Integrand size = 23, antiderivative size = 173 \[ \int \frac {(d+i c d x)^4 (a+b \arctan (c x))}{x^3} \, dx=-\frac {b c d^4}{2 x}-4 i a c^3 d^4 x-\frac {1}{2} b c^3 d^4 x-4 i b c^3 d^4 x \arctan (c x)-\frac {d^4 (a+b \arctan (c x))}{2 x^2}-\frac {4 i c d^4 (a+b \arctan (c x))}{x}+\frac {1}{2} c^4 d^4 x^2 (a+b \arctan (c x))-6 a c^2 d^4 \log (x)+4 i b c^2 d^4 \log (x)-3 i b c^2 d^4 \operatorname {PolyLog}(2,-i c x)+3 i b c^2 d^4 \operatorname {PolyLog}(2,i c x) \]
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Time = 0.14 (sec) , antiderivative size = 173, normalized size of antiderivative = 1.00, number of steps used = 19, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.565, Rules used = {4996, 4930, 266, 4946, 331, 209, 272, 36, 29, 31, 4940, 2438, 327} \[ \int \frac {(d+i c d x)^4 (a+b \arctan (c x))}{x^3} \, dx=\frac {1}{2} c^4 d^4 x^2 (a+b \arctan (c x))-\frac {d^4 (a+b \arctan (c x))}{2 x^2}-\frac {4 i c d^4 (a+b \arctan (c x))}{x}-4 i a c^3 d^4 x-6 a c^2 d^4 \log (x)-4 i b c^3 d^4 x \arctan (c x)-\frac {1}{2} b c^3 d^4 x-3 i b c^2 d^4 \operatorname {PolyLog}(2,-i c x)+3 i b c^2 d^4 \operatorname {PolyLog}(2,i c x)+4 i b c^2 d^4 \log (x)-\frac {b c d^4}{2 x} \]
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Rule 29
Rule 31
Rule 36
Rule 209
Rule 266
Rule 272
Rule 327
Rule 331
Rule 2438
Rule 4930
Rule 4940
Rule 4946
Rule 4996
Rubi steps \begin{align*} \text {integral}& = \int \left (-4 i c^3 d^4 (a+b \arctan (c x))+\frac {d^4 (a+b \arctan (c x))}{x^3}+\frac {4 i c d^4 (a+b \arctan (c x))}{x^2}-\frac {6 c^2 d^4 (a+b \arctan (c x))}{x}+c^4 d^4 x (a+b \arctan (c x))\right ) \, dx \\ & = d^4 \int \frac {a+b \arctan (c x)}{x^3} \, dx+\left (4 i c d^4\right ) \int \frac {a+b \arctan (c x)}{x^2} \, dx-\left (6 c^2 d^4\right ) \int \frac {a+b \arctan (c x)}{x} \, dx-\left (4 i c^3 d^4\right ) \int (a+b \arctan (c x)) \, dx+\left (c^4 d^4\right ) \int x (a+b \arctan (c x)) \, dx \\ & = -4 i a c^3 d^4 x-\frac {d^4 (a+b \arctan (c x))}{2 x^2}-\frac {4 i c d^4 (a+b \arctan (c x))}{x}+\frac {1}{2} c^4 d^4 x^2 (a+b \arctan (c x))-6 a c^2 d^4 \log (x)+\frac {1}{2} \left (b c d^4\right ) \int \frac {1}{x^2 \left (1+c^2 x^2\right )} \, dx-\left (3 i b c^2 d^4\right ) \int \frac {\log (1-i c x)}{x} \, dx+\left (3 i b c^2 d^4\right ) \int \frac {\log (1+i c x)}{x} \, dx+\left (4 i b c^2 d^4\right ) \int \frac {1}{x \left (1+c^2 x^2\right )} \, dx-\left (4 i b c^3 d^4\right ) \int \arctan (c x) \, dx-\frac {1}{2} \left (b c^5 d^4\right ) \int \frac {x^2}{1+c^2 x^2} \, dx \\ & = -\frac {b c d^4}{2 x}-4 i a c^3 d^4 x-\frac {1}{2} b c^3 d^4 x-4 i b c^3 d^4 x \arctan (c x)-\frac {d^4 (a+b \arctan (c x))}{2 x^2}-\frac {4 i c d^4 (a+b \arctan (c x))}{x}+\frac {1}{2} c^4 d^4 x^2 (a+b \arctan (c x))-6 a c^2 d^4 \log (x)-3 i b c^2 d^4 \operatorname {PolyLog}(2,-i c x)+3 i b c^2 d^4 \operatorname {PolyLog}(2,i c x)+\left (2 i b c^2 d^4\right ) \text {Subst}\left (\int \frac {1}{x \left (1+c^2 x\right )} \, dx,x,x^2\right )+\left (4 i b c^4 d^4\right ) \int \frac {x}{1+c^2 x^2} \, dx \\ & = -\frac {b c d^4}{2 x}-4 i a c^3 d^4 x-\frac {1}{2} b c^3 d^4 x-4 i b c^3 d^4 x \arctan (c x)-\frac {d^4 (a+b \arctan (c x))}{2 x^2}-\frac {4 i c d^4 (a+b \arctan (c x))}{x}+\frac {1}{2} c^4 d^4 x^2 (a+b \arctan (c x))-6 a c^2 d^4 \log (x)+2 i b c^2 d^4 \log \left (1+c^2 x^2\right )-3 i b c^2 d^4 \operatorname {PolyLog}(2,-i c x)+3 i b c^2 d^4 \operatorname {PolyLog}(2,i c x)+\left (2 i b c^2 d^4\right ) \text {Subst}\left (\int \frac {1}{x} \, dx,x,x^2\right )-\left (2 i b c^4 d^4\right ) \text {Subst}\left (\int \frac {1}{1+c^2 x} \, dx,x,x^2\right ) \\ & = -\frac {b c d^4}{2 x}-4 i a c^3 d^4 x-\frac {1}{2} b c^3 d^4 x-4 i b c^3 d^4 x \arctan (c x)-\frac {d^4 (a+b \arctan (c x))}{2 x^2}-\frac {4 i c d^4 (a+b \arctan (c x))}{x}+\frac {1}{2} c^4 d^4 x^2 (a+b \arctan (c x))-6 a c^2 d^4 \log (x)+4 i b c^2 d^4 \log (x)-3 i b c^2 d^4 \operatorname {PolyLog}(2,-i c x)+3 i b c^2 d^4 \operatorname {PolyLog}(2,i c x) \\ \end{align*}
Time = 0.13 (sec) , antiderivative size = 163, normalized size of antiderivative = 0.94 \[ \int \frac {(d+i c d x)^4 (a+b \arctan (c x))}{x^3} \, dx=\frac {d^4 \left (-a-8 i a c x-b c x-8 i a c^3 x^3-b c^3 x^3+a c^4 x^4-b \arctan (c x)-8 i b c x \arctan (c x)-8 i b c^3 x^3 \arctan (c x)+b c^4 x^4 \arctan (c x)-12 a c^2 x^2 \log (x)+8 i b c^2 x^2 \log (c x)-6 i b c^2 x^2 \operatorname {PolyLog}(2,-i c x)+6 i b c^2 x^2 \operatorname {PolyLog}(2,i c x)\right )}{2 x^2} \]
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Time = 1.85 (sec) , antiderivative size = 178, normalized size of antiderivative = 1.03
method | result | size |
parts | \(d^{4} a \left (\frac {c^{4} x^{2}}{2}-4 i c^{3} x -\frac {1}{2 x^{2}}-6 c^{2} \ln \left (x \right )-\frac {4 i c}{x}\right )+d^{4} b \,c^{2} \left (-4 i \arctan \left (c x \right ) c x +\frac {c^{2} x^{2} \arctan \left (c x \right )}{2}-6 \arctan \left (c x \right ) \ln \left (c x \right )-\frac {4 i \arctan \left (c x \right )}{c x}-\frac {\arctan \left (c x \right )}{2 c^{2} x^{2}}-\frac {c x}{2}+4 i \ln \left (c x \right )-\frac {1}{2 c x}-3 i \ln \left (c x \right ) \ln \left (i c x +1\right )+3 i \ln \left (c x \right ) \ln \left (-i c x +1\right )-3 i \operatorname {dilog}\left (i c x +1\right )+3 i \operatorname {dilog}\left (-i c x +1\right )\right )\) | \(178\) |
