Integrand size = 23, antiderivative size = 203 \[ \int \frac {(d+i c d x)^4 (a+b \arctan (c x))}{x} \, dx=4 i a c d^4 x+\frac {13}{4} b c d^4 x+\frac {2}{3} i b c^2 d^4 x^2-\frac {1}{12} b c^3 d^4 x^3-\frac {13}{4} b d^4 \arctan (c x)+4 i b c d^4 x \arctan (c x)-3 c^2 d^4 x^2 (a+b \arctan (c x))-\frac {4}{3} i c^3 d^4 x^3 (a+b \arctan (c x))+\frac {1}{4} c^4 d^4 x^4 (a+b \arctan (c x))+a d^4 \log (x)-\frac {8}{3} i b d^4 \log \left (1+c^2 x^2\right )+\frac {1}{2} i b d^4 \operatorname {PolyLog}(2,-i c x)-\frac {1}{2} i b d^4 \operatorname {PolyLog}(2,i c x) \]
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Time = 0.15 (sec) , antiderivative size = 203, normalized size of antiderivative = 1.00, number of steps used = 19, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.478, Rules used = {4996, 4930, 266, 4940, 2438, 4946, 327, 209, 272, 45, 308} \[ \int \frac {(d+i c d x)^4 (a+b \arctan (c x))}{x} \, dx=\frac {1}{4} c^4 d^4 x^4 (a+b \arctan (c x))-\frac {4}{3} i c^3 d^4 x^3 (a+b \arctan (c x))-3 c^2 d^4 x^2 (a+b \arctan (c x))+4 i a c d^4 x+a d^4 \log (x)-\frac {13}{4} b d^4 \arctan (c x)+4 i b c d^4 x \arctan (c x)-\frac {1}{12} b c^3 d^4 x^3+\frac {2}{3} i b c^2 d^4 x^2-\frac {8}{3} i b d^4 \log \left (c^2 x^2+1\right )+\frac {1}{2} i b d^4 \operatorname {PolyLog}(2,-i c x)-\frac {1}{2} i b d^4 \operatorname {PolyLog}(2,i c x)+\frac {13}{4} b c d^4 x \]
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Rule 45
Rule 209
Rule 266
Rule 272
Rule 308
Rule 327
Rule 2438
Rule 4930
Rule 4940
Rule 4946
Rule 4996
Rubi steps \begin{align*} \text {integral}& = \int \left (4 i c d^4 (a+b \arctan (c x))+\frac {d^4 (a+b \arctan (c x))}{x}-6 c^2 d^4 x (a+b \arctan (c x))-4 i c^3 d^4 x^2 (a+b \arctan (c x))+c^4 d^4 x^3 (a+b \arctan (c x))\right ) \, dx \\ & = d^4 \int \frac {a+b \arctan (c x)}{x} \, dx+\left (4 i c d^4\right ) \int (a+b \arctan (c x)) \, dx-\left (6 c^2 d^4\right ) \int x (a+b \arctan (c x)) \, dx-\left (4 i c^3 d^4\right ) \int x^2 (a+b \arctan (c x)) \, dx+\left (c^4 d^4\right ) \int x^3 (a+b \arctan (c x)) \, dx \\ & = 4 i a c d^4 x-3 c^2 d^4 x^2 (a+b \arctan (c x))-\frac {4}{3} i c^3 d^4 x^3 (a+b \arctan (c x))+\frac {1}{4} c^4 d^4 x^4 (a+b \arctan (c x))+a d^4 \log (x)+\frac {1}{2} \left (i b d^4\right ) \int \frac {\log (1-i c x)}{x} \, dx-\frac {1}{2} \left (i b d^4\right ) \int \frac {\log (1+i c x)}{x} \, dx+\left (4 i b c d^4\right ) \int \arctan (c x) \, dx+\left (3 b c^3 d^4\right ) \int \frac {x^2}{1+c^2 x^2} \, dx+\frac {1}{3} \left (4 i b c^4 d^4\right ) \int \frac {x^3}{1+c^2 x^2} \, dx-\frac {1}{4} \left (b c^5 d^4\right ) \int \frac {x^4}{1+c^2 x^2} \, dx \\ & = 4 i a c d^4 x+3 b c d^4 x+4 i b c d^4 x \arctan (c x)-3 c^2 d^4 x^2 (a+b \arctan (c x))-\frac {4}{3} i c^3 d^4 x^3 (a+b \arctan (c x))+\frac {1}{4} c^4 d^4 x^4 (a+b \arctan (c x))+a d^4 \log (x)+\frac {1}{2} i b d^4 \operatorname {PolyLog}(2,-i c