Integrand size = 23, antiderivative size = 197 \[ \int \frac {a+b \arctan (c x)}{x^4 (d+i c d x)} \, dx=-\frac {b c}{6 d x^2}+\frac {i b c^2}{2 d x}+\frac {i b c^3 \arctan (c x)}{2 d}-\frac {a+b \arctan (c x)}{3 d x^3}+\frac {i c (a+b \arctan (c x))}{2 d x^2}+\frac {c^2 (a+b \arctan (c x))}{d x}-\frac {4 b c^3 \log (x)}{3 d}+\frac {2 b c^3 \log \left (1+c^2 x^2\right )}{3 d}+\frac {i c^3 (a+b \arctan (c x)) \log \left (2-\frac {2}{1+i c x}\right )}{d}-\frac {b c^3 \operatorname {PolyLog}\left (2,-1+\frac {2}{1+i c x}\right )}{2 d} \]
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Time = 0.24 (sec) , antiderivative size = 197, normalized size of antiderivative = 1.00, number of steps used = 17, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.478, Rules used = {4990, 4946, 272, 46, 331, 209, 36, 29, 31, 4988, 2497} \[ \int \frac {a+b \arctan (c x)}{x^4 (d+i c d x)} \, dx=\frac {i c^3 \log \left (2-\frac {2}{1+i c x}\right ) (a+b \arctan (c x))}{d}+\frac {c^2 (a+b \arctan (c x))}{d x}-\frac {a+b \arctan (c x)}{3 d x^3}+\frac {i c (a+b \arctan (c x))}{2 d x^2}+\frac {i b c^3 \arctan (c x)}{2 d}-\frac {b c^3 \operatorname {PolyLog}\left (2,\frac {2}{i c x+1}-1\right )}{2 d}-\frac {4 b c^3 \log (x)}{3 d}+\frac {i b c^2}{2 d x}+\frac {2 b c^3 \log \left (c^2 x^2+1\right )}{3 d}-\frac {b c}{6 d x^2} \]
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Rule 29
Rule 31
Rule 36
Rule 46
Rule 209
Rule 272
Rule 331
Rule 2497
Rule 4946
Rule 4988
Rule 4990
Rubi steps \begin{align*} \text {integral}& = -\left ((i c) \int \frac {a+b \arctan (c x)}{x^3 (d+i c d x)} \, dx\right )+\frac {\int \frac {a+b \arctan (c x)}{x^4} \, dx}{d} \\ & = -\frac {a+b \arctan (c x)}{3 d x^3}-c^2 \int \frac {a+b \arctan (c x)}{x^2 (d+i c d x)} \, dx-\frac {(i c) \int \frac {a+b \arctan (c x)}{x^3} \, dx}{d}+\frac {(b c) \int \frac {1}{x^3 \left (1+c^2 x^2\right )} \, dx}{3 d} \\ & = -\frac {a+b \arctan (c x)}{3 d x^3}+\frac {i c (a+b \arctan (c x))}{2 d x^2}+\left (i c^3\right ) \int \frac {a+b \arctan (c x)}{x (d+i c d x)} \, dx+\frac {(b c) \text {Subst}\left (\int \frac {1}{x^2 \left (1+c^2 x\right )} \, dx,x,x^2\right )}{6 d}-\frac {c^2 \int \frac {a+b \arctan (c x)}{x^2} \, dx}{d}-\frac {\left (i b c^2\right ) \int \frac {1}{x^2 \left (1+c^2 x^2\right )} \, dx}{2 d} \\ & = \frac {i b c^2}{2 d x}-\frac {a+b \arctan (c x)}{3 d x^3}+\frac {i c (a+b \arctan (c x))}{2 d x^2}+\frac {c^2 (a+b \arctan (c x))}{d x}+\frac {i c^3 (a+b \arctan (c x)) \log \left (2-\frac {2}{1+i c x}\right )}{d}+\frac {(b c) \text {Subst}\left (\int \left (\frac {1}{x^2}-\frac {c^2}{x}+\frac {c^4}{1+c^2 x}\right ) \, dx,x,x^2\right )}{6 d}-\frac {\left (b c^3\right ) \int \frac {1}{x \left (1+c^2 x^2\right )} \, dx}{d}+\frac {\left (i b c^4\right ) \int \frac {1}{1+c^2 x^2} \, dx}{2 d}-\frac {\left (i b c^4\right ) \int \frac {\log \left (2-\frac {2}{1+i c x}\right )}{1+c^2 x^2} \, dx}{d} \\ & = -\frac {b c}{6 d x^2}+\frac {i b c^2}{2 d x}+\frac {i b c^3 \arctan (c x)}{2 d}-\frac {a+b \arctan (c x)}{3 d