Optimal. Leaf size=338 \[ -\frac {i d \text {Li}_2\left (-\frac {b (d+c x)}{(a+i) c-b d}\right )}{2 c^2}+\frac {i d \text {Li}_2\left (\frac {b (d+c x)}{-a c+i c+b d}\right )}{2 c^2}+\frac {i d \log (c x+d) \log \left (\frac {c (-a-b x+i)}{-a c+b d+i c}\right )}{2 c^2}-\frac {i d \log \left (-\frac {-a-b x+i}{a+b x}\right ) \log (c x+d)}{2 c^2}-\frac {i d \log (c x+d) \log \left (\frac {c (a+b x+i)}{-b d+(a+i) c}\right )}{2 c^2}+\frac {i d \log \left (\frac {a+b x+i}{a+b x}\right ) \log (c x+d)}{2 c^2}+\frac {\log (-a-b x+i)}{2 b c}+\frac {i (a+b x) \log \left (-\frac {-a-b x+i}{a+b x}\right )}{2 b c}+\frac {\log (a+b x+i)}{2 b c}-\frac {i (a+b x) \log \left (\frac {a+b x+i}{a+b x}\right )}{2 b c} \]
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Rubi [A] time = 0.50, antiderivative size = 422, normalized size of antiderivative = 1.25, number of steps used = 37, number of rules used = 10, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.625, Rules used = {5052, 2513, 2409, 2389, 2295, 2394, 2393, 2391, 193, 43} \[ -\frac {i d \text {PolyLog}\left (2,\frac {c (-a-b x+i)}{b d+(-a+i) c}\right )}{2 c^2}+\frac {i d \text {PolyLog}\left (2,\frac {c (a+b x+i)}{-b d+(a+i) c}\right )}{2 c^2}-\frac {i d \left (\log \left (-\frac {-a-b x+i}{a+b x}\right )+\log (a+b x)-\log (a+b x-i)\right ) \log (c x+d)}{2 c^2}+\frac {i d \left (\log (a+b x)-\log (a+b x+i)+\log \left (\frac {a+b x+i}{a+b x}\right )\right ) \log (c x+d)}{2 c^2}+\frac {i d \log (a+b x+i) \log \left (-\frac {b (c x+d)}{-b d+(a+i) c}\right )}{2 c^2}-\frac {i d \log (a+b x-i) \log \left (\frac {b (c x+d)}{b d+(-a+i) c}\right )}{2 c^2}+\frac {i x \left (\log \left (-\frac {-a-b x+i}{a+b x}\right )+\log (a+b x)-\log (a+b x-i)\right )}{2 c}-\frac {i (-a-b x+i) \log (a+b x-i)}{2 b c}-\frac {i (a+b x+i) \log (a+b x+i)}{2 b c}-\frac {i x \left (\log (a+b x)-\log (a+b x+i)+\log \left (\frac {a+b x+i}{a+b x}\right )\right )}{2 c} \]
Warning: Unable to verify antiderivative.
