Optimal. Leaf size=238 \[ \frac {\sqrt {d} \sqrt {\frac {e x^2}{d}+1} \text {Li}_2\left (e^{2 \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}\right )}{2 \sqrt {d+e x^2}}-\frac {\sqrt {d} \sqrt {\frac {e x^2}{d}+1} \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )^2}{2 \sqrt {d+e x^2}}+\frac {\sqrt {d} \sqrt {\frac {e x^2}{d}+1} \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \log \left (1-e^{2 \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}\right )}{\sqrt {d+e x^2}}-\frac {\sqrt {d} \log (x) \sqrt {\frac {e x^2}{d}+1} \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {d+e x^2}}+\log (x) \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right ) \]
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Rubi [A] time = 0.16, antiderivative size = 238, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.348, Rules used = {6219, 2327, 2325, 5659, 3716, 2190, 2279, 2391} \[ \frac {\sqrt {d} \sqrt {\frac {e x^2}{d}+1} \text {PolyLog}\left (2,e^{2 \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}\right )}{2 \sqrt {d+e x^2}}-\frac {\sqrt {d} \sqrt {\frac {e x^2}{d}+1} \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )^2}{2 \sqrt {d+e x^2}}+\frac {\sqrt {d} \sqrt {\frac {e x^2}{d}+1} \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \log \left (1-e^{2 \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}\right )}{\sqrt {d+e x^2}}-\frac {\sqrt {d} \log (x) \sqrt {\frac {e x^2}{d}+1} \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {d+e x^2}}+\log (x) \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right ) \]
Antiderivative was successfully verified.
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Rule 2190
Rule 2279
Rule 2325
Rule 2327
Rule 2391
Rule 3716
Rule 5659
Rule 6219
Rubi steps
\begin {align*} \int \frac {\tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{x} \, dx &=\tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right ) \log (x)-\sqrt {e} \int \frac {\log (x)}{\sqrt {d+e x^2}} \, dx\\ &=\tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right ) \log (x)-\frac {\left (\sqrt {e} \sqrt {1+\frac {e x^2}{d}}\right ) \int \frac {\log (x)}{\sqrt {1+\frac {e x^2}{d}}} \, dx}{\sqrt {d+e x^2}}\\ &=-\frac {\sqrt {d} \sqrt {1+\frac {e x^2}{d}} \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \log (x)}{\sqrt {d+e x^2}}+\tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right ) \log (x)+\frac {\left (\sqrt {d} \sqrt {1+\frac {e x^2}{d}}\right ) \int \frac {\sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{x} \, dx}{\sqrt {d+e x^2}}\\ &=-\frac {\sqrt {d} \sqrt {1+\frac {e x^2}{d}} \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \log (x)}{\sqrt {d+e x^2}}+\tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right ) \log (x)+\frac {\left (\sqrt {d} \sqrt {1+\frac {e x^2}{d}}\right ) \operatorname {Subst}\left (\int x \coth (x) \, dx,x,\sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )\right )}{\sqrt {d+e x^2}}\\ &=-\frac {\sqrt {d} \sqrt {1+\frac {e x^2}{d}} \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )^2}{2 \sqrt {d+e x^2}}-\frac {\sqrt {d} \sqrt {1+\frac {e x^2}{d}} \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \log (x)}{\sqrt {d+e x^2}}+\tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right ) \log (x)-\frac {\left (2 \sqrt {d} \sqrt {1+\frac {e x^2}{d}}\right ) \operatorname {Subst}\left (\int \frac {e^{2 x} x}{1-e^{2 x}} \, dx,x,\sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )\right )}{\sqrt {d+e x^2}}\\ &=-\frac {\sqrt {d} \sqrt {1+\frac {e x^2}{d}} \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )^2}{2 \sqrt {d+e x^2}}+\frac {\sqrt {d} \sqrt {1+\frac {e x^2}{d}} \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \log \left (1-e^{2 \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}\right )}{\sqrt {d+e x^2}}-\frac {\sqrt {d} \sqrt {1+\frac {e x^2}{d}} \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \log (x)}{\sqrt {d+e x^2}}+\tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right ) \log (x)-\frac {\left (\sqrt {d} \sqrt {1+\frac {e x^2}{d}}\right ) \operatorname {Subst}\left (\int \log \left (1-e^{2 x}\right ) \, dx,x,\sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )\right )}{\sqrt {d+e x^2}}\\ &=-\frac {\sqrt {d} \sqrt {1+\frac {e x^2}{d}} \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )^2}{2 \sqrt {d+e x^2}}+\frac {\sqrt {d} \sqrt {1+\frac {e x^2}{d}} \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \log \left (1-e^{2 \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}\right )}{\sqrt {d+e x^2}}-\frac {\sqrt {d} \sqrt {1+\frac {e x^2}{d}} \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \log (x)}{\sqrt {d+e x^2}}+\tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right ) \log (x)-\frac {\left (\sqrt {d} \sqrt {1+\frac {e x^2}{d}}\right ) \operatorname {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{2 \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}\right )}{2 \sqrt {d+e x^2}}\\ &=-\frac {\sqrt {d} \sqrt {1+\frac {e x^2}{d}} \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )^2}{2 \sqrt {d+e x^2}}+\frac {\sqrt {d} \sqrt {1+\frac {e x^2}{d}} \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \log \left (1-e^{2 \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}\right )}{\sqrt {d+e x^2}}-\frac {\sqrt {d} \sqrt {1+\frac {e x^2}{d}} \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \log (x)}{\sqrt {d+e x^2}}+\tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right ) \log (x)+\frac {\sqrt {d} \sqrt {1+\frac {e x^2}{d}} \text {Li}_2\left (e^{2 \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}\right )}{2 \sqrt {d+e x^2}}\\ \end {align*}
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Mathematica [A] time = 2.48, size = 167, normalized size = 0.70 \[ \frac {\sqrt {e} \sqrt {\frac {e x^2}{d}+1} \left (-\text {Li}_2\left (e^{-2 \sinh ^{-1}\left (\sqrt {\frac {e}{d}} x\right )}\right )-2 \log (x) \log \left (\sqrt {\frac {e x^2}{d}+1}+x \sqrt {\frac {e}{d}}\right )+\sinh ^{-1}\left (x \sqrt {\frac {e}{d}}\right )^2+2 \sinh ^{-1}\left (x \sqrt {\frac {e}{d}}\right ) \log \left (1-e^{-2 \sinh ^{-1}\left (x \sqrt {\frac {e}{d}}\right )}\right )\right )}{2 \sqrt {\frac {e}{d}} \sqrt {d+e x^2}}+\log (x) \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right ) \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.57, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\operatorname {artanh}\left (\frac {\sqrt {e} x}{\sqrt {e x^{2} + d}}\right )}{x}, x\right ) \]
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 \[ \int \frac {\operatorname {artanh}\left (\frac {\sqrt {e} x}{\sqrt {e x^{2} + d}}\right )}{x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.31, size = 209, normalized size = 0.88 \[ -\frac {\arctanh \left (\frac {x \sqrt {e}}{\sqrt {e \,x^{2}+d}}\right )^{2}}{2}+\arctanh \left (\frac {x \sqrt {e}}{\sqrt {e \,x^{2}+d}}\right ) \ln \left (1+\frac {\frac {x \sqrt {e}}{\sqrt {e \,x^{2}+d}}+1}{\sqrt {-\frac {x^{2} e}{e \,x^{2}+d}+1}}\right )+\polylog \left (2, -\frac {\frac {x \sqrt {e}}{\sqrt {e \,x^{2}+d}}+1}{\sqrt {-\frac {x^{2} e}{e \,x^{2}+d}+1}}\right )+\arctanh \left (\frac {x \sqrt {e}}{\sqrt {e \,x^{2}+d}}\right ) \ln \left (1-\frac {\frac {x \sqrt {e}}{\sqrt {e \,x^{2}+d}}+1}{\sqrt {-\frac {x^{2} e}{e \,x^{2}+d}+1}}\right )+\polylog \left (2, \frac {\frac {x \sqrt {e}}{\sqrt {e \,x^{2}+d}}+1}{\sqrt {-\frac {x^{2} e}{e \,x^{2}+d}+1}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {artanh}\left (\frac {\sqrt {e} x}{\sqrt {e x^{2} + d}}\right )}{x}\,{d x} \]
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 \[ \int \frac {\mathrm {atanh}\left (\frac {\sqrt {e}\,x}{\sqrt {e\,x^2+d}}\right )}{x} \,d x \]
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 \[ \int \frac {\operatorname {atanh}{\left (\frac {\sqrt {e} x}{\sqrt {d + e x^{2}}} \right )}}{x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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