Optimal. Leaf size=40 \[ -\frac {\left (a^2-b^2\right ) \log (a+b \tanh (x))}{b^3}+\frac {a \tanh (x)}{b^2}-\frac {\tanh ^2(x)}{2 b} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.07, antiderivative size = 40, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {3506, 697} \[ -\frac {\left (a^2-b^2\right ) \log (a+b \tanh (x))}{b^3}+\frac {a \tanh (x)}{b^2}-\frac {\tanh ^2(x)}{2 b} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 697
Rule 3506
Rubi steps
\begin {align*} \int \frac {\text {sech}^4(x)}{a+b \tanh (x)} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {1-\frac {x^2}{b^2}}{a+x} \, dx,x,b \tanh (x)\right )}{b}\\ &=\frac {\operatorname {Subst}\left (\int \left (\frac {a}{b^2}-\frac {x}{b^2}+\frac {-a^2+b^2}{b^2 (a+x)}\right ) \, dx,x,b \tanh (x)\right )}{b}\\ &=-\frac {\left (a^2-b^2\right ) \log (a+b \tanh (x))}{b^3}+\frac {a \tanh (x)}{b^2}-\frac {\tanh ^2(x)}{2 b}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.19, size = 49, normalized size = 1.22 \[ \frac {2 \left (a^2-b^2\right ) (\log (\cosh (x))-\log (a \cosh (x)+b \sinh (x)))+2 a b \tanh (x)+b^2 \text {sech}^2(x)}{2 b^3} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 0.61, size = 430, normalized size = 10.75 \[ -\frac {2 \, {\left (a b - b^{2}\right )} \cosh \relax (x)^{2} + 4 \, {\left (a b - b^{2}\right )} \cosh \relax (x) \sinh \relax (x) + 2 \, {\left (a b - b^{2}\right )} \sinh \relax (x)^{2} + 2 \, a b + {\left ({\left (a^{2} - b^{2}\right )} \cosh \relax (x)^{4} + 4 \, {\left (a^{2} - b^{2}\right )} \cosh \relax (x) \sinh \relax (x)^{3} + {\left (a^{2} - b^{2}\right )} \sinh \relax (x)^{4} + 2 \, {\left (a^{2} - b^{2}\right )} \cosh \relax (x)^{2} + 2 \, {\left (3 \, {\left (a^{2} - b^{2}\right )} \cosh \relax (x)^{2} + a^{2} - b^{2}\right )} \sinh \relax (x)^{2} + a^{2} - b^{2} + 4 \, {\left ({\left (a^{2} - b^{2}\right )} \cosh \relax (x)^{3} + {\left (a^{2} - b^{2}\right )} \cosh \relax (x)\right )} \sinh \relax (x)\right )} \log \left (\frac {2 \, {\left (a \cosh \relax (x) + b \sinh \relax (x)\right )}}{\cosh \relax (x) - \sinh \relax (x)}\right ) - {\left ({\left (a^{2} - b^{2}\right )} \cosh \relax (x)^{4} + 4 \, {\left (a^{2} - b^{2}\right )} \cosh \relax (x) \sinh \relax (x)^{3} + {\left (a^{2} - b^{2}\right )} \sinh \relax (x)^{4} + 2 \, {\left (a^{2} - b^{2}\right )} \cosh \relax (x)^{2} + 2 \, {\left (3 \, {\left (a^{2} - b^{2}\right )} \cosh \relax (x)^{2} + a^{2} - b^{2}\right )} \sinh \relax (x)^{2} + a^{2} - b^{2} + 4 \, {\left ({\left (a^{2} - b^{2}\right )} \cosh \relax (x)^{3} + {\left (a^{2} - b^{2}\right )} \cosh \relax (x)\right )} \sinh \relax (x)\right )} \log \left (\frac {2 \, \cosh \relax (x)}{\cosh \relax (x) - \sinh \relax (x)}\right )}{b^{3} \cosh \relax (x)^{4} + 4 \, b^{3} \cosh \relax (x) \sinh \relax (x)^{3} + b^{3} \sinh \relax (x)^{4} + 2 \, b^{3} \cosh \relax (x)^{2} + b^{3} + 2 \, {\left (3 \, b^{3} \cosh \relax (x)^{2} + b^{3}\right )} \sinh \relax (x)^{2} + 4 \, {\left (b^{3} \cosh \relax (x)^{3} + b^{3} \cosh \relax (x)\right )} \sinh \relax (x)} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [B] time = 0.14, size = 104, normalized size = 2.60 \[ -\frac {{\left (a^{3} + a^{2} b - a b^{2} - b^{3}\right )} \log \left ({\left | a e^{\left (2 \, x\right )} + b e^{\left (2 \, x\right )} + a - b \right |}\right )}{a b^{3} + b^{4}} + \frac {{\left (a^{2} - b^{2}\right )} \log \left (e^{\left (2 \, x\right )} + 1\right )}{b^{3}} - \frac {2 \, {\left (a b + {\left (a b - b^{2}\right )} e^{\left (2 \, x\right )}\right )}}{b^{3} {\left (e^{\left (2 \, x\right )} + 1\right )}^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.12, size = 143, normalized size = 3.58 \[ -\frac {\ln \left (a \left (\tanh ^{2}\left (\frac {x}{2}\right )\right )+2 \tanh \left (\frac {x}{2}\right ) b +a \right ) a^{2}}{b^{3}}+\frac {\ln \left (a \left (\tanh ^{2}\left (\frac {x}{2}\right )\right )+2 \tanh \left (\frac {x}{2}\right ) b +a \right )}{b}+\frac {2 \left (\tanh ^{3}\left (\frac {x}{2}\right )\right ) a}{b^{2} \left (\tanh ^{2}\left (\frac {x}{2}\right )+1\right )^{2}}-\frac {2 \left (\tanh ^{2}\left (\frac {x}{2}\right )\right )}{b \left (\tanh ^{2}\left (\frac {x}{2}\right )+1\right )^{2}}+\frac {2 a \tanh \left (\frac {x}{2}\right )}{b^{2} \left (\tanh ^{2}\left (\frac {x}{2}\right )+1\right )^{2}}+\frac {\ln \left (\tanh ^{2}\left (\frac {x}{2}\right )+1\right ) a^{2}}{b^{3}}-\frac {\ln \left (\tanh ^{2}\left (\frac {x}{2}\right )+1\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [B] time = 0.41, size = 89, normalized size = 2.22 \[ \frac {2 \, {\left ({\left (a + b\right )} e^{\left (-2 \, x\right )} + a\right )}}{2 \, b^{2} e^{\left (-2 \, x\right )} + b^{2} e^{\left (-4 \, x\right )} + b^{2}} - \frac {{\left (a^{2} - b^{2}\right )} \log \left (-{\left (a - b\right )} e^{\left (-2 \, x\right )} - a - b\right )}{b^{3}} + \frac {{\left (a^{2} - b^{2}\right )} \log \left (e^{\left (-2 \, x\right )} + 1\right )}{b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 1.27, size = 88, normalized size = 2.20 \[ \frac {\ln \left ({\mathrm {e}}^{2\,x}+1\right )\,\left (a+b\right )\,\left (a-b\right )}{b^3}-\frac {2\,\left (a-b\right )}{b^2\,\left ({\mathrm {e}}^{2\,x}+1\right )}-\frac {\ln \left (a-b+a\,{\mathrm {e}}^{2\,x}+b\,{\mathrm {e}}^{2\,x}\right )\,\left (a+b\right )\,\left (a-b\right )}{b^3}-\frac {2}{b\,\left (2\,{\mathrm {e}}^{2\,x}+{\mathrm {e}}^{4\,x}+1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {sech}^{4}{\relax (x )}}{a + b \tanh {\relax (x )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________