derivativedivides | \(c^{2} \left (d^{4} a \left (-4 i c x +\frac {c^{2} x^{2}}{2}-6 \ln \left (c x \right )-\frac {4 i}{c x}-\frac {1}{2 c^{2} x^{2}}\right )+d^{4} b \left (-4 i \arctan \left (c x \right ) c x +\frac {c^{2} x^{2} \arctan \left (c x \right )}{2}-6 \arctan \left (c x \right ) \ln \left (c x \right )-\frac {4 i \arctan \left (c x \right )}{c x}-\frac {\arctan \left (c x \right )}{2 c^{2} x^{2}}-\frac {c x}{2}+4 i \ln \left (c x \right )-\frac {1}{2 c x}-3 i \ln \left (c x \right ) \ln \left (i c x +1\right )+3 i \ln \left (c x \right ) \ln \left (-i c x +1\right )-3 i \operatorname {dilog}\left (i c x +1\right )+3 i \operatorname {dilog}\left (-i c x +1\right )\right )\right )\) | \(181\) |
default | \(c^{2} \left (d^{4} a \left (-4 i c x +\frac {c^{2} x^{2}}{2}-6 \ln \left (c x \right )-\frac {4 i}{c x}-\frac {1}{2 c^{2} x^{2}}\right )+d^{4} b \left (-4 i \arctan \left (c x \right ) c x +\frac {c^{2} x^{2} \arctan \left (c x \right )}{2}-6 \arctan \left (c x \right ) \ln \left (c x \right )-\frac {4 i \arctan \left (c x \right )}{c x}-\frac {\arctan \left (c x \right )}{2 c^{2} x^{2}}-\frac {c x}{2}+4 i \ln \left (c x \right )-\frac {1}{2 c x}-3 i \ln \left (c x \right ) \ln \left (i c x +1\right )+3 i \ln \left (c x \right ) \ln \left (-i c x +1\right )-3 i \operatorname {dilog}\left (i c x +1\right )+3 i \operatorname {dilog}\left (-i c x +1\right )\right )\right )\) | \(181\) |
risch | \(\frac {9 a \,c^{2} d^{4}}{2}-\frac {b c \,d^{4}}{2 x}-\frac {b \,c^{3} d^{4} x}{2}-4 i a \,c^{3} d^{4} x +\frac {7 i b \,d^{4} c^{2} \ln \left (i c x \right )}{4}+3 i c^{2} d^{4} b \operatorname {dilog}\left (-i c x +1\right )-\frac {i d^{4} b \ln \left (-i c x +1\right )}{4 x^{2}}-\frac {4 i c \,d^{4} a}{x}+\frac {i b \,d^{4} \ln \left (i c x +1\right )}{4 x^{2}}+\frac {9 i c^{2} d^{4} b \ln \left (-i c x \right )}{4}+\frac {i c^{4} d^{4} b \ln \left (-i c x +1\right ) x^{2}}{4}+\frac {c^{4} d^{4} a \,x^{2}}{2}-\frac {i b \,d^{4} c^{4} \ln \left (i c x +1\right ) x^{2}}{4}-4 i b \,d^{4} c^{2}-6 c^{2} d^{4} a \ln \left (-i c x \right )-3 i b \,d^{4} c^{2} \operatorname {dilog}\left (i c x +1\right )-\frac {d^{4} a}{2 x^{2}}-2 b \,d^{4} c^{3} \ln \left (i c x +1\right ) x -\frac {2 b \,d^{4} c \ln \left (i c x +1\right )}{x}+\frac {2 c \,d^{4} b \ln \left (-i c x +1\right )}{x}+2 c^{3} d^{4} b \ln \left (-i c x +1\right ) x\) | \(317\) |
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\[ \int \frac {(d+i c d x)^4 (a+b \arctan (c x))}{x^3} \, dx=\int { \frac {{\left (i \, c d x + d\right )}^{4} {\left (b \arctan \left (c x\right ) + a\right )}}{x^{3}} \,d x } \]
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Timed out. \[ \int \frac {(d+i c d x)^4 (a+b \arctan (c x))}{x^3} \, dx=\text {Timed out} \]
