Optimal. Leaf size=56 \[ -\frac {3 \sinh (c+d x)}{16 d (3 \cosh (c+d x)+5)}-\frac {5 \tanh ^{-1}\left (\frac {\sinh (c+d x)}{\cosh (c+d x)+3}\right )}{32 d}+\frac {5 x}{64} \]
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Rubi [A] time = 0.04, antiderivative size = 56, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {2664, 12, 2657} \[ -\frac {3 \sinh (c+d x)}{16 d (3 \cosh (c+d x)+5)}-\frac {5 \tanh ^{-1}\left (\frac {\sinh (c+d x)}{\cosh (c+d x)+3}\right )}{32 d}+\frac {5 x}{64} \]
Antiderivative was successfully verified.
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Rule 12
Rule 2657
Rule 2664
Rubi steps
\begin {align*} \int \frac {1}{(5+3 \cosh (c+d x))^2} \, dx &=-\frac {3 \sinh (c+d x)}{16 d (5+3 \cosh (c+d x))}-\frac {1}{16} \int -\frac {5}{5+3 \cosh (c+d x)} \, dx\\ &=-\frac {3 \sinh (c+d x)}{16 d (5+3 \cosh (c+d x))}+\frac {5}{16} \int \frac {1}{5+3 \cosh (c+d x)} \, dx\\ &=\frac {5 x}{64}-\frac {5 \tanh ^{-1}\left (\frac {\sinh (c+d x)}{3+\cosh (c+d x)}\right )}{32 d}-\frac {3 \sinh (c+d x)}{16 d (5+3 \cosh (c+d x))}\\ \end {align*}
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Mathematica [A] time = 0.12, size = 45, normalized size = 0.80 \[ \frac {5 \tanh ^{-1}\left (\frac {1}{2} \tanh \left (\frac {1}{2} (c+d x)\right )\right )-\frac {6 \sinh (c+d x)}{3 \cosh (c+d x)+5}}{32 d} \]
Antiderivative was successfully verified.
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fricas [B] time = 1.03, size = 212, normalized size = 3.79 \[ \frac {5 \, {\left (3 \, \cosh \left (d x + c\right )^{2} + 2 \, {\left (3 \, \cosh \left (d x + c\right ) + 5\right )} \sinh \left (d x + c\right ) + 3 \, \sinh \left (d x + c\right )^{2} + 10 \, \cosh \left (d x + c\right ) + 3\right )} \log \left (3 \, \cosh \left (d x + c\right ) + 3 \, \sinh \left (d x + c\right ) + 1\right ) - 5 \, {\left (3 \, \cosh \left (d x + c\right )^{2} + 2 \, {\left (3 \, \cosh \left (d x + c\right ) + 5\right )} \sinh \left (d x + c\right ) + 3 \, \sinh \left (d x + c\right )^{2} + 10 \, \cosh \left (d x + c\right ) + 3\right )} \log \left (\cosh \left (d x + c\right ) + \sinh \left (d x + c\right ) + 3\right ) + 40 \, \cosh \left (d x + c\right ) + 40 \, \sinh \left (d x + c\right ) + 24}{64 \, {\left (3 \, d \cosh \left (d x + c\right )^{2} + 3 \, d \sinh \left (d x + c\right )^{2} + 10 \, d \cosh \left (d x + c\right ) + 2 \, {\left (3 \, d \cosh \left (d x + c\right ) + 5 \, d\right )} \sinh \left (d x + c\right ) + 3 \, d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.14, size = 65, normalized size = 1.16 \[ \frac {\frac {8 \, {\left (5 \, e^{\left (d x + c\right )} + 3\right )}}{3 \, e^{\left (2 \, d x + 2 \, c\right )} + 10 \, e^{\left (d x + c\right )} + 3} + 5 \, \log \left (3 \, e^{\left (d x + c\right )} + 1\right ) - 5 \, \log \left (e^{\left (d x + c\right )} + 3\right )}{64 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.07, size = 72, normalized size = 1.29 \[ \frac {3}{32 d \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+2\right )}+\frac {5 \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+2\right )}{64 d}+\frac {3}{32 d \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-2\right )}-\frac {5 \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-2\right )}{64 d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 81, normalized size = 1.45 \[ -\frac {5 \, \log \left (3 \, e^{\left (-d x - c\right )} + 1\right )}{64 \, d} + \frac {5 \, \log \left (e^{\left (-d x - c\right )} + 3\right )}{64 \, d} - \frac {5 \, e^{\left (-d x - c\right )} + 3}{8 \, d {\left (10 \, e^{\left (-d x - c\right )} + 3 \, e^{\left (-2 \, d x - 2 \, c\right )} + 3\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.94, size = 77, normalized size = 1.38 \[ \frac {\frac {5\,{\mathrm {e}}^{c+d\,x}}{8\,d}+\frac {3}{8\,d}}{10\,{\mathrm {e}}^{c+d\,x}+3\,{\mathrm {e}}^{2\,c+2\,d\,x}+3}-\frac {5\,\mathrm {atan}\left (\left (\frac {5}{4\,d}+\frac {3\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c}{4\,d}\right )\,\sqrt {-d^2}\right )}{32\,\sqrt {-d^2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.65, size = 199, normalized size = 3.55 \[ \begin {cases} - \frac {5 \log {\left (\tanh {\left (\frac {c}{2} + \frac {d x}{2} \right )} - 2 \right )} \tanh ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{64 d \tanh ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} - 256 d} + \frac {20 \log {\left (\tanh {\left (\frac {c}{2} + \frac {d x}{2} \right )} - 2 \right )}}{64 d \tanh ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} - 256 d} + \frac {5 \log {\left (\tanh {\left (\frac {c}{2} + \frac {d x}{2} \right )} + 2 \right )} \tanh ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{64 d \tanh ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} - 256 d} - \frac {20 \log {\left (\tanh {\left (\frac {c}{2} + \frac {d x}{2} \right )} + 2 \right )}}{64 d \tanh ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} - 256 d} + \frac {12 \tanh {\left (\frac {c}{2} + \frac {d x}{2} \right )}}{64 d \tanh ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} - 256 d} & \text {for}\: d \neq 0 \\\frac {x}{\left (3 \cosh {\relax (c )} + 5\right )^{2}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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