Optimal. Leaf size=120 \[ -\frac {\left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )^2}{b c^4}+\frac {2 \log \left (\frac {2}{1-c \sqrt {x}}\right ) \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )}{c^4}-\frac {x \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )}{c^2}+\frac {b \text {Li}_2\left (1-\frac {2}{1-c \sqrt {x}}\right )}{c^4}+\frac {b \tanh ^{-1}\left (c \sqrt {x}\right )}{c^4}-\frac {b \sqrt {x}}{c^3} \]
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Rubi [A] time = 0.26, antiderivative size = 120, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 9, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {43, 5980, 5916, 321, 206, 5984, 5918, 2402, 2315} \[ \frac {b \text {PolyLog}\left (2,1-\frac {2}{1-c \sqrt {x}}\right )}{c^4}-\frac {\left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )^2}{b c^4}-\frac {x \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )}{c^2}+\frac {2 \log \left (\frac {2}{1-c \sqrt {x}}\right ) \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )}{c^4}-\frac {b \sqrt {x}}{c^3}+\frac {b \tanh ^{-1}\left (c \sqrt {x}\right )}{c^4} \]
Antiderivative was successfully verified.
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Rule 43
Rule 206
Rule 321
Rule 2315
Rule 2402
Rule 5916
Rule 5918
Rule 5980
Rule 5984
Rubi steps
\begin {align*} \int \frac {x \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )}{1-c^2 x} \, dx &=2 \operatorname {Subst}\left (\int \frac {x^3 \left (a+b \tanh ^{-1}(c x)\right )}{1-c^2 x^2} \, dx,x,\sqrt {x}\right )\\ &=-\frac {2 \operatorname {Subst}\left (\int x \left (a+b \tanh ^{-1}(c x)\right ) \, dx,x,\sqrt {x}\right )}{c^2}+\frac {2 \operatorname {Subst}\left (\int \frac {x \left (a+b \tanh ^{-1}(c x)\right )}{1-c^2 x^2} \, dx,x,\sqrt {x}\right )}{c^2}\\ &=-\frac {x \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )}{c^2}-\frac {\left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )^2}{b c^4}+\frac {2 \operatorname {Subst}\left (\int \frac {a+b \tanh ^{-1}(c x)}{1-c x} \, dx,x,\sqrt {x}\right )}{c^3}+\frac {b \operatorname {Subst}\left (\int \frac {x^2}{1-c^2 x^2} \, dx,x,\sqrt {x}\right )}{c}\\ &=-\frac {b \sqrt {x}}{c^3}-\frac {x \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )}{c^2}-\frac {\left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )^2}{b c^4}+\frac {2 \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right ) \log \left (\frac {2}{1-c \sqrt {x}}\right )}{c^4}+\frac {b \operatorname {Subst}\left (\int \frac {1}{1-c^2 x^2} \, dx,x,\sqrt {x}\right )}{c^3}-\frac {(2 b) \operatorname {Subst}\left (\int \frac {\log \left (\frac {2}{1-c x}\right )}{1-c^2 x^2} \, dx,x,\sqrt {x}\right )}{c^3}\\ &=-\frac {b \sqrt {x}}{c^3}+\frac {b \tanh ^{-1}\left (c \sqrt {x}\right )}{c^4}-\frac {x \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )}{c^2}-\frac {\left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )^2}{b c^4}+\frac {2 \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right ) \log \left (\frac {2}{1-c \sqrt {x}}\right )}{c^4}+\frac {(2 b) \operatorname {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1-c \sqrt {x}}\right )}{c^4}\\ &=-\frac {b \sqrt {x}}{c^3}+\frac {b \tanh ^{-1}\left (c \sqrt {x}\right )}{c^4}-\frac {x \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )}{c^2}-\frac {\left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )^2}{b c^4}+\frac {2 \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right ) \log \left (\frac {2}{1-c \sqrt {x}}\right )}{c^4}+\frac {b \text {Li}_2\left (1-\frac {2}{1-c \sqrt {x}}\right )}{c^4}\\ \end {align*}
