Optimal. Leaf size=117 \[ -\frac {b c}{\sqrt {x}}+b c^2 \tanh ^{-1}\left (c \sqrt {x}\right )-\frac {a+b \tanh ^{-1}\left (c \sqrt {x}\right )}{x}+\frac {c^2 \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )^2}{b}+2 c^2 \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right ) \log \left (2-\frac {2}{1+c \sqrt {x}}\right )-b c^2 \text {PolyLog}\left (2,-1+\frac {2}{1+c \sqrt {x}}\right ) \]
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Rubi [A]
time = 0.27, antiderivative size = 117, normalized size of antiderivative = 1.00, number of steps
used = 9, number of rules used = 9, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.346, Rules used = {46, 1607, 6129,
6037, 331, 212, 6135, 6079, 2497} \begin {gather*} \frac {c^2 \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )^2}{b}+2 c^2 \log \left (2-\frac {2}{c \sqrt {x}+1}\right ) \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )-\frac {a+b \tanh ^{-1}\left (c \sqrt {x}\right )}{x}-b c^2 \text {Li}_2\left (\frac {2}{\sqrt {x} c+1}-1\right )+b c^2 \tanh ^{-1}\left (c \sqrt {x}\right )-\frac {b c}{\sqrt {x}} \end {gather*}
Antiderivative was successfully verified.
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Rule 46
Rule 212
Rule 331
Rule 1607
Rule 2497
Rule 6037
Rule 6079
Rule 6129
Rule 6135
Rubi steps
\begin {align*} \int \frac {a+b \tanh ^{-1}\left (c \sqrt {x}\right )}{x^2 \left (1-c^2 x\right )} \, dx &=2 \text {Subst}\left (\int \frac {a+b \tanh ^{-1}(c x)}{x^3-c^2 x^5} \, dx,x,\sqrt {x}\right )\\ &=2 \text {Subst}\left (\int \frac {a+b \tanh ^{-1}(c x)}{x^3 \left (1-c^2 x^2\right )} \, dx,x,\sqrt {x}\right )\\ &=2 \text {Subst}\left (\int \frac {a+b \tanh ^{-1}(c x)}{x^3} \, dx,x,\sqrt {x}\right )+\left (2 c^2\right ) \text {Subst}\left (\int \frac {a+b \tanh ^{-1}(c x)}{x \left (1-c^2 x^2\right )} \, dx,x,\sqrt {x}\right )\\ &=-\frac {a+b \tanh ^{-1}\left (c \sqrt {x}\right )}{x}+\frac {c^2 \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )^2}{b}+(b c) \text {Subst}\left (\int \frac {1}{x^2 \left (1-c^2 x^2\right )} \, dx,x,\sqrt {x}\right )+\left (2 c^2\right ) \text {Subst}\left (\int \frac {a+b \tanh ^{-1}(c x)}{x (1+c x)} \, dx,x,\sqrt {x}\right )\\ &=-\frac {b c}{\sqrt {x}}-\frac {a+b \tanh ^{-1}\left (c \sqrt {x}\right )}{x}+\frac {c^2 \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )^2}{b}+2 c^2 \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right ) \log \left (2-\frac {2}{1+c \sqrt {x}}\right )+\left (b c^3\right ) \text {Subst}\left (\int \frac {1}{1-c^2 x^2} \, dx,x,\sqrt {x}\right )-\left (2 b c^3\right ) \text {Subst}\left (\int \frac {\log \left (2-\frac {2}{1+c x}\right )}{1-c^2 x^2} \, dx,x,\sqrt {x}\right )\\ &=-\frac {b c}{\sqrt {x}}+b c^2 \tanh ^{-1}\left (c \sqrt {x}\right )-\frac {a+b \tanh ^{-1}\left (c \sqrt {x}\right )}{x}+\frac {c^2 \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )^2}{b}+2 c^2 \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right ) \log \left (2-\frac {2}{1+c \sqrt {x}}\right )-b c^2 \text {Li}_2\left (-1+\frac {2}{1+c \sqrt {x}}\right )\\ \end {align*}
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Mathematica [A]
time = 0.22, size = 118, normalized size = 1.01 \begin {gather*} -\frac {a}{x}+2 a c^2 \log \left (\sqrt {x}\right )-a c^2 \log \left (1-c^2 x\right )-b c^2 \left (\frac {1}{c \sqrt {x}}-\tanh ^{-1}\left (c \sqrt {x}\right ) \left (-\frac {1-c^2 x}{c^2 x}+\tanh ^{-1}\left (c \sqrt {x}\right )+2 \log \left (1-e^{-2 \tanh ^{-1}\left (c \sqrt {x}\right )}\right )\right )+\text {PolyLog}\left (2,e^{-2 \tanh ^{-1}\left (c \sqrt {x}\right )}\right )\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. \(277\) vs.
