Optimal. Leaf size=15 \[ a x+\frac {b \cosh (c+d x)}{d} \]
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Rubi [A]
time = 0.01, antiderivative size = 15, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 1, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {2718}
\begin {gather*} a x+\frac {b \cosh (c+d x)}{d} \end {gather*}
Antiderivative was successfully verified.
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Rule 2718
Rubi steps
\begin {align*} \int (a+b \sinh (c+d x)) \, dx &=a x+b \int \sinh (c+d x) \, dx\\ &=a x+\frac {b \cosh (c+d x)}{d}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 26, normalized size = 1.73 \begin {gather*} a x+\frac {b \cosh (c) \cosh (d x)}{d}+\frac {b \sinh (c) \sinh (d x)}{d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.26, size = 16, normalized size = 1.07
method | result | size |
default | \(a x +\frac {b \cosh \left (d x +c \right )}{d}\) | \(16\) |
derivativedivides | \(\frac {\left (d x +c \right ) a +b \cosh \left (d x +c \right )}{d}\) | \(21\) |
risch | \(a x +\frac {b \,{\mathrm e}^{d x +c}}{2 d}+\frac {b \,{\mathrm e}^{-d x -c}}{2 d}\) | \(32\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 15, normalized size = 1.00 \begin {gather*} a x + \frac {b \cosh \left (d x + c\right )}{d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.39, size = 17, normalized size = 1.13 \begin {gather*} \frac {a d x + b \cosh \left (d x + c\right )}{d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.04, size = 17, normalized size = 1.13 \begin {gather*} a x + b \left (\begin {cases} \frac {\cosh {\left (c + d x \right )}}{d} & \text {for}\: d \neq 0 \\x \sinh {\left (c \right )} & \text {otherwise} \end {cases}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 31 vs.
\(2 (15) = 30\).
time = 0.43, size = 31, normalized size = 2.07 \begin {gather*} a x + \frac {1}{2} \, b {\left (\frac {e^{\left (d x + c\right )}}{d} + \frac {e^{\left (-d x - c\right )}}{d}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.43, size = 15, normalized size = 1.00 \begin {gather*} a\,x+\frac {b\,\mathrm {cosh}\left (c+d\,x\right )}{d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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