Optimal. Leaf size=83 \[ i c \left (a+b \cot ^{-1}\left (\frac {x}{c}\right )\right )^2+x \left (a+b \cot ^{-1}\left (\frac {x}{c}\right )\right )^2-2 b c \log \left (\frac {2 c}{c+i x}\right ) \left (a+b \cot ^{-1}\left (\frac {x}{c}\right )\right )+i b^2 c \text {Li}_2\left (1-\frac {2 c}{c+i x}\right ) \]
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Rubi [B] time = 0.44, antiderivative size = 478, normalized size of antiderivative = 5.76, number of steps used = 31, number of rules used = 14, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.167, Rules used = {5029, 2448, 263, 31, 2449, 2391, 2556, 12, 2462, 260, 2416, 2394, 2393, 2315} \[ -\frac {1}{2} i b^2 c \text {PolyLog}\left (2,\frac {c-i x}{2 c}\right )+\frac {1}{2} i b^2 c \text {PolyLog}\left (2,\frac {c+i x}{2 c}\right )-\frac {1}{2} i b^2 c \text {PolyLog}\left (2,-\frac {i c}{x}\right )+\frac {1}{2} i b^2 c \text {PolyLog}\left (2,\frac {i c}{x}\right )+\frac {1}{2} i b^2 c \text {PolyLog}\left (2,1-\frac {i x}{c}\right )-\frac {1}{2} i b^2 c \text {PolyLog}\left (2,1+\frac {i x}{c}\right )+a^2 x+i a b x \log \left (1-\frac {i c}{x}\right )-i a b x \log \left (1+\frac {i c}{x}\right )+a b c \log (c-i x)+a b c \log (c+i x)+\frac {1}{4} b^2 (-x+i c) \log ^2\left (1-\frac {i c}{x}\right )-\frac {1}{4} b^2 (x+i c) \log ^2\left (1+\frac {i c}{x}\right )+\frac {1}{2} b^2 x \log \left (1-\frac {i c}{x}\right ) \log \left (1+\frac {i c}{x}\right )-\frac {1}{2} i b^2 c \log \left (1+\frac {i c}{x}\right ) \log (-c-i x)+\frac {1}{2} i b^2 c \log (-c-i x) \log \left (\frac {c-i x}{2 c}\right )+\frac {1}{2} i b^2 c \log \left (1-\frac {i c}{x}\right ) \log (-c+i x)-\frac {1}{2} i b^2 c \log (-c+i x) \log \left (\frac {c+i x}{2 c}\right )-\frac {1}{2} i b^2 c \log (-c-i x) \log \left (-\frac {i x}{c}\right )+\frac {1}{2} i b^2 c \log (-c+i x) \log \left (\frac {i x}{c}\right ) \]
Warning: Unable to verify antiderivative.
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Rule 12
Rule 31
Rule 260
Rule 263
Rule 2315
Rule 2391
Rule 2393
Rule 2394
Rule 2416
Rule 2448
Rule 2449
Rule 2462
Rule 2556
Rule 5029
Rubi steps
\begin {align*} \int \left (a+b \tan ^{-1}\left (\frac {c}{x}\right )\right )^2 \, dx &=\int \left (a^2+i a b \log \left (1-\frac {i c}{x}\right )-\frac {1}{4} b^2 \log ^2\left (1-\frac {i c}{x}\right )-i a b \log \left (1+\frac {i c}{x}\right )+\frac {1}{2} b^2 \log \left (1-\frac {i c}{x}\right ) \log \left (1+\frac {i c}{x}\right )-\frac {1}{4} b^2 \log ^2\left (1+\frac {i c}{x}\right )\right ) \, dx\\ &=a^2 x+(i a b) \int \log \left (1-\frac {i c}{x}\right ) \, dx-(i a b) \int \log \left (1+\frac {i c}{x}\right ) \, dx-\frac {1}{4} b^2 \int \log ^2\left (1-\frac {i c}{x}\right ) \, dx-\frac {1}{4} b^2 \int \log ^2\left (1+\frac {i c}{x}\right ) \, dx+\frac {1}{2} b^2 \int \log \left (1-\frac {i c}{x}\right ) \log \left (1+\frac {i c}{x}\right ) \, dx\\ &=a^2 x+i a b x \log \left (1-\frac {i c}{x}\right )+\frac {1}{4} b^2 (i c-x) \log ^2\left (1-\frac {i c}{x}\right )-i a b x \log \left (1+\frac {i c}{x}\right )+\frac {1}{2} b^2 x \log \left (1-\frac {i c}{x}\right ) \log \left (1+\frac {i c}{x}\right )-\frac {1}{4} b^2 (i c+x) \log ^2\left (1+\frac {i c}{x}\right )-\frac {1}{2} b^2 \int \frac {c \log \left (1-\frac {i c}{x}\right )}{-c+i x} \, dx-\frac {1}{2} b^2 \int \frac {c \log \left (1+\frac {i c}{x}\right )}{-c-i x} \, dx+(a b c) \int \frac {1}{\left (1-\frac {i c}{x}\right ) x} \, dx+(a b c) \int \frac {1}{\left (1+\frac {i c}{x}\right ) x} \, dx+\frac {1}{2} \left (i b^2 c\right ) \int \frac {\log \left (1-\frac {i c}{x}\right )}{x} \, dx-\frac {1}{2} \left (i b^2 c\right ) \int \frac {\log \left (1+\frac {i c}{x}\right )}{x} \, dx\\ &=a^2 x+i a b x \log \left (1-\frac {i c}{x}\right )+\frac {1}{4} b^2 (i c-x) \log ^2\left (1-\frac {i c}{x}\right )-i a b x \log \left (1+\frac {i c}{x}\right )+\frac {1}{2} b^2 x \log \left (1-\frac {i c}{x}\right ) \log \left (1+\frac {i c}{x}\right )-\frac {1}{4} b^2 (i c+x) \log ^2\left (1+\frac {i c}{x}\right )-\frac {1}{2} i b^2 c \text {Li}_2\left (-\frac {i c}{x}\right )+\frac {1}{2} i b^2 c \text {Li}_2\left (\frac {i c}{x}\right )+(a b c) \int \frac {1}{-i c+x} \, dx+(a b c) \int \frac {1}{i c+x} \, dx-\frac {1}{2} \left (b^2 c\right ) \int \frac {\log \left (1-\frac {i c}{x}\right )}{-c+i x} \, dx-\frac {1}{2} \left (b^2 c\right ) \int \frac {\log \left (1+\frac {i c}{x}\right )}{-c-i x} \, dx\\ &=a^2 x+i a b x \log \left (1-\frac {i c}{x}\right )+\frac {1}{4} b^2 (i c-x) \log ^2\left (1-\frac {i c}{x}\right )-i a b x \log \left (1+\frac {i c}{x}\right )+\frac {1}{2} b^2 x \log \left (1-\frac {i c}{x}\right ) \log \left (1+\frac {i c}{x}\right )-\frac {1}{4} b^2 (i c+x) \log ^2\left (1+\frac {i c}{x}\right )-\frac {1}{2} i b^2 c \log \left (1+\frac {i c}{x}\right ) \log (-c-i x)+a b c \log (c-i x)+\frac {1}{2} i b^2 c \log \left (1-\frac {i c}{x}\right ) \log (-c+i x)+a b c \log (c+i x)-\frac {1}{2} i b^2 c \text {Li}_2\left (-\frac {i c}{x}\right )+\frac {1}{2} i b^2 c \text {Li}_2\left (\frac {i c}{x}\right )+\frac {1}{2} \left (b^2 c^2\right ) \int \frac {\log (-c-i x)}{\left (1+\frac {i c}{x}\right ) x^2} \, dx+\frac {1}{2} \left (b^2 c^2\right ) \int \frac {\log (-c+i x)}{\left (1-\frac {i c}{x}\right ) x^2} \, dx\\ &=a^2 x+i a b x \log \left (1-\frac {i c}{x}\right )+\frac {1}{4} b^2 (i c-x) \log ^2\left (1-\frac {i c}{x}\right )-i a b x \log \left (1+\frac {i c}{x}\right )+\frac {1}{2} b^2 x \log \left (1-\frac {i c}{x}\right ) \log \left (1+\frac {i c}{x}\right )-\frac {1}{4} b^2 (i c+x) \log ^2\left (1+\frac {i c}{x}\right )-\frac {1}{2} i b^2 c \log \left (1+\frac {i c}{x}\right ) \log (-c-i x)+a b c \log (c-i x)+\frac {1}{2} i b^2 c \log \left (1-\frac {i c}{x}\right ) \log (-c+i x)+a b c \log (c+i x)-\frac {1}{2} i b^2 c \text {Li}_2\left (-\frac {i c}{x}\right )+\frac {1}{2} i b^2 