Optimal. Leaf size=28 \[ \frac {(a+b x) \sinh (c+d x)}{d}-\frac {b \cosh (c+d x)}{d^2} \]
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Rubi [A] time = 0.02, antiderivative size = 28, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {3296, 2638} \[ \frac {(a+b x) \sinh (c+d x)}{d}-\frac {b \cosh (c+d x)}{d^2} \]
Antiderivative was successfully verified.
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Rule 2638
Rule 3296
Rubi steps
\begin {align*} \int (a+b x) \cosh (c+d x) \, dx &=\frac {(a+b x) \sinh (c+d x)}{d}-\frac {b \int \sinh (c+d x) \, dx}{d}\\ &=-\frac {b \cosh (c+d x)}{d^2}+\frac {(a+b x) \sinh (c+d x)}{d}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 27, normalized size = 0.96 \[ \frac {d (a+b x) \sinh (c+d x)-b \cosh (c+d x)}{d^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.44, size = 30, normalized size = 1.07 \[ -\frac {b \cosh \left (d x + c\right ) - {\left (b d x + a d\right )} \sinh \left (d x + c\right )}{d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.11, size = 46, normalized size = 1.64 \[ \frac {{\left (b d x + a d - b\right )} e^{\left (d x + c\right )}}{2 \, d^{2}} - \frac {{\left (b d x + a d + b\right )} e^{\left (-d x - c\right )}}{2 \, d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 53, normalized size = 1.89 \[ \frac {\frac {b \left (\left (d x +c \right ) \sinh \left (d x +c \right )-\cosh \left (d x +c \right )\right )}{d}-\frac {b c \sinh \left (d x +c \right )}{d}+a \sinh \left (d x +c \right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.32, size = 68, normalized size = 2.43 \[ \frac {a e^{\left (d x + c\right )}}{2 \, d} + \frac {{\left (d x e^{c} - e^{c}\right )} b e^{\left (d x\right )}}{2 \, d^{2}} - \frac {{\left (d x + 1\right )} b e^{\left (-d x - c\right )}}{2 \, d^{2}} - \frac {a e^{\left (-d x - c\right )}}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.06, size = 35, normalized size = 1.25 \[ \frac {a\,\mathrm {sinh}\left (c+d\,x\right )+b\,x\,\mathrm {sinh}\left (c+d\,x\right )}{d}-\frac {b\,\mathrm {cosh}\left (c+d\,x\right )}{d^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.21, size = 46, normalized size = 1.64 \[ \begin {cases} \frac {a \sinh {\left (c + d x \right )}}{d} + \frac {b x \sinh {\left (c + d x \right )}}{d} - \frac {b \cosh {\left (c + d x \right )}}{d^{2}} & \text {for}\: d \neq 0 \\\left (a x + \frac {b x^{2}}{2}\right ) \cosh {\relax (c )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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