Optimal. Leaf size=403 \[ \frac {i \text {ArcTan}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right ) \log \left (1-\frac {i}{a x}\right )}{2 \sqrt {c} \sqrt {d}}-\frac {i \text {ArcTan}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right ) \log \left (1+\frac {i}{a x}\right )}{2 \sqrt {c} \sqrt {d}}-\frac {i \text {ArcTan}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right ) \log \left (\frac {2 i \sqrt {c} \sqrt {d} (i-a x)}{\left (a \sqrt {c}-\sqrt {d}\right ) \left (\sqrt {c}-i \sqrt {d} x\right )}\right )}{2 \sqrt {c} \sqrt {d}}+\frac {i \text {ArcTan}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right ) \log \left (-\frac {2 i \sqrt {c} \sqrt {d} (i+a x)}{\left (a \sqrt {c}+\sqrt {d}\right ) \left (\sqrt {c}-i \sqrt {d} x\right )}\right )}{2 \sqrt {c} \sqrt {d}}-\frac {\text {PolyLog}\left (2,1-\frac {2 i \sqrt {c} \sqrt {d} (i-a x)}{\left (a \sqrt {c}-\sqrt {d}\right ) \left (\sqrt {c}-i \sqrt {d} x\right )}\right )}{4 \sqrt {c} \sqrt {d}}+\frac {\text {PolyLog}\left (2,1+\frac {2 i \sqrt {c} \sqrt {d} (i+a x)}{\left (a \sqrt {c}+\sqrt {d}\right ) \left (\sqrt {c}-i \sqrt {d} x\right )}\right )}{4 \sqrt {c} \sqrt {d}} \]
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Rubi [A]
time = 0.69, antiderivative size = 403, normalized size of antiderivative = 1.00, number
of steps used = 27, number of rules used = 13, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.929, Rules
used = {5029, 211, 2520, 12, 266, 6820, 4996, 4940, 2438, 4966, 2449, 2352, 2497}
\begin {gather*} \frac {i \log \left (1-\frac {i}{a x}\right ) \text {ArcTan}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{2 \sqrt {c} \sqrt {d}}-\frac {i \log \left (1+\frac {i}{a x}\right ) \text {ArcTan}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{2 \sqrt {c} \sqrt {d}}-\frac {i \text {ArcTan}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right ) \log \left (\frac {2 i \sqrt {c} \sqrt {d} (-a x+i)}{\left (a \sqrt {c}-\sqrt {d}\right ) \left (\sqrt {c}-i \sqrt {d} x\right )}\right )}{2 \sqrt {c} \sqrt {d}}+\frac {i \text {ArcTan}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right ) \log \left (-\frac {2 i \sqrt {c} \sqrt {d} (a x+i)}{\left (a \sqrt {c}+\sqrt {d}\right ) \left (\sqrt {c}-i \sqrt {d} x\right )}\right )}{2 \sqrt {c} \sqrt {d}}-\frac {\text {Li}_2\left (1-\frac {2 i \sqrt {c} \sqrt {d} (i-a x)}{\left (a \sqrt {c}-\sqrt {d}\right ) \left (\sqrt {c}-i \sqrt {d} x\right )}\right )}{4 \sqrt {c} \sqrt {d}}+\frac {\text {Li}_2\left (\frac {2 i \sqrt {c} \sqrt {d} (a x+i)}{\left (\sqrt {c} a+\sqrt {d}\right ) \left (\sqrt {c}-i \sqrt {d} x\right )}+1\right )}{4 \sqrt {c} \sqrt {d}} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 211
Rule 266
Rule 2352
Rule 2438
Rule 2449
Rule 2497
Rule 2520
Rule 4940
Rule 4966
Rule 4996
Rule 5029
Rule 6820
Rubi steps
\begin {align*} \int \frac {\cot ^{-1}(a x)}{c+d x^2} \, dx &=\frac {1}{2} i \int \frac {\log \left (1-\frac {i}{a x}\right )}{c+d x^2} \, dx-\frac {1}{2} i \int \frac {\log \left (1+\frac {i}{a x}\right )}{c+d x^2} \, dx\\ &=\frac {i \tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right ) \log \left (1-\frac {i}{a x}\right )}{2 \sqrt {c} \sqrt {d}}-\frac {i \tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right ) \log \left (1+\frac {i}{a x}\right )}{2 \sqrt {c} \sqrt {d}}+\frac {\int \frac {\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{\sqrt {c} \sqrt {d} \left (1-\frac {i}{a x}\right ) x^2} \, dx}{2 a}+\frac {\int \frac {\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{\sqrt {c} \sqrt {d} \left (1+\frac {i}{a x}\right ) x^2} \, dx}{2 a}\\ &=\frac {i \tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right ) \log \left (1-\frac {i}{a x}\right )}{2 \sqrt {c} \sqrt {d}}-\frac {i \tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right ) \log \left (1+\frac {i}{a x}\right )}{2 \sqrt {c} \sqrt {d}}+\frac {\int \frac {\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{\left (1-\frac {i}{a x}\right ) x^2} \, dx}{2 a \sqrt {c} \sqrt {d}}+\frac {\int \frac {\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{\left (1+\frac {i}{a x}\right ) x^2} \, dx}{2 a \sqrt {c} \sqrt {d}}\\ &=\frac {i \tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right ) \log \left (1-\frac {i}{a x}\right )}{2 \sqrt {c} \sqrt {d}}-\frac {i \tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right ) \log \left (1+\frac {i}{a x}\right )}{2 \sqrt {c} \sqrt {d}}+\frac {\int \frac {a \tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{x (-i+a x)} \, dx}{2 a \sqrt {c} \sqrt {d}}+\frac {\int \frac {a \tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{x (i+a x)} \, dx}{2 a \sqrt {c} \sqrt {d}}\\ &=\frac {i \tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right ) \log \left (1-\frac {i}{a x}\right )}{2 \sqrt {c} \sqrt {d}}-\frac {i \tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right ) \log \left (1+\frac {i}{a x}\right )}{2 \sqrt {c} \sqrt {d}}+\frac {\int \frac {\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{x (-i+a x)} \, dx}{2 \sqrt {c} \sqrt {d}}+\frac {\int \frac {\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{x (i+a x)} \, dx}{2 \sqrt {c} \sqrt {d}}\\ &=\frac {i \tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right ) \log \left (1-\frac {i}{a x}\right )}{2 \sqrt {c} \sqrt {d}}-\frac {i \tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right ) \log \left (1+\frac {i}{a x}\right )}{2 \sqrt {c} \sqrt {d}}+\frac {\int \left (\frac {i \tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{x}-\frac {i a \tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{-i+a x}\right ) \, dx}{2 \sqrt {c} \sqrt {d}}+\frac {\int \left (-\frac {i \tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{x}+\frac {i a \tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{i+a x}\right ) \, dx}{2 \sqrt {c} \sqrt {d}}\\ &=\frac {i \tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right ) \log \left (1-\frac {i}{a x}\right )}{2 \sqrt {c} \sqrt {d}}-\frac {i \tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right ) \log \left (1+\frac {i}{a x}\right )}{2 \sqrt {c} \sqrt {d}}-\frac {(i a) \int \frac {\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{-i+a x} \, dx}{2 \sqrt {c} \sqrt {d}}+\frac {(i a) \int \frac {\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{i+a x} \, dx}{2 \sqrt {c} \sqrt {d}}\\ &=\frac {i \tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right ) \log \left (1-\frac {i}{a x}\right )}{2 \sqrt {c} \sqrt {d}}-\frac {i \tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right ) \log \left (1+\frac {i}{a x}\right )}{2 \sqrt {c} \sqrt {d}}-\frac {i \tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right ) \log \left (\frac {2 i \sqrt {c} \sqrt {d} (i-a x)}{\left (a \sqrt {c}-\sqrt {d}\right ) \left (\sqrt {c}-i \sqrt {d} x\right )}\right )}{2 \sqrt {c} \sqrt {d}}+\frac {i \tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right ) \log \left (-\frac {2 i \sqrt {c} \sqrt {d} (i+a x)}{\left (a \sqrt {c}+\sqrt {d}\right ) \left (\sqrt {c}-i \sqrt {d} x\right )}\right )}{2 \sqrt {c} \sqrt {d}}+\frac {i \int \frac {\log \left (\frac {2 \sqrt {d} (-i+a x)}{\sqrt {c} \left (i a-\frac {i \sqrt {d}}{\sqrt {c}}\right ) \left (1-\frac {i \sqrt {d} x}{\sqrt {c}}\right )}\right )}{1+\frac {d x^2}{c}} \, dx}{2 c}-\frac {i \int \frac {\log \left (\frac {2 \sqrt {d} (i+a x)}{\sqrt {c} \left (i a+\frac {i \sqrt {d}}{\sqrt {c}}\right ) \left (1-\frac {i \sqrt {d} x}{\sqrt {c}}\right )}\right )}{1+\frac {d x^2}{c}} \, dx}{2 c}\\ &=\frac {i \tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right ) \log \left (1-\frac {i}{a x}\right )}{2 \sqrt {c} \sqrt {d}}-\frac {i \tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right ) \log \left (1+\frac {i}{a x}\right )}{2 \sqrt {c} \sqrt {d}}-\frac {i \tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right ) \log \left (\frac {2 i \sqrt {c} \sqrt {d} (i-a x)}{\left (a \sqrt {c}-\sqrt {d}\right ) \left (\sqrt {c}-i \sqrt {d} x\right )}\right )}{2 \sqrt {c} \sqrt {d}}+\frac {i \tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right ) \log \left (-\frac {2 i \sqrt {c} \sqrt {d} (i+a x)}{\left (a \sqrt {c}+\sqrt {d}\right ) \left (\sqrt {c}-i \sqrt {d} x\right )}\right )}{2 \sqrt {c} \sqrt {d}}-\frac {\text {Li}_2\left (1-\frac {2 i \sqrt {c} \sqrt {d} (i-a x)}{\left (a \sqrt {c}-\sqrt {d}\right ) \left (\sqrt {c}-i \sqrt {d} x\right )}\right )}{4 \sqrt {c} \sqrt {d}}+\frac {\text {Li}_2\left (1+\frac {2 i \sqrt {c} \sqrt {d} (i+a x)}{\left (a \sqrt {c}+\sqrt {d}\right ) \left (\sqrt {c}-i \sqrt {d} x\right )}\right )}{4 \sqrt {c} \sqrt {d}}\\ \end {align*}
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Mathematica [A]
time = 0.22, size = 523, normalized size = 1.30 \begin {gather*} \frac {i \left (\log \left (1-\frac {i}{a x}\right ) \log \left (\sqrt {-c}-\sqrt {d} x\right )-\log \left (1+\frac {i}{a x}\right ) \log \left (\sqrt {-c}-\sqrt {d} x\right )-\log \left (\frac {\sqrt {d} (-i+a x)}{a \sqrt {-c}-i \sqrt {d}}\right ) \log \left (\sqrt {-c}-\sqrt {d} x\right )+\log \left (\frac {\sqrt {d} (i+a x)}{a \sqrt {-c}+i \sqrt {d}}\right ) \log \left (\sqrt {-c}-\sqrt {d} x\right )-\log \left (1-\frac {i}{a x}\right ) \log \left (\sqrt {-c}+\sqrt {d} x\right )+\log \left (1+\frac {i}{a x}\right ) \log \left (\sqrt {-c}+\sqrt {d} x\right )+\log \left (\frac {\sqrt {d} (i-a x)}{a \sqrt {-c}+i \sqrt {d}}\right ) \log \left (\sqrt {-c}+\sqrt {d} x\right )-\log \left (-\frac {\sqrt {d} (i+a x)}{a \sqrt {-c}-i \sqrt {d}}\right ) \log \left (\sqrt {-c}+\sqrt {d} x\right )-\text {PolyLog}\left (2,\frac {a \left (\sqrt {-c}-\sqrt {d} x\right )}{a \sqrt {-c}-i \sqrt {d}}\right )+\text {PolyLog}\left (2,\frac {a \left (\sqrt {-c}-\sqrt {d} x\right )}{a \sqrt {-c}+i \sqrt {d}}\right )-\text {PolyLog}\left (2,\frac {a \left (\sqrt {-c}+\sqrt {d} x\right )}{a \sqrt {-c}-i \sqrt {d}}\right )+\text {PolyLog}\left (2,\frac {a \left (\sqrt {-c}+\sqrt {d} x\right )}{a \sqrt {-c}+i \sqrt {d}}\right )\right )}{4 \sqrt {-c} \sqrt {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. 976 vs. \(2 (285 ) = 570\).
