Optimal. Leaf size=12 \[ x-\frac {2 \sinh (x)}{1+\cosh (x)} \]
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Rubi [A]
time = 0.06, antiderivative size = 12, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.444, Rules used = {4477, 2749,
2759, 8} \begin {gather*} x-\frac {2 \sinh (x)}{\cosh (x)+1} \end {gather*}
Antiderivative was successfully verified.
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Rule 8
Rule 2749
Rule 2759
Rule 4477
Rubi steps
\begin {align*} \int (-\coth (x)+\text {csch}(x))^2 \, dx &=-\int (i-i \cosh (x))^2 \text {csch}^2(x) \, dx\\ &=-\int \frac {\sinh ^2(x)}{(i+i \cosh (x))^2} \, dx\\ &=-\frac {2 \sinh (x)}{1+\cosh (x)}+\int 1 \, dx\\ &=x-\frac {2 \sinh (x)}{1+\cosh (x)}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 18, normalized size = 1.50 \begin {gather*} 2 \tanh ^{-1}\left (\tanh \left (\frac {x}{2}\right )\right )-2 \tanh \left (\frac {x}{2}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.66, size = 11, normalized size = 0.92
method | result | size |
risch | \(x +\frac {4}{{\mathrm e}^{x}+1}\) | \(11\) |
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. 25 vs.
\(2 (12) = 24\).
time = 0.28, size = 25, normalized size = 2.08 \begin {gather*} x - \frac {4}{e^{\left (-x\right )} - e^{x}} + \frac {4}{e^{\left (-2 \, x\right )} - 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.38, size = 20, normalized size = 1.67 \begin {gather*} \frac {x \cosh \left (x\right ) + x \sinh \left (x\right ) + x + 4}{\cosh \left (x\right ) + \sinh \left (x\right ) + 1} \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 \left (- \coth {\left (x \right )} + \operatorname {csch}{\left (x \right )}\right )^{2}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.39, size = 10, normalized size = 0.83 \begin {gather*} x + \frac {4}{e^{x} + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.06, size = 10, normalized size = 0.83 \begin {gather*} x+\frac {4}{{\mathrm {e}}^x+1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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