Integrand size = 23, antiderivative size = 150 \[ \int \frac {(d+i c d x)^3 (a+b \arctan (c x))}{x^6} \, dx=-\frac {b c d^3}{20 x^4}-\frac {i b c^2 d^3}{4 x^3}+\frac {3 b c^3 d^3}{5 x^2}+\frac {5 i b c^4 d^3}{4 x}-\frac {d^3 (1+i c x)^4 (a+b \arctan (c x))}{5 x^5}+\frac {i c d^3 (1+i c x)^4 (a+b \arctan (c x))}{20 x^4}+\frac {6}{5} b c^5 d^3 \log (x)-\frac {6}{5} b c^5 d^3 \log (i+c x) \]
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Time = 0.07 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {47, 37, 4992, 12, 153} \[ \int \frac {(d+i c d x)^3 (a+b \arctan (c x))}{x^6} \, dx=-\frac {d^3 (1+i c x)^4 (a+b \arctan (c x))}{5 x^5}+\frac {i c d^3 (1+i c x)^4 (a+b \arctan (c x))}{20 x^4}+\frac {6}{5} b c^5 d^3 \log (x)-\frac {6}{5} b c^5 d^3 \log (c x+i)+\frac {5 i b c^4 d^3}{4 x}+\frac {3 b c^3 d^3}{5 x^2}-\frac {i b c^2 d^3}{4 x^3}-\frac {b c d^3}{20 x^4} \]
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Rule 12
Rule 37
Rule 47
Rule 153
Rule 4992
Rubi steps \begin{align*} \text {integral}& = -\frac {d^3 (1+i c x)^4 (a+b \arctan (c x))}{5 x^5}+\frac {i c d^3 (1+i c x)^4 (a+b \arctan (c x))}{20 x^4}-(b c) \int \frac {d^3 (-4 i-c x) (1+i c x)^3}{20 x^5 (i+c x)} \, dx \\ & = -\frac {d^3 (1+i c x)^4 (a+b \arctan (c x))}{5 x^5}+\frac {i c d^3 (1+i c x)^4 (a+b \arctan (c x))}{20 x^4}-\frac {1}{20} \left (b c d^3\right ) \int \frac {(-4 i-c x) (1+i c x)^3}{x^5 (i+c x)} \, dx \\ & = -\frac {d^3 (1+i c x)^4 (a+b \arctan (c x))}{5 x^5}+\frac {i c d^3 (1+i c x)^4 (a+b \arctan (c x))}{20 x^4}-\frac {1}{20} \left (b c d^3\right ) \int \left (-\frac {4}{x^5}-\frac {15 i c}{x^4}+\frac {24 c^2}{x^3}+\frac {25 i c^3}{x^2}-\frac {24 c^4}{x}+\frac {24 c^5}{i+c x}\right ) \, dx \\ & = -\frac {b c d^3}{20 x^4}-\frac {i b c^2 d^3}{4 x^3}+\frac {3 b c^3 d^3}{5 x^2}+\frac {5 i b c^4 d^3}{4 x}-\frac {d^3 (1+i c x)^4 (a+b \arctan (c x))}{5 x^5}+\frac {i c d^3 (1+i c x)^4 (a+b \arctan (c x))}{20 x^4}+\frac {6}{5} b c^5 d^3 \log (x)-\frac {6}{5} b c^5 d^3 \log (i+c x) \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.07 (sec) , antiderivative size = 185, normalized size of antiderivative = 1.23 \[ \int \frac {(d+i c d x)^3 (a+b \arctan (c x))}{x^6} \, dx=\frac {d^3 \left (-4 a-15 i a c x-b c x+20 a c^2 x^2+10 i a c^3 x^3+12 b c^3 x^3-4 b \arctan (c x)-15 i b c x \arctan (c x)+20 b c^2 x^2 \arctan (c x)+10 i b c^3 x^3 \arctan (c x)-5 i b c^2 x^2 \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},1,-\frac {1}{2},-c^2 x^2\right )+10 i b c^4 x^4 \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},1,\frac {1}{2},-c^2 x^2\right )+24 b c^5 x^5 \log (x)-12 b c^5 x^5 \log \left (1+c^2 x^2\right )\right )}{20 x^5} \]
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Time = 0.95 (sec) , antiderivative size = 153, normalized size of antiderivative = 1.02
method | result | size |
parts | \(a \,d^{3} \left (\frac {i c^{3}}{2 x^{2}}-\frac {3 i c}{4 x^{4}}-\frac {1}{5 x^{5}}+\frac {c^{2}}{x^{3}}\right )+b \,d^{3} c^{5} \left (-\frac {\arctan \left (c x \right )}{5 c^{5} x^{5}}-\frac {3 i \arctan \left (c x \right )}{4 c^{4} x^{4}}+\frac {\arctan \left (c x \right )}{c^{3} x^{3}}+\frac {i \arctan \left (c x \right )}{2 c^{2} x^{2}}-\frac {i}{4 c^{3} x^{3}}+\frac {5 i}{4 c x}-\frac {1}{20 c^{4} x^{4}}+\frac {3}{5 c^{2} x^{2}}+\frac {6 \ln \left (c x \right )}{5}-\frac {3 \ln \left (c^{2} x^{2}+1\right )}{5}+\frac {5 i \arctan \left (c x \right )}{4}\right )\) | \(153\) |
derivativedivides | \(c^{5} \left (a \,d^{3} \left (-\frac {1}{5 c^{5} x^{5}}-\frac {3 i}{4 c^{4} x^{4}}+\frac {1}{c^{3} x^{3}}+\frac {i}{2 c^{2} x^{2}}\right )+b \,d^{3} \left (-\frac {\arctan \left (c x \right )}{5 c^{5} x^{5}}-\frac {3 i \arctan \left (c x \right )}{4 c^{4} x^{4}}+\frac {\arctan \left (c x \right )}{c^{3} x^{3}}+\frac {i \arctan \left (c x \right )}{2 c^{2} x^{2}}-\frac {i}{4 c^{3} x^{3}}+\frac {5 i}{4 c x}-\frac {1}{20 c^{4} x^{4}}+\frac {3}{5 c^{2} x^{2}}+\frac {6 \ln \left (c x \right )}{5}-\frac {3 \ln \left (c^{2} x^{2}+1\right )}{5}+\frac {5 i \arctan \left (c x \right )}{4}\right )\right )\) | \(159\) |
default | \(c^{5} \left (a \,d^{3} \left (-\frac {1}{5 c^{5} x^{5}}-\frac {3 i}{4 c^{4} x^{4}}+\frac {1}{c^{3} x^{3}}+\frac {i}{2 c^{2} x^{2}}\right )+b \,d^{3} \left (-\frac {\arctan \left (c x \right )}{5 c^{5} x^{5}}-\frac {3 i \arctan \left (c x \right )}{4 c^{4} x^{4}}+\frac {\arctan \left (c x \right )}{c^{3} x^{3}}+\frac {i \arctan \left (c x \right )}{2 c^{2} x^{2}}-\frac {i}{4 c^{3} x^{3}}+\frac {5 i}{4 c x}-\frac {1}{20 c^{4} x^{4}}+\frac {3}{5 c^{2} x^{2}}+\frac {6 \ln \left (c x \right )}{5}-\frac {3 \ln \left (c^{2} x^{2}+1\right )}{5}+\frac {5 i \arctan \left (c x \right )}{4}\right )\right )\) | \(159\) |
