Optimal. Leaf size=31 \[ \frac {\coth (c+d x) \log (\cosh (c+d x))}{d \sqrt {b \coth ^2(c+d x)}} \]
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Rubi [A] time = 0.02, antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {3658, 3475} \[ \frac {\coth (c+d x) \log (\cosh (c+d x))}{d \sqrt {b \coth ^2(c+d x)}} \]
Antiderivative was successfully verified.
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Rule 3475
Rule 3658
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {b \coth ^2(c+d x)}} \, dx &=\frac {\coth (c+d x) \int \tanh (c+d x) \, dx}{\sqrt {b \coth ^2(c+d x)}}\\ &=\frac {\coth (c+d x) \log (\cosh (c+d x))}{d \sqrt {b \coth ^2(c+d x)}}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 31, normalized size = 1.00 \[ \frac {\coth (c+d x) \log (\cosh (c+d x))}{d \sqrt {b \coth ^2(c+d x)}} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.41, size = 128, normalized size = 4.13 \[ -\frac {{\left (d x e^{\left (2 \, d x + 2 \, c\right )} - d x - {\left (e^{\left (2 \, d x + 2 \, c\right )} - 1\right )} \log \left (\frac {2 \, \cosh \left (d x + c\right )}{\cosh \left (d x + c\right ) - \sinh \left (d x + c\right )}\right )\right )} \sqrt {\frac {b e^{\left (4 \, d x + 4 \, c\right )} + 2 \, b e^{\left (2 \, d x + 2 \, c\right )} + b}{e^{\left (4 \, d x + 4 \, c\right )} - 2 \, e^{\left (2 \, d x + 2 \, c\right )} + 1}}}{b d e^{\left (2 \, d x + 2 \, c\right )} + b d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.15, size = 60, normalized size = 1.94 \[ -\frac {\frac {d x + c}{\sqrt {b} \mathrm {sgn}\left (e^{\left (4 \, d x + 4 \, c\right )} - 1\right )} - \frac {\log \left (e^{\left (2 \, d x + 2 \, c\right )} + 1\right )}{\sqrt {b} \mathrm {sgn}\left (e^{\left (4 \, d x + 4 \, c\right )} - 1\right )}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.16, size = 52, normalized size = 1.68 \[ -\frac {\coth \left (d x +c \right ) \left (\ln \left (\coth \left (d x +c \right )-1\right )+\ln \left (\coth \left (d x +c \right )+1\right )-2 \ln \left (\coth \left (d x +c \right )\right )\right )}{2 d \sqrt {b \left (\coth ^{2}\left (d x +c \right )\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.43, size = 34, normalized size = 1.10 \[ -\frac {d x + c}{\sqrt {b} d} - \frac {\log \left (e^{\left (-2 \, d x - 2 \, c\right )} + 1\right )}{\sqrt {b} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.26, size = 30, normalized size = 0.97 \[ \frac {\mathrm {atanh}\left (\frac {\sqrt {b}\,\mathrm {coth}\left (c+d\,x\right )}{\sqrt {b\,{\mathrm {coth}\left (c+d\,x\right )}^2}}\right )}{\sqrt {b}\,d} \]
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 \[ \int \frac {1}{\sqrt {b \coth ^{2}{\left (c + d x \right )}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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