Optimal. Leaf size=27 \[ \frac {\sinh (a+2 b x+c)}{4 b}-\frac {1}{2} x \cosh (a-c) \]
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Rubi [A] time = 0.03, antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {5613, 2637} \[ \frac {\sinh (a+2 b x+c)}{4 b}-\frac {1}{2} x \cosh (a-c) \]
Antiderivative was successfully verified.
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Rule 2637
Rule 5613
Rubi steps
\begin {align*} \int \sinh (a+b x) \sinh (c+b x) \, dx &=\int \left (-\frac {1}{2} \cosh (a-c)+\frac {1}{2} \cosh (a+c+2 b x)\right ) \, dx\\ &=-\frac {1}{2} x \cosh (a-c)+\frac {1}{2} \int \cosh (a+c+2 b x) \, dx\\ &=-\frac {1}{2} x \cosh (a-c)+\frac {\sinh (a+c+2 b x)}{4 b}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 26, normalized size = 0.96 \[ \frac {\sinh (a+2 b x+c)-2 b x \cosh (a-c)}{4 b} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.46, size = 87, normalized size = 3.22 \[ -\frac {2 \, b x \cosh \left (-a + c\right ) - 2 \, \cosh \left (b x + c\right ) \cosh \left (-a + c\right ) \sinh \left (b x + c\right ) + \cosh \left (b x + c\right )^{2} \sinh \left (-a + c\right ) + \sinh \left (b x + c\right )^{2} \sinh \left (-a + c\right )}{4 \, {\left (b \cosh \left (-a + c\right )^{2} - b \sinh \left (-a + c\right )^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.12, size = 71, normalized size = 2.63 \[ -\frac {2 \, b x {\left (e^{\left (2 \, a\right )} + e^{\left (2 \, c\right )}\right )} e^{\left (-a - c\right )} - {\left (e^{\left (2 \, b x + 2 \, a\right )} + e^{\left (2 \, b x + 2 \, c\right )} - 1\right )} e^{\left (-2 \, b x - a - c\right )} - e^{\left (2 \, b x + a + c\right )}}{8 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.09, size = 24, normalized size = 0.89 \[ -\frac {x \cosh \left (a -c \right )}{2}+\frac {\sinh \left (2 b x +a +c \right )}{4 b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.32, size = 58, normalized size = 2.15 \[ -\frac {{\left (b x + a\right )} {\left (e^{\left (2 \, a\right )} + e^{\left (2 \, c\right )}\right )} e^{\left (-a - c\right )}}{4 \, b} + \frac {e^{\left (2 \, b x + a + c\right )}}{8 \, b} - \frac {e^{\left (-2 \, b x - a - c\right )}}{8 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.16, size = 23, normalized size = 0.85 \[ \frac {\mathrm {sinh}\left (a+c+2\,b\,x\right )}{4\,b}-\frac {x\,\mathrm {cosh}\left (a-c\right )}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.74, size = 58, normalized size = 2.15 \[ \begin {cases} \frac {x \sinh {\left (a + b x \right )} \sinh {\left (b x + c \right )}}{2} - \frac {x \cosh {\left (a + b x \right )} \cosh {\left (b x + c \right )}}{2} + \frac {\sinh {\left (a + b x \right )} \cosh {\left (b x + c \right )}}{2 b} & \text {for}\: b \neq 0 \\x \sinh {\relax (a )} \sinh {\relax (c )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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