Integrand size = 23, antiderivative size = 162 \[ \int \frac {(d+i c d x)^3 (a+b \arctan (c x))}{x^2} \, dx=-3 a c^2 d^3 x+\frac {1}{2} i b c^2 d^3 x-\frac {1}{2} i b c d^3 \arctan (c x)-3 b c^2 d^3 x \arctan (c x)-\frac {d^3 (a+b \arctan (c x))}{x}-\frac {1}{2} i c^3 d^3 x^2 (a+b \arctan (c x))+3 i a c d^3 \log (x)+b c d^3 \log (x)+b c d^3 \log \left (1+c^2 x^2\right )-\frac {3}{2} b c d^3 \operatorname {PolyLog}(2,-i c x)+\frac {3}{2} b c d^3 \operatorname {PolyLog}(2,i c x) \]
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Time = 0.12 (sec) , antiderivative size = 162, 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, 272, 36, 29, 31, 4940, 2438, 327, 209} \[ \int \frac {(d+i c d x)^3 (a+b \arctan (c x))}{x^2} \, dx=-\frac {1}{2} i c^3 d^3 x^2 (a+b \arctan (c x))-\frac {d^3 (a+b \arctan (c x))}{x}-3 a c^2 d^3 x+3 i a c d^3 \log (x)-3 b c^2 d^3 x \arctan (c x)-\frac {1}{2} i b c d^3 \arctan (c x)+b c d^3 \log \left (c^2 x^2+1\right )+\frac {1}{2} i b c^2 d^3 x-\frac {3}{2} b c d^3 \operatorname {PolyLog}(2,-i c x)+\frac {3}{2} b c d^3 \operatorname {PolyLog}(2,i c x)+b c d^3 \log (x) \]
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Rule 29
Rule 31
Rule 36
Rule 209
Rule 266
Rule 272
Rule 327
Rule 2438
Rule 4930
Rule 4940
Rule 4946
Rule 4996
Rubi steps \begin{align*} \text {integral}& = \int \left (-3 c^2 d^3 (a+b \arctan (c x))+\frac {d^3 (a+b \arctan (c x))}{x^2}+\frac {3 i c d^3 (a+b \arctan (c x))}{x}-i c^3 d^3 x (a+b \arctan (c x))\right ) \, dx \\ & = d^3 \int \frac {a+b \arctan (c x)}{x^2} \, dx+\left (3 i c d^3\right ) \int \frac {a+b \arctan (c x)}{x} \, dx-\left (3 c^2 d^3\right ) \int (a+b \arctan (c x)) \, dx-\left (i c^3 d^3\right ) \int x (a+b \arctan (c x)) \, dx \\ & = -3 a c^2 d^3 x-\frac {d^3 (a+b \arctan (c x))}{x}-\frac {1}{2} i c^3 d^3 x^2 (a+b \arctan (c x))+3 i a c d^3 \log (x)+\left (b c d^3\right ) \int \frac {1}{x \left (1+c^2 x^2\right )} \, dx-\frac {1}{2} \left (3 b c d^3\right ) \int \frac {\log (1-i c x)}{x} \, dx+\frac {1}{2} \left (3 b c d^3\right ) \int \frac {\log (1+i c x)}{x} \, dx-\left (3 b c^2 d^3\right ) \int \arctan (c x) \, dx+\frac {1}{2} \left (i b c^4 d^3\right ) \int \frac {x^2}{1+c^2 x^2} \, dx \\ & = -3 a c^2 d^3 x+\frac {1}{2} i b c^2 d^3 x-3 b c^2 d^3 x \arctan (c x)-\frac {d^3 (a+b \arctan (c x))}{x}-\frac {1}{2} i c^3 d^3 x^2 (a+b \arctan (c x))+3 i a c d^3 \log (x)-\frac {3}{2} b c d^3 \operatorname {PolyLog}(2,-i c x)+\frac {3}{2} b c d^3 \operatorname {PolyLog}(2,i c x)+\frac {1}{2} \left (b c 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 (i b c^2 d^3\right ) \int \frac {1}{1+c^2 x^2} \, dx+\left (3 b c^3 