Optimal. Leaf size=17 \[ a x-\frac {b \tanh ^{-1}(\cosh (c+d x))}{d} \]
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Rubi [A] time = 0.01, antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {3770} \[ a x-\frac {b \tanh ^{-1}(\cosh (c+d x))}{d} \]
Antiderivative was successfully verified.
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Rule 3770
Rubi steps
\begin {align*} \int (a+b \text {csch}(c+d x)) \, dx &=a x+b \int \text {csch}(c+d x) \, dx\\ &=a x-\frac {b \tanh ^{-1}(\cosh (c+d x))}{d}\\ \end {align*}
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Mathematica [B] time = 0.01, size = 43, normalized size = 2.53 \[ a x+\frac {b \log \left (\sinh \left (\frac {c}{2}+\frac {d x}{2}\right )\right )}{d}-\frac {b \log \left (\cosh \left (\frac {c}{2}+\frac {d x}{2}\right )\right )}{d} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.41, size = 44, normalized size = 2.59 \[ \frac {a d x - b \log \left (\cosh \left (d x + c\right ) + \sinh \left (d x + c\right ) + 1\right ) + b \log \left (\cosh \left (d x + c\right ) + \sinh \left (d x + c\right ) - 1\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.11, size = 32, normalized size = 1.88 \[ a x - \frac {b {\left (\log \left (e^{\left (d x + c\right )} + 1\right ) - \log \left ({\left | e^{\left (d x + c\right )} - 1 \right |}\right )\right )}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 20, normalized size = 1.18 \[ a x +\frac {b \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 19, normalized size = 1.12 \[ a x + \frac {b \log \left (\tanh \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.07, size = 42, normalized size = 2.47 \[ a\,x-\frac {2\,\mathrm {atan}\left (\frac {b\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,\sqrt {-d^2}}{d\,\sqrt {b^2}}\right )\,\sqrt {b^2}}{\sqrt {-d^2}} \]
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 \[ \int \left (a + b \operatorname {csch}{\left (c + d x \right )}\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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