Optimal. Leaf size=238 \[ -\frac {\sqrt {d} \sqrt {1+\frac {e x^2}{d}} \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )^2}{2 \sqrt {d+e x^2}}+\frac {\sqrt {d} \sqrt {1+\frac {e x^2}{d}} \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \log \left (1-e^{2 \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}\right )}{\sqrt {d+e x^2}}-\frac {\sqrt {d} \sqrt {1+\frac {e x^2}{d}} \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \log (x)}{\sqrt {d+e x^2}}+\tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right ) \log (x)+\frac {\sqrt {d} \sqrt {1+\frac {e x^2}{d}} \text {PolyLog}\left (2,e^{2 \sinh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}\right )}{2 \sqrt {d+e x^2}} \]
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Rubi [A]
time = 0.10, 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 = {6354, 2364,
2362, 5775, 3797, 2221, 2317, 2438} \begin {gather*} \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 ) \end {gather*}
Antiderivative was successfully verified.
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Rule 2221
Rule 2317
Rule 2362
Rule 2364
Rule 2438
Rule 3797
Rule 5775
Rule 6354
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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 [F]
time = 0.03, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{x} \, dx \end {gather*}
Verification is not applicable to the result.
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Maple [A]
time = 0.11, size = 209, normalized size = 0.88
method | result | size |
default | \(-\frac {\arctanh \left (\frac {x \sqrt {e}}{\sqrt {e \,x^{2}+d}}\right )^{2}}{2}+\arctanh \left (\frac {x \sqrt {e}}{\sqrt {e \,x^{2}+d}}\right ) \ln \left (1+\frac {\frac {x \sqrt {e}}{\sqrt {e \,x^{2}+d}}+1}{\sqrt {-\frac {x^{2} e}{e \,x^{2}+d}+1}}\right )+\polylog \left (2, -\frac {\frac {x \sqrt {e}}{\sqrt {e \,x^{2}+d}}+1}{\sqrt {-\frac {x^{2} e}{e \,x^{2}+d}+1}}\right )+\arctanh \left (\frac {x \sqrt {e}}{\sqrt {e \,x^{2}+d}}\right ) \ln \left (1-\frac {\frac {x \sqrt {e}}{\sqrt {e \,x^{2}+d}}+1}{\sqrt {-\frac {x^{2} e}{e \,x^{2}+d}+1}}\right )+\polylog \left (2, \frac {\frac {x \sqrt {e}}{\sqrt {e \,x^{2}+d}}+1}{\sqrt {-\frac {x^{2} e}{e \,x^{2}+d}+1}}\right )\) | \(209\) |
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 \begin {gather*} \text {Failed to integrate} \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 {atanh}{\left (\frac {\sqrt {e} x}{\sqrt {d + e x^{2}}} \right )}}{x}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \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 {atanh}\left (\frac {\sqrt {e}\,x}{\sqrt {e\,x^2+d}}\right )}{x} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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