Optimal. Leaf size=44 \[ -\frac {x}{8 a}-\frac {\sinh ^3(x)}{3 a}+\frac {\sinh (x) \cosh ^3(x)}{4 a}-\frac {\sinh (x) \cosh (x)}{8 a} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.14, antiderivative size = 44, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.538, Rules used = {3872, 2839, 2564, 30, 2568, 2635, 8} \[ -\frac {x}{8 a}-\frac {\sinh ^3(x)}{3 a}+\frac {\sinh (x) \cosh ^3(x)}{4 a}-\frac {\sinh (x) \cosh (x)}{8 a} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 8
Rule 30
Rule 2564
Rule 2568
Rule 2635
Rule 2839
Rule 3872
Rubi steps
\begin {align*} \int \frac {\sinh ^4(x)}{a+a \text {sech}(x)} \, dx &=-\int \frac {\cosh (x) \sinh ^4(x)}{-a-a \cosh (x)} \, dx\\ &=-\frac {\int \cosh (x) \sinh ^2(x) \, dx}{a}+\frac {\int \cosh ^2(x) \sinh ^2(x) \, dx}{a}\\ &=\frac {\cosh ^3(x) \sinh (x)}{4 a}-\frac {i \operatorname {Subst}\left (\int x^2 \, dx,x,i \sinh (x)\right )}{a}-\frac {\int \cosh ^2(x) \, dx}{4 a}\\ &=-\frac {\cosh (x) \sinh (x)}{8 a}+\frac {\cosh ^3(x) \sinh (x)}{4 a}-\frac {\sinh ^3(x)}{3 a}-\frac {\int 1 \, dx}{8 a}\\ &=-\frac {x}{8 a}-\frac {\cosh (x) \sinh (x)}{8 a}+\frac {\cosh ^3(x) \sinh (x)}{4 a}-\frac {\sinh ^3(x)}{3 a}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.10, size = 28, normalized size = 0.64 \[ \frac {24 \sinh (x)-8 \sinh (3 x)+3 (\sinh (4 x)-4 x)}{96 a} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.41, size = 36, normalized size = 0.82 \[ \frac {{\left (3 \, \cosh \relax (x) - 2\right )} \sinh \relax (x)^{3} + 3 \, {\left (\cosh \relax (x)^{3} - 2 \, \cosh \relax (x)^{2} + 2\right )} \sinh \relax (x) - 3 \, x}{24 \, a} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.11, size = 42, normalized size = 0.95 \[ -\frac {{\left (24 \, e^{\left (3 \, x\right )} - 8 \, e^{x} + 3\right )} e^{\left (-4 \, x\right )} + 24 \, x - 3 \, e^{\left (4 \, x\right )} + 8 \, e^{\left (3 \, x\right )} - 24 \, e^{x}}{192 \, a} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.14, size = 130, normalized size = 2.95 \[ \frac {1}{4 a \left (\tanh \left (\frac {x}{2}\right )-1\right )^{4}}+\frac {5}{6 a \left (\tanh \left (\frac {x}{2}\right )-1\right )^{3}}+\frac {7}{8 a \left (\tanh \left (\frac {x}{2}\right )-1\right )^{2}}+\frac {1}{8 a \left (\tanh \left (\frac {x}{2}\right )-1\right )}+\frac {\ln \left (\tanh \left (\frac {x}{2}\right )-1\right )}{8 a}-\frac {1}{4 a \left (\tanh \left (\frac {x}{2}\right )+1\right )^{4}}+\frac {5}{6 a \left (\tanh \left (\frac {x}{2}\right )+1\right )^{3}}-\frac {7}{8 a \left (\tanh \left (\frac {x}{2}\right )+1\right )^{2}}+\frac {1}{8 a \left (\tanh \left (\frac {x}{2}\right )+1\right )}-\frac {\ln \left (\tanh \left (\frac {x}{2}\right )+1\right )}{8 a} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.32, size = 54, normalized size = 1.23 \[ -\frac {{\left (8 \, e^{\left (-x\right )} - 24 \, e^{\left (-3 \, x\right )} - 3\right )} e^{\left (4 \, x\right )}}{192 \, a} - \frac {x}{8 \, a} - \frac {24 \, e^{\left (-x\right )} - 8 \, e^{\left (-3 \, x\right )} + 3 \, e^{\left (-4 \, x\right )}}{192 \, a} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 1.48, size = 59, normalized size = 1.34 \[ \frac {{\mathrm {e}}^{-3\,x}}{24\,a}-\frac {{\mathrm {e}}^{-x}}{8\,a}-\frac {{\mathrm {e}}^{3\,x}}{24\,a}-\frac {{\mathrm {e}}^{-4\,x}}{64\,a}+\frac {{\mathrm {e}}^{4\,x}}{64\,a}-\frac {x}{8\,a}+\frac {{\mathrm {e}}^x}{8\,a} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {\int \frac {\sinh ^{4}{\relax (x )}}{\operatorname {sech}{\relax (x )} + 1}\, dx}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________