Integrand size = 23, antiderivative size = 168 \[ \int \frac {(d+i c d x)^4 (a+b \arctan (c x))}{x^7} \, dx=-\frac {b c d^4}{30 x^5}-\frac {i b c^2 d^4}{5 x^4}+\frac {5 b c^3 d^4}{9 x^3}+\frac {16 i b c^4 d^4}{15 x^2}-\frac {13 b c^5 d^4}{6 x}-\frac {d^4 (1+i c x)^5 (a+b \arctan (c x))}{6 x^6}+\frac {i c d^4 (1+i c x)^5 (a+b \arctan (c x))}{30 x^5}+\frac {32}{15} i b c^6 d^4 \log (x)-\frac {32}{15} i b c^6 d^4 \log (i+c x) \]
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Time = 0.08 (sec) , antiderivative size = 168, 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} \[ \int \frac {(d+i c d x)^4 (a+b \arctan (c x))}{x^7} \, dx=-\frac {d^4 (1+i c x)^5 (a+b \arctan (c x))}{6 x^6}+\frac {i c d^4 (1+i c x)^5 (a+b \arctan (c x))}{30 x^5}+\frac {32}{15} i b c^6 d^4 \log (x)-\frac {32}{15} i b c^6 d^4 \log (c x+i)-\frac {13 b c^5 d^4}{6 x}+\frac {16 i b c^4 d^4}{15 x^2}+\frac {5 b c^3 d^4}{9 x^3}-\frac {i b c^2 d^4}{5 x^4}-\frac {b c d^4}{30 x^5} \]
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Rule 12
Rule 37
Rule 47
Rule 153
Rule 4992
Rubi steps \begin{align*} \text {integral}& = -\frac {d^4 (1+i c x)^5 (a+b \arctan (c x))}{6 x^6}+\frac {i c d^4 (1+i c x)^5 (a+b \arctan (c x))}{30 x^5}-(b c) \int \frac {d^4 (i-c x)^4 (-5 i-c x)}{30 x^6 (i+c x)} \, dx \\ & = -\frac {d^4 (1+i c x)^5 (a+b \arctan (c x))}{6 x^6}+\frac {i c d^4 (1+i c x)^5 (a+b \arctan (c x))}{30 x^5}-\frac {1}{30} \left (b c d^4\right ) \int \frac {(i-c x)^4 (-5 i-c x)}{x^6 (i+c x)} \, dx \\ & = -\frac {d^4 (1+i c x)^5 (a+b \arctan (c x))}{6 x^6}+\frac {i c d^4 (1+i c x)^5 (a+b \arctan (c x))}{30 x^5}-\frac {1}{30} \left (b c d^4\right ) \int \left (-\frac {5}{x^6}-\frac {24 i c}{x^5}+\frac {50 c^2}{x^4}+\frac {64 i c^3}{x^3}-\frac {65 c^4}{x^2}-\frac {64 i c^5}{x}+\frac {64 i c^6}{i+c x}\right ) \, dx \\ & = -\frac {b c d^4}{30 x^5}-\frac {i b c^2 d^4}{5 x^4}+\frac {5 b c^3 d^4}{9 x^3}+\frac {16 i b c^4 d^4}{15 x^2}-\frac {13 b c^5 d^4}{6 x}-\frac {d^4 (1+i c x)^5 (a+b \arctan (c x))}{6 x^6}+\frac {i c d^4 (1+i c x)^5 (a+b \arctan (c x))}{30 x^5}+\frac {32}{15} i b c^6 d^4 \log (x)-\frac {32}{15} i b c^6 d^4 \log (i+c x) \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.09 (sec) , antiderivative size = 235, normalized size of antiderivative = 1.40 \[ \int \frac {(d+i c d x)^4 (a+b \arctan (c x))}{x^7} \, dx=-\frac {d^4 \left (5 a+24 i a c x-45 a c^2 x^2+6 i b c^2 x^2-40 i a c^3 x^3+15 a c^4 x^4-32 i b c^4 x^4+5 b \arctan (c x)+24 i b c x \arctan (c x)-45 b c^2 x^2 \arctan (c x)-40 i b c^3 x^3 \arctan (c x)+15 b c^4 x^4 \arctan (c x)+b c x \operatorname {Hypergeometric2F1}\left (-\frac {5}{2},1,-\frac {3}{2},-c^2 x^2\right )-15 b c^3 x^3 \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},1,-\frac {1}{2},-c^2 x^2\right )+15 b c^5 x^5 \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},1,\frac {1}{2},-c^2 x^2\right )-64 i b c^6 x^6 \log (x)+32 i b c^6 x^6 \log \left (1+c^2 x^2\right )\right )}{30 x^6} \]
