Integrand size = 23, antiderivative size = 194 \[ \int \frac {a+b \arctan (c x)}{x^2 (d+i c d x)^2} \, dx=-\frac {i b c}{2 d^2 (i-c x)}+\frac {i b c \arctan (c x)}{2 d^2}-\frac {a+b \arctan (c x)}{d^2 x}+\frac {c (a+b \arctan (c x))}{d^2 (i-c x)}-\frac {2 i a c \log (x)}{d^2}+\frac {b c \log (x)}{d^2}-\frac {2 i c (a+b \arctan (c x)) \log \left (\frac {2}{1+i c x}\right )}{d^2}-\frac {b c \log \left (1+c^2 x^2\right )}{2 d^2}+\frac {b c \operatorname {PolyLog}(2,-i c x)}{d^2}-\frac {b c \operatorname {PolyLog}(2,i c x)}{d^2}+\frac {b c \operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )}{d^2} \]
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Time = 0.17 (sec) , antiderivative size = 194, normalized size of antiderivative = 1.00, number of steps used = 18, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.652, Rules used = {4996, 4946, 272, 36, 29, 31, 4940, 2438, 4972, 641, 46, 209, 4964, 2449, 2352} \[ \int \frac {a+b \arctan (c x)}{x^2 (d+i c d x)^2} \, dx=\frac {c (a+b \arctan (c x))}{d^2 (-c x+i)}-\frac {a+b \arctan (c x)}{d^2 x}-\frac {2 i c \log \left (\frac {2}{1+i c x}\right ) (a+b \arctan (c x))}{d^2}-\frac {2 i a c \log (x)}{d^2}+\frac {i b c \arctan (c x)}{2 d^2}-\frac {b c \log \left (c^2 x^2+1\right )}{2 d^2}+\frac {b c \operatorname {PolyLog}(2,-i c x)}{d^2}-\frac {b c \operatorname {PolyLog}(2,i c x)}{d^2}+\frac {b c \operatorname {PolyLog}\left (2,1-\frac {2}{i c x+1}\right )}{d^2}-\frac {i b c}{2 d^2 (-c x+i)}+\frac {b c \log (x)}{d^2} \]
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Rule 29
Rule 31
Rule 36
Rule 46
Rule 209
Rule 272
Rule 641
Rule 2352
Rule 2438
Rule 2449
Rule 4940
Rule 4946
Rule 4964
Rule 4972
Rule 4996
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {a+b \arctan (c x)}{d^2 x^2}-\frac {2 i c (a+b \arctan (c x))}{d^2 x}+\frac {c^2 (a+b \arctan (c x))}{d^2 (-i+c x)^2}+\frac {2 i c^2 (a+b \arctan (c x))}{d^2 (-i+c x)}\right ) \, dx \\ & = \frac {\int \frac {a+b \arctan (c x)}{x^2} \, dx}{d^2}-\frac {(2 i c) \int \frac {a+b \arctan (c x)}{x} \, dx}{d^2}+\frac {\left (2 i c^2\right ) \int \frac {a+b \arctan (c x)}{-i+c x} \, dx}{d^2}+\frac {c^2 \int \frac {a+b \arctan (c x)}{(-i+c x)^2} \, dx}{d^2} \\ & = -\frac {a+b \arctan (c x)}{d^2 x}+\frac {c (a+b \arctan (c x))}{d^2 (i-c x)}-\frac {2 i a c \log (x)}{d^2}-\frac {2 i c (a+b \arctan (c x)) \log \left (\frac {2}{1+i c x}\right )}{d^2}+\frac {(b c) \int \frac {1}{x \left (1+c^2 x^2\right )} \, dx}{d^2}+\frac {(b c) \int \frac {\log (1-i c x)}{x} \, dx}{d^2}-\frac {(b c) \int \frac {\log (1+i c x)}{x} \, dx}{d^2}+\frac {\left (2 i b c^2\right ) \int \frac {\log \left (\frac {2}{1+i c x}\right )}{1+c^2 x^2} \, dx}{d^2}+\frac {\left (b c^2\right ) \int \frac {1}{(-i+c x) \left (1+c^2 x^2\right )} \, dx}{d^2} \\ & = -\frac {a+b \arctan (c x)}{d^2 x}+\frac {c (a+b \arctan (c x))}{d^2 (i-c x)}-\frac {2 i a c \log (x)}{d^2}-\frac {2 i c (a+b \arctan (c x)) \log \left (\frac {2}{1+i c x}\right )}{d^2}+\frac {b c \operatorname {PolyLog}(2,-i c x)}{d^2}-\frac {b c \operatorname {PolyLog}(2,i c