Optimal. Leaf size=14 \[ \frac {1}{b (\tanh (a+b x)+1)} \]
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Rubi [A] time = 0.21, antiderivative size = 14, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.032, Rules used = {32} \[ \frac {1}{b (\tanh (a+b x)+1)} \]
Antiderivative was successfully verified.
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Rule 32
Rubi steps
\begin {align*} \int \frac {-\text {csch}(a+b x)+\text {sech}(a+b x)}{\text {csch}(a+b x)+\text {sech}(a+b x)} \, dx &=-\frac {\operatorname {Subst}\left (\int \frac {1}{(1+x)^2} \, dx,x,\tanh (a+b x)\right )}{b}\\ &=\frac {1}{b (1+\tanh (a+b x))}\\ \end {align*}
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Mathematica [B] time = 0.02, size = 65, normalized size = 4.64 \[ \frac {\sinh (2 a) \sinh (2 b x)}{2 b}+\frac {\cosh (2 a) \cosh (2 b x)}{2 b}-\frac {\sinh (2 a) \cosh (2 b x)}{2 b}-\frac {\cosh (2 a) \sinh (2 b x)}{2 b} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.41, size = 40, normalized size = 2.86 \[ \frac {1}{2 \, {\left (b \cosh \left (b x + a\right )^{2} + 2 \, b \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + b \sinh \left (b x + a\right )^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 14, normalized size = 1.00 \[ \frac {e^{\left (-2 \, b x - 2 \, a\right )}}{2 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 1.08, size = 36, normalized size = 2.57 \[ \frac {-\frac {2}{\tanh \left (\frac {b x}{2}+\frac {a}{2}\right )+1}+\frac {2}{\left (\tanh \left (\frac {b x}{2}+\frac {a}{2}\right )+1\right )^{2}}}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.35, size = 14, normalized size = 1.00 \[ \frac {e^{\left (-2 \, b x - 2 \, a\right )}}{2 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.08, size = 14, normalized size = 1.00 \[ \frac {{\mathrm {e}}^{-2\,a-2\,b\,x}}{2\,b} \]
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 \[ - \int \frac {\operatorname {csch}{\left (a + b x \right )}}{\operatorname {csch}{\left (a + b x \right )} + \operatorname {sech}{\left (a + b x \right )}}\, dx - \int \left (- \frac {\operatorname {sech}{\left (a + b x \right )}}{\operatorname {csch}{\left (a + b x \right )} + \operatorname {sech}{\left (a + b x \right )}}\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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