Optimal. Leaf size=15 \[ \frac {\sinh ^2(a+b x)}{2 b} \]
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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 = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {2644, 30}
\begin {gather*} \frac {\sinh ^2(a+b x)}{2 b} \end {gather*}
Antiderivative was successfully verified.
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Rule 30
Rule 2644
Rubi steps
\begin {align*} \int \cosh (a+b x) \sinh (a+b x) \, dx &=-\frac {\text {Subst}(\int x \, dx,x,i \sinh (a+b x))}{b}\\ &=\frac {\sinh ^2(a+b x)}{2 b}\\ \end {align*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(37\) vs. \(2(15)=30\).
time = 0.01, size = 37, normalized size = 2.47 \begin {gather*} \frac {1}{2} \left (\frac {\cosh (2 a) \cosh (2 b x)}{2 b}+\frac {\sinh (2 a) \sinh (2 b x)}{2 b}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.36, size = 14, normalized size = 0.93
method | result | size |
derivativedivides | \(\frac {\cosh ^{2}\left (b x +a \right )}{2 b}\) | \(14\) |
default | \(\frac {\cosh ^{2}\left (b x +a \right )}{2 b}\) | \(14\) |
risch | \(\frac {{\mathrm e}^{2 b x +2 a}}{8 b}+\frac {{\mathrm e}^{-2 b x -2 a}}{8 b}\) | \(30\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 13, normalized size = 0.87 \begin {gather*} \frac {\cosh \left (b x + a\right )^{2}}{2 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 22, normalized size = 1.47 \begin {gather*} \frac {\cosh \left (b x + a\right )^{2} + \sinh \left (b x + a\right )^{2}}{4 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.07, size = 19, normalized size = 1.27 \begin {gather*} \begin {cases} \frac {\cosh ^{2}{\left (a + b x \right )}}{2 b} & \text {for}\: b \neq 0 \\x \sinh {\left (a \right )} \cosh {\left (a \right )} & \text {otherwise} \end {cases} \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. 29 vs.
\(2 (13) = 26\).
time = 0.39, size = 29, normalized size = 1.93 \begin {gather*} \frac {e^{\left (2 \, b x + 2 \, a\right )}}{8 \, b} + \frac {e^{\left (-2 \, b x - 2 \, a\right )}}{8 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.07, size = 13, normalized size = 0.87 \begin {gather*} \frac {{\mathrm {cosh}\left (a+b\,x\right )}^2}{2\,b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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