Optimal. Leaf size=390 \[ -\frac {\text {ArcTan}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right ) \log \left (1-\frac {1}{a x}\right )}{2 \sqrt {c} \sqrt {d}}+\frac {\text {ArcTan}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right ) \log \left (1+\frac {1}{a x}\right )}{2 \sqrt {c} \sqrt {d}}+\frac {\text {ArcTan}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right ) \log \left (-\frac {2 \sqrt {c} \sqrt {d} (1-a x)}{\left (i a \sqrt {c}-\sqrt {d}\right ) \left (\sqrt {c}-i \sqrt {d} x\right )}\right )}{2 \sqrt {c} \sqrt {d}}-\frac {\text {ArcTan}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right ) \log \left (\frac {2 \sqrt {c} \sqrt {d} (1+a x)}{\left (i a \sqrt {c}+\sqrt {d}\right ) \left (\sqrt {c}-i \sqrt {d} x\right )}\right )}{2 \sqrt {c} \sqrt {d}}-\frac {i \text {PolyLog}\left (2,1+\frac {2 \sqrt {c} \sqrt {d} (1-a x)}{\left (i a \sqrt {c}-\sqrt {d}\right ) \left (\sqrt {c}-i \sqrt {d} x\right )}\right )}{4 \sqrt {c} \sqrt {d}}+\frac {i \text {PolyLog}\left (2,1-\frac {2 \sqrt {c} \sqrt {d} (1+a x)}{\left (i a \sqrt {c}+\sqrt {d}\right ) \left (\sqrt {c}-i \sqrt {d} x\right )}\right )}{4 \sqrt {c} \sqrt {d}} \]
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Rubi [A]
time = 0.69, antiderivative size = 390, 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 = {6120, 211, 2520, 12, 266, 6820, 4996, 4940, 2438, 4966, 2449, 2352, 2497}
\begin {gather*} -\frac {\log \left (1-\frac {1}{a x}\right ) \text {ArcTan}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{2 \sqrt {c} \sqrt {d}}+\frac {\log \left (\frac {1}{a x}+1\right ) \text {ArcTan}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{2 \sqrt {c} \sqrt {d}}+\frac {\text {ArcTan}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right ) \log \left (-\frac {2 \sqrt {c} \sqrt {d} (1-a x)}{\left (-\sqrt {d}+i a \sqrt {c}\right ) \left (\sqrt {c}-i \sqrt {d} x\right )}\right )}{2 \sqrt {c} \sqrt {d}}-\frac {\text {ArcTan}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right ) \log \left (\frac {2 \sqrt {c} \sqrt {d} (a x+1)}{\left (\sqrt {d}+i a \sqrt {c}\right ) \left (\sqrt {c}-i \sqrt {d} x\right )}\right )}{2 \sqrt {c} \sqrt {d}}-\frac {i \text {Li}_2\left (\frac {2 \sqrt {c} \sqrt {d} (1-a x)}{\left (i a \sqrt {c}-\sqrt {d}\right ) \left (\sqrt {c}-i \sqrt {d} x\right )}+1\right )}{4 \sqrt {c} \sqrt {d}}+\frac {i \text {Li}_2\left (1-\frac {2 \sqrt {c} \sqrt {d} (a x+1)}{\left (i \sqrt {c} a+\sqrt {d}\right ) \left (\sqrt {c}-i \sqrt {d} x\right )}\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 6120
Rule 6820
Rubi steps
\begin {align*} \int \frac {\coth ^{-1}(a x)}{c+d x^2} \, dx &=-\left (\frac {1}{2} \int \frac {\log \left (1-\frac {1}{a x}\right )}{c+d x^2} \, dx\right )+\frac {1}{2} \int \frac {\log \left (1+\frac {1}{a x}\right )}{c+d x^2} \, dx\\ &=-\frac {\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right ) \log \left (1-\frac {1}{a x}\right )}{2 \sqrt {c} \sqrt {d}}+\frac {\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right ) \log \left (1+\frac {1}{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 {1}{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 {1}{a x}\right ) x^2} \, dx}{2 a}\\ &=-\frac {\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right ) \log \left (1-\frac {1}{a x}\right )}{2 \sqrt {c} \sqrt {d}}+\frac {\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right ) \log \left (1+\frac {1}{a x}\right )}{2 \sqrt {c} \sqrt {d}}+\frac {\int \frac {\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{\left (1-\frac {1}{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 {1}{a x}\right ) x^2} \, dx}{2 a \sqrt {c} \sqrt {d}}\\ &=-\frac {\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right ) \log \left (1-\frac {1}{a x}\right )}{2 \sqrt {c} \sqrt {d}}+\frac {\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right ) \log \left (1+\frac {1}{a x}\right )}{2 \sqrt {c} \sqrt {d}}+\frac {\int \frac {a \tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{x (-1+a x)} \, dx}{2 a \sqrt {c} \sqrt {d}}+\frac {\int \frac {a \tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{x (1+a x)} \, dx}{2 a \sqrt {c} \sqrt {d}}\\ &=-\frac {\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right ) \log \left (1-\frac {1}{a x}\right )}{2 \sqrt {c} \sqrt {d}}+\frac {\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right ) \log \left (1+\frac {1}{a x}\right )}{2 \sqrt {c} \sqrt {d}}+\frac {\int \frac {\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{x (-1+a x)} \, dx}{2 \sqrt {c} \sqrt {d}}+\frac {\int \frac {\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{x (1+a x)} \, dx}{2 \sqrt {c} \sqrt {d}}\\ &=-\frac {\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right ) \log \left (1-\frac {1}{a x}\right )}{2 \sqrt {c} \sqrt {d}}+\frac {\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right ) \log \left (1+\frac {1}{a x}\right )}{2 \sqrt {c} \sqrt {d}}+\frac {\int \left (-\frac {\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{x}+\frac {a \tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{-1+a x}\right ) \, dx}{2 \sqrt {c} \sqrt {d}}+\frac {\int \left (\frac {\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{x}-\frac {a \tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{1+a x}\right ) \, dx}{2 \sqrt {c} \sqrt {d}}\\ &=-\frac {\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right ) \log \left (1-\frac {1}{a x}\right )}{2 \sqrt {c} \sqrt {d}}+\frac {\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right ) \log \left (1+\frac {1}{a x}\right )}{2 \sqrt {c} \sqrt {d}}+\frac {a \int \frac {\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{-1+a x} \, dx}{2 \sqrt {c} \sqrt {d}}-\frac {a \int \frac {\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{1+a x} \, dx}{2 \sqrt {c} \sqrt {d}}\\ &=-\frac {\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right ) \log \left (1-\frac {1}{a x}\right )}{2 \sqrt {c} \sqrt {d}}+\frac {\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right ) \log \left (1+\frac {1}{a x}\right )}{2 \sqrt {c} \sqrt {d}}+\frac {\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right ) \log \left (-\frac {2 \sqrt {c} \sqrt {d} (1-a x)}{\left (i a \sqrt {c}-\sqrt {d}\right ) \left (\sqrt {c}-i \sqrt {d} x\right )}\right )}{2 \sqrt {c} \sqrt {d}}-\frac {\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right ) \log \left (\frac {2 \sqrt {c} \sqrt {d} (1+a x)}{\left (i a \sqrt {c}+\sqrt {d}\right ) \left (\sqrt {c}-i \sqrt {d} x\right )}\right )}{2 \sqrt {c} \sqrt {d}}-\frac {\int \frac {\log \left (\frac {2 \sqrt {d} (-1+a x)}{\sqrt {c} \left (i a-\frac {\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 {\int \frac {\log \left (\frac {2 \sqrt {d} (1+a x)}{\sqrt {c} \left (i a+\frac {\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 {\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right ) \log \left (1-\frac {1}{a x}\right )}{2 \sqrt {c} \sqrt {d}}+\frac {\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right ) \log \left (1+\frac {1}{a x}\right )}{2 \sqrt {c} \sqrt {d}}+\frac {\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right ) \log \left (-\frac {2 \sqrt {c} \sqrt {d} (1-a x)}{\left (i a \sqrt {c}-\sqrt {d}\right ) \left (\sqrt {c}-i \sqrt {d} x\right )}\right )}{2 \sqrt {c} \sqrt {d}}-\frac {\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right ) \log \left (\frac {2 \sqrt {c} \sqrt {d} (1+a x)}{\left (i a \sqrt {c}+\sqrt {d}\right ) \left (\sqrt {c}-i \sqrt {d} x\right )}\right )}{2 \sqrt {c} \sqrt {d}}-\frac {i \text {Li}_2\left (1+\frac {2 \sqrt {c} \sqrt {d} (1-a x)}{\left (i a \sqrt {c}-\sqrt {d}\right ) \left (\sqrt {c}-i \sqrt {d} x\right )}\right )}{4 \sqrt {c} \sqrt {d}}+\frac {i \text {Li}_2\left (1-\frac {2 \sqrt {c} \sqrt {d} (1+a