Optimal. Leaf size=268 \[ -\frac {2 \left (a+b \tanh ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^2 \tanh ^{-1}\left (1-\frac {2}{1-\frac {\sqrt {1-c x}}{\sqrt {1+c x}}}\right )}{c}+\frac {b \left (a+b \tanh ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right ) \text {PolyLog}\left (2,1-\frac {2}{1-\frac {\sqrt {1-c x}}{\sqrt {1+c x}}}\right )}{c}-\frac {b \left (a+b \tanh ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right ) \text {PolyLog}\left (2,-1+\frac {2}{1-\frac {\sqrt {1-c x}}{\sqrt {1+c x}}}\right )}{c}-\frac {b^2 \text {PolyLog}\left (3,1-\frac {2}{1-\frac {\sqrt {1-c x}}{\sqrt {1+c x}}}\right )}{2 c}+\frac {b^2 \text {PolyLog}\left (3,-1+\frac {2}{1-\frac {\sqrt {1-c x}}{\sqrt {1+c x}}}\right )}{2 c} \]
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Rubi [A]
time = 0.22, antiderivative size = 268, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 6, integrand size = 40, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {6813, 6033,
6199, 6095, 6205, 6745} \begin {gather*} \frac {b \text {Li}_2\left (1-\frac {2}{1-\frac {\sqrt {1-c x}}{\sqrt {c x+1}}}\right ) \left (a+b \tanh ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )\right )}{c}-\frac {b \text {Li}_2\left (\frac {2}{1-\frac {\sqrt {1-c x}}{\sqrt {c x+1}}}-1\right ) \left (a+b \tanh ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )\right )}{c}-\frac {2 \tanh ^{-1}\left (1-\frac {2}{1-\frac {\sqrt {1-c x}}{\sqrt {c x+1}}}\right ) \left (a+b \tanh ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )\right )^2}{c}-\frac {b^2 \text {Li}_3\left (1-\frac {2}{1-\frac {\sqrt {1-c x}}{\sqrt {c x+1}}}\right )}{2 c}+\frac {b^2 \text {Li}_3\left (\frac {2}{1-\frac {\sqrt {1-c x}}{\sqrt {c x+1}}}-1\right )}{2 c} \end {gather*}
Antiderivative was successfully verified.
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Rule 6033
Rule 6095
Rule 6199
Rule 6205
Rule 6745
Rule 6813
Rubi steps
\begin {align*} \int \frac {\left (a+b \tanh ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^2}{1-c^2 x^2} \, dx &=-\frac {\text {Subst}\left (\int \frac {\left (a+b \tanh ^{-1}(x)\right )^2}{x} \, dx,x,\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )}{c}\\ &=-\frac {2 \left (a+b \tanh ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^2 \tanh ^{-1}\left (1-\frac {2}{1-\frac {\sqrt {1-c x}}{\sqrt {1+c x}}}\right )}{c}+\frac {(4 b) \text {Subst}\left (\int \frac {\tanh ^{-1}\left (1-\frac {2}{1-x}\right ) \left (a+b \tanh ^{-1}(x)\right )}{1-x^2} \, dx,x,\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )}{c}\\ &=-\frac {2 \left (a+b \tanh ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^2 \tanh ^{-1}\left (1-\frac {2}{1-\frac {\sqrt {1-c x}}{\sqrt {1+c x}}}\right )}{c}+\frac {(2 b) \text {Subst}\left (\int \frac {\left (a+b \tanh ^{-1}(x)\right ) \log \left (2-\frac {2}{1-x}\right )}{1-x^2} \, dx,x,\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )}{c}-\frac {(2 b) \text {Subst}\left (\int \frac {\left (a+b \tanh ^{-1}(x)\right ) \log \left (\frac {2}{1-x}\right )}{1-x^2} \, dx,x,\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )}{c}\\ &=-\frac {2 \left (a+b \tanh ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^2 \tanh ^{-1}\left (1-\frac {2}{1-\frac {\sqrt {1-c x}}{\sqrt {1+c x}}}\right )}{c}+\frac {b \left (a+b \tanh ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right ) \text {Li}_2\left (1-\frac {2}{1-\frac {\sqrt {1-c x}}{\sqrt {1+c x}}}\right )}{c}-\frac {b \left (a+b \tanh ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right ) \text {Li}_2\left (-1+\frac {2}{1-\frac {\sqrt {1-c x}}{\sqrt {1+c x}}}\right )}{c}-\frac {b^2 \text {Subst}\left (\int \frac {\text {Li}_2\left (1-\frac {2}{1-x}\right )}{1-x^2} \, dx,x,\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )}{c}+\frac {b^2 \text {Subst}\left (\int \frac {\text {Li}_2\left (-1+\frac {2}{1-x}\right )}{1-x^2} \, dx,x,\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )}{c}\\ &=-\frac {2 \left (a+b \tanh ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^2 \tanh ^{-1}\left (1-\frac {2}{1-\frac {\sqrt {1-c x}}{\sqrt {1+c x}}}\right )}{c}+\frac {b \left (a+b \tanh ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right ) \text {Li}_2\left (1-\frac {2}{1-\frac {\sqrt {1-c x}}{\sqrt {1+c x}}}\right )}{c}-\frac {b \left (a+b \tanh ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right ) \text {Li}_2\left (-1+\frac {2}{1-\frac {\sqrt {1-c x}}{\sqrt {1+c x}}}\right )}{c}-\frac {b^2 \text {Li}_3\left (1-\frac {2}{1-\frac {\sqrt {1-c x}}{\sqrt {1+c x}}}\right )}{2 c}+\frac {b^2 \text {Li}_3\left (-1+\frac {2}{1-\frac {\sqrt {1-c x}}{\sqrt {1+c x}}}\right )}{2 c}\\ \end {align*}