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Time = 1.79 (sec) , antiderivative size = 184, normalized size of antiderivative = 1.10
method | result | size |
parts | \(d^{4} a \left (-\frac {c^{4}}{2 x^{2}}+\frac {3 c^{2}}{2 x^{4}}-\frac {4 i c}{5 x^{5}}-\frac {1}{6 x^{6}}+\frac {4 i c^{3}}{3 x^{3}}\right )+d^{4} b \,c^{6} \left (-\frac {4 i \arctan \left (c x \right )}{5 c^{5} x^{5}}+\frac {3 \arctan \left (c x \right )}{2 c^{4} x^{4}}+\frac {4 i \arctan \left (c x \right )}{3 c^{3} x^{3}}-\frac {\arctan \left (c x \right )}{2 c^{2} x^{2}}-\frac {\arctan \left (c x \right )}{6 c^{6} x^{6}}+\frac {32 i \ln \left (c x \right )}{15}-\frac {i}{5 c^{4} x^{4}}+\frac {16 i}{15 c^{2} x^{2}}-\frac {1}{30 c^{5} x^{5}}+\frac {5}{9 c^{3} x^{3}}-\frac {13}{6 c x}-\frac {16 i \ln \left (c^{2} x^{2}+1\right )}{15}-\frac {13 \arctan \left (c x \right )}{6}\right )\) | \(184\) |
derivativedivides | \(c^{6} \left (d^{4} a \left (-\frac {4 i}{5 c^{5} x^{5}}+\frac {3}{2 c^{4} x^{4}}+\frac {4 i}{3 c^{3} x^{3}}-\frac {1}{2 c^{2} x^{2}}-\frac {1}{6 c^{6} x^{6}}\right )+d^{4} b \left (-\frac {4 i \arctan \left (c x \right )}{5 c^{5} x^{5}}+\frac {3 \arctan \left (c x \right )}{2 c^{4} x^{4}}+\frac {4 i \arctan \left (c x \right )}{3 c^{3} x^{3}}-\frac {\arctan \left (c x \right )}{2 c^{2} x^{2}}-\frac {\arctan \left (c x \right )}{6 c^{6} x^{6}}+\frac {32 i \ln \left (c x \right )}{15}-\frac {i}{5 c^{4} x^{4}}+\frac {16 i}{15 c^{2} x^{2}}-\frac {1}{30 c^{5} x^{5}}+\frac {5}{9 c^{3} x^{3}}-\frac {13}{6 c x}-\frac {16 i \ln \left (c^{2} x^{2}+1\right )}{15}-\frac {13 \arctan \left (c x \right )}{6}\right )\right )\) | \(190\) |
default | \(c^{6} \left (d^{4} a \left (-\frac {4 i}{5 c^{5} x^{5}}+\frac {3}{2 c^{4} x^{4}}+\frac {4 i}{3 c^{3} x^{3}}-\frac {1}{2 c^{2} x^{2}}-\frac {1}{6 c^{6} x^{6}}\right )+d^{4} b \left (-\frac {4 i \arctan \left (c x \right )}{5 c^{5} x^{5}}+\frac {3 \arctan \left (c x \right )}{2 c^{4} x^{4}}+\frac {4 i \arctan \left (c x \right )}{3 c^{3} x^{3}}-\frac {\arctan \left (c x \right )}{2 c^{2} x^{2}}-\frac {\arctan \left (c x \right )}{6 c^{6} x^{6}}+\frac {32 i \ln \left (c x \right )}{15}-\frac {i}{5 c^{4} x^{4}}+\frac {16 i}{15 c^{2} x^{2}}-\frac {1}{30 c^{5} x^{5}}+\frac {5}{9 c^{3} x^{3}}-\frac {13}{6 c x}-\frac {16 i \ln \left (c^{2} x^{2}+1\right )}{15}-\frac {13 \arctan \left (c x \right )}{6}\right )\right )\) | \(190\) |
parallelrisch | \(-\frac {96 i c^{6} b \,d^{4} \ln \left (c^{2} x^{2}+1\right ) x^{6}+96 i x^{6} b \,c^{6} d^{4}-96 i x^{4} b \,c^{4} d^{4}+195 b \,c^{6} d^{4} \arctan \left (c x \right ) x^{6}-45 a \,c^{6} d^{4} x^{6}+195 b \,c^{5} d^{4} x^{5}-120 i x^{3} \arctan \left (c x \right ) b \,c^{3} d^{4}+45 x^{4} \arctan \left (c x \right ) b \,c^{4} d^{4}+72 i x \arctan \left (c x \right ) b c \,d^{4}+45 a \,c^{4} d^{4} x^{4}+72 i a c \,d^{4} x -50 b \,c^{3} d^{4} x^{3}-192 i c^{6} b \,d^{4} \ln \left (x \right ) x^{6}-135 x^{2} \arctan \left (c x \right ) b \,c^{2} d^{4}-120 i x^{3} a \,c^{3} d^{4}-135 x^{2} d^{4} c^{2} a +18 i x^{2} b \,c^{2} d^{4}+3 b c \,d^{4} x +15 b \,d^{4} \arctan \left (c x \right )+15 d^{4} a}{90 x^{6}}\) | \(268\) |
