Optimal. Leaf size=26 \[ \frac {(a \cosh (c+d x)+a \sinh (c+d x))^2}{2 d} \]
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Rubi [A]
time = 0.01, antiderivative size = 26, normalized size of antiderivative = 1.00, number of steps
used = 1, number of rules used = 1, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {3150}
\begin {gather*} \frac {(a \sinh (c+d x)+a \cosh (c+d x))^2}{2 d} \end {gather*}
Antiderivative was successfully verified.
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Rule 3150
Rubi steps
\begin {align*} \int (a \cosh (c+d x)+a \sinh (c+d x))^2 \, dx &=\frac {(a \cosh (c+d x)+a \sinh (c+d x))^2}{2 d}\\ \end {align*}
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Mathematica [A]
time = 0.04, size = 25, normalized size = 0.96 \begin {gather*} \frac {a^2 (\cosh (c+d x)+\sinh (c+d x))^2}{2 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(69\) vs.
\(2(24)=48\).
time = 3.43, size = 70, normalized size = 2.69
method | result | size |
risch | \(\frac {a^{2} {\mathrm e}^{2 d x +2 c}}{2 d}\) | \(18\) |
gosper | \(\frac {a^{2} \left (\cosh \left (d x +c \right )+\sinh \left (d x +c \right )\right )^{2}}{2 d}\) | \(24\) |
derivativedivides | \(\frac {a^{2} \left (\frac {\cosh \left (d x +c \right ) \sinh \left (d x +c \right )}{2}-\frac {d x}{2}-\frac {c}{2}\right )+a^{2} \left (\cosh ^{2}\left (d x +c \right )\right )+a^{2} \left (\frac {\cosh \left (d x +c \right ) \sinh \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )}{d}\) | \(70\) |
default | \(\frac {a^{2} \left (\frac {\cosh \left (d x +c \right ) \sinh \left (d x +c \right )}{2}-\frac {d x}{2}-\frac {c}{2}\right )+a^{2} \left (\cosh ^{2}\left (d x +c \right )\right )+a^{2} \left (\frac {\cosh \left (d x +c \right ) \sinh \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )}{d}\) | \(70\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 88 vs.
\(2 (24) = 48\).
time = 0.26, size = 88, normalized size = 3.38 \begin {gather*} \frac {1}{8} \, a^{2} {\left (4 \, x + \frac {e^{\left (2 \, d x + 2 \, c\right )}}{d} - \frac {e^{\left (-2 \, d x - 2 \, c\right )}}{d}\right )} - \frac {1}{8} \, a^{2} {\left (4 \, x - \frac {e^{\left (2 \, d x + 2 \, c\right )}}{d} + \frac {e^{\left (-2 \, d x - 2 \, c\right )}}{d}\right )} + \frac {a^{2} \cosh \left (d x + c\right )^{2}}{d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.35, size = 43, normalized size = 1.65 \begin {gather*} \frac {a^{2} \cosh \left (d x + c\right ) + a^{2} \sinh \left (d x + c\right )}{2 \, {\left (d \cosh \left (d x + c\right ) - d \sinh \left (d x + c\right )\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 44 vs.
\(2 (20) = 40\).
time = 0.08, size = 44, normalized size = 1.69 \begin {gather*} \begin {cases} \frac {a^{2} \sinh {\left (c + d x \right )} \cosh {\left (c + d x \right )}}{d} + \frac {a^{2} \cosh ^{2}{\left (c + d x \right )}}{d} & \text {for}\: d \neq 0 \\x \left (a \sinh {\left (c \right )} + a \cosh {\left (c \right )}\right )^{2} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.40, size = 17, normalized size = 0.65 \begin {gather*} \frac {a^{2} e^{\left (2 \, d x + 2 \, c\right )}}{2 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.07, size = 17, normalized size = 0.65 \begin {gather*} \frac {a^2\,{\mathrm {e}}^{2\,c+2\,d\,x}}{2\,d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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