Optimal. Leaf size=258 \[ -\frac {\text {Li}_2\left (1-\frac {2 \left (b+2 c x-\sqrt {b^2-4 a c}\right )}{\left (b+2 c-\sqrt {b^2-4 a c}\right ) (x+1)}\right )}{2 \sqrt {b^2-4 a c}}+\frac {\text {Li}_2\left (1-\frac {2 \left (b+2 c x+\sqrt {b^2-4 a c}\right )}{\left (b+2 c+\sqrt {b^2-4 a c}\right ) (x+1)}\right )}{2 \sqrt {b^2-4 a c}}+\frac {\tanh ^{-1}(x) \log \left (\frac {2 \left (-\sqrt {b^2-4 a c}+b+2 c x\right )}{(x+1) \left (-\sqrt {b^2-4 a c}+b+2 c\right )}\right )}{\sqrt {b^2-4 a c}}-\frac {\tanh ^{-1}(x) \log \left (\frac {2 \left (\sqrt {b^2-4 a c}+b+2 c x\right )}{(x+1) \left (\sqrt {b^2-4 a c}+b+2 c\right )}\right )}{\sqrt {b^2-4 a c}} \]
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Rubi [A] time = 0.33, antiderivative size = 258, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 7, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.467, Rules used = {618, 206, 6728, 5920, 2402, 2315, 2447} \[ -\frac {\text {PolyLog}\left (2,1-\frac {2 \left (-\sqrt {b^2-4 a c}+b+2 c x\right )}{(x+1) \left (-\sqrt {b^2-4 a c}+b+2 c\right )}\right )}{2 \sqrt {b^2-4 a c}}+\frac {\text {PolyLog}\left (2,1-\frac {2 \left (\sqrt {b^2-4 a c}+b+2 c x\right )}{(x+1) \left (\sqrt {b^2-4 a c}+b+2 c\right )}\right )}{2 \sqrt {b^2-4 a c}}+\frac {\tanh ^{-1}(x) \log \left (\frac {2 \left (-\sqrt {b^2-4 a c}+b+2 c x\right )}{(x+1) \left (-\sqrt {b^2-4 a c}+b+2 c\right )}\right )}{\sqrt {b^2-4 a c}}-\frac {\tanh ^{-1}(x) \log \left (\frac {2 \left (\sqrt {b^2-4 a c}+b+2 c x\right )}{(x+1) \left (\sqrt {b^2-4 a c}+b+2 c\right )}\right )}{\sqrt {b^2-4 a c}} \]
Antiderivative was successfully verified.
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Rule 206
Rule 618
Rule 2315
Rule 2402
Rule 2447
Rule 5920
Rule 6728
Rubi steps
\begin {align*} \int \frac {\tanh ^{-1}(x)}{a+b x+c x^2} \, dx &=\int \left (\frac {2 c \tanh ^{-1}(x)}{\sqrt {b^2-4 a c} \left (b-\sqrt {b^2-4 a c}+2 c x\right )}-\frac {2 c \tanh ^{-1}(x)}{\sqrt {b^2-4 a c} \left (b+\sqrt {b^2-4 a c}+2 c x\right )}\right ) \, dx\\ &=\frac {(2 c) \int \frac {\tanh ^{-1}(x)}{b-\sqrt {b^2-4 a c}+2 c x} \, dx}{\sqrt {b^2-4 a c}}-\frac {(2 c) \int \frac {\tanh ^{-1}(x)}{b+\sqrt {b^2-4 a c}+2 c x} \, dx}{\sqrt {b^2-4 a c}}\\ &=\frac {\tanh ^{-1}(x) \log \left (\frac {2 \left (b-\sqrt {b^2-4 a c}+2 c x\right )}{\left (b+2 c-\sqrt {b^2-4 a c}\right ) (1+x)}\right )}{\sqrt {b^2-4 a c}}-\frac {\tanh ^{-1}(x) \log \left (\frac {2 \left (b+\sqrt {b^2-4 a c}+2 c x\right )}{\left (b+2 c+\sqrt {b^2-4 a c}\right ) (1+x)}\right )}{\sqrt {b^2-4 a c}}-\frac {\int \frac {\log \left (\frac {2 \left (b-\sqrt {b^2-4 a c}+2 c x\right )}{\left (b+2 c-\sqrt {b^2-4 a c}\right ) (1+x)}\right )}{1-x^2} \, dx}{\sqrt {b^2-4 a c}}+\frac {\int \frac {\log \left (\frac {2 \left (b+\sqrt {b^2-4 a c}+2 c x\right )}{\left (b+2 c+\sqrt {b^2-4 a c}\right ) (1+x)}\right )}{1-x^2} \, dx}{\sqrt {b^2-4 a c}}\\ &=\frac {\tanh ^{-1}(x) \log \left (\frac {2 \left (b-\sqrt {b^2-4 a c}+2 c x\right )}{\left (b+2 c-\sqrt {b^2-4 a c}\right ) (1+x)}\right )}{\sqrt {b^2-4 a c}}-\frac {\tanh ^{-1}(x) \log \left (\frac {2 \left (b+\sqrt {b^2-4 a c}+2 c x\right )}{\left (b+2 c+\sqrt {b^2-4 a c}\right ) (1+x)}\right )}{\sqrt {b^2-4 a c}}-\frac {\text {Li}_2\left (1-\frac {2 \left (b-\sqrt {b^2-4 a c}+2 c x\right )}{\left (b+2 c-\sqrt {b^2-4 a c}\right ) (1+x)}\right )}{2 \sqrt {b^2-4 a c}}+\frac {\text {Li}_2\left (1-\frac {2 \left (b+\sqrt {b^2-4 a c}+2 c x\right )}{\left (b+2 c+\sqrt {b^2-4 a c}\right ) (1+x)}\right )}{2 \sqrt {b^2-4 a c}}\\ \end {align*}
