Optimal. Leaf size=38 \[ x \left (a^2+b^2\right )+\frac {2 a b \log (\cosh (c+d x))}{d}-\frac {b^2 \tanh (c+d x)}{d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.02, antiderivative size = 38, 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} \[ x \left (a^2+b^2\right )+\frac {2 a b \log (\cosh (c+d x))}{d}-\frac {b^2 \tanh (c+d x)}{d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3475
Rule 3477
Rubi steps
\begin {align*} \int (a+b \tanh (c+d x))^2 \, dx &=\left (a^2+b^2\right ) x-\frac {b^2 \tanh (c+d x)}{d}+(2 a b) \int \tanh (c+d x) \, dx\\ &=\left (a^2+b^2\right ) x+\frac {2 a b \log (\cosh (c+d x))}{d}-\frac {b^2 \tanh (c+d x)}{d}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.11, size = 54, normalized size = 1.42 \[ \frac {(a-b)^2 \log (\tanh (c+d x)+1)-(a+b)^2 \log (1-\tanh (c+d x))-2 b^2 \tanh (c+d x)}{2 d} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 0.57, size = 201, normalized size = 5.29 \[ \frac {{\left (a^{2} - 2 \, a b + b^{2}\right )} d x \cosh \left (d x + c\right )^{2} + 2 \, {\left (a^{2} - 2 \, a b + b^{2}\right )} d x \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + {\left (a^{2} - 2 \, a b + b^{2}\right )} d x \sinh \left (d x + c\right )^{2} + {\left (a^{2} - 2 \, a b + b^{2}\right )} d x + 2 \, b^{2} + 2 \, {\left (a b \cosh \left (d x + c\right )^{2} + 2 \, a b \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + a b \sinh \left (d x + c\right )^{2} + a b\right )} \log \left (\frac {2 \, \cosh \left (d x + c\right )}{\cosh \left (d x + c\right ) - \sinh \left (d x + c\right )}\right )}{d \cosh \left (d x + c\right )^{2} + 2 \, d \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + d \sinh \left (d x + c\right )^{2} + d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.12, size = 56, normalized size = 1.47 \[ \frac {2 \, a b \log \left (e^{\left (2 \, d x + 2 \, c\right )} + 1\right ) + {\left (a^{2} - 2 \, a b + b^{2}\right )} {\left (d x + c\right )} + \frac {2 \, b^{2}}{e^{\left (2 \, d x + 2 \, c\right )} + 1}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.01, size = 116, normalized size = 3.05 \[ -\frac {b^{2} \tanh \left (d x +c \right )}{d}-\frac {a^{2} \ln \left (\tanh \left (d x +c \right )-1\right )}{2 d}-\frac {\ln \left (\tanh \left (d x +c \right )-1\right ) a b}{d}-\frac {\ln \left (\tanh \left (d x +c \right )-1\right ) b^{2}}{2 d}+\frac {\ln \left (1+\tanh \left (d x +c \right )\right ) a^{2}}{2 d}-\frac {\ln \left (1+\tanh \left (d x +c \right )\right ) a b}{d}+\frac {\ln \left (1+\tanh \left (d x +c \right )\right ) b^{2}}{2 d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.31, size = 49, normalized size = 1.29 \[ b^{2} {\left (x + \frac {c}{d} - \frac {2}{d {\left (e^{\left (-2 \, d x - 2 \, c\right )} + 1\right )}}\right )} + a^{2} x + \frac {2 \, a b \log \left (\cosh \left (d x + c\right )\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 1.04, size = 44, normalized size = 1.16 \[ x\,\left (a^2+2\,a\,b+b^2\right )-\frac {b^2\,\mathrm {tanh}\left (c+d\,x\right )}{d}-\frac {2\,a\,b\,\ln \left (\mathrm {tanh}\left (c+d\,x\right )+1\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 0.19, size = 54, normalized size = 1.42 \[ \begin {cases} a^{2} x + 2 a b x - \frac {2 a b \log {\left (\tanh {\left (c + d x \right )} + 1 \right )}}{d} + b^{2} x - \frac {b^{2} \tanh {\left (c + d x \right )}}{d} & \text {for}\: d \neq 0 \\x \left (a + b \tanh {\relax (c )}\right )^{2} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________