x)-\frac {1}{2} i b d^4 \operatorname {PolyLog}(2,i c x)-\left (3 b c d^4\right ) \int \frac {1}{1+c^2 x^2} \, dx-\left (4 i b c^2 d^4\right ) \int \frac {x}{1+c^2 x^2} \, dx+\frac {1}{3} \left (2 i b c^4 d^4\right ) \text {Subst}\left (\int \frac {x}{1+c^2 x} \, dx,x,x^2\right )-\frac {1}{4} \left (b c^5 d^4\right ) \int \left (-\frac {1}{c^4}+\frac {x^2}{c^2}+\frac {1}{c^4 \left (1+c^2 x^2\right )}\right ) \, dx \\ & = 4 i a c d^4 x+\frac {13}{4} b c d^4 x-\frac {1}{12} b c^3 d^4 x^3-3 b d^4 \arctan (c x)+4 i b c d^4 x \arctan (c x)-3 c^2 d^4 x^2 (a+b \arctan (c x))-\frac {4}{3} i c^3 d^4 x^3 (a+b \arctan (c x))+\frac {1}{4} c^4 d^4 x^4 (a+b \arctan (c x))+a d^4 \log (x)-2 i b d^4 \log \left (1+c^2 x^2\right )+\frac {1}{2} i b d^4 \operatorname {PolyLog}(2,-i c x)-\frac {1}{2} i b d^4 \operatorname {PolyLog}(2,i c x)-\frac {1}{4} \left (b c d^4\right ) \int \frac {1}{1+c^2 x^2} \, dx+\frac {1}{3} \left (2 i b c^4 d^4\right ) \text {Subst}\left (\int \left (\frac {1}{c^2}-\frac {1}{c^2 \left (1+c^2 x\right )}\right ) \, dx,x,x^2\right ) \\ & = 4 i a c d^4 x+\frac {13}{4} b c d^4 x+\frac {2}{3} i b c^2 d^4 x^2-\frac {1}{12} b c^3 d^4 x^3-\frac {13}{4} b d^4 \arctan (c x)+4 i b c d^4 x \arctan (c x)-3 c^2 d^4 x^2 (a+b \arctan (c x))-\frac {4}{3} i c^3 d^4 x^3 (a+b \arctan (c x))+\frac {1}{4} c^4 d^4 x^4 (a+b \arctan (c x))+a d^4 \log (x)-\frac {8}{3} i b d^4 \log \left (1+c^2 x^2\right )+\frac {1}{2} i b d^4 \operatorname {PolyLog}(2,-i c x)-\frac {1}{2} i b d^4 \operatorname {PolyLog}(2,i c x) \\ \end{align*}
Time = 0.12 (sec) , antiderivative size = 174, normalized size of antiderivative = 0.86 \[ \int \frac {(d+i c d x)^4 (a+b \arctan (c x))}{x} \, dx=\frac {1}{12} d^4 \left (48 i a c x+39 b c x-36 a c^2 x^2+8 i b c^2 x^2-16 i a c^3 x^3-b c^3 x^3+3 a c^4 x^4-39 b \arctan (c x)+48 i b c x \arctan (c x)-36 b c^2 x^2 \arctan (c x)-16 i b c^3 x^3 \arctan (c x)+3 b c^4 x^4 \arctan (c x)+12 a \log (x)-32 i b \log \left (1+c^2 x^2\right )+6 i b \operatorname {PolyLog}(2,-i c x)-6 i b \operatorname {PolyLog}(2,i c x)\right ) \]
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Time = 1.42 (sec) , antiderivative size = 193, normalized size of antiderivative = 0.95
method | result | size |
parts | \(d^{4} a \left (\frac {c^{4} x^{4}}{4}-\frac {4 i c^{3} x^{3}}{3}-3 c^{2} x^{2}+4 i c x +\ln \left (x \right )\right )+d^{4} b \left (4 i \arctan \left (c x \right ) c x +\frac {c^{4} x^{4} \arctan \left (c x \right )}{4}-\frac {4 i \arctan \left (c x \right ) c^{3} x^{3}}{3}-3 c^{2} x^{2} \arctan \left (c x \right )+\arctan \left (c x \right ) \ln \left (c x \right )+\frac {i \ln \left (c x \right ) \ln \left (i c x +1\right )}{2}-\frac {i \ln \left (c x \right ) \ln \left (-i c x +1\right )}{2}+\frac {i \operatorname {dilog}\left (i c x +1\right )}{2}-\frac {i \operatorname {dilog}\left (-i c x +1\right )}{2}+\frac {13 c x}{4}-\frac {c^{3} x^{3}}{12}+\frac {2 i c^{2} x^{2}}{3}-\frac {8 i \ln \left (c^{2} x^{2}+1\right )}{3}-\frac {13 \arctan \left (c x \right )}{4}\right )\) | \(193\) |