x)}{\left (i 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 = 0.93, size = 671, normalized size = 1.72 \begin {gather*} \frac {a \left (-2 i \text {ArcCos}\left (\frac {a^2 c-d}{a^2 c+d}\right ) \text {ArcTan}\left (\frac {a c}{\sqrt {a^2 c d} x}\right )+4 \coth ^{-1}(a x) \text {ArcTan}\left (\frac {a d x}{\sqrt {a^2 c d}}\right )-\left (\text {ArcCos}\left (\frac {a^2 c-d}{a^2 c+d}\right )+2 \text {ArcTan}\left (\frac {a c}{\sqrt {a^2 c d} x}\right )\right ) \log \left (\frac {2 d \left (a^2 c-i \sqrt {a^2 c d}\right ) (-1+a x)}{\left (a^2 c+d\right ) \left (i \sqrt {a^2 c d}+a d x\right )}\right )-\left (\text {ArcCos}\left (\frac {a^2 c-d}{a^2 c+d}\right )-2 \text {ArcTan}\left (\frac {a c}{\sqrt {a^2 c d} x}\right )\right ) \log \left (\frac {2 d \left (a^2 c+i \sqrt {a^2 c d}\right ) (1+a x)}{\left (a^2 c+d\right ) \left (i \sqrt {a^2 c d}+a d x\right )}\right )+\left (\text {ArcCos}\left (\frac {a^2 c-d}{a^2 c+d}\right )+2 \left (\text {ArcTan}\left (\frac {a c}{\sqrt {a^2 c d} x}\right )+\text {ArcTan}\left (\frac {a d x}{\sqrt {a^2 c d}}\right )\right )\right ) \log \left (\frac {\sqrt {2} \sqrt {a^2 c d} e^{-\coth ^{-1}(a x)}}{\sqrt {a^2 c+d} \sqrt {-a^2 c+d+\left (a^2 c+d\right ) \cosh \left (2 \coth ^{-1}(a x)\right )}}\right )+\left (\text {ArcCos}\left (\frac {a^2 c-d}{a^2 c+d}\right )-2 \left (\text {ArcTan}\left (\frac {a c}{\sqrt {a^2 c d} x}\right )+\text {ArcTan}\left (\frac {a d x}{\sqrt {a^2 c d}}\right )\right )\right ) \log \left (\frac {\sqrt {2} \sqrt {a^2 c d} e^{\coth ^{-1}(a x)}}{\sqrt {a^2 c+d} \sqrt {-a^2 c+d+\left (a^2 c+d\right ) \cosh \left (2 \coth ^{-1}(a x)\right )}}\right )+i \left (-\text {PolyLog}\left (2,\frac {\left (a^2 c-d-2 i \sqrt {a^2 c d}\right ) \left (\sqrt {a^2 c d}+i a d x\right )}{\left (a^2 c+d\right ) \left (\sqrt {a^2 c d}-i a d x\right )}\right )+\text {PolyLog}\left (2,\frac {\left (a^2 c-d+2 i \sqrt {a^2 c d}\right ) \left (\sqrt {a^2 c d}+i a d x\right )}{\left (a^2 c+d\right ) \left (\sqrt {a^2 c d}-i a d x\right )}\right )\right )\right )}{4 \sqrt {a^2 c d}} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(953\) vs.
\(2(280)=560\).
time = 0.66, size = 954, normalized size = 2.45 Too large to display
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.56, size = 406, normalized size = 1.04 \begin {gather*} \frac {\operatorname {arcoth}\left (a x\right ) \arctan \left (\frac {d x}{\sqrt {c d}}\right )}{\sqrt {c d}} + \frac {{\left (\arctan \left (\frac {{\left (a^{2} x + a\right )} \sqrt {c} \sqrt {d}}{a^{2} c + d}, \frac {a d x + d}{a^{2} c + d}\right ) - \arctan \left (\frac {{\left (a^{2} x - a\right )} \sqrt {c} \sqrt {d}}{a^{2} c + d}, -\frac {a d x - d}{a^{2} c + d}\right )\right )} \log \left (d x^{2} + c\right ) - \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ) \log \left (\frac {a^{2} d x^{2} + 2 \, a d x + d}{a^{2} c + d}\right ) + \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ) \log \left (\frac {a^{2} d x^{2} - 2 \, a d x + d}{a^{2} c + d}\right ) - i \, {\rm Li}_2\left (\frac {a^{2} c + a d x - {\left (i \, a^{2} x - i \, a\right )} \sqrt {c} \sqrt {d}}{a^{2} c + 2 i \, a \sqrt {c} \sqrt {d} - d}\right ) - i \, {\rm Li}_2\left (\frac {a^{2} c - a d x + {\left (i \, a^{2} x + i \, a\right )} \sqrt {c} \sqrt {d}}{a^{2} c + 2 i \, a \sqrt {c} \sqrt {d} - d}\right ) + i \, {\rm Li}_2\left (\frac {a^{2} c + a d x + {\left (i \, a^{2} x - i \, a\right )} \sqrt {c} \sqrt {d}}{a^{2} c - 2 i \, a \sqrt {c} \sqrt {d} - d}\right ) + i \, {\rm Li}_2\left (\frac {a^{2} c - a d x - {\left (i \, a^{2} x + i \, a\right )} \sqrt {c} \sqrt {d}}{a^{2} c - 2 i \, a \sqrt {c} \sqrt {d} - d}\right )}{4 \, \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 {acoth}{\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 {acoth}\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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