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Mathematica [A]
time = 0.07, size = 324, normalized size = 1.21 \begin {gather*} -\frac {2 \left (a+b \tanh ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^2 \tanh ^{-1}\left (1-\frac {2}{1-\frac {\sqrt {1-c x}}{\sqrt {1+c x}}}\right )+b \left (a+b \tanh ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right ) \text {PolyLog}\left (2,-\frac {\sqrt {1-c x}+\sqrt {1+c x}}{\sqrt {1-c x}-\sqrt {1+c x}}\right )-b \left (a+b \tanh ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right ) \text {PolyLog}\left (2,\frac {\sqrt {1-c x}+\sqrt {1+c x}}{\sqrt {1-c x}-\sqrt {1+c x}}\right )-\frac {1}{2} b^2 \text {PolyLog}\left (3,-\frac {\sqrt {1-c x}+\sqrt {1+c x}}{\sqrt {1-c x}-\sqrt {1+c x}}\right )+\frac {1}{2} b^2 \text {PolyLog}\left (3,\frac {\sqrt {1-c x}+\sqrt {1+c x}}{\sqrt {1-c x}-\sqrt {1+c x}}\right )}{c} \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. \(673\) vs.
\(2(232)=464\).
time = 0.02, size = 674, normalized size = 2.51
method | result | size |
default | \(\frac {a^{2} \ln \left (c x +1\right )}{2 c}-\frac {a^{2} \ln \left (c x -1\right )}{2 c}-\frac {b^{2} \arctanh \left (\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}\right )^{2} \ln \left (1-\frac {\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}+1}{\sqrt {-\frac {-c x +1}{c x +1}+1}}\right )}{c}-\frac {2 b^{2} \arctanh \left (\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}\right ) \polylog \left (2, \frac {\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}+1}{\sqrt {-\frac {-c x +1}{c x +1}+1}}\right )}{c}+\frac {2 b^{2} \polylog \left (3, \frac {\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}+1}{\sqrt {-\frac {-c x +1}{c x +1}+1}}\right )}{c}-\frac {b^{2} \arctanh \left (\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}\right )^{2} \ln \left (1+\frac {\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}+1}{\sqrt {-\frac {-c x +1}{c x +1}+1}}\right )}{c}-\frac {2 b^{2} \arctanh \left (\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}\right ) \polylog \left (2, -\frac {\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}+1}{\sqrt {-\frac {-c x +1}{c x +1}+1}}\right )}{c}+\frac {2 b^{2} \polylog \left (3, -\frac {\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}+1}{\sqrt {-\frac {-c x +1}{c x +1}+1}}\right )}{c}+\frac {b^{2} \arctanh \left (\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}\right )^{2} \ln \left (\frac {\left (\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}+1\right )^{2}}{-\frac {-c x +1}{c x +1}+1}+1\right )}{c}+\frac {b^{2} \arctanh \left (\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}\right ) \polylog \left (2, -\frac {\left (\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}+1\right )^{2}}{-\frac {-c x +1}{c x +1}+1}\right )}{c}-\frac {b^{2} \polylog \left (3, -\frac {\left (\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}+1\right )^{2}}{-\frac {-c x +1}{c x +1}+1}\right )}{2 c}-\frac {a b \left (4 \dilog \left (\frac {-\frac {-c x +1}{c x +1}+1}{\left (\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}+1\right )^{2}}\right )-\dilog \left (\frac {\left (-\frac {-c x +1}{c x +1}+1\right )^{2}}{\left (\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}+1\right )^{4}}\right )\right )}{2 c}\) | \(674\) |
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 {a^{2}}{c^{2} x^{2} - 1}\, dx - \int \frac {b^{2} \operatorname {atanh}^{2}{\left (\frac {\sqrt {- c x + 1}}{\sqrt {c x + 1}} \right )}}{c^{2} x^{2} - 1}\, dx - \int \frac {2 a b \operatorname {atanh}{\left (\frac {\sqrt {- c x + 1}}{\sqrt {c x + 1}} \right )}}{c^{2} x^{2} - 1}\, 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 = -0.00 \begin {gather*} \int -\frac {{\left (a+b\,\mathrm {atanh}\left (\frac {\sqrt {1-c\,x}}{\sqrt {c\,x+1}}\right )\right )}^2}{c^2\,x^2-1} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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