derivativedivides | \(d^{4} a \left (4 i c x +\frac {c^{4} x^{4}}{4}-\frac {4 i c^{3} x^{3}}{3}-3 c^{2} x^{2}+\ln \left (c x \right )\right )+d^{4} b \left (4 i \arctan \left (c x \right ) c x +\frac {c^{4} x^{4} \arctan \left (c x \right )}{4}-\frac {4 i \arctan \left (c x \right ) c^{3} x^{3}}{3}-3 c^{2} x^{2} \arctan \left (c x \right )+\arctan \left (c x \right ) \ln \left (c x \right )+\frac {i \ln \left (c x \right ) \ln \left (i c x +1\right )}{2}-\frac {i \ln \left (c x \right ) \ln \left (-i c x +1\right )}{2}+\frac {i \operatorname {dilog}\left (i c x +1\right )}{2}-\frac {i \operatorname {dilog}\left (-i c x +1\right )}{2}+\frac {13 c x}{4}-\frac {c^{3} x^{3}}{12}+\frac {2 i c^{2} x^{2}}{3}-\frac {8 i \ln \left (c^{2} x^{2}+1\right )}{3}-\frac {13 \arctan \left (c x \right )}{4}\right )\) | \(195\) |
default | \(d^{4} a \left (4 i c x +\frac {c^{4} x^{4}}{4}-\frac {4 i c^{3} x^{3}}{3}-3 c^{2} x^{2}+\ln \left (c x \right )\right )+d^{4} b \left (4 i \arctan \left (c x \right ) c x +\frac {c^{4} x^{4} \arctan \left (c x \right )}{4}-\frac {4 i \arctan \left (c x \right ) c^{3} x^{3}}{3}-3 c^{2} x^{2} \arctan \left (c x \right )+\arctan \left (c x \right ) \ln \left (c x \right )+\frac {i \ln \left (c x \right ) \ln \left (i c x +1\right )}{2}-\frac {i \ln \left (c x \right ) \ln \left (-i c x +1\right )}{2}+\frac {i \operatorname {dilog}\left (i c x +1\right )}{2}-\frac {i \operatorname {dilog}\left (-i c x +1\right )}{2}+\frac {13 c x}{4}-\frac {c^{3} x^{3}}{12}+\frac {2 i c^{2} x^{2}}{3}-\frac {8 i \ln \left (c^{2} x^{2}+1\right )}{3}-\frac {13 \arctan \left (c x \right )}{4}\right )\) | \(195\) |
risch | \(-3 x^{2} d^{4} c^{2} a +\frac {13 b c \,d^{4} x}{4}-\frac {103 d^{4} a}{12}-\frac {b \,c^{3} d^{4} x^{3}}{12}-\frac {i b \,d^{4} \ln \left (i c x +1\right ) c^{4} x^{4}}{8}+\frac {i b \,d^{4} \operatorname {dilog}\left (i c x +1\right )}{2}+\frac {a \,c^{4} d^{4} x^{4}}{4}-\frac {103 i d^{4} b \ln \left (-i c x +1\right )}{24}+\frac {58 i b \,d^{4}}{9}-\frac {2 b \,d^{4} \ln \left (i c x +1\right ) c^{3} x^{3}}{3}+\frac {3 i b \,d^{4} \ln \left (i c x +1\right ) c^{2} x^{2}}{2}+4 i a c \,d^{4} x -\frac {3 i d^{4} b \ln \left (-i c x +1\right ) c^{2} x^{2}}{2}+d^{4} a \ln \left (-i c x \right )+\frac {2 d^{4} b \,c^{3} x^{3} \ln \left (-i c x +1\right )}{3}-2 d^{4} b c x \ln \left (-i c x +1\right )-\frac {4 i x^{3} a \,c^{3} d^{4}}{3}+2 b \,d^{4} \ln \left (i c x +1\right ) c x -\frac {25 i b \,d^{4} \ln \left (i c x +1\right )}{24}+\frac {2 i x^{2} b \,c^{2} d^{4}}{3}+\frac {i d^{4} b \ln \left (-i c x +1\right ) c^{4} x^{4}}{8}-\frac {i d^{4} b \operatorname {dilog}\left (-i c x +1\right )}{2}\) | \(321\) |