risch | \(\frac {i d^{4} b \left (15 c^{4} x^{4}-40 i c^{3} x^{3}-45 c^{2} x^{2}+24 i c x +5\right ) \ln \left (i c x +1\right )}{60 x^{6}}-\frac {i d^{4} \left (387 b \,c^{6} \ln \left (-16705 c x -16705 i\right ) x^{6}-3 b \,c^{6} \ln \left (8255 c x -8255 i\right ) x^{6}-384 b \,c^{6} \ln \left (-32639 c x \right ) x^{6}+45 x^{4} b \ln \left (-i c x +1\right ) c^{4}-30 i a -192 b \,c^{4} x^{4}+270 i a \,c^{2} x^{2}-240 a \,c^{3} x^{3}-120 i b \,c^{3} x^{3} \ln \left (-i c x +1\right )-135 x^{2} b \ln \left (-i c x +1\right ) c^{2}+72 i b c x \ln \left (-i c x +1\right )+36 b \,c^{2} x^{2}-390 i b \,c^{5} x^{5}+144 c x a -90 i a \,c^{4} x^{4}+15 b \ln \left (-i c x +1\right )-6 i b c x +100 i b \,c^{3} x^{3}\right )}{180 x^{6}}\) | \(269\) |
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Time = 0.27 (sec) , antiderivative size = 216, normalized size of antiderivative = 1.29 \[ \int \frac {(d+i c d x)^4 (a+b \arctan (c x))}{x^7} \, dx=\frac {384 i \, b c^{6} d^{4} x^{6} \log \left (x\right ) - 387 i \, b c^{6} d^{4} x^{6} \log \left (\frac {c x + i}{c}\right ) + 3 i \, b c^{6} d^{4} x^{6} \log \left (\frac {c x - i}{c}\right ) - 390 \, b c^{5} d^{4} x^{5} - 6 \, {\left (15 \, a - 32 i \, b\right )} c^{4} d^{4} x^{4} - 20 \, {\left (-12 i \, a - 5 \, b\right )} c^{3} d^{4} x^{3} + 18 \, {\left (15 \, a - 2 i \, b\right )} c^{2} d^{4} x^{2} - 6 \, {\left (24 i \, a + b\right )} c d^{4} x - 30 \, a d^{4} - 3 \, {\left (15 i \, b c^{4} d^{4} x^{4} + 40 \, b c^{3} d^{4} x^{3} - 45 i \, b c^{2} d^{4} x^{2} - 24 \, b c d^{4} x + 5 i \, b d^{4}\right )} \log \left (-\frac {c x + i}{c x - i}\right )}{180 \, x^{6}} \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 388 vs. \(2 (163) = 326\).
Time = 69.41 (sec) , antiderivative size = 388, normalized size of antiderivative = 2.31 \[ \int \frac {(d+i c d x)^4 (a+b \arctan (c x))}{x^7} \, dx=\frac {32 i b c^{6} d^{4} \log {\left (2121535 b^{2} c^{13} d^{8} x \right )}}{15} + \frac {i b c^{6} d^{4} \log {\left (2121535 b^{2} c^{13} d^{8} x - 2121535 i b^{2} c^{12} d^{8} \right )}}{60} - \frac {43 i b c^{6} d^{4} \log {\left (2121535 b^{2} c^{13} d^{8} x + 2121535 i b^{2} c^{12} d^{8} \right )}}{20} + \frac {\left (- 15 i b c^{4} d^{4} x^{4} - 40 b c^{3} d^{4} x^{3} + 45 i b c^{2} d^{4} x^{2} + 24 b c d^{4} x - 5 i b d^{4}\right ) \log {\left (- i c x + 1 \right )}}{60 x^{6}} + \frac {\left (15 i b c^{4} d^{4} x^{4} + 40 b c^{3} d^{4} x^{3} - 45 i b c^{2} d^{4} x^{2} - 24 b c d^{4} x + 5 i b d^{4}\right ) \log {\left (i c x + 1 \right )}}{60 x^{6}} + \frac {- 15 a d^{4} - 195 b c^{5} d^{4} x^{5} + x^{4} \left (- 45 a c^{4} d^{4} + 96 i b c^{4} d^{4}\right ) + x^{3} \cdot \left (120 i a c^{3} d^{4} + 50 b c^{3} d^{4}\right ) + x^{2} \cdot \left (135 a c^{2} d^{4} - 18 i b c^{2} d^{4}\right ) + x \left (- 72 i a c d^{4} - 3 b c d^{4}\right )}{90 x^{6}} \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 290 vs. \(2 (134) = 268\).