x^3}+\frac {i c (a+b \arctan (c x))}{2 d x^2}+\frac {c^2 (a+b \arctan (c x))}{d x}-\frac {b c^3 \log (x)}{3 d}+\frac {b c^3 \log \left (1+c^2 x^2\right )}{6 d}+\frac {i c^3 (a+b \arctan (c x)) \log \left (2-\frac {2}{1+i c x}\right )}{d}-\frac {b c^3 \operatorname {PolyLog}\left (2,-1+\frac {2}{1+i c x}\right )}{2 d}-\frac {\left (b c^3\right ) \text {Subst}\left (\int \frac {1}{x \left (1+c^2 x\right )} \, dx,x,x^2\right )}{2 d} \\ & = -\frac {b c}{6 d x^2}+\frac {i b c^2}{2 d x}+\frac {i b c^3 \arctan (c x)}{2 d}-\frac {a+b \arctan (c x)}{3 d x^3}+\frac {i c (a+b \arctan (c x))}{2 d x^2}+\frac {c^2 (a+b \arctan (c x))}{d x}-\frac {b c^3 \log (x)}{3 d}+\frac {b c^3 \log \left (1+c^2 x^2\right )}{6 d}+\frac {i c^3 (a+b \arctan (c x)) \log \left (2-\frac {2}{1+i c x}\right )}{d}-\frac {b c^3 \operatorname {PolyLog}\left (2,-1+\frac {2}{1+i c x}\right )}{2 d}-\frac {\left (b c^3\right ) \text {Subst}\left (\int \frac {1}{x} \, dx,x,x^2\right )}{2 d}+\frac {\left (b c^5\right ) \text {Subst}\left (\int \frac {1}{1+c^2 x} \, dx,x,x^2\right )}{2 d} \\ & = -\frac {b c}{6 d x^2}+\frac {i b c^2}{2 d x}+\frac {i b c^3 \arctan (c x)}{2 d}-\frac {a+b \arctan (c x)}{3 d x^3}+\frac {i c (a+b \arctan (c x))}{2 d x^2}+\frac {c^2 (a+b \arctan (c x))}{d x}-\frac {4 b c^3 \log (x)}{3 d}+\frac {2 b c^3 \log \left (1+c^2 x^2\right )}{3 d}+\frac {i c^3 (a+b \arctan (c x)) \log \left (2-\frac {2}{1+i c x}\right )}{d}-\frac {b c^3 \operatorname {PolyLog}\left (2,-1+\frac {2}{1+i c x}\right )}{2 d} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 0.10 (sec) , antiderivative size = 254, normalized size of antiderivative = 1.29 \[ \int \frac {a+b \arctan (c x)}{x^4 (d+i c d x)} \, dx=\frac {-2 a+3 i a c x-b c x+6 a c^2 x^2-2 b \arctan (c x)+3 i b c x \arctan (c x)+6 b c^2 x^2 \arctan (c x)+3 i b c^2 x^2 \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},1,\frac {1}{2},-c^2 x^2\right )+6 i a c^3 x^3 \log (x)-8 b c^3 x^3 \log (x)+6 i a c^3 x^3 \log \left (\frac {2 i}{i-c x}\right )+6 i b c^3 x^3 \arctan (c x) \log \left (\frac {2 i}{i-c x}\right )+4 b c^3 x^3 \log \left (1+c^2 x^2\right )-3 b c^3 x^3 \operatorname {PolyLog}(2,-i c x)+3 b c^3 x^3 \operatorname {PolyLog}(2,i c x)-3 b c^3 x^3 \operatorname {PolyLog}\left (2,\frac {i+c x}{-i+c x}\right )}{6 d x^3} \]
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Time = 1.03 (sec) , antiderivative size = 278, normalized size of antiderivative = 1.41
method | result | size |
derivativedivides | \(c^{3} \left (-\frac {a}{3 d \,c^{3} x^{3}}+\frac {i a}{2 d \,c^{2} x^{2}}+\frac {i a \ln \left (c x \right )}{d}+\frac {a}{d c x}-\frac {i a \ln \left (c^{2} x^{2}+1\right )}{2 d}+\frac {a \arctan \left (c x \right )}{d}+\frac {b \left (-\frac {\arctan \left (c x \right )}{3 c^{3} x^{3}}+\frac {i \arctan \left (c x \right )}{2 c^{2} x^{2}}+\frac {i}{2 c x}+\frac {\arctan \left (c x \right )}{c x}+i \arctan \left (c x \right ) \ln \left (c x \right )-\frac {\ln \left (c x -i\right ) \ln \left (-\frac {i \left (c x +i\right )}{2}\right )}{2}-\frac {\operatorname {dilog}\left (-\frac {i \left (c x +i\right )}{2}\right )}{2}+\frac {\ln \left (c x -i\right )^{2}}{4}-\frac {\ln \left (c x \right ) \ln \left (i c x +1\right )}{2}+\frac {\ln \left (c x \right ) \ln \left (-i c x +1\right )}{2}-\frac {\operatorname {dilog}\left (i c x +1\right )}{2}+\frac {\operatorname {dilog}\left (-i c x +1\right )}{2}+\frac {i \arctan \left (c x \right )}{2}-\frac {1}{6 c^{2} x^{2}}-\frac {4 \ln \left (c x \right )}{3}+\frac {2 \ln \left (c^{2} x^{2}+1\right )}{3}-i \arctan \left (c x \right ) \ln \left (c x -i\right )\right )}{d}\right )\) | \(278\) |
default | \(c^{3} \left (-\frac {a}{3 d \,c^{3} x^{3}}+\frac {i a}{2 d \,c^{2} x^{2}}+\frac {i a \ln \left (c x \right )}{d}+\frac {a}{d c x}-\frac {i a \ln \left (c^{2} x^{2}+1\right )}{2 d}+\frac {a \arctan \left (c x \right )}{d}+\frac {b \left (-\frac {\arctan \left (c x \right )}{3 c^{3} x^{3}}+\frac {i \arctan \left (c x \right )}{2 c^{2} x^{2}}+\frac {i}{2 c x}+\frac {\arctan \left (c x \right )}{c x}+i \arctan \left (c x \right ) \ln \left (c x \right )-\frac {\ln \left (c x -i\right ) \ln \left (-\frac {i \left (c x +i\right )}{2}\right )}{2}-\frac {\operatorname {dilog}\left (-\frac {i \left (c x +i\right )}{2}\right )}{2}+\frac {\ln \left (c x -i\right )^{2}}{4}-\frac {\ln \left (c x \right ) \ln \left (i c x +1\right )}{2}+\frac {\ln \left (c x \right ) \ln \left (-i c x +1\right )}{2}-\frac {\operatorname {dilog}\left (i c x +1\right )}{2}+\frac {\operatorname {dilog}\left (-i c x +1\right )}{2}+\frac {i \arctan \left (c x \right )}{2}-\frac {1}{6 c^{2} x^{2}}-\frac {4 \ln \left (c x \right )}{3}+\frac {2 \ln \left (c^{2} x^{2}+1\right )}{3}-i \arctan \left (c x \right ) \ln \left (c x -i\right )\right )}{d}\right )\) | \(278\) |
parts | \(-\frac {a}{3 d \,x^{3}}+\frac {i c a}{2 d \,x^{2}}+\frac {i a \,c^{3} \ln \left (x \right )}{d}+\frac {c^{2} a}{d x}-\frac {i c^{3} a \ln \left (c^{2} x^{2}+1\right )}{2 d}+\frac {c^{3} a \arctan \left (c x \right )}{d}+\frac {b \,c^{3} \left (-\frac {\arctan \left (c x \right )}{3 c^{3} x^{3}}+\frac {i \arctan \left (c x \right )}{2 c^{2} x^{2}}+\frac {i}{2 c x}+\frac {\arctan \left (c x \right )}{c x}+i \arctan \left (c x \right ) \ln \left (c x \right )-\frac {\ln \left (c x -i\right ) \ln \left (-\frac {i \left (c x +i\right )}{2}\right )}{2}-\frac {\operatorname {dilog}\left (-\frac {i \left (c x +i\right )}{2}\right )}{2}+\frac {\ln \left (c x -i\right )^{2}}{4}-\frac {\ln \left (c x \right ) \ln \left (i c x +1\right )}{2}+\frac {\ln \left (c x \right ) \ln \left (-i c x +1\right )}{2}-\frac {\operatorname {dilog}\left (i c x +1\right )}{2}+\frac {\operatorname {dilog}\left (-i c x +1\right )}{2}+\frac {i \arctan \left (c x \right )}{2}-\frac {1}{6 c^{2} x^{2}}-\frac {4 \ln \left (c x \right )}{3}+\frac {2 \ln \left (c^{2} x^{2}+1\right )}{3}-i \arctan \left (c x \right ) \ln \left (c x -i\right )\right )}{d}\) | \(279\) |