d^3\right ) \int \frac {x}{1+c^2 x^2} \, dx \\ & = -3 a c^2 d^3 x+\frac {1}{2} i b c^2 d^3 x-\frac {1}{2} i b c d^3 \arctan (c x)-3 b c^2 d^3 x \arctan (c x)-\frac {d^3 (a+b \arctan (c x))}{x}-\frac {1}{2} i c^3 d^3 x^2 (a+b \arctan (c x))+3 i a c d^3 \log (x)+\frac {3}{2} b c d^3 \log \left (1+c^2 x^2\right )-\frac {3}{2} b c d^3 \operatorname {PolyLog}(2,-i c x)+\frac {3}{2} b c d^3 \operatorname {PolyLog}(2,i c x)+\frac {1}{2} \left (b c d^3\right ) \text {Subst}\left (\int \frac {1}{x} \, dx,x,x^2\right )-\frac {1}{2} \left (b c^3 d^3\right ) \text {Subst}\left (\int \frac {1}{1+c^2 x} \, dx,x,x^2\right ) \\ & = -3 a c^2 d^3 x+\frac {1}{2} i b c^2 d^3 x-\frac {1}{2} i b c d^3 \arctan (c x)-3 b c^2 d^3 x \arctan (c x)-\frac {d^3 (a+b \arctan (c x))}{x}-\frac {1}{2} i c^3 d^3 x^2 (a+b \arctan (c x))+3 i a c d^3 \log (x)+b c d^3 \log (x)+b c d^3 \log \left (1+c^2 x^2\right )-\frac {3}{2} b c d^3 \operatorname {PolyLog}(2,-i c x)+\frac {3}{2} b c d^3 \operatorname {PolyLog}(2,i c x) \\ \end{align*}
Time = 0.14 (sec) , antiderivative size = 150, normalized size of antiderivative = 0.93 \[ \int \frac {(d+i c d x)^3 (a+b \arctan (c x))}{x^2} \, dx=\frac {d^3 \left (-2 a-6 a c^2 x^2+i b c^2 x^2-i a c^3 x^3-2 b \arctan (c x)-i b c x \arctan (c x)-6 b c^2 x^2 \arctan (c x)-i b c^3 x^3 \arctan (c x)+6 i a c x \log (x)+2 b c x \log (c x)+2 b c x \log \left (1+c^2 x^2\right )-3 b c x \operatorname {PolyLog}(2,-i c x)+3 b c x \operatorname {PolyLog}(2,i c x)\right )}{2 x} \]
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Time = 1.11 (sec) , antiderivative size = 159, normalized size of antiderivative = 0.98
method | result | size |
parts | \(a \,d^{3} \left (-\frac {i c^{3} x^{2}}{2}-3 c^{2} x +3 i c \ln \left (x \right )-\frac {1}{x}\right )+b \,d^{3} c \left (-3 c x \arctan \left (c x \right )-\frac {i \arctan \left (c x \right ) c^{2} x^{2}}{2}+3 i \arctan \left (c x \right ) \ln \left (c x \right )-\frac {\arctan \left (c x \right )}{c x}-\frac {3 \ln \left (c x \right ) \ln \left (i c x +1\right )}{2}+\frac {3 \ln \left (c x \right ) \ln \left (-i c x +1\right )}{2}-\frac {3 \operatorname {dilog}\left (i c x +1\right )}{2}+\frac {3 \operatorname {dilog}\left (-i c x +1\right )}{2}+\frac {i c x}{2}+\ln \left (c x \right )+\ln \left (c^{2} x^{2}+1\right )-\frac {i \arctan \left (c x \right )}{2}\right )\) | \(159\) |
