Optimal. Leaf size=194 \[ \frac {b \text {Li}_2\left (e^{-2 \sinh ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )}\right ) \left (a+b \sinh ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )\right )}{c}-\frac {\left (a+b \sinh ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )\right )^3}{3 b c}-\frac {\log \left (1-e^{-2 \sinh ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )}\right ) \left (a+b \sinh ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )\right )^2}{c}+\frac {b^2 \text {Li}_3\left (e^{-2 \sinh ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )}\right )}{2 c} \]
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Rubi [A] time = 0.22, antiderivative size = 195, normalized size of antiderivative = 1.01, number of steps used = 7, number of rules used = 7, integrand size = 40, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.175, Rules used = {6681, 5659, 3716, 2190, 2531, 2282, 6589} \[ -\frac {b \text {PolyLog}\left (2,e^{2 \sinh ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )}\right ) \left (a+b \sinh ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )\right )}{c}+\frac {b^2 \text {PolyLog}\left (3,e^{2 \sinh ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )}\right )}{2 c}+\frac {\left (a+b \sinh ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )\right )^3}{3 b c}-\frac {\log \left (1-e^{2 \sinh ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )}\right ) \left (a+b \sinh ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )\right )^2}{c} \]
Warning: Unable to verify antiderivative.
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Rule 2190
Rule 2282
Rule 2531
Rule 3716
Rule 5659
Rule 6589
Rule 6681
Rubi steps
\begin {align*} \int \frac {\left (a+b \sinh ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^2}{1-c^2 x^2} \, dx &=-\frac {\operatorname {Subst}\left (\int \frac {\left (a+b \sinh ^{-1}(x)\right )^2}{x} \, dx,x,\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )}{c}\\ &=-\frac {\operatorname {Subst}\left (\int (a+b x)^2 \coth (x) \, dx,x,\sinh ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )}{c}\\ &=\frac {\left (a+b \sinh ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^3}{3 b c}+\frac {2 \operatorname {Subst}\left (\int \frac {e^{2 x} (a+b x)^2}{1-e^{2 x}} \, dx,x,\sinh ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )}{c}\\ &=\frac {\left (a+b \sinh ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^3}{3 b c}-\frac {\left (a+b \sinh ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^2 \log \left (1-e^{2 \sinh ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )}\right )}{c}+\frac {(2 b) \operatorname {Subst}\left (\int (a+b x) \log \left (1-e^{2 x}\right ) \, dx,x,\sinh ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )}{c}\\ &=\frac {\left (a+b \sinh ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^3}{3 b c}-\frac {\left (a+b \sinh ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^2 \log \left (1-e^{2 \sinh ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )}\right )}{c}-\frac {b \left (a+b \sinh ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right ) \text {Li}_2\left (e^{2 \sinh ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )}\right )}{c}+\frac {b^2 \operatorname {Subst}\left (\int \text {Li}_2\left (e^{2 x}\right ) \, dx,x,\sinh ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )}{c}\\ &=\frac {\left (a+b \sinh ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^3}{3 b c}-\frac {\left (a+b \sinh ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^2 \log \left (1-e^{2 \sinh ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )}\right )}{c}-\frac {b \left (a+b \sinh ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right ) \text {Li}_2\left (e^{2 \sinh ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )}\right )}{c}+\frac {b^2 \operatorname {Subst}\left (\int \frac {\text {Li}_2(x)}{x} \, dx,x,e^{2 \sinh ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )}\right )}{2 c}\\ &=\frac {\left (a+b \sinh ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^3}{3 b c}-\frac {\left (a+b \sinh ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^2 \log \left (1-e^{2 \sinh ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )}\right )}{c}-\frac {b \left (a+b \sinh ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right ) \text {Li}_2\left (e^{2 \sinh ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )}\right )}{c}+\frac {b^2 \text {Li}_3\left (e^{2 \sinh ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )}\right )}{2 c}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 187, normalized size = 0.96 \[ \frac {-6 b^2 \text {Li}_2\left (e^{2 \sinh ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )}\right ) \left (a+b \sinh ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )\right )+2 \left (a+b \sinh ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )\right )^2 \left (a+b \sinh ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )-3 b \log \left (1-e^{2 \sinh ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )}\right )\right )+3 b^3 \text {Li}_3\left (e^{2 \sinh ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )}\right )}{6 b c} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.47, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {b^{2} \operatorname {arsinh}\left (\frac {\sqrt {-c x + 1}}{\sqrt {c x + 1}}\right )^{2} + 2 \, a b \operatorname {arsinh}\left (\frac {\sqrt {-c x + 1}}{\sqrt {c x + 1}}\right ) + a^{2}}{c^{2} x^{2} - 1}, x\right ) \]
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 \[ \int -\frac {{\left (b \operatorname {arsinh}\left (\frac {\sqrt {-c x + 1}}{\sqrt {c x + 1}}\right ) + a\right )}^{2}}{c^{2} x^{2} - 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.01, size = 649, normalized size = 3.35 \[ \frac {a^{2} \ln \left (c x +1\right )}{2 c}-\frac {a^{2} \ln \left (c x -1\right )}{2 c}+\frac {b^{2} \arcsinh \left (\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}\right )^{3}}{3 c}-\frac {b^{2} \arcsinh \left (\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}\right )^{2} \ln \left (1+\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}+\sqrt {1+\frac {-c x +1}{c x +1}}\right )}{c}-\frac {2 b^{2} \arcsinh \left (\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}\right ) \polylog \left (2, -\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}-\sqrt {1+\frac {-c x +1}{c x +1}}\right )}{c}+\frac {2 b^{2} \polylog \left (3, -\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}-\sqrt {1+\frac {-c x +1}{c x +1}}\right )}{c}-\frac {b^{2} \arcsinh \left (\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}\right )^{2} \ln \left (1-\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}-\sqrt {1+\frac {-c x +1}{c x +1}}\right )}{c}-\frac {2 b^{2} \arcsinh \left (\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}\right ) \polylog \left (2, \frac {\sqrt {-c x +1}}{\sqrt {c x +1}}+\sqrt {1+\frac {-c x +1}{c x +1}}\right )}{c}+\frac {2 b^{2} \polylog \left (3, \frac {\sqrt {-c x +1}}{\sqrt {c x +1}}+\sqrt {1+\frac {-c x +1}{c x +1}}\right )}{c}+\frac {a b \arcsinh \left (\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}\right )^{2}}{c}-\frac {2 a b \arcsinh \left (\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}\right ) \ln \left (1+\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}+\sqrt {1+\frac {-c x +1}{c x +1}}\right )}{c}-\frac {2 a b \polylog \left (2, -\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}-\sqrt {1+\frac {-c x +1}{c x +1}}\right )}{c}-\frac {2 a b \arcsinh \left (\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}\right ) \ln \left (1-\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}-\sqrt {1+\frac {-c x +1}{c x +1}}\right )}{c}-\frac {2 a b \polylog \left (2, \frac {\sqrt {-c x +1}}{\sqrt {c x +1}}+\sqrt {1+\frac {-c x +1}{c x +1}}\right )}{c} \]
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 \[ \frac {1}{2} \, a^{2} {\left (\frac {\log \left (c x + 1\right )}{c} - \frac {\log \left (c x - 1\right )}{c}\right )} + \frac {{\left (b^{2} \log \left (c x + 1\right ) - b^{2} \log \left (-c x + 1\right )\right )} \log \left (\sqrt {2} + \sqrt {-c x + 1}\right )^{2}}{2 \, c} + \int -\frac {{\left (\sqrt {2} b^{2} + \sqrt {-c x + 1} b^{2}\right )} \log \left (c x + 1\right )^{2} - 4 \, {\left (\sqrt {2} a b + \sqrt {-c x + 1} a b\right )} \log \left (c x + 1\right ) + 2 \, {\left (4 \, \sqrt {2} a b - 2 \, {\left (\sqrt {2} b^{2} + \sqrt {-c x + 1} b^{2}\right )} \log \left (c x + 1\right ) + {\left (4 \, a b + {\left (b^{2} c x + b^{2}\right )} \log \left (c x + 1\right ) - {\left (b^{2} c x + b^{2}\right )} \log \left (-c x + 1\right )\right )} \sqrt {-c x + 1}\right )} \log \left (\sqrt {2} + \sqrt {-c x + 1}\right )}{4 \, {\left (\sqrt {2} c^{2} x^{2} + {\left (c^{2} x^{2} - 1\right )} \sqrt {-c x + 1} - \sqrt {2}\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int -\frac {{\left (a+b\,\mathrm {asinh}\left (\frac {\sqrt {1-c\,x}}{\sqrt {c\,x+1}}\right )\right )}^2}{c^2\,x^2-1} \,d x \]
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 \[ - \int \frac {a^{2}}{c^{2} x^{2} - 1}\, dx - \int \frac {b^{2} \operatorname {asinh}^{2}{\left (\frac {\sqrt {- c x + 1}}{\sqrt {c x + 1}} \right )}}{c^{2} x^{2} - 1}\, dx - \int \frac {2 a b \operatorname {asinh}{\left (\frac {\sqrt {- c x + 1}}{\sqrt {c x + 1}} \right )}}{c^{2} x^{2} - 1}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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