Integrand size = 23, antiderivative size = 180 \[ \int \frac {(d+i c d x)^3 (a+b \arctan (c x))}{x^3} \, dx=-\frac {b c d^3}{2 x}-i a c^3 d^3 x-\frac {1}{2} b c^2 d^3 \arctan (c x)-i b c^3 d^3 x \arctan (c x)-\frac {d^3 (a+b \arctan (c x))}{2 x^2}-\frac {3 i c d^3 (a+b \arctan (c x))}{x}-3 a c^2 d^3 \log (x)+3 i b c^2 d^3 \log (x)-i b c^2 d^3 \log \left (1+c^2 x^2\right )-\frac {3}{2} i b c^2 d^3 \operatorname {PolyLog}(2,-i c x)+\frac {3}{2} i b c^2 d^3 \operatorname {PolyLog}(2,i c x) \]
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Time = 0.13 (sec) , antiderivative size = 180, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.522, Rules used = {4996, 4930, 266, 4946, 331, 209, 272, 36, 29, 31, 4940, 2438} \[ \int \frac {(d+i c d x)^3 (a+b \arctan (c x))}{x^3} \, dx=-\frac {d^3 (a+b \arctan (c x))}{2 x^2}-\frac {3 i c d^3 (a+b \arctan (c x))}{x}-i a c^3 d^3 x-3 a c^2 d^3 \log (x)-i b c^3 d^3 x \arctan (c x)-\frac {1}{2} b c^2 d^3 \arctan (c x)-\frac {3}{2} i b c^2 d^3 \operatorname {PolyLog}(2,-i c x)+\frac {3}{2} i b c^2 d^3 \operatorname {PolyLog}(2,i c x)-i b c^2 d^3 \log \left (c^2 x^2+1\right )+3 i b c^2 d^3 \log (x)-\frac {b c d^3}{2 x} \]
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Rule 29
Rule 31
Rule 36
Rule 209
Rule 266
Rule 272
Rule 331
Rule 2438
Rule 4930
Rule 4940
Rule 4946
Rule 4996
Rubi steps \begin{align*} \text {integral}& = \int \left (-i c^3 d^3 (a+b \arctan (c x))+\frac {d^3 (a+b \arctan (c x))}{x^3}+\frac {3 i c d^3 (a+b \arctan (c x))}{x^2}-\frac {3 c^2 d^3 (a+b \arctan (c x))}{x}\right ) \, dx \\ & = d^3 \int \frac {a+b \arctan (c x)}{x^3} \, dx+\left (3 i c d^3\right ) \int \frac {a+b \arctan (c x)}{x^2} \, dx-\left (3 c^2 d^3\right ) \int \frac {a+b \arctan (c x)}{x} \, dx-\left (i c^3 d^3\right ) \int (a+b \arctan (c x)) \, dx \\ & = -i a c^3 d^3 x-\frac {d^3 (a+b \arctan (c x))}{2 x^2}-\frac {3 i c d^3 (a+b \arctan (c x))}{x}-3 a c^2 d^3 \log (x)+\frac {1}{2} \left (b c d^3\right ) \int \frac {1}{x^2 \left (1+c^2 x^2\right )} \, dx-\frac {1}{2} \left (3 i b c^2 d^3\right ) \int \frac {\log (1-i c x)}{x} \, dx+\frac {1}{2} \left (3 i b c^2 d^3\right ) \int \frac {\log (1+i c x)}{x} \, dx+\left (3 i b c^2 d^3\right ) \int \frac {1}{x \left (1+c^2 x^2\right )} \, dx-\left (i b c^3 d^3\right ) \int \arctan (c x) \, dx \\ & = -\frac {b c d^3}{2 x}-i a c^3 d^3 x-i b c^3 d^3 x \arctan (c x)-\frac {d^3 (a+b \arctan (c x))}{2 x^2}-\frac {3 i c d^3 (a+b \arctan (c x))}{x}-3 a c^2 d^3 \log (x)-\frac {3}{2} i b c^2 d^3 \operatorname {PolyLog}(2,-i c x)+\frac {3}{2} i b c^2 d^3 \operatorname {PolyLog}(2,i c x)+\frac {1}{2} \left (3 i b c^2 d^3\right ) \text {Subst}\left (\int \frac {1}{x \left (1+c^2 x\right )} \, dx,x,x^2\right )-\frac {1}{2} \left (b c^3 d^3\right ) \int \frac {1}{1+c^2 x^2} \, dx+\left (i b c^4 d^3\right ) \int \frac {x}{1+c^2 x^2} \, dx \\ & = -\frac {b c d^3}{2 x}-i a c^3 d^3 x-\frac {1}{2} b c^2 d^3 \arctan (c x)-i b c^3 d^3 x \arctan (c x)-\frac {d^3 (a+b \arctan (c x))}{2 x^2}-\frac {3 i c d^3 (a+b \arctan (c x))}{x}-3 a c^2 d^3 \log (x)+\frac {1}{2} i b c^2 d^3 \log \left (1+c^2 x^2\right )-\frac {3}{2} i b c^2 d^3 \operatorname {PolyLog}(2,-i c x)+\frac {3}{2} i b c^2 d^3 \operatorname {PolyLog}(2,i c x)+\frac {1}{2} \left (3 i b c^2 d^3\right ) \text {Subst}\left (\int \frac {1}{x} \, dx,x,x^2\right )-\frac {1}{2} \left (3 i b c^4 d^3\right ) \text {Subst}\left (\int \frac {1}{1+c^2 x} \, dx,x,x^2\right ) \\ & = -\frac {b c d^3}{2 x}-i a c^3 d^3 x-\frac {1}{2} b c^2 d^3 \arctan (c x)-i b c^3 d^3 x \arctan (c x)-\frac {d^3 (a+b \arctan (c x))}{2 x^2}-\frac {3 i c d^3 (a+b \arctan (c x))}{x}-3 a c^2 d^3 \log (x)+3 i b c^2 d^3 \log (x)-i b c^2 d^3 \log \left (1+c^2 x^2\right )-\frac {3}{2} i b c^2 d^3 \operatorname {PolyLog}(2,-i c x)+\frac {3}{2} i b c^2 d^3 \operatorname {PolyLog}(2,i c x) \\ \end{align*}
Time = 0.16 (sec) , antiderivative size = 164, normalized size of antiderivative = 0.91 \[ \int \frac {(d+i c d x)^3 (a+b \arctan (c x))}{x^3} \, dx=-\frac {i d^3 \left (-i a+6 a c x-i b c x+2 a c^3 x^3-i b \arctan (c x)+6 b c x \arctan (c x)-i b c^2 x^2 \arctan (c x)+2 b c^3 x^3 \arctan (c x)-6 i a c^2 x^2 \log (x)-6 b c^2 x^2 \log (c x)+2 b c^2 x^2 \log \left (1+c^2 x^2\right )+3 b c^2 x^2 \operatorname {PolyLog}(2,-i c x)-3 b c^2 x^2 \operatorname {PolyLog}(2,i c x)\right )}{2 x^2} \]
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Time = 1.39 (sec) , antiderivative size = 173, normalized size of antiderivative = 0.96
method | result | size |
parts | \(a \,d^{3} \left (-i c^{3} x -\frac {1}{2 x^{2}}-3 c^{2} \ln \left (x \right )-\frac {3 i c}{x}\right )+b \,d^{3} c^{2} \left (-i \arctan \left (c x \right ) c x -3 \arctan \left (c x \right ) \ln \left (c x \right )-\frac {3 i \arctan \left (c x \right )}{c x}-\frac {\arctan \left (c x \right )}{2 c^{2} x^{2}}-\frac {3 i \ln \left (c x \right ) \ln \left (i c x +1\right )}{2}+\frac {3 i \ln \left (c x \right ) \ln \left (-i c x +1\right )}{2}-\frac {3 i \operatorname {dilog}\left (i c x +1\right )}{2}+\frac {3 i \operatorname {dilog}\left (-i c x +1\right )}{2}-\frac {1}{2 c x}+3 i \ln \left (c x \right )-i \ln \left (c^{2} x^{2}+1\right )-\frac {\arctan \left (c x \right )}{2}\right )\) | \(173\) |
derivativedivides | \(c^{2} \left (a \,d^{3} \left (-i c x -3 \ln \left (c x \right )-\frac {3 i}{c x}-\frac {1}{2 c^{2} x^{2}}\right )+b \,d^{3} \left (-i \arctan \left (c x \right ) c x -3 \arctan \left (c x \right ) \ln \left (c x \right )-\frac {3 i \arctan \left (c x \right )}{c x}-\frac {\arctan \left (c x \right )}{2 c^{2} x^{2}}-\frac {3 i \ln \left (c x \right ) \ln \left (i c x +1\right )}{2}+\frac {3 i \ln \left (c x \right ) \ln \left (-i c x +1\right )}{2}-\frac {3 i \operatorname {dilog}\left (i c x +1\right )}{2}+\frac {3 i \operatorname {dilog}\left (-i c x +1\right )}{2}-\frac {1}{2 c x}+3 i \ln \left (c x \right )-i \ln \left (c^{2} x^{2}+1\right )-\frac {\arctan \left (c x \right )}{2}\right )\right )\) | \(176\) |
default | \(c^{2} \left (a \,d^{3} \left (-i c x -3 \ln \left (c x \right )-\frac {3 i}{c x}-\frac {1}{2 c^{2} x^{2}}\right )+b \,d^{3} \left (-i \arctan \left (c x \right ) c x -3 \arctan \left (c x \right ) \ln \left (c x \right )-\frac {3 i \arctan \left (c x \right )}{c x}-\frac {\arctan \left (c x \right )}{2 c^{2} x^{2}}-\frac {3 i \ln \left (c x \right ) \ln \left (i c x +1\right )}{2}+\frac {3 i \ln \left (c x \right ) \ln \left (-i c x +1\right )}{2}-\frac {3 i \operatorname {dilog}\left (i c x +1\right )}{2}+\frac {3 i \operatorname {dilog}\left (-i c x +1\right )}{2}-\frac {1}{2 c x}+3 i \ln \left (c x \right )-i \ln \left (c^{2} x^{2}+1\right )-\frac {\arctan \left (c x \right )}{2}\right )\right )\) | \(176\) |