derivativedivides | \(c \left (a \,d^{3} \left (-3 c x -\frac {i c^{2} x^{2}}{2}+3 i \ln \left (c x \right )-\frac {1}{c x}\right )+b \,d^{3} \left (-3 c x \arctan \left (c x \right )-\frac {i \arctan \left (c x \right ) c^{2} x^{2}}{2}+3 i \arctan \left (c x \right ) \ln \left (c x \right )-\frac {\arctan \left (c x \right )}{c x}-\frac {3 \ln \left (c x \right ) \ln \left (i c x +1\right )}{2}+\frac {3 \ln \left (c x \right ) \ln \left (-i c x +1\right )}{2}-\frac {3 \operatorname {dilog}\left (i c x +1\right )}{2}+\frac {3 \operatorname {dilog}\left (-i c x +1\right )}{2}+\frac {i c x}{2}+\ln \left (c x \right )+\ln \left (c^{2} x^{2}+1\right )-\frac {i \arctan \left (c x \right )}{2}\right )\right )\) | \(162\) |
default | \(c \left (a \,d^{3} \left (-3 c x -\frac {i c^{2} x^{2}}{2}+3 i \ln \left (c x \right )-\frac {1}{c x}\right )+b \,d^{3} \left (-3 c x \arctan \left (c x \right )-\frac {i \arctan \left (c x \right ) c^{2} x^{2}}{2}+3 i \arctan \left (c x \right ) \ln \left (c x \right )-\frac {\arctan \left (c x \right )}{c x}-\frac {3 \ln \left (c x \right ) \ln \left (i c x +1\right )}{2}+\frac {3 \ln \left (c x \right ) \ln \left (-i c x +1\right )}{2}-\frac {3 \operatorname {dilog}\left (i c x +1\right )}{2}+\frac {3 \operatorname {dilog}\left (-i c x +1\right )}{2}+\frac {i c x}{2}+\ln \left (c x \right )+\ln \left (c^{2} x^{2}+1\right )-\frac {i \arctan \left (c x \right )}{2}\right )\right )\) | \(162\) |
risch | \(-\frac {b \,c^{3} d^{3} \ln \left (i c x +1\right ) x^{2}}{4}-\frac {7 i d^{3} c a}{2}+\frac {3 b c \,d^{3} \ln \left (i c x +1\right )}{4}+\frac {i b \,d^{3} \ln \left (i c x +1\right )}{2 x}-3 b c \,d^{3}-\frac {3 b c \,d^{3} \operatorname {dilog}\left (i c x +1\right )}{2}+\frac {b c \,d^{3} \ln \left (i c x \right )}{2}-\frac {3 i c^{2} d^{3} b x \ln \left (-i c x +1\right )}{2}+\frac {3 i b \,c^{2} d^{3} \ln \left (i c x +1\right ) x}{2}-3 c^{2} x \,d^{3} a +\frac {i b \,c^{2} d^{3} x}{2}+3 i c \,d^{3} \ln \left (-i c x \right ) a -\frac {d^{3} a}{x}+\frac {c^{3} d^{3} x^{2} b \ln \left (-i c x +1\right )}{4}-\frac {i c^{3} d^{3} a \,x^{2}}{2}+\frac {5 c \,d^{3} \ln \left (-i c x +1\right ) b}{4}-\frac {i d^{3} b \ln \left (-i c x +1\right )}{2 x}+\frac {3 c \,d^{3} \operatorname {dilog}\left (-i c x +1\right ) b}{2}+\frac {c \,d^{3} b \ln \left (-i c x \right )}{2}\) | \(274\) |
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\[ \int \frac {(d+i c d x)^3 (a+b \arctan (c x))}{x^2} \, dx=\int { \frac {{\left (i \, c d x + d\right )}^{3} {\left (b \arctan \left (c x\right ) + a\right )}}{x^{2}} \,d x } \]
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\[ \int \frac {(d+i c d x)^3 (a+b \arctan (c x))}{x^2} \, dx=- i d^{3} \left (\int \left (- 3 i a c^{2}\right )\, dx + \int \frac {i a}{x^{2}}\, dx + \int \left (- \frac {3 a c}{x}\right )\, dx + \int a c^{3} x\, dx + \int \left (- 3 i b c^{2} \operatorname {atan}{\left (c x \right )}\right )\, dx + \int \frac {i b \operatorname {atan}{\left (c x \right )}}{x^{2}}\, dx + \int \left (- \frac {3 b c \operatorname {atan}{\left (c x \right )}}{x}\right )\, dx + \int b c^{3} x \operatorname {atan}{\left (c x \right )}\, dx\right ) \]