c \text {Li}_2\left (\frac {i c}{x}\right )+\frac {1}{2} \left (b^2 c^2\right ) \int \left (\frac {\log (-c-i x)}{c (c-i x)}-\frac {i \log (-c-i x)}{c x}\right ) \, dx+\frac {1}{2} \left (b^2 c^2\right ) \int \left (\frac {\log (-c+i x)}{c (c+i x)}+\frac {i \log (-c+i x)}{c x}\right ) \, dx\\ &=a^2 x+i a b x \log \left (1-\frac {i c}{x}\right )+\frac {1}{4} b^2 (i c-x) \log ^2\left (1-\frac {i c}{x}\right )-i a b x \log \left (1+\frac {i c}{x}\right )+\frac {1}{2} b^2 x \log \left (1-\frac {i c}{x}\right ) \log \left (1+\frac {i c}{x}\right )-\frac {1}{4} b^2 (i c+x) \log ^2\left (1+\frac {i c}{x}\right )-\frac {1}{2} i b^2 c \log \left (1+\frac {i c}{x}\right ) \log (-c-i x)+a b c \log (c-i x)+\frac {1}{2} i b^2 c \log \left (1-\frac {i c}{x}\right ) \log (-c+i x)+a b c \log (c+i x)-\frac {1}{2} i b^2 c \text {Li}_2\left (-\frac {i c}{x}\right )+\frac {1}{2} i b^2 c \text {Li}_2\left (\frac {i c}{x}\right )-\frac {1}{2} \left (i b^2 c\right ) \int \frac {\log (-c-i x)}{x} \, dx+\frac {1}{2} \left (i b^2 c\right ) \int \frac {\log (-c+i x)}{x} \, dx+\frac {1}{2} \left (b^2 c\right ) \int \frac {\log (-c-i x)}{c-i x} \, dx+\frac {1}{2} \left (b^2 c\right ) \int \frac {\log (-c+i x)}{c+i x} \, dx\\ &=a^2 x+i a b x \log \left (1-\frac {i c}{x}\right )+\frac {1}{4} b^2 (i c-x) \log ^2\left (1-\frac {i c}{x}\right )-i a b x \log \left (1+\frac {i c}{x}\right )+\frac {1}{2} b^2 x \log \left (1-\frac {i c}{x}\right ) \log \left (1+\frac {i c}{x}\right )-\frac {1}{4} b^2 (i c+x) \log ^2\left (1+\frac {i c}{x}\right )-\frac {1}{2} i b^2 c \log \left (1+\frac {i c}{x}\right ) \log (-c-i x)+a b c \log (c-i x)+\frac {1}{2} i b^2 c \log (-c-i x) \log \left (\frac {c-i x}{2 c}\right )+\frac {1}{2} i b^2 c \log \left (1-\frac {i c}{x}\right ) \log (-c+i x)+a b c \log (c+i x)-\frac {1}{2} i b^2 c \log (-c+i x) \log \left (\frac {c+i x}{2 c}\right )-\frac {1}{2} i b^2 c \log (-c-i x) \log \left (-\frac {i x}{c}\right )+\frac {1}{2} i b^2 c \log (-c+i x) \log \left (\frac {i x}{c}\right )-\frac {1}{2} i b^2 c \text {Li}_2\left (-\frac {i c}{x}\right )+\frac {1}{2} i b^2 c \text {Li}_2\left (\frac {i c}{x}\right )-\frac {1}{2} \left (b^2 c\right ) \int \frac {\log \left (\frac {c-i x}{2 c}\right )}{-c-i x} \, dx-\frac {1}{2} \left (b^2 c\right ) \int \frac {\log \left (\frac {c+i x}{2 c}\right )}{-c+i x} \, dx+\frac {1}{2} \left (b^2 c\right ) \int \frac {\log \left (-\frac {i x}{c}\right )}{-c-i x} \, dx+\frac {1}{2} \left (b^2 c\right ) \int \frac {\log \left (\frac {i x}{c}\right )}{-c+i x} \, dx\\ &=a^2 x+i a b x \log \left (1-\frac {i c}{x}\right )+\frac {1}{4} b^2 (i c-x) \log ^2\left (1-\frac {i c}{x}\right )-i a b x \log \left (1+\frac {i c}{x}\right )+\frac {1}{2} b^2 x \log \left (1-\frac {i c}{x}\right ) \log \left (1+\frac {i c}{x}\right )-\frac {1}{4} b^2 (i c+x) \log ^2\left (1+\frac {i c}{x}\right )-\frac {1}{2} i b^2 c \log \left (1+\frac {i c}{x}\right ) \log (-c-i