\(2(105)=210\).
time = 0.26, size = 278, normalized size = 2.38
method | result | size |
derivativedivides | \(-2 c^{2} \left (\frac {a}{2 c^{2} x}-a \ln \left (c \sqrt {x}\right )+\frac {a \ln \left (1+c \sqrt {x}\right )}{2}+\frac {a \ln \left (c \sqrt {x}-1\right )}{2}+\frac {b \arctanh \left (c \sqrt {x}\right )}{2 c^{2} x}-b \arctanh \left (c \sqrt {x}\right ) \ln \left (c \sqrt {x}\right )+\frac {b \arctanh \left (c \sqrt {x}\right ) \ln \left (1+c \sqrt {x}\right )}{2}+\frac {b \arctanh \left (c \sqrt {x}\right ) \ln \left (c \sqrt {x}-1\right )}{2}+\frac {b}{2 c \sqrt {x}}-\frac {b \ln \left (1+c \sqrt {x}\right )}{4}+\frac {b \ln \left (c \sqrt {x}-1\right )}{4}+\frac {b \dilog \left (1+c \sqrt {x}\right )}{2}+\frac {b \ln \left (c \sqrt {x}\right ) \ln \left (1+c \sqrt {x}\right )}{2}+\frac {b \dilog \left (c \sqrt {x}\right )}{2}+\frac {b \ln \left (c \sqrt {x}-1\right )^{2}}{8}-\frac {b \dilog \left (\frac {c \sqrt {x}}{2}+\frac {1}{2}\right )}{2}-\frac {b \ln \left (c \sqrt {x}-1\right ) \ln \left (\frac {c \sqrt {x}}{2}+\frac {1}{2}\right )}{4}-\frac {b \ln \left (1+c \sqrt {x}\right )^{2}}{8}-\frac {b \ln \left (-\frac {c \sqrt {x}}{2}+\frac {1}{2}\right ) \ln \left (\frac {c \sqrt {x}}{2}+\frac {1}{2}\right )}{4}+\frac {b \ln \left (-\frac {c \sqrt {x}}{2}+\frac {1}{2}\right ) \ln \left (1+c \sqrt {x}\right )}{4}\right )\) | \(278\) |
default | \(-2 c^{2} \left (\frac {a}{2 c^{2} x}-a \ln \left (c \sqrt {x}\right )+\frac {a \ln \left (1+c \sqrt {x}\right )}{2}+\frac {a \ln \left (c \sqrt {x}-1\right )}{2}+\frac {b \arctanh \left (c \sqrt {x}\right )}{2 c^{2} x}-b \arctanh \left (c \sqrt {x}\right ) \ln \left (c \sqrt {x}\right )+\frac {b \arctanh \left (c \sqrt {x}\right ) \ln \left (1+c \sqrt {x}\right )}{2}+\frac {b \arctanh \left (c \sqrt {x}\right ) \ln \left (c \sqrt {x}-1\right )}{2}+\frac {b}{2 c \sqrt {x}}-\frac {b \ln \left (1+c \sqrt {x}\right )}{4}+\frac {b \ln \left (c \sqrt {x}-1\right )}{4}+\frac {b \dilog \left (1+c \sqrt {x}\right )}{2}+\frac {b \ln \left (c \sqrt {x}\right ) \ln \left (1+c \sqrt {x}\right )}{2}+\frac {b \dilog \left (c \sqrt {x}\right )}{2}+\frac {b \ln \left (c \sqrt {x}-1\right )^{2}}{8}-\frac {b \dilog \left (\frac {c \sqrt {x}}{2}+\frac {1}{2}\right )}{2}-\frac {b \ln \left (c \sqrt {x}-1\right ) \ln \left (\frac {c \sqrt {x}}{2}+\frac {1}{2}\right )}{4}-\frac {b \ln \left (1+c \sqrt {x}\right )^{2}}{8}-\frac {b \ln \left (-\frac {c \sqrt {x}}{2}+\frac {1}{2}\right ) \ln \left (\frac {c \sqrt {x}}{2}+\frac {1}{2}\right )}{4}+\frac {b \ln \left (-\frac {c \sqrt {x}}{2}+\frac {1}{2}\right ) \ln \left (1+c \sqrt {x}\right )}{4}\right )\) | \(278\) |
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. 248 vs.
\(2 (102) = 204\).
time = 0.41, size = 248, normalized size = 2.12 \begin {gather*} -{\left (\log \left (c \sqrt {x} + 1\right ) \log \left (-\frac {1}{2} \, c \sqrt {x} + \frac {1}{2}\right ) + {\rm Li}_2\left (\frac {1}{2} \, c \sqrt {x} + \frac {1}{2}\right )\right )} b c^{2} - {\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 c^{2} + {\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 c^{2} + \frac {1}{2} \, b c^{2} \log \left (c \sqrt {x} + 1\right ) - \frac {1}{2} \, b c^{2} \log \left (c \sqrt {x} - 1\right ) - {\left (c^{2} \log \left (c \sqrt {x} + 1\right ) + c^{2} \log \left (c \sqrt {x} - 1\right ) - c^{2} \log \left (x\right ) + \frac {1}{x}\right )} a - \frac {b c^{2} x \log \left (c \sqrt {x} + 1\right )^{2} - b c^{2} x \log \left (-c \sqrt {x} + 1\right )^{2} + 4 \, b c \sqrt {x} + 2 \, b \log \left (c \sqrt {x} + 1\right ) - 2 \, {\left (b c^{2} x \log \left (c \sqrt {x} + 1\right ) + b\right )} \log \left (-c \sqrt {x} + 1\right )}{4 \, x} \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}{c^{2} x^{3} - x^{2}}\, dx - \int \frac {b \operatorname {atanh}{\left (c \sqrt {x} \right )}}{c^{2} x^{3} - 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 = -0.01 \begin {gather*} -\int \frac {a+b\,\mathrm {atanh}\left (c\,\sqrt {x}\right )}{x^2\,\left (c^2\,x-1\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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