Optimal. Leaf size=22 \[ \frac {\text {ArcTan}\left (\frac {1}{2} \tanh \left (\frac {1}{2} (c+d x)\right )\right )}{2 d} \]
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Rubi [A]
time = 0.01, antiderivative size = 22, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {2738, 212}
\begin {gather*} \frac {\text {ArcTan}\left (\frac {1}{2} \tanh \left (\frac {1}{2} (c+d x)\right )\right )}{2 d} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 2738
Rubi steps
\begin {align*} \int \frac {1}{3+5 \cosh (c+d x)} \, dx &=-\frac {(2 i) \text {Subst}\left (\int \frac {1}{8-2 x^2} \, dx,x,\tan \left (\frac {1}{2} (i c+i d x)\right )\right )}{d}\\ &=\frac {\tan ^{-1}\left (\frac {1}{2} \tanh \left (\frac {1}{2} (c+d x)\right )\right )}{2 d}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 23, normalized size = 1.05 \begin {gather*} -\frac {\text {ArcTan}\left (2 \coth \left (\frac {c}{2}+\frac {d x}{2}\right )\right )}{2 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.87, size = 18, normalized size = 0.82
method | result | size |
derivativedivides | \(\frac {\arctan \left (\frac {\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{2}\right )}{2 d}\) | \(18\) |
default | \(\frac {\arctan \left (\frac {\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{2}\right )}{2 d}\) | \(18\) |
risch | \(\frac {i \ln \left ({\mathrm e}^{d x +c}+\frac {3}{5}+\frac {4 i}{5}\right )}{4 d}-\frac {i \ln \left ({\mathrm e}^{d x +c}+\frac {3}{5}-\frac {4 i}{5}\right )}{4 d}\) | \(36\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.48, size = 19, normalized size = 0.86 \begin {gather*} -\frac {\arctan \left (\frac {5}{4} \, e^{\left (-d x - c\right )} + \frac {3}{4}\right )}{2 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.44, size = 24, normalized size = 1.09 \begin {gather*} \frac {\arctan \left (\frac {5}{4} \, \cosh \left (d x + c\right ) + \frac {5}{4} \, \sinh \left (d x + c\right ) + \frac {3}{4}\right )}{2 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.44, size = 24, normalized size = 1.09 \begin {gather*} \begin {cases} \frac {\operatorname {atan}{\left (\frac {\tanh {\left (\frac {c}{2} + \frac {d x}{2} \right )}}{2} \right )}}{2 d} & \text {for}\: d \neq 0 \\\frac {x}{5 \cosh {\left (c \right )} + 3} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.41, size = 16, normalized size = 0.73 \begin {gather*} \frac {\arctan \left (\frac {5}{4} \, e^{\left (d x + c\right )} + \frac {3}{4}\right )}{2 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.13, size = 34, normalized size = 1.55 \begin {gather*} \frac {\mathrm {atan}\left (\frac {3\,\sqrt {d^2}+5\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,\sqrt {d^2}}{4\,d}\right )}{2\,\sqrt {d^2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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