Optimal. Leaf size=55 \[ \frac {x}{a}-\frac {\coth ^5(x) (1-\text {sech}(x))}{5 a}-\frac {\coth ^3(x) (5-4 \text {sech}(x))}{15 a}-\frac {\coth (x) (15-8 \text {sech}(x))}{15 a} \]
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Rubi [A] time = 0.12, antiderivative size = 55, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {3888, 3882, 8} \[ \frac {x}{a}-\frac {\coth ^5(x) (1-\text {sech}(x))}{5 a}-\frac {\coth ^3(x) (5-4 \text {sech}(x))}{15 a}-\frac {\coth (x) (15-8 \text {sech}(x))}{15 a} \]
Antiderivative was successfully verified.
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Rule 8
Rule 3882
Rule 3888
Rubi steps
\begin {align*} \int \frac {\coth ^4(x)}{a+a \text {sech}(x)} \, dx &=-\frac {\int \coth ^6(x) (-a+a \text {sech}(x)) \, dx}{a^2}\\ &=-\frac {\coth ^5(x) (1-\text {sech}(x))}{5 a}+\frac {\int \coth ^4(x) (5 a-4 a \text {sech}(x)) \, dx}{5 a^2}\\ &=-\frac {\coth ^3(x) (5-4 \text {sech}(x))}{15 a}-\frac {\coth ^5(x) (1-\text {sech}(x))}{5 a}-\frac {\int \coth ^2(x) (-15 a+8 a \text {sech}(x)) \, dx}{15 a^2}\\ &=-\frac {\coth (x) (15-8 \text {sech}(x))}{15 a}-\frac {\coth ^3(x) (5-4 \text {sech}(x))}{15 a}-\frac {\coth ^5(x) (1-\text {sech}(x))}{5 a}+\frac {\int 15 a \, dx}{15 a^2}\\ &=\frac {x}{a}-\frac {\coth (x) (15-8 \text {sech}(x))}{15 a}-\frac {\coth ^3(x) (5-4 \text {sech}(x))}{15 a}-\frac {\coth ^5(x) (1-\text {sech}(x))}{5 a}\\ \end {align*}
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Mathematica [A] time = 0.10, size = 69, normalized size = 1.25 \[ \frac {\text {csch}^3(x) \text {sech}(x) (-90 x \sinh (x)-30 x \sinh (2 x)+30 x \sinh (3 x)+15 x \sinh (4 x)+8 \cosh (x)+16 \cosh (2 x)-16 \cosh (3 x)-23 \cosh (4 x)-25)}{120 a (\text {sech}(x)+1)} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.38, size = 151, normalized size = 2.75 \[ -\frac {23 \, \cosh \relax (x)^{4} - 2 \, {\left (2 \, {\left (15 \, x + 23\right )} \cosh \relax (x) + 15 \, x + 23\right )} \sinh \relax (x)^{3} + 23 \, \sinh \relax (x)^{4} + 16 \, \cosh \relax (x)^{3} + 2 \, {\left (69 \, \cosh \relax (x)^{2} + 24 \, \cosh \relax (x) - 8\right )} \sinh \relax (x)^{2} - 16 \, \cosh \relax (x)^{2} - 2 \, {\left (2 \, {\left (15 \, x + 23\right )} \cosh \relax (x)^{3} + 3 \, {\left (15 \, x + 23\right )} \cosh \relax (x)^{2} - 2 \, {\left (15 \, x + 23\right )} \cosh \relax (x) - 45 \, x - 69\right )} \sinh \relax (x) - 8 \, \cosh \relax (x) + 25}{30 \, {\left ({\left (2 \, a \cosh \relax (x) + a\right )} \sinh \relax (x)^{3} + {\left (2 \, a \cosh \relax (x)^{3} + 3 \, a \cosh \relax (x)^{2} - 2 \, a \cosh \relax (x) - 3 \, a\right )} \sinh \relax (x)\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.13, size = 64, normalized size = 1.16 \[ \frac {x}{a} - \frac {21 \, e^{\left (2 \, x\right )} - 36 \, e^{x} + 19}{24 \, a {\left (e^{x} - 1\right )}^{3}} + \frac {115 \, e^{\left (4 \, x\right )} + 380 \, e^{\left (3 \, x\right )} + 530 \, e^{\left (2 \, x\right )} + 340 \, e^{x} + 91}{40 \, a {\left (e^{x} + 1\right )}^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.17, size = 78, normalized size = 1.42 \[ -\frac {\tanh ^{5}\left (\frac {x}{2}\right )}{80 a}-\frac {\tanh ^{3}\left (\frac {x}{2}\right )}{8 a}-\frac {\tanh \left (\frac {x}{2}\right )}{a}-\frac {\ln \left (\tanh \left (\frac {x}{2}\right )-1\right )}{a}+\frac {\ln \left (\tanh \left (\frac {x}{2}\right )+1\right )}{a}-\frac {1}{48 a \tanh \left (\frac {x}{2}\right )^{3}}-\frac {3}{8 a \tanh \left (\frac {x}{2}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.35, size = 105, normalized size = 1.91 \[ \frac {x}{a} - \frac {2 \, {\left (31 \, e^{\left (-x\right )} - 31 \, e^{\left (-2 \, x\right )} - 73 \, e^{\left (-3 \, x\right )} + 25 \, e^{\left (-4 \, x\right )} + 65 \, e^{\left (-5 \, x\right )} + 15 \, e^{\left (-6 \, x\right )} - 15 \, e^{\left (-7 \, x\right )} + 23\right )}}{15 \, {\left (2 \, a e^{\left (-x\right )} - 2 \, a e^{\left (-2 \, x\right )} - 6 \, a e^{\left (-3 \, x\right )} + 6 \, a e^{\left (-5 \, x\right )} + 2 \, a e^{\left (-6 \, x\right )} - 2 \, a e^{\left (-7 \, x\right )} - a e^{\left (-8 \, x\right )} + a\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.53, size = 264, normalized size = 4.80 \[ \frac {\frac {9\,{\mathrm {e}}^{2\,x}}{4\,a}+\frac {3\,{\mathrm {e}}^{3\,x}}{2\,a}+\frac {23\,{\mathrm {e}}^{4\,x}}{40\,a}+\frac {23}{40\,a}+\frac {3\,{\mathrm {e}}^x}{2\,a}}{10\,{\mathrm {e}}^{2\,x}+10\,{\mathrm {e}}^{3\,x}+5\,{\mathrm {e}}^{4\,x}+{\mathrm {e}}^{5\,x}+5\,{\mathrm {e}}^x+1}+\frac {\frac {9\,{\mathrm {e}}^{2\,x}}{8\,a}+\frac {23\,{\mathrm {e}}^{3\,x}}{40\,a}+\frac {3}{8\,a}+\frac {9\,{\mathrm {e}}^x}{8\,a}}{6\,{\mathrm {e}}^{2\,x}+4\,{\mathrm {e}}^{3\,x}+{\mathrm {e}}^{4\,x}+4\,{\mathrm {e}}^x+1}+\frac {\frac {23\,{\mathrm {e}}^{2\,x}}{40\,a}+\frac {3}{8\,a}+\frac {3\,{\mathrm {e}}^x}{4\,a}}{3\,{\mathrm {e}}^{2\,x}+{\mathrm {e}}^{3\,x}+3\,{\mathrm {e}}^x+1}+\frac {\frac {3}{8\,a}+\frac {23\,{\mathrm {e}}^x}{40\,a}}{{\mathrm {e}}^{2\,x}+2\,{\mathrm {e}}^x+1}+\frac {1}{6\,a\,\left (3\,{\mathrm {e}}^{2\,x}-{\mathrm {e}}^{3\,x}-3\,{\mathrm {e}}^x+1\right )}-\frac {1}{4\,a\,\left ({\mathrm {e}}^{2\,x}-2\,{\mathrm {e}}^x+1\right )}+\frac {x}{a}-\frac {7}{8\,a\,\left ({\mathrm {e}}^x-1\right )}+\frac {23}{40\,a\,\left ({\mathrm {e}}^x+1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {\int \frac {\coth ^{4}{\relax (x )}}{\operatorname {sech}{\relax (x )} + 1}\, dx}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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