Optimal. Leaf size=31 \[ \frac {\coth (c+d x) \log (\cosh (c+d x)) \sqrt {-\tanh ^2(c+d x)}}{d} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.01, 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 = {3739, 3556}
\begin {gather*} \frac {\sqrt {-\tanh ^2(c+d x)} \coth (c+d x) \log (\cosh (c+d x))}{d} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 3556
Rule 3739
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*}
________________________________________________________________________________________
Mathematica [A]
time = 0.03, size = 31, normalized size = 1.00 \begin {gather*} \frac {\coth (c+d x) \log (\cosh (c+d x)) \sqrt {-\tanh ^2(c+d x)}}{d} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 1.45, size = 45, normalized size = 1.45
method | result | size |
derivativedivides | \(-\frac {\sqrt {-\left (\tanh ^{2}\left (d x +c \right )\right )}\, \left (\ln \left (\tanh \left (d x +c \right )-1\right )+\ln \left (\tanh \left (d x +c \right )+1\right )\right )}{2 d \tanh \left (d x +c \right )}\) | \(45\) |
default | \(-\frac {\sqrt {-\left (\tanh ^{2}\left (d x +c \right )\right )}\, \left (\ln \left (\tanh \left (d x +c \right )-1\right )+\ln \left (\tanh \left (d x +c \right )+1\right )\right )}{2 d \tanh \left (d x +c \right )}\) | \(45\) |
risch | \(\frac {\left (1+{\mathrm e}^{2 d x +2 c}\right ) \sqrt {-\frac {\left ({\mathrm e}^{2 d x +2 c}-1\right )^{2}}{\left (1+{\mathrm e}^{2 d x +2 c}\right )^{2}}}\, x}{{\mathrm e}^{2 d x +2 c}-1}-\frac {2 \left (1+{\mathrm e}^{2 d x +2 c}\right ) \sqrt {-\frac {\left ({\mathrm e}^{2 d x +2 c}-1\right )^{2}}{\left (1+{\mathrm e}^{2 d x +2 c}\right )^{2}}}\, \left (d x +c \right )}{\left ({\mathrm e}^{2 d x +2 c}-1\right ) d}+\frac {\left (1+{\mathrm e}^{2 d x +2 c}\right ) \sqrt {-\frac {\left ({\mathrm e}^{2 d x +2 c}-1\right )^{2}}{\left (1+{\mathrm e}^{2 d x +2 c}\right )^{2}}}\, \ln \left (1+{\mathrm e}^{2 d x +2 c}\right )}{\left ({\mathrm e}^{2 d x +2 c}-1\right ) d}\) | \(192\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [C] Result contains complex when optimal does not.
time = 0.50, size = 28, normalized size = 0.90 \begin {gather*} -\frac {i \, {\left (d x + c\right )}}{d} - \frac {i \, \log \left (e^{\left (-2 \, d x - 2 \, c\right )} + 1\right )}{d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [C] Result contains complex when optimal does not.
time = 0.36, size = 23, normalized size = 0.74 \begin {gather*} \frac {-i \, d x + i \, \log \left (e^{\left (2 \, d x + 2 \, c\right )} + 1\right )}{d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \sqrt {- \tanh ^{2}{\left (c + d x \right )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [C] Result contains complex when optimal does not.
time = 0.42, size = 54, normalized size = 1.74 \begin {gather*} \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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int \sqrt {-{\mathrm {tanh}\left (c+d\,x\right )}^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________