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 = {2564, 30} \[ \frac {\sinh ^2(a+b x)}{2 b} \]
Antiderivative was successfully verified.
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Rule 30
Rule 2564
Rubi steps
\begin {align*} \int \cosh (a+b x) \sinh (a+b x) \, dx &=-\frac {\operatorname {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] time = 0.01, size = 37, normalized size = 2.47 \[ \frac {1}{2} \left (\frac {\sinh (2 a) \sinh (2 b x)}{2 b}+\frac {\cosh (2 a) \cosh (2 b x)}{2 b}\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.60, size = 22, normalized size = 1.47 \[ \frac {\cosh \left (b x + a\right )^{2} + \sinh \left (b x + a\right )^{2}}{4 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.13, size = 29, normalized size = 1.93 \[ \frac {e^{\left (2 \, b x + 2 \, a\right )}}{8 \, b} + \frac {e^{\left (-2 \, b x - 2 \, a\right )}}{8 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 14, normalized size = 0.93 \[ \frac {\cosh ^{2}\left (b x +a \right )}{2 b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.38, size = 13, normalized size = 0.87 \[ \frac {\cosh \left (b x + a\right )^{2}}{2 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.43, size = 13, normalized size = 0.87 \[ \frac {{\mathrm {cosh}\left (a+b\,x\right )}^2}{2\,b} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.18, size = 19, normalized size = 1.27 \[ \begin {cases} \frac {\sinh ^{2}{\left (a + b x \right )}}{2 b} & \text {for}\: b \neq 0 \\x \sinh {\relax (a )} \cosh {\relax (a )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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