parallelrisch | \(\frac {-15 i x \arctan \left (c x \right ) b c \,d^{3}-10 i x^{5} a \,c^{5} d^{3}-12 b \,c^{5} d^{3} \ln \left (c^{2} x^{2}+1\right ) x^{5}+24 b \,c^{5} d^{3} \ln \left (x \right ) x^{5}-12 b \,c^{5} d^{3} x^{5}+10 i x^{3} a \,c^{3} d^{3}-5 i x^{2} b \,c^{2} d^{3}-15 i a c \,d^{3} x +12 b \,c^{3} d^{3} x^{3}+25 i c^{5} b \,d^{3} \arctan \left (c x \right ) x^{5}+20 x^{2} \arctan \left (c x \right ) b \,c^{2} d^{3}+10 i x^{3} \arctan \left (c x \right ) b \,c^{3} d^{3}+20 x^{2} d^{3} c^{2} a +25 i x^{4} b \,c^{4} d^{3}-b c \,d^{3} x -4 b \,d^{3} \arctan \left (c x \right )-4 a \,d^{3}}{20 x^{5}}\) | \(227\) |
risch | \(\frac {\left (10 b \,c^{3} d^{3} x^{3}-20 i x^{2} b \,c^{2} d^{3}-15 b c \,d^{3} x +4 i b \,d^{3}\right ) \ln \left (i c x +1\right )}{40 x^{5}}-\frac {d^{3} \left (49 b \,c^{5} \ln \left (-c x -i\right ) x^{5}-b \,c^{5} \ln \left (c x -i\right ) x^{5}-48 b \,c^{5} \ln \left (-x \right ) x^{5}-50 i b \,c^{4} x^{4}-20 i a \,c^{3} x^{3}+10 b \,c^{3} x^{3} \ln \left (-i c x +1\right )-20 i b \,x^{2} \ln \left (-i c x +1\right ) c^{2}-24 b \,c^{3} x^{3}+10 i b \,c^{2} x^{2}-40 c^{2} x^{2} a +30 i x a c -15 b c x \ln \left (-i c x +1\right )+4 i b \ln \left (-i c x +1\right )+2 x b c +8 a \right )}{40 x^{5}}\) | \(233\) |
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Time = 0.25 (sec) , antiderivative size = 185, normalized size of antiderivative = 1.23 \[ \int \frac {(d+i c d x)^3 (a+b \arctan (c x))}{x^6} \, dx=\frac {48 \, b c^{5} d^{3} x^{5} \log \left (x\right ) - 49 \, b c^{5} d^{3} x^{5} \log \left (\frac {c x + i}{c}\right ) + b c^{5} d^{3} x^{5} \log \left (\frac {c x - i}{c}\right ) + 50 i \, b c^{4} d^{3} x^{4} - 4 \, {\left (-5 i \, a - 6 \, b\right )} c^{3} d^{3} x^{3} + 10 \, {\left (4 \, a - i \, b\right )} c^{2} d^{3} x^{2} - 2 \, {\left (15 i \, a + b\right )} c d^{3} x - 8 \, a d^{3} - {\left (10 \, b c^{3} d^{3} x^{3} - 20 i \, b c^{2} d^{3} x^{2} - 15 \, b c d^{3} x + 4 i \, b d^{3}\right )} \log \left (-\frac {c x + i}{c x - i}\right )}{40 \, x^{5}} \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 326 vs. \(2 (144) = 288\).
