Optimal. Leaf size=152 \[ -\frac {\cot ^{-1}(a+b x) \log \left (\frac {2}{1-i (a+b x)}\right )}{d}+\frac {\cot ^{-1}(a+b x) \log \left (\frac {2 b (c+d x)}{(b c+i d-a d) (1-i (a+b x))}\right )}{d}-\frac {i \text {PolyLog}\left (2,1-\frac {2}{1-i (a+b x)}\right )}{2 d}+\frac {i \text {PolyLog}\left (2,1-\frac {2 b (c+d x)}{(b c+i d-a d) (1-i (a+b x))}\right )}{2 d} \]
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Rubi [A]
time = 0.11, antiderivative size = 152, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.357, Rules used = {5156, 4967,
2449, 2352, 2497} \begin {gather*} \frac {i \text {Li}_2\left (1-\frac {2 b (c+d x)}{(b c-a d+i d) (1-i (a+b x))}\right )}{2 d}+\frac {\cot ^{-1}(a+b x) \log \left (\frac {2 b (c+d x)}{(1-i (a+b x)) (-a d+b c+i d)}\right )}{d}-\frac {i \text {Li}_2\left (1-\frac {2}{1-i (a+b x)}\right )}{2 d}-\frac {\log \left (\frac {2}{1-i (a+b x)}\right ) \cot ^{-1}(a+b x)}{d} \end {gather*}
Antiderivative was successfully verified.
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Rule 2352
Rule 2449
Rule 2497
Rule 4967
Rule 5156
Rubi steps
\begin {align*} \int \frac {\cot ^{-1}(a+b x)}{c+d x} \, dx &=\frac {\text {Subst}\left (\int \frac {\cot ^{-1}(x)}{\frac {b c-a d}{b}+\frac {d x}{b}} \, dx,x,a+b x\right )}{b}\\ &=-\frac {\cot ^{-1}(a+b x) \log \left (\frac {2}{1-i (a+b x)}\right )}{d}+\frac {\cot ^{-1}(a+b x) \log \left (\frac {2 b (c+d x)}{(b c+i d-a d) (1-i (a+b x))}\right )}{d}-\frac {\text {Subst}\left (\int \frac {\log \left (\frac {2}{1-i x}\right )}{1+x^2} \, dx,x,a+b x\right )}{d}+\frac {\text {Subst}\left (\int \frac {\log \left (\frac {2 \left (\frac {b c-a d}{b}+\frac {d x}{b}\right )}{\left (\frac {i d}{b}+\frac {b c-a d}{b}\right ) (1-i x)}\right )}{1+x^2} \, dx,x,a+b x\right )}{d}\\ &=-\frac {\cot ^{-1}(a+b x) \log \left (\frac {2}{1-i (a+b x)}\right )}{d}+\frac {\cot ^{-1}(a+b x) \log \left (\frac {2 b (c+d x)}{(b c+i d-a d) (1-i (a+b x))}\right )}{d}+\frac {i \text {Li}_2\left (1-\frac {2 b (c+d x)}{(b c+i d-a d) (1-i (a+b x))}\right )}{2 d}-\frac {i \text {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1-i (a+b x)}\right )}{d}\\ &=-\frac {\cot ^{-1}(a+b x) \log \left (\frac {2}{1-i (a+b x)}\right )}{d}+\frac {\cot ^{-1}(a+b x) \log \left (\frac {2 b (c+d x)}{(b c+i d-a d) (1-i (a+b x))}\right )}{d}-\frac {i \text {Li}_2\left (1-\frac {2}{1-i (a+b x)}\right )}{2 d}+\frac {i \text {Li}_2\left (1-\frac {2 b (c+d x)}{(b c+i d-a d) (1-i (a+b x))}\right )}{2 d}\\ \end {align*}
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Mathematica [B] Both result and optimal contain complex but leaf count is larger than twice
the leaf count of optimal. \(345\) vs. \(2(152)=304\).
