Optimal. Leaf size=28 \[ \frac {(c+d x) \cosh (a+b x)}{b}-\frac {d \sinh (a+b x)}{b^2} \]
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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, 2637} \[ \frac {(c+d x) \cosh (a+b x)}{b}-\frac {d \sinh (a+b x)}{b^2} \]
Antiderivative was successfully verified.
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Rule 2637
Rule 3296
Rubi steps
\begin {align*} \int (c+d x) \sinh (a+b x) \, dx &=\frac {(c+d x) \cosh (a+b x)}{b}-\frac {d \int \cosh (a+b x) \, dx}{b}\\ &=\frac {(c+d x) \cosh (a+b x)}{b}-\frac {d \sinh (a+b x)}{b^2}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 27, normalized size = 0.96 \[ \frac {b (c+d x) \cosh (a+b x)-d \sinh (a+b x)}{b^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.39, size = 29, normalized size = 1.04 \[ \frac {{\left (b d x + b c\right )} \cosh \left (b x + a\right ) - d \sinh \left (b x + a\right )}{b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.21, size = 46, normalized size = 1.64 \[ \frac {{\left (b d x + b c - d\right )} e^{\left (b x + a\right )}}{2 \, b^{2}} + \frac {{\left (b d x + b c + d\right )} e^{\left (-b x - a\right )}}{2 \, b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 53, normalized size = 1.89 \[ \frac {\frac {d \left (\left (b x +a \right ) \cosh \left (b x +a \right )-\sinh \left (b x +a \right )\right )}{b}-\frac {d a \cosh \left (b x +a \right )}{b}+c \cosh \left (b x +a \right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.43, size = 68, normalized size = 2.43 \[ \frac {c e^{\left (b x + a\right )}}{2 \, b} + \frac {{\left (b x e^{a} - e^{a}\right )} d e^{\left (b x\right )}}{2 \, b^{2}} + \frac {c e^{\left (-b x - a\right )}}{2 \, b} + \frac {{\left (b x + 1\right )} d e^{\left (-b x - a\right )}}{2 \, b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.10, size = 35, normalized size = 1.25 \[ \frac {c\,\mathrm {cosh}\left (a+b\,x\right )+d\,x\,\mathrm {cosh}\left (a+b\,x\right )}{b}-\frac {d\,\mathrm {sinh}\left (a+b\,x\right )}{b^2} \]
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 \[ \begin {cases} \frac {c \cosh {\left (a + b x \right )}}{b} + \frac {d x \cosh {\left (a + b x \right )}}{b} - \frac {d \sinh {\left (a + b x \right )}}{b^{2}} & \text {for}\: b \neq 0 \\\left (c x + \frac {d x^{2}}{2}\right ) \sinh {\relax (a )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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