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Rule 43
Rule 193
Rule 2295
Rule 2389
Rule 2391
Rule 2393
Rule 2394
Rule 2409
Rule 2513
Rule 5052
Rubi steps
\begin {align*} \int \frac {\cot ^{-1}(a+b x)}{c+\frac {d}{x}} \, dx &=\frac {1}{2} i \int \frac {\log \left (\frac {-i+a+b x}{a+b x}\right )}{c+\frac {d}{x}} \, dx-\frac {1}{2} i \int \frac {\log \left (\frac {i+a+b x}{a+b x}\right )}{c+\frac {d}{x}} \, dx\\ &=\frac {1}{2} i \int \frac {\log (-i+a+b x)}{c+\frac {d}{x}} \, dx-\frac {1}{2} i \int \frac {\log (i+a+b x)}{c+\frac {d}{x}} \, dx-\frac {1}{2} \left (i \left (-\log (a+b x)+\log (-i+a+b x)-\log \left (\frac {-i+a+b x}{a+b x}\right )\right )\right ) \int \frac {1}{c+\frac {d}{x}} \, dx+\frac {1}{2} \left (i \left (-\log (a+b x)+\log (i+a+b x)-\log \left (\frac {i+a+b x}{a+b x}\right )\right )\right ) \int \frac {1}{c+\frac {d}{x}} \, dx\\ &=\frac {1}{2} i \int \left (\frac {\log (-i+a+b x)}{c}-\frac {d \log (-i+a+b x)}{c (d+c x)}\right ) \, dx-\frac {1}{2} i \int \left (\frac {\log (i+a+b x)}{c}-\frac {d \log (i+a+b x)}{c (d+c x)}\right ) \, dx-\frac {1}{2} \left (i \left (-\log (a+b x)+\log (-i+a+b x)-\log \left (\frac {-i+a+b x}{a+b x}\right )\right )\right ) \int \frac {x}{d+c x} \, dx+\frac {1}{2} \left (i \left (-\log (a+b x)+\log (i+a+b x)-\log \left (\frac {i+a+b x}{a+b x}\right )\right )\right ) \int \frac {x}{d+c x} \, dx\\ &=\frac {i \int \log (-i+a+b x) \, dx}{2 c}-\frac {i \int \log (i+a+b x) \, dx}{2 c}-\frac {(i d) \int \frac {\log (-i+a+b x)}{d+c x} \, dx}{2 c}+\frac {(i d) \int \frac {\log (i+a+b x)}{d+c x} \, dx}{2 c}-\frac {1}{2} \left (i \left (-\log (a+b x)+\log (-i+a+b x)-\log \left (\frac {-i+a+b x}{a+b x}\right )\right )\right ) \int \left (\frac {1}{c}-\frac {d}{c (d+c x)}\right ) \, dx+\frac {1}{2} \left (i \left (-\log (a+b x)+\log (i+a+b x)-\log \left (\frac {i+a+b x}{a+b x}\right )\right )\right ) \int \left (\frac {1}{c}-\frac {d}{c (d+c x)}\right ) \, dx\\ &=\frac {i x \left (\log \left (-\frac {i-a-b x}{a+b x}\right )+\log (a+b x)-\log (-i+a+b x)\right )}{2 c}-\frac {i x \left (\log (a+b x)-\log (i+a+b x)+\log \left (\frac {i+a+b x}{a+b x}\right )\right )}{2 c}-\frac {i d \left (\log \left (-\frac {i-a-b x}{a+b x}\right )+\log (a+b x)-\log (-i+a+b x)\right ) \log (d+c x)}{2 c^2}+\frac {i d \left (\log (a+b x)-\log (i+a+b x)+\log \left (\frac {i+a+b x}{a+b x}\right )\right ) \log (d+c x)}{2 c^2}+\frac {i d \log (i+a+b x) \log \left (-\frac {b (d+c x)}{(i+a) c-b d}\right )}{2 c^2}-\frac {i d \log (-i+a+b x) \log \left (\frac {b (d+c x)}{(i-a) c+b d}\right )}{2 c^2}+\frac {i \operatorname {Subst}(\int \log (x) \, dx,x,-i+a+b x)}{2 b c}-\frac {i \operatorname {Subst}(\int \log (x) \, dx,x,i+a+b x)}{2 b c}+\frac {(i b d) \int \frac {\log \left (\frac {b (d+c x)}{-(-i+a) c+b d}\right )}{-i+a+b x} \, dx}{2 c^2}-\frac {(i b d) \int \frac {\log \left (\frac {b (d+c x)}{-(i+a) c+b d}\right )}{i+a+b x} \, dx}{2 c^2}\\ &=\frac {i x \left (\log \left (-\frac {i-a-b x}{a+b x}\right )+\log (a+b x)-\log (-i+a+b