Optimal. Leaf size=321 \[ -\frac {2 \left (a+b \cot ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^2 \coth ^{-1}\left (1-\frac {2}{1+\frac {i \sqrt {1-c x}}{\sqrt {1+c x}}}\right )}{c}+\frac {i b \left (a+b \cot ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right ) \text {PolyLog}\left (2,1-\frac {2 i}{i+\frac {\sqrt {1-c x}}{\sqrt {1+c x}}}\right )}{c}-\frac {i b \left (a+b \cot ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right ) \text {PolyLog}\left (2,1-\frac {2 \sqrt {1-c x}}{\sqrt {1+c x} \left (i+\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )}\right )}{c}+\frac {b^2 \text {PolyLog}\left (3,1-\frac {2 i}{i+\frac {\sqrt {1-c x}}{\sqrt {1+c x}}}\right )}{2 c}-\frac {b^2 \text {PolyLog}\left (3,1-\frac {2 \sqrt {1-c x}}{\sqrt {1+c x} \left (i+\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )}\right )}{2 c} \]
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Rubi [A]
time = 0.24, antiderivative size = 321, 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, 4943,
5109, 5005, 5113, 6745} \begin {gather*} \frac {i b \text {Li}_2\left (1-\frac {2 i}{\frac {\sqrt {1-c x}}{\sqrt {c x+1}}+i}\right ) \left (a+b \cot ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )\right )}{c}-\frac {i b \text {Li}_2\left (1-\frac {2 \sqrt {1-c x}}{\sqrt {c x+1} \left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}+i\right )}\right ) \left (a+b \cot ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )\right )}{c}-\frac {2 \coth ^{-1}\left (1-\frac {2}{1+\frac {i \sqrt {1-c x}}{\sqrt {c x+1}}}\right ) \left (a+b \cot ^{-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 i}{\frac {\sqrt {1-c x}}{\sqrt {c x+1}}+i}\right )}{2 c}-\frac {b^2 \text {Li}_3\left (1-\frac {2 \sqrt {1-c x}}{\sqrt {c x+1} \left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}+i\right )}\right )}{2 c} \end {gather*}
Antiderivative was successfully verified.
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Rule 4943
Rule 5005
Rule 5109
Rule 5113
Rule 6745
Rule 6813
Rubi steps
\begin {align*} \int \frac {\left (a+b \cot ^{-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 \cot ^{-1}(x)\right )^2}{x} \, dx,x,\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )}{c}\\ &=-\frac {2 \left (a+b \cot ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^2 \coth ^{-1}\left (1-\frac {2}{1+\frac {i \sqrt {1-c x}}{\sqrt {1+c x}}}\right )}{c}-\frac {(4 b) \text {Subst}\left (\int \frac {\left (a+b \cot ^{-1}(x)\right ) \coth ^{-1}\left (1-\frac {2}{1+i x}\right )}{1+x^2} \, dx,x,\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )}{c}\\ &=-\frac {2 \left (a+b \cot ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^2 \coth ^{-1}\left (1-\frac {2}{1+\frac {i \sqrt {1-c x}}{\sqrt {1+c x}}}\right )}{c}+\frac {(2 b) \text {Subst}\left (\int \frac {\left (a+b \cot ^{-1}(x)\right ) \log \left (\frac {2 i}{i+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 \cot ^{-1}(x)\right ) \log \left (\frac {2 x}{i+x}\right )}{1+x^2} \, dx,x,\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )}{c}\\ &=-\frac {2 \left (a+b \cot ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^2 \coth ^{-1}\left (1-\frac {2}{1+\frac {i \sqrt {1-c x}}{\sqrt {1+c x}}}\right )}{c}+\frac {i b \left (a+b \cot ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right ) \text {Li}_2\left (1-\frac {2 i}{i+\frac {\sqrt {1-c x}}{\sqrt {1+c x}}}\right )}{c}-\frac {i b \left (a+b \cot ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right ) \text {Li}_2\left (1-\frac {2 \sqrt {1-c x}}{\sqrt {1+c x} \left (i+\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )}\right )}{c}+\frac {\left (i b^2\right ) \text {Subst}\left (\int \frac {\text {Li}_2\left (1-\frac {2 i}{i+x}\right )}{1+x^2} \, dx,x,\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )}{c}-\frac {\left (i b^2\right ) \text {Subst}\left (\int \frac {\text {Li}_2\left (1-\frac {2 x}{i+x}\right )}{1+x^2} \, dx,x,\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )}{c}\\ &=-\frac {2 \left (a+b \cot ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^2 \coth ^{-1}\left (1-\frac {2}{1+\frac {i \sqrt {1-c x}}{\sqrt {1+c x}}}\right )}{c}+\frac {i b \left (a+b \cot ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right ) \text {Li}_2\left (1-\frac {2 i}{i+\frac {\sqrt {1-c x}}{\sqrt {1+c x}}}\right )}{c}-\frac {i b \left (a+b \cot ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right ) \text {Li}_2\left (1-\frac {2 \sqrt {1-c x}}{\sqrt {1+c x} \left (i+\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )}\right )}{c}+\frac {b^2 \text {Li}_3\left (1-\frac {2 i}{i+\frac {\sqrt {1-c x}}{\sqrt {1+c x}}}\right )}{2 c}-\frac {b^2 \text {Li}_3\left (1-\frac {2 \sqrt {1-c x}}{\sqrt {1+c x} \left (i+\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )}\right )}{2 c}\\ \end {align*}
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Mathematica [F]
time = 0.30, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a+b \cot ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^2}{1-c^2 x^2} \, dx \end {gather*}
Verification is not applicable to the result.
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Maple [B] Both result and optimal contain complex but leaf count of result is larger than twice
the leaf count of optimal. 822 vs. \(2 (268 ) = 536\).
time = 0.39, size = 823, normalized size = 2.56
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} \mathrm {arccot}\left (\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}\right )^{2} \ln \left (1+\frac {i+\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}}{\sqrt {\frac {-c x +1}{c x +1}+1}}\right )}{c}-\frac {2 i b^{2} \mathrm {arccot}\left (\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}\right ) \polylog \left (2, -\frac {i+\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}}{\sqrt {\frac {-c x +1}{c x +1}+1}}\right )}{c}+\frac {2 b^{2} \polylog \left (3, -\frac {i+\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}}{\sqrt {\frac {-c x +1}{c x +1}+1}}\right )}{c}-\frac {b^{2} \mathrm {arccot}\left (\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}\right )^{2} \ln \left (\frac {\left (i+\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}\right )^{2}}{\frac {-c x +1}{c x +1}+1}+1\right )}{c}+\frac {i b^{2} \mathrm {arccot}\left (\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}\right ) \polylog \left (2, -\frac {\left (i+\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}\right )^{2}}{\frac {-c x +1}{c x +1}+1}\right )}{c}-\frac {b^{2} \polylog \left (3, -\frac {\left (i+\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}\right )^{2}}{\frac {-c x +1}{c x +1}+1}\right )}{2 c}+\frac {b^{2} \mathrm {arccot}\left (\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}\right )^{2} \ln \left (1-\frac {i+\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}}{\sqrt {\frac {-c x +1}{c x +1}+1}}\right )}{c}-\frac {2 i b^{2} \mathrm {arccot}\left (\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}\right ) \polylog \left (2, \frac {i+\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}}{\sqrt {\frac {-c x +1}{c x +1}+1}}\right )}{c}+\frac {2 b^{2} \polylog \left (3, \frac {i+\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}}{\sqrt {\frac {-c x +1}{c x +1}+1}}\right )}{c}+\frac {2 a b \,\mathrm {arccot}\left (\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}\right ) \ln \left (1-\frac {\left (i+\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}\right )^{2}}{\frac {-c x +1}{c x +1}+1}\right )}{c}-\frac {2 a b \,\mathrm {arccot}\left (\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}\right ) \ln \left (\frac {\left (i+\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}\right )^{2}}{\frac {-c x +1}{c x +1}+1}+1\right )}{c}+\frac {i a b \dilog \left (\frac {\left (i+\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}\right )^{2}}{\frac {-c x +1}{c x +1}+1}+1\right )}{c}-\frac {i a b \dilog \left (1-\frac {\left (i+\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}\right )^{2}}{\frac {-c x +1}{c x +1}+1}\right )}{c}\) | \(823\) |
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(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \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 {acot}\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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