Optimal. Leaf size=29 \[ a \log (x)-b \text {PolyLog}\left (2,-c \sqrt {x}\right )+b \text {PolyLog}\left (2,c \sqrt {x}\right ) \]
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Rubi [A]
time = 0.02, antiderivative size = 29, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {6035, 6031}
\begin {gather*} a \log (x)-b \text {Li}_2\left (-c \sqrt {x}\right )+b \text {Li}_2\left (c \sqrt {x}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 6031
Rule 6035
Rubi steps
\begin {align*} \int \frac {a+b \tanh ^{-1}\left (c \sqrt {x}\right )}{x} \, dx &=2 \text {Subst}\left (\int \frac {a+b \tanh ^{-1}(c x)}{x} \, dx,x,\sqrt {x}\right )\\ &=a \log (x)-b \text {Li}_2\left (-c \sqrt {x}\right )+b \text {Li}_2\left (c \sqrt {x}\right )\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 29, normalized size = 1.00 \begin {gather*} a \log (x)-b \text {PolyLog}\left (2,-c \sqrt {x}\right )+b \text {PolyLog}\left (2,c \sqrt {x}\right ) \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. \(62\) vs.
\(2(25)=50\).
time = 0.08, size = 63, normalized size = 2.17
method | result | size |
derivativedivides | \(2 a \ln \left (c \sqrt {x}\right )+2 b \ln \left (c \sqrt {x}\right ) \arctanh \left (c \sqrt {x}\right )-b \dilog \left (c \sqrt {x}\right )-b \dilog \left (1+c \sqrt {x}\right )-b \ln \left (c \sqrt {x}\right ) \ln \left (1+c \sqrt {x}\right )\) | \(63\) |
default | \(2 a \ln \left (c \sqrt {x}\right )+2 b \ln \left (c \sqrt {x}\right ) \arctanh \left (c \sqrt {x}\right )-b \dilog \left (c \sqrt {x}\right )-b \dilog \left (1+c \sqrt {x}\right )-b \ln \left (c \sqrt {x}\right ) \ln \left (1+c \sqrt {x}\right )\) | \(63\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 61 vs.
\(2 (23) = 46\).
time = 0.36, size = 61, normalized size = 2.10 \begin {gather*} -{\left (\log \left (c \sqrt {x}\right ) \log \left (-c \sqrt {x} + 1\right ) + {\rm Li}_2\left (-c \sqrt {x} + 1\right )\right )} b + {\left (\log \left (c \sqrt {x} + 1\right ) \log \left (-c \sqrt {x}\right ) + {\rm Li}_2\left (c \sqrt {x} + 1\right )\right )} b + a \log \left (x\right ) \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 {a + b \operatorname {atanh}{\left (c \sqrt {x} \right )}}{x}\, 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 = -0.03 \begin {gather*} \int \frac {a+b\,\mathrm {atanh}\left (c\,\sqrt {x}\right )}{x} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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