time = 0.03, size = 345, normalized size = 2.27 \begin {gather*} -\frac {i \log \left (\frac {d (-i+a+b x)}{b \left (-\frac {i d}{b}-\frac {b c-a d}{b}\right )}\right ) \log \left (\frac {b c-a d}{b}+\frac {d (a+b x)}{b}\right )}{2 d}+\frac {i \log \left (\frac {-i+a+b x}{a+b x}\right ) \log \left (\frac {b c-a d}{b}+\frac {d (a+b x)}{b}\right )}{2 d}+\frac {i \log \left (\frac {d (i+a+b x)}{b \left (\frac {i d}{b}-\frac {b c-a d}{b}\right )}\right ) \log \left (\frac {b c-a d}{b}+\frac {d (a+b x)}{b}\right )}{2 d}-\frac {i \log \left (\frac {i+a+b x}{a+b x}\right ) \log \left (\frac {b c-a d}{b}+\frac {d (a+b x)}{b}\right )}{2 d}+\frac {i \text {PolyLog}\left (2,\frac {b \left (\frac {b c-a d}{b}+\frac {d (a+b x)}{b}\right )}{b c-i d-a d}\right )}{2 d}-\frac {i \text {PolyLog}\left (2,\frac {b \left (\frac {b c-a d}{b}+\frac {d (a+b x)}{b}\right )}{b c+i d-a d}\right )}{2 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.14, size = 187, normalized size = 1.23
method | result | size |
derivativedivides | \(\frac {\frac {b \ln \left (a d -b c -d \left (b x +a \right )\right ) \mathrm {arccot}\left (b x +a \right )}{d}-b \left (-\frac {i \ln \left (a d -b c -d \left (b x +a \right )\right ) \left (\ln \left (\frac {i d +d \left (b x +a \right )}{a d -b c +i d}\right )-\ln \left (\frac {i d -d \left (b x +a \right )}{-a d +b c +i d}\right )\right )}{2 d}-\frac {i \left (\dilog \left (\frac {i d +d \left (b x +a \right )}{a d -b c +i d}\right )-\dilog \left (\frac {i d -d \left (b x +a \right )}{-a d +b c +i d}\right )\right )}{2 d}\right )}{b}\) | \(187\) |
default | \(\frac {\frac {b \ln \left (a d -b c -d \left (b x +a \right )\right ) \mathrm {arccot}\left (b x +a \right )}{d}-b \left (-\frac {i \ln \left (a d -b c -d \left (b x +a \right )\right ) \left (\ln \left (\frac {i d +d \left (b x +a \right )}{a d -b c +i d}\right )-\ln \left (\frac {i d -d \left (b x +a \right )}{-a d +b c +i d}\right )\right )}{2 d}-\frac {i \left (\dilog \left (\frac {i d +d \left (b x +a \right )}{a d -b c +i d}\right )-\dilog \left (\frac {i d -d \left (b x +a \right )}{-a d +b c +i d}\right )\right )}{2 d}\right )}{b}\) | \(187\) |
risch | \(-\frac {i \dilog \left (\frac {i a d -i c b +\left (-i b x -i a +1\right ) d -d}{i a d -i c b -d}\right )}{2 d}-\frac {i \ln \left (-i b x -i a +1\right ) \ln \left (\frac {i a d -i c b +\left (-i b x -i a +1\right ) d -d}{i a d -i c b -d}\right )}{2 d}+\frac {\pi \ln \left (i a d -i c b +\left (-i b x -i a +1\right ) d -d \right )}{2 d}+\frac {i \dilog \left (\frac {-i a d +i c b +\left (i b x +i a +1\right ) d -d}{-i a d +i c b -d}\right )}{2 d}+\frac {i \ln \left (i b x +i a +1\right ) \ln \left (\frac {-i a d +i c b +\left (i b x +i a +1\right ) d -d}{-i a d +i c b -d}\right )}{2 d}\) | \(264\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Both result and optimal contain complex but leaf count of result is larger than
twice the leaf count of optimal. 283 vs. \(2 (130) = 260\).
time = 0.55, size = 283, normalized size = 1.86 \begin {gather*} \frac {\operatorname {arccot}\left (b x + a\right ) \log \left (d x + c\right )}{d} + \frac {\arctan \left (\frac {b^{2} x + a b}{b}\right ) \log \left (d x + c\right )}{d} + \frac {\arctan \left (\frac {b d^{2} x + b c d}{b^{2} c^{2} - 2 \, a b c d + {\left (a^{2} + 1\right )} d^{2}}, \frac {b^{2} c^{2} - a b c d + {\left (b^{2} c d - a b d^{2}\right )} x}{b^{2} c^{2} - 2 \, a b c d + {\left (a^{2} + 1\right )} d^{2}}\right ) \log \left (b^{2} x^{2} + 2 \, a b x + a^{2} + 1\right ) - \arctan \left (b x + a\right ) \log \left (\frac {b^{2} d^{2} x^{2} + 2 \, b^{2} c d x + b^{2} c^{2}}{b^{2} c^{2} - 2 \, a b c d + {\left (a^{2} + 1\right )} d^{2}}\right ) + i \, {\rm Li}_2\left (\frac {i \, b d x + {\left (i \, a + 1\right )} d}{-i \, b c + {\left (i \, a + 1\right )} d}\right ) - i \, {\rm Li}_2\left (\frac {i \, b d x + {\left (i \, a - 1\right )} d}{-i \, b c + {\left (i \, a - 1\right )} d}\right )}{2 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {\mathrm {acot}\left (a+b\,x\right )}{c+d\,x} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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