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 = 0.0199479, antiderivative size = 28, normalized size of antiderivative = 1., 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 3296
Rule 2637
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*}
Mathematica [A] time = 0.0625134, 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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Maple [A] time = 0.004, size = 53, normalized size = 1.9 \begin{align*}{\frac{1}{b} \left ({\frac{d \left ( \left ( bx+a \right ) \cosh \left ( bx+a \right ) -\sinh \left ( bx+a \right ) \right ) }{b}}-{\frac{da\cosh \left ( bx+a \right ) }{b}}+c\cosh \left ( bx+a \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.08766, size = 92, normalized size = 3.29 \begin{align*} \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}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.52931, size = 72, normalized size = 2.57 \begin{align*} \frac{{\left (b d x + b c\right )} \cosh \left (b x + a\right ) - d \sinh \left (b x + a\right )}{b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.265362, size = 46, normalized size = 1.64 \begin{align*} \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{\left (a \right )} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.14771, size = 62, normalized size = 2.21 \begin{align*} \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}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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