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Time = 0.43 (sec) , antiderivative size = 251, normalized size of antiderivative = 1.45 \[ \int \frac {(d+i c d x)^4 (a+b \arctan (c x))}{x^3} \, dx=\frac {1}{2} \, a c^{4} d^{4} x^{2} - 4 i \, a c^{3} d^{4} x - \frac {1}{2} \, b c^{3} d^{4} x + \frac {3}{2} \, \pi b c^{2} d^{4} \log \left (c^{2} x^{2} + 1\right ) - 6 \, b c^{2} d^{4} \arctan \left (c x\right ) \log \left (c x\right ) - 2 i \, {\left (2 \, c x \arctan \left (c x\right ) - \log \left (c^{2} x^{2} + 1\right )\right )} b c^{2} d^{4} + 3 i \, b c^{2} d^{4} {\rm Li}_2\left (i \, c x + 1\right ) - 3 i \, b c^{2} d^{4} {\rm Li}_2\left (-i \, c x + 1\right ) - 6 \, a c^{2} d^{4} \log \left (x\right ) - 2 i \, {\left (c {\left (\log \left (c^{2} x^{2} + 1\right ) - \log \left (x^{2}\right )\right )} + \frac {2 \, \arctan \left (c x\right )}{x}\right )} b c d^{4} - \frac {1}{2} \, {\left ({\left (c \arctan \left (c x\right ) + \frac {1}{x}\right )} c + \frac {\arctan \left (c x\right )}{x^{2}}\right )} b d^{4} - \frac {4 i \, a c d^{4}}{x} - \frac {a d^{4}}{2 \, x^{2}} + \frac {1}{2} \, {\left (b c^{4} d^{4} x^{2} + b c^{2} d^{4}\right )} \arctan \left (c x\right ) \]
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\[ \int \frac {(d+i c d x)^4 (a+b \arctan (c x))}{x^3} \, dx=\int { \frac {{\left (i \, c d x + d\right )}^{4} {\left (b \arctan \left (c x\right ) + a\right )}}{x^{3}} \,d x } \]
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Time = 0.94 (sec) , antiderivative size = 258, normalized size of antiderivative = 1.49 \[ \int \frac {(d+i c d x)^4 (a+b \arctan (c x))}{x^3} \, dx=\left \{\begin {array}{cl} -\frac {a\,d^4}{2\,x^2} & \text {\ if\ \ }c=0\\ \frac {a\,c^4\,d^4\,x^2}{2}-\frac {\frac {a\,d^4}{2}+a\,c\,d^4\,x\,4{}\mathrm {i}}{x^2}-6\,a\,c^2\,d^4\,\ln \left (x\right )-\frac {b\,d^4\,\left (c^3\,\mathrm {atan}\left (c\,x\right )+\frac {c^2}{x}\right )}{2\,c}-\frac {b\,c^3\,d^4\,x}{2}-\frac {b\,d^4\,\mathrm {atan}\left (c\,x\right )}{2\,x^2}+b\,c^4\,d^4\,\mathrm {atan}\left (c\,x\right )\,\left (\frac {1}{2\,c^2}+\frac {x^2}{2}\right )+b\,d^4\,\left (c^2\,\ln \left (x\right )-\frac {c^2\,\ln \left (c^2\,x^2+1\right )}{2}\right )\,4{}\mathrm {i}+b\,c^2\,d^4\,\ln \left (c^2\,x^2+1\right )\,2{}\mathrm {i}+b\,c^2\,d^4\,{\mathrm {Li}}_{\mathrm {2}}\left (1-c\,x\,1{}\mathrm {i}\right )\,3{}\mathrm {i}-b\,c^2\,d^4\,{\mathrm {Li}}_{\mathrm {2}}\left (1+c\,x\,1{}\mathrm {i}\right )\,3{}\mathrm {i}-a\,c^3\,d^4\,x\,4{}\mathrm {i}-\frac {b\,c\,d^4\,\mathrm {atan}\left (c\,x\right )\,4{}\mathrm {i}}{x}-b\,c^3\,d^4\,x\,\mathrm {atan}\left (c\,x\right )\,4{}\mathrm {i} & \text {\ if\ \ }c\neq 0 \end {array}\right . \]
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