x)}{d^2}+\frac {(b c) \text {Subst}\left (\int \frac {1}{x \left (1+c^2 x\right )} \, dx,x,x^2\right )}{2 d^2}+\frac {(2 b c) \text {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1+i c x}\right )}{d^2}+\frac {\left (b c^2\right ) \int \frac {1}{(-i+c x)^2 (i+c x)} \, dx}{d^2} \\ & = -\frac {a+b \arctan (c x)}{d^2 x}+\frac {c (a+b \arctan (c x))}{d^2 (i-c x)}-\frac {2 i a c \log (x)}{d^2}-\frac {2 i c (a+b \arctan (c x)) \log \left (\frac {2}{1+i c x}\right )}{d^2}+\frac {b c \operatorname {PolyLog}(2,-i c x)}{d^2}-\frac {b c \operatorname {PolyLog}(2,i c x)}{d^2}+\frac {b c \operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )}{d^2}+\frac {(b c) \text {Subst}\left (\int \frac {1}{x} \, dx,x,x^2\right )}{2 d^2}+\frac {\left (b c^2\right ) \int \left (-\frac {i}{2 (-i+c x)^2}+\frac {i}{2 \left (1+c^2 x^2\right )}\right ) \, dx}{d^2}-\frac {\left (b c^3\right ) \text {Subst}\left (\int \frac {1}{1+c^2 x} \, dx,x,x^2\right )}{2 d^2} \\ & = -\frac {i b c}{2 d^2 (i-c x)}-\frac {a+b \arctan (c x)}{d^2 x}+\frac {c (a+b \arctan (c x))}{d^2 (i-c x)}-\frac {2 i a c \log (x)}{d^2}+\frac {b c \log (x)}{d^2}-\frac {2 i c (a+b \arctan (c x)) \log \left (\frac {2}{1+i c x}\right )}{d^2}-\frac {b c \log \left (1+c^2 x^2\right )}{2 d^2}+\frac {b c \operatorname {PolyLog}(2,-i c x)}{d^2}-\frac {b c \operatorname {PolyLog}(2,i c x)}{d^2}+\frac {b c \operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )}{d^2}+\frac {\left (i b c^2\right ) \int \frac {1}{1+c^2 x^2} \, dx}{2 d^2} \\ & = -\frac {i b c}{2 d^2 (i-c x)}+\frac {i b c \arctan (c x)}{2 d^2}-\frac {a+b \arctan (c x)}{d^2 x}+\frac {c (a+b \arctan (c x))}{d^2 (i-c x)}-\frac {2 i a c \log (x)}{d^2}+\frac {b c \log (x)}{d^2}-\frac {2 i c (a+b \arctan (c x)) \log \left (\frac {2}{1+i c x}\right )}{d^2}-\frac {b c \log \left (1+c^2 x^2\right )}{2 d^2}+\frac {b c \operatorname {PolyLog}(2,-i c x)}{d^2}-\frac {b c \operatorname {PolyLog}(2,i c x)}{d^2}+\frac {b c \operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )}{d^2} \\ \end{align*}
Time = 0.22 (sec) , antiderivative size = 165, normalized size of antiderivative = 0.85 \[ \int \frac {a+b \arctan (c x)}{x^2 (d+i c d x)^2} \, dx=-\frac {i b c \left (\frac {1}{i-c x}-\arctan (c x)\right )+\frac {2 (a+b \arctan (c x))}{x}+\frac {2 c (a+b \arctan (c x))}{-i+c x}+4 i a c \log (x)+4 i c (a+b \arctan (c x)) \log \left (\frac {2 i}{i-c x}\right )+b c \left (-2 \log (x)+\log \left (1+c^2 x^2\right )\right )-2 b c \operatorname {PolyLog}(2,-i c x)+2 b c \operatorname {PolyLog}(2,i c x)-2 b c \operatorname {PolyLog}\left (2,\frac {i+c x}{-i+c x}\right )}{2 d^2} \]
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Time = 0.88 (sec) , antiderivative size = 257, normalized size of antiderivative = 1.32
method | result | size |
