Optimal. Leaf size=31 \[ \frac {\sqrt {-\tanh ^2(c+d x)} \coth (c+d x) \log (\cosh (c+d x))}{d} \]
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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} \[ \frac {\sqrt {-\tanh ^2(c+d x)} \coth (c+d x) \log (\cosh (c+d x))}{d} \]
Antiderivative was successfully verified.
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Rule 3475
Rule 3658
Rubi steps
\begin {align*} \int \sqrt {-\tanh ^2(c+d x)} \, dx &=\left (\coth (c+d x) \sqrt {-\tanh ^2(c+d x)}\right ) \int \tanh (c+d x) \, dx\\ &=\frac {\coth (c+d x) \log (\cosh (c+d x)) \sqrt {-\tanh ^2(c+d x)}}{d}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 31, normalized size = 1.00 \[ \frac {\sqrt {-\tanh ^2(c+d x)} \coth (c+d x) \log (\cosh (c+d x))}{d} \]
Antiderivative was successfully verified.
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fricas [C] time = 0.46, 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 = 0.15, size = 54, normalized size = 1.74 \[ \frac {i \, {\left (d x + c\right )} \mathrm {sgn}\left (-e^{\left (4 \, d x + 4 \, c\right )} + 1\right ) - i \, \log \left (e^{\left (2 \, d x + 2 \, c\right )} + 1\right ) \mathrm {sgn}\left (-e^{\left (4 \, d x + 4 \, c\right )} + 1\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.13, size = 45, normalized size = 1.45 \[ -\frac {\sqrt {-\left (\tanh ^{2}\left (d x +c \right )\right )}\, \left (\ln \left (\tanh \left (d x +c \right )-1\right )+\ln \left (1+\tanh \left (d x +c \right )\right )\right )}{2 d \tanh \left (d x +c \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [C] time = 0.42, size = 28, normalized size = 0.90 \[ -\frac {i \, {\left (d x + c\right )}}{d} - \frac {i \, \log \left (e^{\left (-2 \, d x - 2 \, c\right )} + 1\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.03 \[ \int \sqrt {-{\mathrm {tanh}\left (c+d\,x\right )}^2} \,d x \]
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 \[ \int \sqrt {- \tanh ^{2}{\left (c + d x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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