Optimal. Leaf size=31 \[ \frac {x}{a}-\frac {\text {ArcTan}(\sinh (x))}{2 a}-\frac {(2-\text {sech}(x)) \tanh (x)}{2 a} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.05, antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {3973, 3966,
3855} \begin {gather*} -\frac {\text {ArcTan}(\sinh (x))}{2 a}+\frac {x}{a}-\frac {\tanh (x) (2-\text {sech}(x))}{2 a} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 3855
Rule 3966
Rule 3973
Rubi steps
\begin {align*} \int \frac {\tanh ^4(x)}{a+a \text {sech}(x)} \, dx &=-\frac {\int (-a+a \text {sech}(x)) \tanh ^2(x) \, dx}{a^2}\\ &=-\frac {(2-\text {sech}(x)) \tanh (x)}{2 a}-\frac {\int (-2 a+a \text {sech}(x)) \, dx}{2 a^2}\\ &=\frac {x}{a}-\frac {(2-\text {sech}(x)) \tanh (x)}{2 a}-\frac {\int \text {sech}(x) \, dx}{2 a}\\ &=\frac {x}{a}-\frac {\tan ^{-1}(\sinh (x))}{2 a}-\frac {(2-\text {sech}(x)) \tanh (x)}{2 a}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.04, size = 41, normalized size = 1.32 \begin {gather*} \frac {\cosh ^2\left (\frac {x}{2}\right ) \text {sech}(x) \left (2 \left (x-\text {ArcTan}\left (\tanh \left (\frac {x}{2}\right )\right )\right )+(-2+\text {sech}(x)) \tanh (x)\right )}{a (1+\text {sech}(x))} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(58\) vs.
\(2(27)=54\).
time = 0.67, size = 59, normalized size = 1.90
method | result | size |
default | \(\frac {-\ln \left (\tanh \left (\frac {x}{2}\right )-1\right )+\ln \left (\tanh \left (\frac {x}{2}\right )+1\right )+\frac {2 \left (-\frac {3 \left (\tanh ^{3}\left (\frac {x}{2}\right )\right )}{2}-\frac {\tanh \left (\frac {x}{2}\right )}{2}\right )}{\left (\tanh ^{2}\left (\frac {x}{2}\right )+1\right )^{2}}-\arctan \left (\tanh \left (\frac {x}{2}\right )\right )}{a}\) | \(59\) |
risch | \(\frac {x}{a}+\frac {{\mathrm e}^{3 x}+2 \,{\mathrm e}^{2 x}-{\mathrm e}^{x}+2}{\left (1+{\mathrm e}^{2 x}\right )^{2} a}+\frac {i \ln \left ({\mathrm e}^{x}-i\right )}{2 a}-\frac {i \ln \left ({\mathrm e}^{x}+i\right )}{2 a}\) | \(59\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 51 vs.
\(2 (25) = 50\).
time = 0.48, size = 51, normalized size = 1.65 \begin {gather*} \frac {x}{a} + \frac {e^{\left (-x\right )} - 2 \, e^{\left (-2 \, x\right )} - e^{\left (-3 \, x\right )} - 2}{2 \, a e^{\left (-2 \, x\right )} + a e^{\left (-4 \, x\right )} + a} + \frac {\arctan \left (e^{\left (-x\right )}\right )}{a} \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. 210 vs.
\(2 (25) = 50\).
time = 0.38, size = 210, normalized size = 6.77 \begin {gather*} \frac {x \cosh \left (x\right )^{4} + x \sinh \left (x\right )^{4} + {\left (4 \, x \cosh \left (x\right ) + 1\right )} \sinh \left (x\right )^{3} + 2 \, {\left (x + 1\right )} \cosh \left (x\right )^{2} + \cosh \left (x\right )^{3} + {\left (6 \, x \cosh \left (x\right )^{2} + 2 \, x + 3 \, \cosh \left (x\right ) + 2\right )} \sinh \left (x\right )^{2} - {\left (\cosh \left (x\right )^{4} + 4 \, \cosh \left (x\right ) \sinh \left (x\right )^{3} + \sinh \left (x\right )^{4} + 2 \, {\left (3 \, \cosh \left (x\right )^{2} + 1\right )} \sinh \left (x\right )^{2} + 2 \, \cosh \left (x\right )^{2} + 4 \, {\left (\cosh \left (x\right )^{3} + \cosh \left (x\right )\right )} \sinh \left (x\right ) + 1\right )} \arctan \left (\cosh \left (x\right ) + \sinh \left (x\right )\right ) + {\left (4 \, x \cosh \left (x\right )^{3} + 4 \, {\left (x + 1\right )} \cosh \left (x\right ) + 3 \, \cosh \left (x\right )^{2} - 1\right )} \sinh \left (x\right ) + x - \cosh \left (x\right ) + 2}{a \cosh \left (x\right )^{4} + 4 \, a \cosh \left (x\right ) \sinh \left (x\right )^{3} + a \sinh \left (x\right )^{4} + 2 \, a \cosh \left (x\right )^{2} + 2 \, {\left (3 \, a \cosh \left (x\right )^{2} + a\right )} \sinh \left (x\right )^{2} + 4 \, {\left (a \cosh \left (x\right )^{3} + a \cosh \left (x\right )\right )} \sinh \left (x\right ) + a} \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*} \frac {\int \frac {\tanh ^{4}{\left (x \right )}}{\operatorname {sech}{\left (x \right )} + 1}\, dx}{a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 0.38, size = 42, normalized size = 1.35 \begin {gather*} \frac {x}{a} - \frac {\arctan \left (e^{x}\right )}{a} + \frac {e^{\left (3 \, x\right )} + 2 \, e^{\left (2 \, x\right )} - e^{x} + 2}{a {\left (e^{\left (2 \, x\right )} + 1\right )}^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 1.44, size = 67, normalized size = 2.16 \begin {gather*} \frac {x}{a}+\frac {\frac {2}{a}+\frac {{\mathrm {e}}^x}{a}}{{\mathrm {e}}^{2\,x}+1}-\frac {\mathrm {atan}\left (\frac {{\mathrm {e}}^x\,\sqrt {a^2}}{a}\right )}{\sqrt {a^2}}-\frac {2\,{\mathrm {e}}^x}{a\,\left (2\,{\mathrm {e}}^{2\,x}+{\mathrm {e}}^{4\,x}+1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________