Optimal. Leaf size=45 \[ -\frac {i \text {PolyLog}\left (2,-\frac {i}{a+b x}\right )}{2 d}+\frac {i \text {PolyLog}\left (2,\frac {i}{a+b x}\right )}{2 d} \]
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Rubi [A]
time = 0.03, antiderivative size = 45, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 4, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.210, Rules used = {5152, 12, 4941,
2438} \begin {gather*} \frac {i \text {Li}_2\left (\frac {i}{a+b x}\right )}{2 d}-\frac {i \text {Li}_2\left (-\frac {i}{a+b x}\right )}{2 d} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 2438
Rule 4941
Rule 5152
Rubi steps
\begin {align*} \int \frac {\cot ^{-1}(a+b x)}{\frac {a d}{b}+d x} \, dx &=\frac {\text {Subst}\left (\int \frac {b \cot ^{-1}(x)}{d x} \, dx,x,a+b x\right )}{b}\\ &=\frac {\text {Subst}\left (\int \frac {\cot ^{-1}(x)}{x} \, dx,x,a+b x\right )}{d}\\ &=\frac {i \text {Subst}\left (\int \frac {\log \left (1-\frac {i}{x}\right )}{x} \, dx,x,a+b x\right )}{2 d}-\frac {i \text {Subst}\left (\int \frac {\log \left (1+\frac {i}{x}\right )}{x} \, dx,x,a+b x\right )}{2 d}\\ &=-\frac {i \text {Li}_2\left (-\frac {i}{a+b x}\right )}{2 d}+\frac {i \text {Li}_2\left (\frac {i}{a+b x}\right )}{2 d}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 38, normalized size = 0.84 \begin {gather*} -\frac {i \left (\text {PolyLog}\left (2,-\frac {i}{a+b x}\right )-\text {PolyLog}\left (2,\frac {i}{a+b x}\right )\right )}{2 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Both result and optimal contain complex but leaf count of result is larger than twice
the leaf count of optimal. 96 vs. \(2 (37 ) = 74\).
time = 0.11, size = 97, normalized size = 2.16
method | result | size |
risch | \(\frac {i \dilog \left (-i b x -i a +1\right )}{2 d}+\frac {\pi \ln \left (-i b x -i a \right )}{2 d}-\frac {i \dilog \left (i b x +i a +1\right )}{2 d}\) | \(55\) |
derivativedivides | \(\frac {\frac {b \ln \left (b x +a \right ) \mathrm {arccot}\left (b x +a \right )}{d}+\frac {b \left (-\frac {i \ln \left (b x +a \right ) \ln \left (1+i \left (b x +a \right )\right )}{2}+\frac {i \ln \left (b x +a \right ) \ln \left (1-i \left (b x +a \right )\right )}{2}-\frac {i \dilog \left (1+i \left (b x +a \right )\right )}{2}+\frac {i \dilog \left (1-i \left (b x +a \right )\right )}{2}\right )}{d}}{b}\) | \(97\) |
default | \(\frac {\frac {b \ln \left (b x +a \right ) \mathrm {arccot}\left (b x +a \right )}{d}+\frac {b \left (-\frac {i \ln \left (b x +a \right ) \ln \left (1+i \left (b x +a \right )\right )}{2}+\frac {i \ln \left (b x +a \right ) \ln \left (1-i \left (b x +a \right )\right )}{2}-\frac {i \dilog \left (1+i \left (b x +a \right )\right )}{2}+\frac {i \dilog \left (1-i \left (b x +a \right )\right )}{2}\right )}{d}}{b}\) | \(97\) |
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. 122 vs. \(2 (31) = 62\).
time = 0.51, size = 122, normalized size = 2.71 \begin {gather*} \frac {\operatorname {arccot}\left (b x + a\right ) \log \left (d x + \frac {a d}{b}\right )}{d} + \frac {\arctan \left (\frac {b^{2} x + a b}{b}\right ) \log \left (d x + \frac {a d}{b}\right )}{d} + \frac {\arctan \left (b x + a, 0\right ) \log \left (b^{2} x^{2} + 2 \, a b x + a^{2} + 1\right ) - 2 \, \arctan \left (b x + a\right ) \log \left ({\left | b x + a \right |}\right ) + i \, {\rm Li}_2\left (i \, b x + i \, a + 1\right ) - i \, {\rm Li}_2\left (-i \, b x - i \, a + 1\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]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \frac {b \int \frac {\operatorname {acot}{\left (a + b x \right )}}{a + b x}\, dx}{d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 103 vs.
\(2 (31) = 62\).
time = 0.68, size = 103, normalized size = 2.29 \begin {gather*} -\frac {\arctan \left (\frac {1}{b x + a}\right ) \tan \left (\frac {1}{2} \, \arctan \left (\frac {1}{b x + a}\right )\right )^{4} + 2 \, \arctan \left (\frac {1}{b x + a}\right ) \tan \left (\frac {1}{2} \, \arctan \left (\frac {1}{b x + a}\right )\right )^{2} - 2 \, \tan \left (\frac {1}{2} \, \arctan \left (\frac {1}{b x + a}\right )\right )^{3} + \arctan \left (\frac {1}{b x + a}\right ) + 2 \, \tan \left (\frac {1}{2} \, \arctan \left (\frac {1}{b x + a}\right )\right )}{8 \, b d \tan \left (\frac {1}{2} \, \arctan \left (\frac {1}{b x + a}\right )\right )^{2}} \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 = -0.02 \begin {gather*} \int \frac {\mathrm {acot}\left (a+b\,x\right )}{d\,x+\frac {a\,d}{b}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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