Optimal. Leaf size=31 \[ \frac {\coth (c+d x) \log (\cosh (c+d x))}{d \sqrt {b \coth ^2(c+d x)}} \]
[Out]
________________________________________________________________________________________
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 = {3739, 3556}
\begin {gather*} \frac {\coth (c+d x) \log (\cosh (c+d x))}{d \sqrt {b \coth ^2(c+d x)}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 3556
Rule 3739
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*}
________________________________________________________________________________________
Mathematica [A]
time = 0.04, size = 31, normalized size = 1.00 \begin {gather*} \frac {\coth (c+d x) \log (\cosh (c+d x))}{d \sqrt {b \coth ^2(c+d x)}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 1.57, size = 56, normalized size = 1.81
method | result | size |
derivativedivides | \(\frac {\coth \left (d x +c \right ) \left (2 \ln \left (\coth \left (d x +c \right )\right )-\ln \left (\coth \left (d x +c \right )+1\right )-\ln \left (\coth \left (d x +c \right )-1\right )\right )}{2 d \sqrt {b \left (\coth ^{2}\left (d x +c \right )\right )}}\) | \(56\) |
default | \(\frac {\coth \left (d x +c \right ) \left (2 \ln \left (\coth \left (d x +c \right )\right )-\ln \left (\coth \left (d x +c \right )+1\right )-\ln \left (\coth \left (d x +c \right )-1\right )\right )}{2 d \sqrt {b \left (\coth ^{2}\left (d x +c \right )\right )}}\) | \(56\) |
risch | \(\frac {\left (1+{\mathrm e}^{2 d x +2 c}\right ) x}{\sqrt {\frac {b \left (1+{\mathrm e}^{2 d x +2 c}\right )^{2}}{\left ({\mathrm e}^{2 d x +2 c}-1\right )^{2}}}\, \left ({\mathrm e}^{2 d x +2 c}-1\right )}-\frac {2 \left (1+{\mathrm e}^{2 d x +2 c}\right ) \left (d x +c \right )}{\sqrt {\frac {b \left (1+{\mathrm e}^{2 d x +2 c}\right )^{2}}{\left ({\mathrm e}^{2 d x +2 c}-1\right )^{2}}}\, \left ({\mathrm e}^{2 d x +2 c}-1\right ) d}+\frac {\left (1+{\mathrm e}^{2 d x +2 c}\right ) \ln \left (1+{\mathrm e}^{2 d x +2 c}\right )}{\sqrt {\frac {b \left (1+{\mathrm e}^{2 d x +2 c}\right )^{2}}{\left ({\mathrm e}^{2 d x +2 c}-1\right )^{2}}}\, \left ({\mathrm e}^{2 d x +2 c}-1\right ) d}\) | \(192\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.50, size = 34, normalized size = 1.10 \begin {gather*} -\frac {d x + c}{\sqrt {b} d} - \frac {\log \left (e^{\left (-2 \, d x - 2 \, c\right )} + 1\right )}{\sqrt {b} d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 128 vs.
\(2 (29) = 58\).
time = 0.43, size = 128, normalized size = 4.13 \begin {gather*} -\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} \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 \frac {1}{\sqrt {b \coth ^{2}{\left (c + d x \right )}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 60 vs.
\(2 (29) = 58\).
time = 0.42, size = 60, normalized size = 1.94 \begin {gather*} -\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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 1.26, size = 30, normalized size = 0.97 \begin {gather*} \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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________