Optimal. Leaf size=56 \[ \frac {x}{32}-\frac {1}{32 (\coth (x)+1)}-\frac {1}{32 (\coth (x)+1)^2}-\frac {1}{24 (\coth (x)+1)^3}-\frac {1}{16 (\coth (x)+1)^4}-\frac {1}{10 (\coth (x)+1)^5} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.05, antiderivative size = 56, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 2, integrand size = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {3479, 8} \[ \frac {x}{32}-\frac {1}{32 (\coth (x)+1)}-\frac {1}{32 (\coth (x)+1)^2}-\frac {1}{24 (\coth (x)+1)^3}-\frac {1}{16 (\coth (x)+1)^4}-\frac {1}{10 (\coth (x)+1)^5} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 8
Rule 3479
Rubi steps
\begin {align*} \int \frac {1}{(1+\coth (x))^5} \, dx &=-\frac {1}{10 (1+\coth (x))^5}+\frac {1}{2} \int \frac {1}{(1+\coth (x))^4} \, dx\\ &=-\frac {1}{10 (1+\coth (x))^5}-\frac {1}{16 (1+\coth (x))^4}+\frac {1}{4} \int \frac {1}{(1+\coth (x))^3} \, dx\\ &=-\frac {1}{10 (1+\coth (x))^5}-\frac {1}{16 (1+\coth (x))^4}-\frac {1}{24 (1+\coth (x))^3}+\frac {1}{8} \int \frac {1}{(1+\coth (x))^2} \, dx\\ &=-\frac {1}{10 (1+\coth (x))^5}-\frac {1}{16 (1+\coth (x))^4}-\frac {1}{24 (1+\coth (x))^3}-\frac {1}{32 (1+\coth (x))^2}+\frac {1}{16} \int \frac {1}{1+\coth (x)} \, dx\\ &=-\frac {1}{10 (1+\coth (x))^5}-\frac {1}{16 (1+\coth (x))^4}-\frac {1}{24 (1+\coth (x))^3}-\frac {1}{32 (1+\coth (x))^2}-\frac {1}{32 (1+\coth (x))}+\frac {\int 1 \, dx}{32}\\ &=\frac {x}{32}-\frac {1}{10 (1+\coth (x))^5}-\frac {1}{16 (1+\coth (x))^4}-\frac {1}{24 (1+\coth (x))^3}-\frac {1}{32 (1+\coth (x))^2}-\frac {1}{32 (1+\coth (x))}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.14, size = 62, normalized size = 1.11 \[ \frac {(\cosh (5 x)-\sinh (5 x)) (-500 \sinh (x)+375 \sinh (3 x)+120 x \sinh (5 x)-12 \sinh (5 x)-100 \cosh (x)+225 \cosh (3 x)+120 x \cosh (5 x)+12 \cosh (5 x))}{3840} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 0.39, size = 159, normalized size = 2.84 \[ \frac {12 \, {\left (10 \, x + 1\right )} \cosh \relax (x)^{5} + 60 \, {\left (10 \, x + 1\right )} \cosh \relax (x) \sinh \relax (x)^{4} + 12 \, {\left (10 \, x - 1\right )} \sinh \relax (x)^{5} + 15 \, {\left (8 \, {\left (10 \, x - 1\right )} \cosh \relax (x)^{2} + 25\right )} \sinh \relax (x)^{3} + 225 \, \cosh \relax (x)^{3} + 15 \, {\left (8 \, {\left (10 \, x + 1\right )} \cosh \relax (x)^{3} + 45 \, \cosh \relax (x)\right )} \sinh \relax (x)^{2} + 5 \, {\left (12 \, {\left (10 \, x - 1\right )} \cosh \relax (x)^{4} + 225 \, \cosh \relax (x)^{2} - 100\right )} \sinh \relax (x) - 100 \, \cosh \relax (x)}{3840 \, {\left (\cosh \relax (x)^{5} + 5 \, \cosh \relax (x)^{4} \sinh \relax (x) + 10 \, \cosh \relax (x)^{3} \sinh \relax (x)^{2} + 10 \, \cosh \relax (x)^{2} \sinh \relax (x)^{3} + 5 \, \cosh \relax (x) \sinh \relax (x)^{4} + \sinh \relax (x)^{5}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.12, size = 36, normalized size = 0.64 \[ \frac {1}{3840} \, {\left (300 \, e^{\left (8 \, x\right )} - 300 \, e^{\left (6 \, x\right )} + 200 \, e^{\left (4 \, x\right )} - 75 \, e^{\left (2 \, x\right )} + 12\right )} e^{\left (-10 \, x\right )} + \frac {1}{32} \, x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.05, size = 56, normalized size = 1.00 \[ -\frac {\ln \left (\coth \relax (x )-1\right )}{64}-\frac {1}{10 \left (1+\coth \relax (x )\right )^{5}}-\frac {1}{16 \left (1+\coth \relax (x )\right )^{4}}-\frac {1}{24 \left (1+\coth \relax (x )\right )^{3}}-\frac {1}{32 \left (1+\coth \relax (x )\right )^{2}}-\frac {1}{32 \left (1+\coth \relax (x )\right )}+\frac {\ln \left (1+\coth \relax (x )\right )}{64} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.32, size = 34, normalized size = 0.61 \[ \frac {1}{32} \, x + \frac {5}{64} \, e^{\left (-2 \, x\right )} - \frac {5}{64} \, e^{\left (-4 \, x\right )} + \frac {5}{96} \, e^{\left (-6 \, x\right )} - \frac {5}{256} \, e^{\left (-8 \, x\right )} + \frac {1}{320} \, e^{\left (-10 \, x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 1.15, size = 34, normalized size = 0.61 \[ \frac {x}{32}+\frac {5\,{\mathrm {e}}^{-2\,x}}{64}-\frac {5\,{\mathrm {e}}^{-4\,x}}{64}+\frac {5\,{\mathrm {e}}^{-6\,x}}{96}-\frac {5\,{\mathrm {e}}^{-8\,x}}{256}+\frac {{\mathrm {e}}^{-10\,x}}{320} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [B] time = 1.80, size = 444, normalized size = 7.93 \[ \frac {15 x \tanh ^{5}{\relax (x )}}{480 \tanh ^{5}{\relax (x )} + 2400 \tanh ^{4}{\relax (x )} + 4800 \tanh ^{3}{\relax (x )} + 4800 \tanh ^{2}{\relax (x )} + 2400 \tanh {\relax (x )} + 480} + \frac {75 x \tanh ^{4}{\relax (x )}}{480 \tanh ^{5}{\relax (x )} + 2400 \tanh ^{4}{\relax (x )} + 4800 \tanh ^{3}{\relax (x )} + 4800 \tanh ^{2}{\relax (x )} + 2400 \tanh {\relax (x )} + 480} + \frac {150 x \tanh ^{3}{\relax (x )}}{480 \tanh ^{5}{\relax (x )} + 2400 \tanh ^{4}{\relax (x )} + 4800 \tanh ^{3}{\relax (x )} + 4800 \tanh ^{2}{\relax (x )} + 2400 \tanh {\relax (x )} + 480} + \frac {150 x \tanh ^{2}{\relax (x )}}{480 \tanh ^{5}{\relax (x )} + 2400 \tanh ^{4}{\relax (x )} + 4800 \tanh ^{3}{\relax (x )} + 4800 \tanh ^{2}{\relax (x )} + 2400 \tanh {\relax (x )} + 480} + \frac {75 x \tanh {\relax (x )}}{480 \tanh ^{5}{\relax (x )} + 2400 \tanh ^{4}{\relax (x )} + 4800 \tanh ^{3}{\relax (x )} + 4800 \tanh ^{2}{\relax (x )} + 2400 \tanh {\relax (x )} + 480} + \frac {15 x}{480 \tanh ^{5}{\relax (x )} + 2400 \tanh ^{4}{\relax (x )} + 4800 \tanh ^{3}{\relax (x )} + 4800 \tanh ^{2}{\relax (x )} + 2400 \tanh {\relax (x )} + 480} + \frac {465 \tanh ^{4}{\relax (x )}}{480 \tanh ^{5}{\relax (x )} + 2400 \tanh ^{4}{\relax (x )} + 4800 \tanh ^{3}{\relax (x )} + 4800 \tanh ^{2}{\relax (x )} + 2400 \tanh {\relax (x )} + 480} + \frac {1125 \tanh ^{3}{\relax (x )}}{480 \tanh ^{5}{\relax (x )} + 2400 \tanh ^{4}{\relax (x )} + 4800 \tanh ^{3}{\relax (x )} + 4800 \tanh ^{2}{\relax (x )} + 2400 \tanh {\relax (x )} + 480} + \frac {1205 \tanh ^{2}{\relax (x )}}{480 \tanh ^{5}{\relax (x )} + 2400 \tanh ^{4}{\relax (x )} + 4800 \tanh ^{3}{\relax (x )} + 4800 \tanh ^{2}{\relax (x )} + 2400 \tanh {\relax (x )} + 480} + \frac {625 \tanh {\relax (x )}}{480 \tanh ^{5}{\relax (x )} + 2400 \tanh ^{4}{\relax (x )} + 4800 \tanh ^{3}{\relax (x )} + 4800 \tanh ^{2}{\relax (x )} + 2400 \tanh {\relax (x )} + 480} + \frac {128}{480 \tanh ^{5}{\relax (x )} + 2400 \tanh ^{4}{\relax (x )} + 4800 \tanh ^{3}{\relax (x )} + 4800 \tanh ^{2}{\relax (x )} + 2400 \tanh {\relax (x )} + 480} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________