parts | \(-\frac {a}{d^{2} x}-\frac {2 i a c \ln \left (x \right )}{d^{2}}+\frac {a c}{d^{2} \left (-c x +i\right )}-\frac {2 c a \arctan \left (c x \right )}{d^{2}}+\frac {i c a \ln \left (c^{2} x^{2}+1\right )}{d^{2}}+\frac {b c \left (-\frac {\arctan \left (c x \right )}{c x}-2 i \arctan \left (c x \right ) \ln \left (c x \right )-\frac {\arctan \left (c x \right )}{c x -i}+2 i \arctan \left (c x \right ) \ln \left (c x -i\right )-\operatorname {dilog}\left (-i \left (c x +i\right )\right )-\ln \left (c x \right ) \ln \left (-i \left (c x +i\right )\right )+\left (\ln \left (c x \right )-\ln \left (-i c x \right )\right ) \ln \left (-i \left (-c x +i\right )\right )-\operatorname {dilog}\left (-i c x \right )+\operatorname {dilog}\left (-\frac {i \left (c x +i\right )}{2}\right )+\ln \left (c x -i\right ) \ln \left (-\frac {i \left (c x +i\right )}{2}\right )-\frac {\ln \left (c x -i\right )^{2}}{2}-\frac {\ln \left (c^{2} x^{2}+1\right )}{2}+\frac {i \arctan \left (c x \right )}{2}+\ln \left (c x \right )+\frac {i}{2 c x -2 i}\right )}{d^{2}}\) | \(257\) |
derivativedivides | \(c \left (-\frac {a}{d^{2} c x}-\frac {2 i a \ln \left (c x \right )}{d^{2}}-\frac {a}{d^{2} \left (c x -i\right )}-\frac {2 a \arctan \left (c x \right )}{d^{2}}+\frac {i a \ln \left (c^{2} x^{2}+1\right )}{d^{2}}+\frac {b \left (-\frac {\arctan \left (c x \right )}{c x}-2 i \arctan \left (c x \right ) \ln \left (c x \right )-\frac {\arctan \left (c x \right )}{c x -i}+2 i \arctan \left (c x \right ) \ln \left (c x -i\right )-\operatorname {dilog}\left (-i \left (c x +i\right )\right )-\ln \left (c x \right ) \ln \left (-i \left (c x +i\right )\right )+\left (\ln \left (c x \right )-\ln \left (-i c x \right )\right ) \ln \left (-i \left (-c x +i\right )\right )-\operatorname {dilog}\left (-i c x \right )+\operatorname {dilog}\left (-\frac {i \left (c x +i\right )}{2}\right )+\ln \left (c x -i\right ) \ln \left (-\frac {i \left (c x +i\right )}{2}\right )-\frac {\ln \left (c x -i\right )^{2}}{2}-\frac {\ln \left (c^{2} x^{2}+1\right )}{2}+\frac {i \arctan \left (c x \right )}{2}+\ln \left (c x \right )+\frac {i}{2 c x -2 i}\right )}{d^{2}}\right )\) | \(259\) |
default | \(c \left (-\frac {a}{d^{2} c x}-\frac {2 i a \ln \left (c x \right )}{d^{2}}-\frac {a}{d^{2} \left (c x -i\right )}-\frac {2 a \arctan \left (c x \right )}{d^{2}}+\frac {i a \ln \left (c^{2} x^{2}+1\right )}{d^{2}}+\frac {b \left (-\frac {\arctan \left (c x \right )}{c x}-2 i \arctan \left (c x \right ) \ln \left (c x \right )-\frac {\arctan \left (c x \right )}{c x -i}+2 i \arctan \left (c x \right ) \ln \left (c x -i\right )-\operatorname {dilog}\left (-i \left (c x +i\right )\right )-\ln \left (c x \right ) \ln \left (-i \left (c x +i\right )\right )+\left (\ln \left (c x \right )-\ln \left (-i c x \right )\right ) \ln \left (-i \left (-c x +i\right )\right )-\operatorname {dilog}\left (-i c x \right )+\operatorname {dilog}\left (-\frac {i \left (c x +i\right )}{2}\right )+\ln \left (c x -i\right ) \ln \left (-\frac {i \left (c x +i\right )}{2}\right )-\frac {\ln \left (c x -i\right )^{2}}{2}-\frac {\ln \left (c^{2} x^{2}+1\right )}{2}+\frac {i \arctan \left (c x \right )}{2}+\ln \left (c x \right )+\frac {i}{2 c x -2 i}\right )}{d^{2}}\right )\) | \(259\) |