Time = 0.27 (sec) , antiderivative size = 290, normalized size of antiderivative = 1.73 \[ \int \frac {(d+i c d x)^4 (a+b \arctan (c x))}{x^7} \, dx=-\frac {1}{2} \, {\left ({\left (c \arctan \left (c x\right ) + \frac {1}{x}\right )} c + \frac {\arctan \left (c x\right )}{x^{2}}\right )} b c^{4} d^{4} - \frac {2}{3} i \, {\left ({\left (c^{2} \log \left (c^{2} x^{2} + 1\right ) - c^{2} \log \left (x^{2}\right ) - \frac {1}{x^{2}}\right )} c - \frac {2 \, \arctan \left (c x\right )}{x^{3}}\right )} b c^{3} d^{4} - \frac {1}{2} \, {\left ({\left (3 \, c^{3} \arctan \left (c x\right ) + \frac {3 \, c^{2} x^{2} - 1}{x^{3}}\right )} c - \frac {3 \, \arctan \left (c x\right )}{x^{4}}\right )} b c^{2} d^{4} - \frac {1}{5} i \, {\left ({\left (2 \, c^{4} \log \left (c^{2} x^{2} + 1\right ) - 2 \, c^{4} \log \left (x^{2}\right ) - \frac {2 \, c^{2} x^{2} - 1}{x^{4}}\right )} c + \frac {4 \, \arctan \left (c x\right )}{x^{5}}\right )} b c d^{4} - \frac {a c^{4} d^{4}}{2 \, x^{2}} - \frac {1}{90} \, {\left ({\left (15 \, c^{5} \arctan \left (c x\right ) + \frac {15 \, c^{4} x^{4} - 5 \, c^{2} x^{2} + 3}{x^{5}}\right )} c + \frac {15 \, \arctan \left (c x\right )}{x^{6}}\right )} b d^{4} + \frac {4 i \, a c^{3} d^{4}}{3 \, x^{3}} + \frac {3 \, a c^{2} d^{4}}{2 \, x^{4}} - \frac {4 i \, a c d^{4}}{5 \, x^{5}} - \frac {a d^{4}}{6 \, x^{6}} \]
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\[ \int \frac {(d+i c d x)^4 (a+b \arctan (c x))}{x^7} \, dx=\int { \frac {{\left (i \, c d x + d\right )}^{4} {\left (b \arctan \left (c x\right ) + a\right )}}{x^{7}} \,d x } \]
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Time = 0.98 (sec) , antiderivative size = 208, normalized size of antiderivative = 1.24 \[ \int \frac {(d+i c d x)^4 (a+b \arctan (c x))}{x^7} \, dx=-\frac {d^4\,\left (195\,b\,c^5\,\mathrm {atan}\left (x\,\sqrt {c^2}\right )\,\sqrt {c^2}+b\,c^6\,\ln \left (c^2\,x^2+1\right )\,96{}\mathrm {i}-b\,c^6\,\ln \left (x\right )\,192{}\mathrm {i}\right )}{90}-\frac {\frac {d^4\,\left (15\,a+15\,b\,\mathrm {atan}\left (c\,x\right )\right )}{90}+\frac {d^4\,x\,\left (a\,c\,72{}\mathrm {i}+3\,b\,c+b\,c\,\mathrm {atan}\left (c\,x\right )\,72{}\mathrm {i}\right )}{90}+\frac {d^4\,x^4\,\left (45\,a\,c^4+45\,b\,c^4\,\mathrm {atan}\left (c\,x\right )-b\,c^4\,96{}\mathrm {i}\right )}{90}-\frac {d^4\,x^2\,\left (135\,a\,c^2+135\,b\,c^2\,\mathrm {atan}\left (c\,x\right )-b\,c^2\,18{}\mathrm {i}\right )}{90}-\frac {d^4\,x^3\,\left (a\,c^3\,120{}\mathrm {i}+50\,b\,c^3+b\,c^3\,\mathrm {atan}\left (c\,x\right )\,120{}\mathrm {i}\right )}{90}+\frac {13\,b\,c^5\,d^4\,x^5}{6}}{x^6} \]
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