x)+a b c \log (c-i x)+\frac {1}{2} i b^2 c \log (-c-i x) \log \left (\frac {c-i x}{2 c}\right )+\frac {1}{2} i b^2 c \log \left (1-\frac {i c}{x}\right ) \log (-c+i x)+a b c \log (c+i x)-\frac {1}{2} i b^2 c \log (-c+i x) \log \left (\frac {c+i x}{2 c}\right )-\frac {1}{2} i b^2 c \log (-c-i x) \log \left (-\frac {i x}{c}\right )+\frac {1}{2} i b^2 c \log (-c+i x) \log \left (\frac {i x}{c}\right )-\frac {1}{2} i b^2 c \text {Li}_2\left (-\frac {i c}{x}\right )+\frac {1}{2} i b^2 c \text {Li}_2\left (\frac {i c}{x}\right )+\frac {1}{2} i b^2 c \text {Li}_2\left (1-\frac {i x}{c}\right )-\frac {1}{2} i b^2 c \text {Li}_2\left (1+\frac {i x}{c}\right )-\frac {1}{2} \left (i b^2 c\right ) \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {x}{2 c}\right )}{x} \, dx,x,-c-i x\right )+\frac {1}{2} \left (i b^2 c\right ) \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {x}{2 c}\right )}{x} \, dx,x,-c+i x\right )\\ &=a^2 x+i a b x \log \left (1-\frac {i c}{x}\right )+\frac {1}{4} b^2 (i c-x) \log ^2\left (1-\frac {i c}{x}\right )-i a b x \log \left (1+\frac {i c}{x}\right )+\frac {1}{2} b^2 x \log \left (1-\frac {i c}{x}\right ) \log \left (1+\frac {i c}{x}\right )-\frac {1}{4} b^2 (i c+x) \log ^2\left (1+\frac {i c}{x}\right )-\frac {1}{2} i b^2 c \log \left (1+\frac {i c}{x}\right ) \log (-c-i x)+a b c \log (c-i x)+\frac {1}{2} i b^2 c \log (-c-i x) \log \left (\frac {c-i x}{2 c}\right )+\frac {1}{2} i b^2 c \log \left (1-\frac {i c}{x}\right ) \log (-c+i x)+a b c \log (c+i x)-\frac {1}{2} i b^2 c \log (-c+i x) \log \left (\frac {c+i x}{2 c}\right )-\frac {1}{2} i b^2 c \log (-c-i x) \log \left (-\frac {i x}{c}\right )+\frac {1}{2} i b^2 c \log (-c+i x) \log \left (\frac {i x}{c}\right )-\frac {1}{2} i b^2 c \text {Li}_2\left (\frac {c-i x}{2 c}\right )+\frac {1}{2} i b^2 c \text {Li}_2\left (\frac {c+i x}{2 c}\right )-\frac {1}{2} i b^2 c \text {Li}_2\left (-\frac {i c}{x}\right )+\frac {1}{2} i b^2 c \text {Li}_2\left (\frac {i c}{x}\right )+\frac {1}{2} i b^2 c \text {Li}_2\left (1-\frac {i x}{c}\right )-\frac {1}{2} i b^2 c \text {Li}_2\left (1+\frac {i x}{c}\right )\\ \end {align*}
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Mathematica [A] time = 0.11, size = 105, normalized size = 1.27 \[ a \left (a x+b c \log \left (\frac {c^2}{x^2}+1\right )-2 b c \log \left (\frac {c}{x}\right )\right )+2 b \tan ^{-1}\left (\frac {c}{x}\right ) \left (a x-b c \log \left (1-e^{2 i \tan ^{-1}\left (\frac {c}{x}\right )}\right )\right )+i b^2 c \text {Li}_2\left (e^{2 i \tan ^{-1}\left (\frac {c}{x}\right )}\right )+b^2 (x+i c) \tan ^{-1}\left (\frac {c}{x}\right )^2 \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 1.10, size = 0, normalized size = 0.00 \[ {\rm integral}\left (b^{2} \arctan \left (\frac {c}{x}\right )^{2} + 2 \, a b \arctan \left (\frac {c}{x}\right ) + a^{2}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (b \arctan \left (\frac {c}{x}\right ) + a\right )}^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.10, size = 357, normalized size = 4.30 \[ a^{2} x +b^{2} x \arctan \left (\frac {c}{x}\right )^{2}-2 c \,b^{2} \ln \left (\frac {c}{x}\right ) \arctan \left (\frac {c}{x}\right )+c \,b^{2} \arctan \left (\frac {c}{x}\right ) \ln \left (1+\frac {c^{2}}{x^{2}}\right )+i c \,b^{2} \dilog \left (1-\frac {i c}{x}\right )+i c \,b^{2} \ln \left (\frac {c}{x}\right ) \ln \left (1-\frac {i c}{x}\right )+\frac {i c \,b^{2} \ln \left (i+\frac {c}{x}\right ) \ln \left (\frac {i \left (\frac {c}{x}-i\right )}{2}\right )}{2}-\frac {i c \,b^{2} \ln \left (\frac {c}{x}-i\right ) \ln \left (-\frac {i \left (i+\frac {c}{x}\right )}{2}\right )}{2}-\frac {i c \,b^{2} \ln \left (i+\frac {c}{x}\right ) \ln \left (1+\frac {c^{2}}{x^{2}}\right )}{2}+\frac {i c \,b^{2} \ln \left (\frac {c}{x}-i\right ) \ln \left (1+\frac {c^{2}}{x^{2}}\right )}{2}-i c \,b^{2} \ln \left (\frac {c}{x}\right ) \ln \left (1+\frac {i c}{x}\right )+\frac {i c \,b^{2} \ln \left (i+\frac {c}{x}\right )^{2}}{4}+\frac {i c \,b^{2} \dilog \left (\frac {i \left (\frac {c}{x}-i\right )}{2}\right )}{2}-i c \,b^{2} \dilog \left (1+\frac {i c}{x}\right )-\frac {i c \,b^{2} \ln \left (\frac {c}{x}-i\right )^{2}}{4}-\frac {i c \,b^{2} \dilog \left (-\frac {i \left (i+\frac {c}{x}\right )}{2}\right )}{2}+2 a b x \arctan \left (\frac {c}{x}\right )-2 c a b \ln \left (\frac {c}{x}\right )+c a b \ln \left (1+\frac {c^{2}}{x^{2}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ {\left (2 \, x \arctan \left (\frac {c}{x}\right ) + c \log \left (c^{2} + x^{2}\right )\right )} a b + \frac {1}{16} \, {\left (12 \, c \arctan \left (\frac {c}{x}\right )^{2} \arctan \left (\frac {x}{c}\right ) + 4 \, {\left (\frac {3 \, \arctan \left (\frac {c}{x}\right ) \arctan \left (\frac {x}{c}\right )^{2}}{c} + \frac {\arctan \left (\frac {x}{c}\right )^{3}}{c}\right )} c^{2} + 4 \, x \arctan \left (c, x\right )^{2} + 16 \, c^{2} \int \frac {\log \left (c^{2} + x^{2}\right )^{2}}{16 \, {\left (c^{2} + x^{2}\right )}}\,{d x} - x \log \left (c^{2} + x^{2}\right )^{2} + 128 \, c \int \frac {x \arctan \left (\frac {c}{x}\right )}{16 \, {\left (c^{2} + x^{2}\right )}}\,{d x} + 192 \, \int \frac {x^{2} \arctan \left (\frac {c}{x}\right )^{2}}{16 \, {\left (c^{2} + x^{2}\right )}}\,{d x} + 16 \, \int \frac {x^{2} \log \left (c^{2} + x^{2}\right )^{2}}{16 \, {\left (c^{2} + x^{2}\right )}}\,{d x} + 64 \, \int \frac {x^{2} \log \left (c^{2} + x^{2}\right )}{16 \, {\left (c^{2} + x^{2}\right )}}\,{d x}\right )} b^{2} + a^{2} x \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int {\left (a+b\,\mathrm {atan}\left (\frac {c}{x}\right )\right )}^2 \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (a + b \operatorname {atan}{\left (\frac {c}{x} \right )}\right )^{2}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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