Optimal. Leaf size=20 \[ a x+\frac {b \sinh ^2(c+d x)}{2 d} \]
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Rubi [A] time = 0.02, antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {2564, 30} \[ a x+\frac {b \sinh ^2(c+d x)}{2 d} \]
Antiderivative was successfully verified.
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Rule 30
Rule 2564
Rubi steps
\begin {align*} \int (a+b \cosh (c+d x) \sinh (c+d x)) \, dx &=a x+b \int \cosh (c+d x) \sinh (c+d x) \, dx\\ &=a x-\frac {b \operatorname {Subst}(\int x \, dx,x,i \sinh (c+d x))}{d}\\ &=a x+\frac {b \sinh ^2(c+d x)}{2 d}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 38, normalized size = 1.90 \[ a x+\frac {b \sinh (2 c) \sinh (2 d x)}{4 d}+\frac {b \cosh (2 c) \cosh (2 d x)}{4 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.56, size = 31, normalized size = 1.55 \[ \frac {4 \, a d x + b \cosh \left (d x + c\right )^{2} + b \sinh \left (d x + c\right )^{2}}{4 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.13, size = 34, normalized size = 1.70 \[ a x + \frac {1}{8} \, b {\left (\frac {e^{\left (2 \, d x + 2 \, c\right )}}{d} + \frac {e^{\left (-2 \, d x - 2 \, c\right )}}{d}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 19, normalized size = 0.95 \[ a x +\frac {b \left (\cosh ^{2}\left (d x +c \right )\right )}{2 d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.30, size = 18, normalized size = 0.90 \[ a x + \frac {b \cosh \left (d x + c\right )^{2}}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.64, size = 18, normalized size = 0.90 \[ a\,x+\frac {b\,{\mathrm {cosh}\left (c+d\,x\right )}^2}{2\,d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.19, size = 24, normalized size = 1.20 \[ a x + b \left (\begin {cases} \frac {\sinh ^{2}{\left (c + d x \right )}}{2 d} & \text {for}\: d \neq 0 \\x \sinh {\relax (c )} \cosh {\relax (c )} & \text {otherwise} \end {cases}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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