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Mathematica [A] time = 0.22, size = 96, normalized size = 0.80 \[ -\frac {a c^2 x+a \log \left (1-c^2 x\right )+b \tanh ^{-1}\left (c \sqrt {x}\right ) \left (c^2 x-2 \log \left (e^{-2 \tanh ^{-1}\left (c \sqrt {x}\right )}+1\right )-1\right )+b \text {Li}_2\left (-e^{-2 \tanh ^{-1}\left (c \sqrt {x}\right )}\right )+b c \sqrt {x}-b \tanh ^{-1}\left (c \sqrt {x}\right )^2}{c^4} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.95, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {b x \operatorname {artanh}\left (c \sqrt {x}\right ) + a x}{c^{2} x - 1}, x\right ) \]
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 \[ \int -\frac {{\left (b \operatorname {artanh}\left (c \sqrt {x}\right ) + a\right )} x}{c^{2} x - 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.06, size = 243, normalized size = 2.02 \[ -\frac {x a}{c^{2}}-\frac {a \ln \left (c \sqrt {x}-1\right )}{c^{4}}-\frac {a \ln \left (1+c \sqrt {x}\right )}{c^{4}}-\frac {b \arctanh \left (c \sqrt {x}\right ) x}{c^{2}}-\frac {b \arctanh \left (c \sqrt {x}\right ) \ln \left (c \sqrt {x}-1\right )}{c^{4}}-\frac {b \arctanh \left (c \sqrt {x}\right ) \ln \left (1+c \sqrt {x}\right )}{c^{4}}-\frac {b \sqrt {x}}{c^{3}}-\frac {b \ln \left (c \sqrt {x}-1\right )}{2 c^{4}}+\frac {b \ln \left (1+c \sqrt {x}\right )}{2 c^{4}}-\frac {b \ln \left (c \sqrt {x}-1\right )^{2}}{4 c^{4}}+\frac {b \dilog \left (\frac {1}{2}+\frac {c \sqrt {x}}{2}\right )}{c^{4}}+\frac {b \ln \left (c \sqrt {x}-1\right ) \ln \left (\frac {1}{2}+\frac {c \sqrt {x}}{2}\right )}{2 c^{4}}+\frac {b \ln \left (1+c \sqrt {x}\right )^{2}}{4 c^{4}}-\frac {b \ln \left (-\frac {c \sqrt {x}}{2}+\frac {1}{2}\right ) \ln \left (1+c \sqrt {x}\right )}{2 c^{4}}+\frac {b \ln \left (-\frac {c \sqrt {x}}{2}+\frac {1}{2}\right ) \ln \left (\frac {1}{2}+\frac {c \sqrt {x}}{2}\right )}{2 c^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.55, size = 166, normalized size = 1.38 \[ -a {\left (\frac {x}{c^{2}} + \frac {\log \left (c^{2} x - 1\right )}{c^{4}}\right )} - \frac {{\left (\log \left (c \sqrt {x} + 1\right ) \log \left (-\frac {1}{2} \, c \sqrt {x} + \frac {1}{2}\right ) + {\rm Li}_2\left (\frac {1}{2} \, c \sqrt {x} + \frac {1}{2}\right )\right )} b}{c^{4}} + \frac {b \log \left (c \sqrt {x} + 1\right )}{2 \, c^{4}} - \frac {b \log \left (c \sqrt {x} - 1\right )}{2 \, c^{4}} - \frac {2 \, b c^{2} x \log \left (c \sqrt {x} + 1\right ) + b \log \left (c \sqrt {x} + 1\right )^{2} - b \log \left (-c \sqrt {x} + 1\right )^{2} + 4 \, b c \sqrt {x} - 2 \, {\left (b c^{2} x + b \log \left (c \sqrt {x} + 1\right )\right )} \log \left (-c \sqrt {x} + 1\right )}{4 \, c^{4}} \]
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 \[ \int -\frac {x\,\left (a+b\,\mathrm {atanh}\left (c\,\sqrt {x}\right )\right )}{c^2\,x-1} \,d x \]
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 \[ - \int \frac {a x}{c^{2} x - 1}\, dx - \int \frac {b x \operatorname {atanh}{\left (c \sqrt {x} \right )}}{c^{2} x - 1}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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