risch | \(-\frac {b \,c^{3} d^{3} \ln \left (i c x +1\right ) x}{2}+\frac {3 i d^{3} c^{2} b \operatorname {dilog}\left (-i c x +1\right )}{2}+\frac {7 i d^{3} c^{2} b \ln \left (-i c x \right )}{4}-\frac {i d^{3} b \ln \left (-i c x +1\right )}{4 x^{2}}-i c^{2} b \,d^{3}-\frac {3 b c \,d^{3} \ln \left (i c x +1\right )}{2 x}-\frac {b c \,d^{3}}{2 x}-i a \,c^{3} d^{3} x +a \,c^{2} d^{3}+\frac {i b \,d^{3} \ln \left (i c x +1\right )}{4 x^{2}}-3 d^{3} c^{2} a \ln \left (-i c x \right )-\frac {3 i d^{3} c a}{x}-\frac {b \,c^{2} d^{3} \arctan \left (c x \right )}{2}-\frac {d^{3} a}{2 x^{2}}+\frac {d^{3} c^{3} b \ln \left (-i c x +1\right ) x}{2}+\frac {5 i b \,c^{2} d^{3} \ln \left (i c x \right )}{4}-\frac {3 i b \,c^{2} d^{3} \operatorname {dilog}\left (i c x +1\right )}{2}-i b \,c^{2} d^{3} \ln \left (c^{2} x^{2}+1\right )+\frac {3 d^{3} c b \ln \left (-i c x +1\right )}{2 x}\) | \(285\) |
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\[ \int \frac {(d+i c d x)^3 (a+b \arctan (c x))}{x^3} \, dx=\int { \frac {{\left (i \, c d x + d\right )}^{3} {\left (b \arctan \left (c x\right ) + a\right )}}{x^{3}} \,d x } \]
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Timed out. \[ \int \frac {(d+i c d x)^3 (a+b \arctan (c x))}{x^3} \, dx=\text {Timed out} \]
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\[ \int \frac {(d+i c d x)^3 (a+b \arctan (c x))}{x^3} \, dx=\int { \frac {{\left (i \, c d x + d\right )}^{3} {\left (b \arctan \left (c x\right ) + a\right )}}{x^{3}} \,d x } \]
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\[ \int \frac {(d+i c d x)^3 (a+b \arctan (c x))}{x^3} \, dx=\int { \frac {{\left (i \, c d x + d\right )}^{3} {\left (b \arctan \left (c x\right ) + a\right )}}{x^{3}} \,d x } \]
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Time = 0.80 (sec) , antiderivative size = 205, normalized size of antiderivative = 1.14 \[ \int \frac {(d+i c d x)^3 (a+b \arctan (c x))}{x^3} \, dx=\left \{\begin {array}{cl} -\frac {a\,d^3}{2\,x^2} & \text {\ if\ \ }c=0\\ b\,d^3\,\left (c^2\,\ln \left (x\right )-\frac {c^2\,\ln \left (c^2\,x^2+1\right )}{2}\right )\,3{}\mathrm {i}+\frac {b\,c^2\,d^3\,\ln \left (c^2\,x^2+1\right )\,1{}\mathrm {i}}{2}+\frac {b\,c^2\,d^3\,{\mathrm {Li}}_{\mathrm {2}}\left (1-c\,x\,1{}\mathrm {i}\right )\,3{}\mathrm {i}}{2}-\frac {b\,c^2\,d^3\,{\mathrm {Li}}_{\mathrm {2}}\left (1+c\,x\,1{}\mathrm {i}\right )\,3{}\mathrm {i}}{2}-\frac {b\,d^3\,\left (c^3\,\mathrm {atan}\left (c\,x\right )+\frac {c^2}{x}\right )}{2\,c}-\frac {a\,d^3\,\left (6\,c^2\,x^2\,\ln \left (x\right )+1+c\,x\,6{}\mathrm {i}+c^3\,x^3\,2{}\mathrm {i}\right )}{2\,x^2}-\frac {b\,d^3\,\mathrm {atan}\left (c\,x\right )}{2\,x^2}-\frac {b\,c\,d^3\,\mathrm {atan}\left (c\,x\right )\,3{}\mathrm {i}}{x}-b\,c^3\,d^3\,x\,\mathrm {atan}\left (c\,x\right )\,1{}\mathrm {i} & \text {\ if\ \ }c\neq 0 \end {array}\right . \]
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