Optimal. Leaf size=36 \[ -\frac {a^2 \tanh (c+d x)}{d}+\frac {2 a^2 \log (\cosh (c+d x))}{d}+2 a^2 x \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.02, antiderivative size = 36, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {3477, 3475} \[ -\frac {a^2 \tanh (c+d x)}{d}+\frac {2 a^2 \log (\cosh (c+d x))}{d}+2 a^2 x \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3475
Rule 3477
Rubi steps
\begin {align*} \int (a+a \tanh (c+d x))^2 \, dx &=2 a^2 x-\frac {a^2 \tanh (c+d x)}{d}+\left (2 a^2\right ) \int \tanh (c+d x) \, dx\\ &=2 a^2 x+\frac {2 a^2 \log (\cosh (c+d x))}{d}-\frac {a^2 \tanh (c+d x)}{d}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.54, size = 58, normalized size = 1.61 \[ \frac {a^2 \text {sech}(c) \text {sech}(c+d x) (\cosh (d x) (\log (\cosh (c+d x))+d x)+\cosh (2 c+d x) (\log (\cosh (c+d x))+d x)-\sinh (d x))}{d} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 0.84, size = 117, normalized size = 3.25 \[ \frac {2 \, {\left (a^{2} + {\left (a^{2} \cosh \left (d x + c\right )^{2} + 2 \, a^{2} \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + a^{2} \sinh \left (d x + c\right )^{2} + a^{2}\right )} \log \left (\frac {2 \, \cosh \left (d x + c\right )}{\cosh \left (d x + c\right ) - \sinh \left (d x + c\right )}\right )\right )}}{d \cosh \left (d x + c\right )^{2} + 2 \, d \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + d \sinh \left (d x + c\right )^{2} + d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.12, size = 39, normalized size = 1.08 \[ \frac {2 \, {\left (a^{2} \log \left (e^{\left (2 \, d x + 2 \, c\right )} + 1\right ) + \frac {a^{2}}{e^{\left (2 \, d x + 2 \, c\right )} + 1}\right )}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.01, size = 33, normalized size = 0.92 \[ -\frac {a^{2} \tanh \left (d x +c \right )}{d}-\frac {2 a^{2} \ln \left (\tanh \left (d x +c \right )-1\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.31, size = 50, normalized size = 1.39 \[ a^{2} {\left (x + \frac {c}{d} - \frac {2}{d {\left (e^{\left (-2 \, d x - 2 \, c\right )} + 1\right )}}\right )} + a^{2} x + \frac {2 \, a^{2} \log \left (\cosh \left (d x + c\right )\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 1.07, size = 33, normalized size = 0.92 \[ 4\,a^2\,x-\frac {a^2\,\left (2\,\ln \left (\mathrm {tanh}\left (c+d\,x\right )+1\right )+\mathrm {tanh}\left (c+d\,x\right )\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 0.18, size = 44, normalized size = 1.22 \[ \begin {cases} 4 a^{2} x - \frac {2 a^{2} \log {\left (\tanh {\left (c + d x \right )} + 1 \right )}}{d} - \frac {a^{2} \tanh {\left (c + d x \right )}}{d} & \text {for}\: d \neq 0 \\x \left (a \tanh {\relax (c )} + a\right )^{2} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________