x)\right )}{2 c}-\frac {i (i-a-b x) \log (-i+a+b x)}{2 b c}-\frac {i (i+a+b x) \log (i+a+b x)}{2 b c}-\frac {i x \left (\log (a+b x)-\log (i+a+b x)+\log \left (\frac {i+a+b x}{a+b x}\right )\right )}{2 c}-\frac {i d \left (\log \left (-\frac {i-a-b x}{a+b x}\right )+\log (a+b x)-\log (-i+a+b x)\right ) \log (d+c x)}{2 c^2}+\frac {i d \left (\log (a+b x)-\log (i+a+b x)+\log \left (\frac {i+a+b x}{a+b x}\right )\right ) \log (d+c x)}{2 c^2}+\frac {i d \log (i+a+b x) \log \left (-\frac {b (d+c x)}{(i+a) c-b d}\right )}{2 c^2}-\frac {i d \log (-i+a+b x) \log \left (\frac {b (d+c x)}{(i-a) c+b d}\right )}{2 c^2}+\frac {(i d) \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {c x}{-(-i+a) c+b d}\right )}{x} \, dx,x,-i+a+b x\right )}{2 c^2}-\frac {(i d) \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {c x}{-(i+a) c+b d}\right )}{x} \, dx,x,i+a+b x\right )}{2 c^2}\\ &=\frac {i x \left (\log \left (-\frac {i-a-b x}{a+b x}\right )+\log (a+b x)-\log (-i+a+b x)\right )}{2 c}-\frac {i (i-a-b x) \log (-i+a+b x)}{2 b c}-\frac {i (i+a+b x) \log (i+a+b x)}{2 b c}-\frac {i x \left (\log (a+b x)-\log (i+a+b x)+\log \left (\frac {i+a+b x}{a+b x}\right )\right )}{2 c}-\frac {i d \left (\log \left (-\frac {i-a-b x}{a+b x}\right )+\log (a+b x)-\log (-i+a+b x)\right ) \log (d+c x)}{2 c^2}+\frac {i d \left (\log (a+b x)-\log (i+a+b x)+\log \left (\frac {i+a+b x}{a+b x}\right )\right ) \log (d+c x)}{2 c^2}+\frac {i d \log (i+a+b x) \log \left (-\frac {b (d+c x)}{(i+a) c-b d}\right )}{2 c^2}-\frac {i d \log (-i+a+b x) \log \left (\frac {b (d+c x)}{(i-a) c+b d}\right )}{2 c^2}-\frac {i d \text {Li}_2\left (\frac {c (i-a-b x)}{(i-a) c+b d}\right )}{2 c^2}+\frac {i d \text {Li}_2\left (\frac {c (i+a+b x)}{(i+a) c-b d}\right )}{2 c^2}\\ \end {align*}
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Mathematica [A] time = 11.12, size = 602, normalized size = 1.78 \[ -\frac {\left ((a+b x)^2+1\right ) \left (a b c d \sqrt {\frac {\left (a^2+1\right ) c^2-2 a b c d+b^2 d^2}{(a c-b d)^2}} \cot ^{-1}(a+b x)^2 e^{-i \tan ^{-1}\left (\frac {c}{a c-b d}\right )}-b^2 d^2 \sqrt {\frac {\left (a^2+1\right ) c^2-2 a b c d+b^2 d^2}{(a c-b d)^2}} \cot ^{-1}(a+b x)^2 e^{-i \tan ^{-1}\left (\frac {c}{a c-b d}\right )}+b^2 d^2 \cot ^{-1}(a+b x)^2+2 c^2 \log \left (\frac {1}{(a+b x) \sqrt {\frac {1}{(a+b x)^2}+1}}\right )-2 c^2 (a+b x) \cot ^{-1}(a+b x)-i b c d \text {Li}_2\left (\exp \left (2 i \left (\cot ^{-1}(a+b x)-\tan ^{-1}\left (\frac {c}{a c-b d}\right )\right )\right )\right )+2 b c d \cot ^{-1}(a+b x) \log \left (1-\exp \left (2 i \left (\cot ^{-1}(a+b x)-\tan ^{-1}\left (\frac {c}{a c-b d}\right )\right )\right )\right )-2 b c d \tan ^{-1}\left (\frac {c}{a c-b d}\right ) \log \left (1-\exp \left (2 i \left (\cot ^{-1}(a+b x)-\tan ^{-1}\left (\frac {c}{a c-b d}\right )\right )\right )\right )+i b c d \text {Li}_2\left (e^{2 i \cot ^{-1}(a+b x)}\right )-\pi b c d \log \left (\frac {1}{\sqrt {\frac {1}{(a+b x)^2}+1}}\right )-a b c d \cot ^{-1}(a+b x)^2+i b c d \cot ^{-1}(a+b