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\[ \int \frac {(d+i c d x)^4 (a+b \arctan (c x))}{x} \, dx=\int { \frac {{\left (i \, c d x + d\right )}^{4} {\left (b \arctan \left (c x\right ) + a\right )}}{x} \,d x } \]
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Timed out. \[ \int \frac {(d+i c d x)^4 (a+b \arctan (c x))}{x} \, dx=\text {Timed out} \]
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Time = 0.45 (sec) , antiderivative size = 220, normalized size of antiderivative = 1.08 \[ \int \frac {(d+i c d x)^4 (a+b \arctan (c x))}{x} \, dx=\frac {1}{4} \, a c^{4} d^{4} x^{4} - \frac {4}{3} i \, a c^{3} d^{4} x^{3} - \frac {1}{12} \, b c^{3} d^{4} x^{3} - 3 \, a c^{2} d^{4} x^{2} + \frac {2}{3} i \, b c^{2} d^{4} x^{2} + 4 i \, a c d^{4} x + \frac {13}{4} \, b c d^{4} x - \frac {1}{12} \, {\left (3 \, \pi + 8 i\right )} b d^{4} \log \left (c^{2} x^{2} + 1\right ) + b d^{4} \arctan \left (c x\right ) \log \left (c x\right ) + 2 i \, {\left (2 \, c x \arctan \left (c x\right ) - \log \left (c^{2} x^{2} + 1\right )\right )} b d^{4} - \frac {1}{2} i \, b d^{4} {\rm Li}_2\left (i \, c x + 1\right ) + \frac {1}{2} i \, b d^{4} {\rm Li}_2\left (-i \, c x + 1\right ) + a d^{4} \log \left (x\right ) + \frac {1}{12} \, {\left (3 \, b c^{4} d^{4} x^{4} - 16 i \, b c^{3} d^{4} x^{3} - 36 \, b c^{2} d^{4} x^{2} - 39 \, b d^{4}\right )} \arctan \left (c x\right ) \]
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\[ \int \frac {(d+i c d x)^4 (a+b \arctan (c x))}{x} \, dx=\int { \frac {{\left (i \, c d x + d\right )}^{4} {\left (b \arctan \left (c x\right ) + a\right )}}{x} \,d x } \]
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Time = 1.05 (sec) , antiderivative size = 248, normalized size of antiderivative = 1.22 \[ \int \frac {(d+i c d x)^4 (a+b \arctan (c x))}{x} \, dx=\left \{\begin {array}{cl} a\,d^4\,\ln \left (x\right ) & \text {\ if\ \ }c=0\\ a\,d^4\,\ln \left (x\right )-b\,d^4\,\ln \left (c^2\,x^2+1\right )\,2{}\mathrm {i}-\frac {b\,d^4\,\left (3\,\mathrm {atan}\left (c\,x\right )-3\,c\,x+c^3\,x^3\right )}{12}-\frac {b\,d^4\,{\mathrm {Li}}_{\mathrm {2}}\left (1-c\,x\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{2}+\frac {b\,d^4\,{\mathrm {Li}}_{\mathrm {2}}\left (1+c\,x\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{2}-3\,a\,c^2\,d^4\,x^2-\frac {a\,c^3\,d^4\,x^3\,4{}\mathrm {i}}{3}+\frac {a\,c^4\,d^4\,x^4}{4}+a\,c\,d^4\,x\,4{}\mathrm {i}+3\,b\,c\,d^4\,x+\frac {b\,c^2\,d^4\,\left (\frac {x^2}{2}-\frac {\ln \left (c^2\,x^2+1\right )}{2\,c^2}\right )\,4{}\mathrm {i}}{3}-6\,b\,c^2\,d^4\,\mathrm {atan}\left (c\,x\right )\,\left (\frac {1}{2\,c^2}+\frac {x^2}{2}\right )-\frac {b\,c^3\,d^4\,x^3\,\mathrm {atan}\left (c\,x\right )\,4{}\mathrm {i}}{3}+\frac {b\,c^4\,d^4\,x^4\,\mathrm {atan}\left (c\,x\right )}{4}+b\,c\,d^4\,x\,\mathrm {atan}\left (c\,x\right )\,4{}\mathrm {i} & \text {\ if\ \ }c\neq 0 \end {array}\right . \]
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