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Time = 0.44 (sec) , antiderivative size = 201, normalized size of antiderivative = 1.24 \[ \int \frac {(d+i c d x)^3 (a+b \arctan (c x))}{x^2} \, dx=-\frac {1}{2} i \, a c^{3} d^{3} x^{2} - 3 \, a c^{2} d^{3} x + \frac {1}{2} i \, b c^{2} d^{3} x - \frac {3}{4} i \, \pi b c d^{3} \log \left (c^{2} x^{2} + 1\right ) + 3 i \, b c d^{3} \arctan \left (c x\right ) \log \left (c x\right ) - \frac {3}{2} \, {\left (2 \, c x \arctan \left (c x\right ) - \log \left (c^{2} x^{2} + 1\right )\right )} b c d^{3} + \frac {3}{2} \, b c d^{3} {\rm Li}_2\left (i \, c x + 1\right ) - \frac {3}{2} \, b c d^{3} {\rm Li}_2\left (-i \, c x + 1\right ) + 3 i \, a c d^{3} \log \left (x\right ) - \frac {1}{2} \, {\left (c {\left (\log \left (c^{2} x^{2} + 1\right ) - \log \left (x^{2}\right )\right )} + \frac {2 \, \arctan \left (c x\right )}{x}\right )} b d^{3} - \frac {a d^{3}}{x} + \frac {1}{2} \, {\left (-i \, b c^{3} d^{3} x^{2} - i \, b c d^{3}\right )} \arctan \left (c x\right ) \]
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\[ \int \frac {(d+i c d x)^3 (a+b \arctan (c x))}{x^2} \, dx=\int { \frac {{\left (i \, c d x + d\right )}^{3} {\left (b \arctan \left (c x\right ) + a\right )}}{x^{2}} \,d x } \]
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Time = 0.78 (sec) , antiderivative size = 195, normalized size of antiderivative = 1.20 \[ \int \frac {(d+i c d x)^3 (a+b \arctan (c x))}{x^2} \, dx=\left \{\begin {array}{cl} -\frac {a\,d^3}{x} & \text {\ if\ \ }c=0\\ \frac {b\,d^3\,\left (c^2\,\ln \left (x\right )-\frac {c^2\,\ln \left (c^2\,x^2+1\right )}{2}\right )}{c}-\frac {a\,c^3\,d^3\,x^2\,1{}\mathrm {i}}{2}-\frac {a\,d^3}{x}+\frac {3\,b\,c\,d^3\,\left ({\mathrm {Li}}_{\mathrm {2}}\left (1-c\,x\,1{}\mathrm {i}\right )-{\mathrm {Li}}_{\mathrm {2}}\left (1+c\,x\,1{}\mathrm {i}\right )\right )}{2}+\frac {3\,b\,c\,d^3\,\ln \left (c^2\,x^2+1\right )}{2}-3\,a\,c^2\,d^3\,x+\frac {b\,c^2\,d^3\,x\,1{}\mathrm {i}}{2}+a\,c\,d^3\,\ln \left (x\right )\,3{}\mathrm {i}-\frac {b\,d^3\,\mathrm {atan}\left (c\,x\right )}{x}-3\,b\,c^2\,d^3\,x\,\mathrm {atan}\left (c\,x\right )-b\,c^3\,d^3\,\mathrm {atan}\left (c\,x\right )\,\left (\frac {1}{2\,c^2}+\frac {x^2}{2}\right )\,1{}\mathrm {i} & \text {\ if\ \ }c\neq 0 \end {array}\right . \]
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