left (1+c^2 x^2\right )}{35 c^4}\\ \end {align*}
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Mathematica [A] time = 0.17, size = 290, normalized size = 1.22 \[ \frac {1}{8} a c^4 d^4 x^8-\frac {4}{7} i a c^3 d^4 x^7-a c^2 d^4 x^6+\frac {4}{5} i a c d^4 x^5+\frac {1}{4} a d^4 x^4+\frac {1}{8} b c^4 d^4 x^8 \tan ^{-1}(c x)-\frac {11 b d^4 \tan ^{-1}(c x)}{8 c^4}-\frac {1}{56} b c^3 d^4 x^7-\frac {4}{7} i b c^3 d^4 x^7 \tan ^{-1}(c x)+\frac {11 b d^4 x}{8 c^3}+\frac {2}{21} i b c^2 d^4 x^6-b c^2 d^4 x^6 \tan ^{-1}(c x)+\frac {24 i b d^4 x^2}{35 c^2}-\frac {24 i b d^4 \log \left (c^2 x^2+1\right )}{35 c^4}+\frac {9}{40} b c d^4 x^5+\frac {4}{5} i b c d^4 x^5 \tan ^{-1}(c x)+\frac {1}{4} b d^4 x^4 \tan ^{-1}(c x)-\frac {11 b d^4 x^3}{24 c}-\frac {12}{35} i b d^4 x^4 \]
Antiderivative was successfully verified.
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fricas [A] time = 0.50, size = 229, normalized size = 0.96 \[ \frac {210 \, a c^{8} d^{4} x^{8} + {\left (-960 i \, a - 30 \, b\right )} c^{7} d^{4} x^{7} - 80 \, {\left (21 \, a - 2 i \, b\right )} c^{6} d^{4} x^{6} + {\left (1344 i \, a + 378 \, b\right )} c^{5} d^{4} x^{5} + 12 \, {\left (35 \, a - 48 i \, b\right )} c^{4} d^{4} x^{4} - 770 \, b c^{3} d^{4} x^{3} + 1152 i \, b c^{2} d^{4} x^{2} + 2310 \, b c d^{4} x - 2307 i \, b d^{4} \log \left (\frac {c x + i}{c}\right ) + 3 i \, b d^{4} \log \left (\frac {c x - i}{c}\right ) + {\left (105 i \, b c^{8} d^{4} x^{8} + 480 \, b c^{7} d^{4} x^{7} - 840 i \, b c^{6} d^{4} x^{6} - 672 \, b c^{5} d^{4} x^{5} + 210 i \, b c^{4} d^{4} x^{4}\right )} \log \left (-\frac {c x + i}{c x - i}\right )}{1680 \, c^{4}} \]
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 = 0.04, size = 249, normalized size = 1.05 \[ \frac {c^{4} d^{4} a \,x^{8}}{8}-\frac {4 i c^{3} d^{4} b \arctan \left (c x \right ) x^{7}}{7}-c^{2} d^{4} a \,x^{6}+\frac {4 i c \,d^{4} b \arctan \left (c x \right ) x^{5}}{5}+\frac {d^{4} a \,x^{4}}{4}+\frac {c^{4} d^{4} b \arctan \left (c x \right ) x^{8}}{8}+\frac {24 i b \,d^{4} x^{2}}{35 c^{2}}-c^{2} d^{4} b \arctan \left (c x \right ) x^{6}-\frac {12 i b \,d^{4} x^{4}}{35}+\frac {d^{4} b \arctan \left (c x \right ) x^{4}}{4}+\frac {11 b \,d^{4} x}{8 c^{3}}-\frac {b \,c^{3} d^{4} x^{7}}{56}+\frac {2 i b \,c^{2} d^{4} x^{6}}{21}+\frac {9 b c \,d^{4} x^{5}}{40}+\frac {4 i c \,d^{4} a \,x^{5}}{5}-\frac {11 b \,d^{4} x^{3}}{24 c}-\frac {4 i c^{3} d^{4} a \,x^{7}}{7}-\frac {24 i b \,d^{4} \ln \left (c^{2} x^{2}+1\right )}{35 c^{4}}-\frac {11 b \,d^{4} \arctan \left (c x \right )}{8 c^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.41, size = 337, normalized size = 1.42 \[ \frac {1}{8} \, a c^{4} d^{4} x^{8} - \frac {4}{7} i \, a c^{3} d^{4} x^{7} - a c^{2} d^{4} x^{6} + \frac {4}{5} i \, a c d^{4} x^{5} + \frac {1}{840} \, {\left (105 \, x^{8} \arctan \left (c x\right ) - c {\left (\frac {15 \, c^{6} x^{7} - 21 \, c^{4} x^{5} + 35 \, c^{2} x^{3} - 105 \, x}{c^{8}} + \frac {105 \, \arctan \left (c x\right )}{c^{9}}\right )}\right )} b c^{4} d^{4} - \frac {1}{21} i \, {\left (12 \, x^{7} \arctan \left (c x\right ) - c {\left (\frac {2 \, c^{4} x^{6} - 3 \, c^{2} x^{4} + 6 \, x^{2}}{c^{6}} - \frac {6 \, \log \left (c^{2} x^{2} + 1\right )}{c^{8}}\right )}\right )} b c^{3} d^{4} + \frac {1}{4} \, a d^{4} x^{4} - \frac {1}{15} \, {\left (15 \, x^{6} \arctan \left (c x\right ) - c {\left (\frac {3 \, c^{4} x^{5} - 5 \, c^{2} x^{3} + 15 \, x}{c^{6}} - \frac {15 \, \arctan \left (c x\right )}{c^{7}}\right )}\right )} b c^{2} d^{4} + \frac {1}{5} i \, {\left (4 \, x^{5} \arctan \left (c x\right ) - c {\left (\frac {c^{2} x^{4} - 2 \, x^{2}}{c^{4}} + \frac {2 \, \log \left (c^{2} x^{2} + 1\right )}{c^{6}}\right )}\right )} b c d^{4} + \frac {1}{12} \, {\left (3 \, x^{4} \arctan \left (c x\right ) - c {\left (\frac {c^{2} x^{3} - 3 \, x}{c^{4}} + \frac {3 \, \arctan \left (c x\right )}{c^{5}}\right )}\right )} b d^{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.59, size = 217, normalized size = 0.91 \[ \frac {c^4\,d^4\,\left (105\,a\,x^8+105\,b\,x^8\,\mathrm {atan}\left (c\,x\right )\right )}{840}+\frac {d^4\,\left (210\,a\,x^4+210\,b\,x^4\,\mathrm {atan}\left (c\,x\right )-b\,x^4\,288{}\mathrm {i}\right )}{840}-\frac {\frac {d^4\,\left (1155\,b\,\mathrm {atan}\left (c\,x\right )+b\,\ln \left (c^2\,x^2+1\right )\,576{}\mathrm {i}\right )}{840}+\frac {11\,b\,c^3\,d^4\,x^3}{24}-\frac {11\,b\,c\,d^4\,x}{8}-\frac {b\,c^2\,d^4\,x^2\,24{}\mathrm {i}}{35}}{c^4}+\frac {c\,d^4\,\left (a\,x^5\,672{}\mathrm {i}+189\,b\,x^5+b\,x^5\,\mathrm {atan}\left (c\,x\right )\,672{}\mathrm {i}\right )}{840}-\frac {c^3\,d^4\,\left (a\,x^7\,480{}\mathrm {i}+15\,b\,x^7+b\,x^7\,\mathrm {atan}\left (c\,x\right )\,480{}\mathrm {i}\right )}{840}-\frac {c^2\,d^4\,\left (840\,a\,x^6+840\,b\,x^6\,\mathrm {atan}\left (c\,x\right )-b\,x^6\,80{}\mathrm {i}\right )}{840} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 6.70, size = 389, normalized size = 1.63 \[ \frac {a c^{4} d^{4} x^{8}}{8} - \frac {11 b d^{4} x^{3}}{24 c} + \frac {24 i b d^{4} x^{2}}{35 c^{2}} + \frac {11 b d^{4} x}{8 c^{3}} + \frac {b d^{4} \left (\frac {i \log {\left (5893 b c d^{4} x - 5893 i b d^{4} \right )}}{560} - \frac {1471 i \log {\left (5893 b c d^{4} x + 5893 i b d^{4} \right )}}{1260}\right )}{c^{4}} + x^{7} \left (- \frac {4 i a c^{3} d^{4}}{7} - \frac {b c^{3} d^{4}}{56}\right ) + x^{6} \left (- a c^{2} d^{4} + \frac {2 i b c^{2} d^{4}}{21}\right ) + x^{5} \left (\frac {4 i a c d^{4}}{5} + \frac {9 b c d^{4}}{40}\right ) + x^{4} \left (\frac {a d^{4}}{4} - \frac {12 i b d^{4}}{35}\right ) + \left (- \frac {i b c^{4} d^{4} x^{8}}{16} - \frac {2 b c^{3} d^{4} x^{7}}{7} + \frac {i b c^{2} d^{4} x^{6}}{2} + \frac {2 b c d^{4} x^{5}}{5} - \frac {i b d^{4} x^{4}}{8}\right ) \log {\left (i c x + 1 \right )} - \frac {\left (- 315 i b c^{8} d^{4} x^{8} - 1440 b c^{7} d^{4} x^{7} + 2520 i b c^{6} d^{4} x^{6} + 2016 b c^{5} d^{4} x^{5} - 630 i b c^{4} d^{4} x^{4} + 1037 i b d^{4}\right ) \log {\left (- i c x + 1 \right )}}{5040 c^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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