Time = 25.71 (sec) , antiderivative size = 326, normalized size of antiderivative = 2.17 \[ \int \frac {(d+i c d x)^3 (a+b \arctan (c x))}{x^6} \, dx=\frac {6 b c^{5} d^{3} \log {\left (113975 b^{2} c^{11} d^{6} x \right )}}{5} + \frac {b c^{5} d^{3} \log {\left (113975 b^{2} c^{11} d^{6} x - 113975 i b^{2} c^{10} d^{6} \right )}}{40} - \frac {49 b c^{5} d^{3} \log {\left (113975 b^{2} c^{11} d^{6} x + 113975 i b^{2} c^{10} d^{6} \right )}}{40} + \frac {\left (- 10 b c^{3} d^{3} x^{3} + 20 i b c^{2} d^{3} x^{2} + 15 b c d^{3} x - 4 i b d^{3}\right ) \log {\left (- i c x + 1 \right )}}{40 x^{5}} + \frac {\left (10 b c^{3} d^{3} x^{3} - 20 i b c^{2} d^{3} x^{2} - 15 b c d^{3} x + 4 i b d^{3}\right ) \log {\left (i c x + 1 \right )}}{40 x^{5}} - \frac {4 a d^{3} - 25 i b c^{4} d^{3} x^{4} + x^{3} \left (- 10 i a c^{3} d^{3} - 12 b c^{3} d^{3}\right ) + x^{2} \left (- 20 a c^{2} d^{3} + 5 i b c^{2} d^{3}\right ) + x \left (15 i a c d^{3} + b c d^{3}\right )}{20 x^{5}} \]
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Time = 0.27 (sec) , antiderivative size = 224, normalized size of antiderivative = 1.49 \[ \int \frac {(d+i c d x)^3 (a+b \arctan (c x))}{x^6} \, dx=\frac {1}{2} i \, {\left ({\left (c \arctan \left (c x\right ) + \frac {1}{x}\right )} c + \frac {\arctan \left (c x\right )}{x^{2}}\right )} b c^{3} d^{3} - \frac {1}{2} \, {\left ({\left (c^{2} \log \left (c^{2} x^{2} + 1\right ) - c^{2} \log \left (x^{2}\right ) - \frac {1}{x^{2}}\right )} c - \frac {2 \, \arctan \left (c x\right )}{x^{3}}\right )} b c^{2} d^{3} + \frac {1}{4} i \, {\left ({\left (3 \, c^{3} \arctan \left (c x\right ) + \frac {3 \, c^{2} x^{2} - 1}{x^{3}}\right )} c - \frac {3 \, \arctan \left (c x\right )}{x^{4}}\right )} b c d^{3} - \frac {1}{20} \, {\left ({\left (2 \, c^{4} \log \left (c^{2} x^{2} + 1\right ) - 2 \, c^{4} \log \left (x^{2}\right ) - \frac {2 \, c^{2} x^{2} - 1}{x^{4}}\right )} c + \frac {4 \, \arctan \left (c x\right )}{x^{5}}\right )} b d^{3} + \frac {i \, a c^{3} d^{3}}{2 \, x^{2}} + \frac {a c^{2} d^{3}}{x^{3}} - \frac {3 i \, a c d^{3}}{4 \, x^{4}} - \frac {a d^{3}}{5 \, x^{5}} \]
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\[ \int \frac {(d+i c d x)^3 (a+b \arctan (c x))}{x^6} \, dx=\int { \frac {{\left (i \, c d x + d\right )}^{3} {\left (b \arctan \left (c x\right ) + a\right )}}{x^{6}} \,d x } \]
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Time = 1.03 (sec) , antiderivative size = 174, normalized size of antiderivative = 1.16 \[ \int \frac {(d+i c d x)^3 (a+b \arctan (c x))}{x^6} \, dx=\frac {d^3\,\left (24\,b\,c^5\,\ln \left (x\right )-12\,b\,c^5\,\ln \left (c^2\,x^2+1\right )+b\,c^4\,\mathrm {atan}\left (x\,\sqrt {c^2}\right )\,\sqrt {c^2}\,25{}\mathrm {i}\right )}{20}+\frac {-\frac {d^3\,\left (4\,a+4\,b\,\mathrm {atan}\left (c\,x\right )\right )}{20}-\frac {d^3\,x\,\left (a\,c\,15{}\mathrm {i}+b\,c+b\,c\,\mathrm {atan}\left (c\,x\right )\,15{}\mathrm {i}\right )}{20}+\frac {d^3\,x^3\,\left (a\,c^3\,10{}\mathrm {i}+12\,b\,c^3+b\,c^3\,\mathrm {atan}\left (c\,x\right )\,10{}\mathrm {i}\right )}{20}+\frac {d^3\,x^2\,\left (20\,a\,c^2+20\,b\,c^2\,\mathrm {atan}\left (c\,x\right )-b\,c^2\,5{}\mathrm {i}\right )}{20}+\frac {b\,c^4\,d^3\,x^4\,5{}\mathrm {i}}{4}}{x^5} \]
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