risch | \(-\frac {b c}{6 d \,x^{2}}+\frac {11 b \,c^{3} \ln \left (c^{2} x^{2}+1\right )}{24 d}+\frac {i b \,c^{2}}{2 d x}+\frac {i b \ln \left (i c x +1\right )}{6 d \,x^{3}}+\frac {i c^{3} \ln \left (-i c x \right ) a}{d}-\frac {c^{3} \ln \left (\frac {1}{2}-\frac {i c x}{2}\right ) \ln \left (\frac {1}{2}+\frac {i c x}{2}\right ) b}{2 d}+\frac {c^{3} b \ln \left (\frac {1}{2}+\frac {i c x}{2}\right ) \ln \left (-i c x +1\right )}{2 d}-\frac {c b \ln \left (-i c x +1\right )}{4 d \,x^{2}}-\frac {i b \ln \left (-i c x +1\right )}{6 d \,x^{3}}+\frac {11 i c^{3} b \arctan \left (c x \right )}{12 d}+\frac {b c \ln \left (i c x +1\right )}{4 d \,x^{2}}+\frac {i c a}{2 d \,x^{2}}-\frac {i c^{3} a \ln \left (c^{2} x^{2}+1\right )}{2 d}+\frac {i c^{2} b \ln \left (-i c x +1\right )}{2 d x}-\frac {i b \,c^{2} \ln \left (i c x +1\right )}{2 d x}+\frac {c^{3} \operatorname {dilog}\left (-i c x +1\right ) b}{2 d}-\frac {5 c^{3} b \ln \left (-i c x \right )}{12 d}+\frac {5 c^{3} b \ln \left (-i c x +1\right )}{12 d}-\frac {c^{3} b \operatorname {dilog}\left (\frac {1}{2}-\frac {i c x}{2}\right )}{2 d}-\frac {b \,c^{3} \ln \left (i c x +1\right )^{2}}{4 d}-\frac {b \,c^{3} \operatorname {dilog}\left (i c x +1\right )}{2 d}-\frac {11 b \,c^{3} \ln \left (i c x \right )}{12 d}-\frac {a}{3 d \,x^{3}}+\frac {c^{2} a}{d x}+\frac {c^{3} a \arctan \left (c x \right )}{d}\) | \(418\) |
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Time = 0.26 (sec) , antiderivative size = 155, normalized size of antiderivative = 0.79 \[ \int \frac {a+b \arctan (c x)}{x^4 (d+i c d x)} \, dx=\frac {6 \, b c^{3} x^{3} {\rm Li}_2\left (\frac {c x + i}{c x - i} + 1\right ) - 4 \, {\left (-3 i \, a + 4 \, b\right )} c^{3} x^{3} \log \left (x\right ) + 5 \, b c^{3} x^{3} \log \left (\frac {c x + i}{c}\right ) + {\left (-12 i \, a + 11 \, b\right )} c^{3} x^{3} \log \left (\frac {c x - i}{c}\right ) + 6 \, {\left (2 \, a + i \, b\right )} c^{2} x^{2} - 2 \, {\left (-3 i \, a + b\right )} c x + {\left (6 i \, b c^{2} x^{2} - 3 \, b c x - 2 i \, b\right )} \log \left (-\frac {c x + i}{c x - i}\right ) - 4 \, a}{12 \, d x^{3}} \]
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\[ \int \frac {a+b \arctan (c x)}{x^4 (d+i c d x)} \, dx=- \frac {i \left (\int \frac {a}{c x^{5} - i x^{4}}\, dx + \int \frac {b \operatorname {atan}{\left (c x \right )}}{c x^{5} - i x^{4}}\, dx\right )}{d} \]
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\[ \int \frac {a+b \arctan (c x)}{x^4 (d+i c d x)} \, dx=\int { \frac {b \arctan \left (c x\right ) + a}{{\left (i \, c d x + d\right )} x^{4}} \,d x } \]
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\[ \int \frac {a+b \arctan (c x)}{x^4 (d+i c d x)} \, dx=\int { \frac {b \arctan \left (c x\right ) + a}{{\left (i \, c d x + d\right )} x^{4}} \,d x } \]
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Timed out. \[ \int \frac {a+b \arctan (c x)}{x^4 (d+i c d x)} \, dx=\int \frac {a+b\,\mathrm {atan}\left (c\,x\right )}{x^4\,\left (d+c\,d\,x\,1{}\mathrm {i}\right )} \,d x \]
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