x)^2+i \pi b c d \cot ^{-1}(a+b x)+\pi b c d \log \left (1+e^{-2 i \cot ^{-1}(a+b x)}\right )-2 b c d \cot ^{-1}(a+b x) \log \left (1-e^{2 i \cot ^{-1}(a+b x)}\right )+2 i b c d \cot ^{-1}(a+b x) \tan ^{-1}\left (\frac {c}{a c-b d}\right )+2 b c d \tan ^{-1}\left (\frac {c}{a c-b d}\right ) \log \left (\sin \left (\cot ^{-1}(a+b x)-\tan ^{-1}\left (\frac {c}{a c-b d}\right )\right )\right )\right )}{2 b c^3 (a+b x)^2 \sqrt {\frac {a^2+2 a b x+b^2 x^2+1}{(a+b x)^2}} \sqrt {\frac {1}{(a+b x)^2}+1}} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 1.69, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {x \operatorname {arccot}\left (b x + a\right )}{c x + d}, x\right ) \]
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 \[ \int \frac {\operatorname {arccot}\left (b x + a\right )}{c + \frac {d}{x}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.07, size = 317, normalized size = 0.94 \[ \frac {\mathrm {arccot}\left (b x +a \right ) x}{c}+\frac {\mathrm {arccot}\left (b x +a \right ) a}{b c}-\frac {\mathrm {arccot}\left (b x +a \right ) d \ln \left (c \left (b x +a \right )-a c +b d \right )}{c^{2}}+\frac {\ln \left (a^{2} c^{2}-2 a b c d +b^{2} d^{2}+2 a c \left (c \left (b x +a \right )-a c +b d \right )-2 \left (c \left (b x +a \right )-a c +b d \right ) b d +\left (c \left (b x +a \right )-a c +b d \right )^{2}+c^{2}\right )}{2 b c}+\frac {i d \ln \left (c \left (b x +a \right )-a c +b d \right ) \ln \left (\frac {i c -c \left (b x +a \right )}{-a c +b d +i c}\right )}{2 c^{2}}-\frac {i d \ln \left (c \left (b x +a \right )-a c +b d \right ) \ln \left (\frac {i c +c \left (b x +a \right )}{a c -b d +i c}\right )}{2 c^{2}}+\frac {i d \dilog \left (\frac {i c -c \left (b x +a \right )}{-a c +b d +i c}\right )}{2 c^{2}}-\frac {i d \dilog \left (\frac {i c +c \left (b x +a \right )}{a c -b d +i c}\right )}{2 c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.55, size = 280, normalized size = 0.83 \[ \frac {2 \, b c x \arctan \left (1, b x + a\right ) - b d \arctan \left (1, b x + a\right ) \log \left (-\frac {b^{2} c^{2} x^{2} + 2 \, b^{2} c d x + b^{2} d^{2}}{2 \, a b c d - b^{2} d^{2} - {\left (a^{2} + 1\right )} c^{2}}\right ) - 2 \, a c \arctan \left (b x + a\right ) + i \, b d {\rm Li}_2\left (\frac {b c x + {\left (a + i\right )} c}{{\left (a + i\right )} c - b d}\right ) - i \, b d {\rm Li}_2\left (\frac {b c x + {\left (a - i\right )} c}{{\left (a - i\right )} c - b d}\right ) - {\left (b d \arctan \left (-\frac {b c^{2} x + b c d}{2 \, a b c d - b^{2} d^{2} - {\left (a^{2} + 1\right )} c^{2}}, \frac {a b c d - b^{2} d^{2} + {\left (a b c^{2} - b^{2} c d\right )} x}{2 \, a b c d - b^{2} d^{2} - {\left (a^{2} + 1\right )} c^{2}}\right ) - c\right )} \log \left (b^{2} x^{2} + 2 \, a b x + a^{2} + 1\right )}{2 \, b c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {\mathrm {acot}\left (a+b\,x\right )}{c+\frac {d}{x}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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