time = 0.33, size = 977, normalized size = 2.42 Too large to display
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.55, size = 528, normalized size = 1.31 \begin {gather*} -\frac {a {\left (\frac {8 \, \arctan \left (a x\right ) \arctan \left (\frac {d x}{\sqrt {c d}}\right )}{a} - \frac {4 \, \arctan \left (a x\right ) \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ) + 4 \, \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ) \arctan \left (-\frac {a \sqrt {d} x}{a \sqrt {c} - \sqrt {d}}, -\frac {\sqrt {d}}{a \sqrt {c} - \sqrt {d}}\right ) + \log \left (d x^{2} + c\right ) \log \left (\frac {a^{2} c d + {\left (a^{4} c d + a^{2} d^{2}\right )} x^{2} + 2 \, {\left (a^{3} d x^{2} + a d\right )} \sqrt {c} \sqrt {d} + d^{2}}{a^{4} c^{2} + 6 \, a^{2} c d + 4 \, {\left (a^{3} c + a d\right )} \sqrt {c} \sqrt {d} + d^{2}}\right ) - \log \left (d x^{2} + c\right ) \log \left (\frac {a^{2} c d + {\left (a^{4} c d + a^{2} d^{2}\right )} x^{2} - 2 \, {\left (a^{3} d x^{2} + a d\right )} \sqrt {c} \sqrt {d} + d^{2}}{a^{4} c^{2} + 6 \, a^{2} c d - 4 \, {\left (a^{3} c + a d\right )} \sqrt {c} \sqrt {d} + d^{2}}\right ) + 2 \, {\rm Li}_2\left (\frac {a^{2} c + i \, a d x + {\left (i \, a^{2} x + a\right )} \sqrt {c} \sqrt {d}}{a^{2} c + 2 \, a \sqrt {c} \sqrt {d} + d}\right ) + 2 \, {\rm Li}_2\left (\frac {a^{2} c - i \, a d x - {\left (i \, a^{2} x - a\right )} \sqrt {c} \sqrt {d}}{a^{2} c + 2 \, a \sqrt {c} \sqrt {d} + d}\right ) - 2 \, {\rm Li}_2\left (\frac {a^{2} c + i \, a d x - {\left (i \, a^{2} x + a\right )} \sqrt {c} \sqrt {d}}{a^{2} c - 2 \, a \sqrt {c} \sqrt {d} + d}\right ) - 2 \, {\rm Li}_2\left (\frac {a^{2} c - i \, a d x + {\left (i \, a^{2} x - a\right )} \sqrt {c} \sqrt {d}}{a^{2} c - 2 \, a \sqrt {c} \sqrt {d} + d}\right )}{a}\right )}}{8 \, \sqrt {c d}} + \frac {\operatorname {arccot}\left (a x\right ) \arctan \left (\frac {d x}{\sqrt {c d}}\right )}{\sqrt {c d}} + \frac {\arctan \left (a x\right ) \arctan \left (\frac {d x}{\sqrt {c d}}\right )}{\sqrt {c 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*} \int \frac {\operatorname {acot}{\left (a x \right )}}{c + d x^{2}}\, dx \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 = -0.00 \begin {gather*} \int \frac {\mathrm {acot}\left (a\,x\right )}{d\,x^2+c} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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