risch | \(-\frac {c b \ln \left (-i c x +1\right )}{4 d^{2} \left (-i c x -1\right )}+\frac {b c \ln \left (c^{2} x^{2}+1\right )}{8 d^{2}}-\frac {c b \ln \left (\frac {1}{2}+\frac {i c x}{2}\right ) \ln \left (-i c x +1\right )}{d^{2}}+\frac {i c^{2} b \ln \left (-i c x +1\right ) x}{4 d^{2} \left (-i c x -1\right )}+\frac {c b \ln \left (\frac {1}{2}+\frac {i c x}{2}\right ) \ln \left (\frac {1}{2}-\frac {i c x}{2}\right )}{d^{2}}+\frac {c b \operatorname {dilog}\left (\frac {1}{2}-\frac {i c x}{2}\right )}{d^{2}}-\frac {a}{d^{2} x}-\frac {c b \operatorname {dilog}\left (-i c x +1\right )}{d^{2}}+\frac {i c a \ln \left (c^{2} x^{2}+1\right )}{d^{2}}-\frac {2 i c a \ln \left (-i c x \right )}{d^{2}}-\frac {2 c a \arctan \left (c x \right )}{d^{2}}+\frac {c b \ln \left (-i c x \right )}{2 d^{2}}-\frac {i b \ln \left (-i c x +1\right )}{2 d^{2} x}+\frac {i b \arctan \left (c x \right ) c}{4 d^{2}}+\frac {i c a}{d^{2} \left (-i c x -1\right )}-\frac {c \ln \left (-i c x +1\right ) b}{2 d^{2}}+\frac {b c \ln \left (i c x +1\right )^{2}}{2 d^{2}}+\frac {b c \operatorname {dilog}\left (i c x +1\right )}{d^{2}}+\frac {b c \ln \left (i c x \right )}{2 d^{2}}-\frac {b c \ln \left (i c x +1\right )}{2 d^{2}}+\frac {i b \ln \left (i c x +1\right )}{2 d^{2} x}-\frac {b c \ln \left (i c x +1\right )}{2 d^{2} \left (i c x +1\right )}-\frac {b c}{2 d^{2} \left (i c x +1\right )}\) | \(389\) |
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Time = 0.26 (sec) , antiderivative size = 182, normalized size of antiderivative = 0.94 \[ \int \frac {a+b \arctan (c x)}{x^2 (d+i c d x)^2} \, dx=-\frac {2 \, {\left (4 \, a - i \, b\right )} c x + 4 \, {\left (b c^{2} x^{2} - i \, b c x\right )} {\rm Li}_2\left (\frac {c x + i}{c x - i} + 1\right ) + 4 \, {\left ({\left (2 i \, a - b\right )} c^{2} x^{2} + {\left (2 \, a + i \, b\right )} c x\right )} \log \left (x\right ) + 2 \, {\left (2 i \, b c x + b\right )} \log \left (-\frac {c x + i}{c x - i}\right ) + 3 \, {\left (b c^{2} x^{2} - i \, b c x\right )} \log \left (\frac {c x + i}{c}\right ) - {\left ({\left (8 i \, a - b\right )} c^{2} x^{2} + {\left (8 \, a + i \, b\right )} c x\right )} \log \left (\frac {c x - i}{c}\right ) - 4 i \, a}{4 \, {\left (c d^{2} x^{2} - i \, d^{2} x\right )}} \]
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Timed out. \[ \int \frac {a+b \arctan (c x)}{x^2 (d+i c d x)^2} \, dx=\text {Timed out} \]
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\[ \int \frac {a+b \arctan (c x)}{x^2 (d+i c d x)^2} \, dx=\int { \frac {b \arctan \left (c x\right ) + a}{{\left (i \, c d x + d\right )}^{2} x^{2}} \,d x } \]
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\[ \int \frac {a+b \arctan (c x)}{x^2 (d+i c d x)^2} \, dx=\int { \frac {b \arctan \left (c x\right ) + a}{{\left (i \, c d x + d\right )}^{2} x^{2}} \,d x } \]
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Timed out. \[ \int \frac {a+b \arctan (c x)}{x^2 (d+i c d x)^2} \, dx=\int \frac {a+b\,\mathrm {atan}\left (c\,x\right )}{x^2\,{\left (d+c\,d\,x\,1{}\mathrm {i}\right )}^2} \,d x \]
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