Optimal. Leaf size=31 \[ \frac {\tanh (c+d x) \log (\sinh (c+d x))}{d \sqrt {-\tanh ^2(c+d x)}} \]
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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} \[ \frac {\tanh (c+d x) \log (\sinh (c+d x))}{d \sqrt {-\tanh ^2(c+d x)}} \]
Antiderivative was successfully verified.
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Rule 3475
Rule 3658
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {-\tanh ^2(c+d x)}} \, dx &=\frac {\tanh (c+d x) \int \coth (c+d x) \, dx}{\sqrt {-\tanh ^2(c+d x)}}\\ &=\frac {\log (\sinh (c+d x)) \tanh (c+d x)}{d \sqrt {-\tanh ^2(c+d x)}}\\ \end {align*}
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Mathematica [A] time = 0.08, size = 39, normalized size = 1.26 \[ \frac {\tanh (c+d x) (\log (\tanh (c+d x))+\log (\cosh (c+d x)))}{d \sqrt {-\tanh ^2(c+d x)}} \]
Antiderivative was successfully verified.
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fricas [C] time = 0.50, size = 23, normalized size = 0.74 \[ \frac {i \, d x - i \, \log \left (e^{\left (2 \, d x + 2 \, c\right )} - 1\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [C] time = 0.14, size = 63, normalized size = 2.03 \[ \frac {\frac {-2 i \, d x - 2 i \, c}{\mathrm {sgn}\left (-e^{\left (4 \, d x + 4 \, c\right )} + 1\right )} + \frac {2 i \, \log \left (-i \, e^{\left (2 \, d x + 2 \, c\right )} + i\right )}{\mathrm {sgn}\left (-e^{\left (4 \, d x + 4 \, c\right )} + 1\right )}}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.14, size = 52, normalized size = 1.68 \[ -\frac {\tanh \left (d x +c \right ) \left (\ln \left (\tanh \left (d x +c \right )-1\right )+\ln \left (1+\tanh \left (d x +c \right )\right )-2 \ln \left (\tanh \left (d x +c \right )\right )\right )}{2 d \sqrt {-\left (\tanh ^{2}\left (d x +c \right )\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [C] time = 0.45, size = 45, normalized size = 1.45 \[ \frac {i \, {\left (d x + c\right )}}{d} + \frac {i \, \log \left (e^{\left (-d x - c\right )} + 1\right )}{d} + \frac {i \, \log \left (e^{\left (-d x - c\right )} - 1\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.22, size = 24, normalized size = 0.77 \[ \frac {\mathrm {atan}\left (\frac {\mathrm {tanh}\left (c+d\,x\right )}{\sqrt {-{\mathrm {tanh}\left (c+d\,x\right )}^2}}\right )}{d} \]
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 \[ \int \frac {1}{\sqrt {- \tanh ^{2}{\left (c + d x \right )}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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