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Mathematica [C] time = 19.32, size = 874, normalized size = 3.39 \[ \frac {\frac {2 \sqrt {4 a c-b^2} \left (b \left (\sqrt {\frac {c (a+b+c)}{4 a c-b^2}} e^{i \tan ^{-1}\left (\frac {-b-2 c}{\sqrt {4 a c-b^2}}\right )}-\sqrt {\frac {c (a-b+c)}{4 a c-b^2}} e^{i \tan ^{-1}\left (\frac {2 c-b}{\sqrt {4 a c-b^2}}\right )}\right )-2 c \left (e^{i \tan ^{-1}\left (\frac {2 c-b}{\sqrt {4 a c-b^2}}\right )} \sqrt {\frac {c (a-b+c)}{4 a c-b^2}}+\sqrt {\frac {c (a+b+c)}{4 a c-b^2}} e^{i \tan ^{-1}\left (\frac {-b-2 c}{\sqrt {4 a c-b^2}}\right )}-1\right )\right ) \tan ^{-1}\left (\frac {b+2 c x}{\sqrt {4 a c-b^2}}\right )^2}{b^2-4 c^2}+2 \left (-i \tan ^{-1}\left (\frac {-b-2 c}{\sqrt {4 a c-b^2}}\right )+i \tan ^{-1}\left (\frac {2 c-b}{\sqrt {4 a c-b^2}}\right )+2 \tanh ^{-1}(x)+\log \left (1-e^{2 i \left (\tan ^{-1}\left (\frac {-b-2 c}{\sqrt {4 a c-b^2}}\right )+\tan ^{-1}\left (\frac {b+2 c x}{\sqrt {4 a c-b^2}}\right )\right )}\right )-\log \left (1-e^{2 i \left (\tan ^{-1}\left (\frac {2 c-b}{\sqrt {4 a c-b^2}}\right )+\tan ^{-1}\left (\frac {b+2 c x}{\sqrt {4 a c-b^2}}\right )\right )}\right )\right ) \tan ^{-1}\left (\frac {b+2 c x}{\sqrt {4 a c-b^2}}\right )+2 \left (\tan ^{-1}\left (\frac {-b-2 c}{\sqrt {4 a c-b^2}}\right ) \left (\log \left (1-e^{2 i \left (\tan ^{-1}\left (\frac {-b-2 c}{\sqrt {4 a c-b^2}}\right )+\tan ^{-1}\left (\frac {b+2 c x}{\sqrt {4 a c-b^2}}\right )\right )}\right )-\log \left (\sin \left (\tan ^{-1}\left (\frac {-b-2 c}{\sqrt {4 a c-b^2}}\right )+\tan ^{-1}\left (\frac {b+2 c x}{\sqrt {4 a c-b^2}}\right )\right )\right )\right )+\tan ^{-1}\left (\frac {2 c-b}{\sqrt {4 a c-b^2}}\right ) \left (\log \left (\sin \left (\tan ^{-1}\left (\frac {2 c-b}{\sqrt {4 a c-b^2}}\right )+\tan ^{-1}\left (\frac {b+2 c x}{\sqrt {4 a c-b^2}}\right )\right )\right )-\log \left (1-e^{2 i \left (\tan ^{-1}\left (\frac {2 c-b}{\sqrt {4 a c-b^2}}\right )+\tan ^{-1}\left (\frac {b+2 c x}{\sqrt {4 a c-b^2}}\right )\right )}\right )\right )\right )-i \text {Li}_2\left (e^{2 i \left (\tan ^{-1}\left (\frac {-b-2 c}{\sqrt {4 a c-b^2}}\right )+\tan ^{-1}\left (\frac {b+2 c x}{\sqrt {4 a c-b^2}}\right )\right )}\right )+i \text {Li}_2\left (e^{2 i \left (\tan ^{-1}\left (\frac {2 c-b}{\sqrt {4 a c-b^2}}\right )+\tan ^{-1}\left (\frac {b+2 c x}{\sqrt {4 a c-b^2}}\right )\right )}\right )}{2 \sqrt {4 a c-b^2}} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.88, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\operatorname {artanh}\relax (x)}{c x^{2} + b x + a}, x\right ) \]
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 \[ \int \frac {\operatorname {artanh}\relax (x)}{c x^{2} + b x + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.61, size = 1599, normalized size = 6.20 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]
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 \[ \int \frac {\mathrm {atanh}\relax (x)}{c\,x^2+b\,x+a} \,d x \]
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 \[ \int \frac {\operatorname {atanh}{\relax (x )}}{a + b x + c x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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