Optimal. Leaf size=27 \[ -\frac {1}{2} x \cosh (a-c)+\frac {\sinh (a+c+2 b x)}{4 b} \]
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Rubi [A]
time = 0.02, 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 = {5732, 2717}
\begin {gather*} \frac {\sinh (a+2 b x+c)}{4 b}-\frac {1}{2} x \cosh (a-c) \end {gather*}
Antiderivative was successfully verified.
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Rule 2717
Rule 5732
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 \begin {gather*} \frac {-2 b x \cosh (a-c)+\sinh (a+c+2 b x)}{4 b} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.70, size = 24, normalized size = 0.89
method | result | size |
default | \(-\frac {x \cosh \left (a -c \right )}{2}+\frac {\sinh \left (2 b x +a +c \right )}{4 b}\) | \(24\) |
risch | \(-\frac {x \,{\mathrm e}^{-a -c} {\mathrm e}^{2 a}}{4}-\frac {x \,{\mathrm e}^{-a -c} {\mathrm e}^{2 c}}{4}+\frac {{\mathrm e}^{2 b x +a +c}}{8 b}-\frac {{\mathrm e}^{-2 b x -a -c}}{8 b}\) | \(62\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 58 vs.
\(2 (23) = 46\).
time = 0.27, size = 58, normalized size = 2.15 \begin {gather*} -\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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 87 vs.
\(2 (23) = 46\).
time = 0.35, size = 87, normalized size = 3.22 \begin {gather*} -\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 )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 58 vs.
\(2 (20) = 40\).
time = 0.18, size = 58, normalized size = 2.15 \begin {gather*} \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 {\left (a \right )} \sinh {\left (c \right )} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 71 vs.
\(2 (23) = 46\).
time = 0.39, size = 71, normalized size = 2.63 \begin {gather*} -\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} \end {gather*}
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 \begin {gather*} \frac {\mathrm {sinh}\left (a+c+2\,b\,x\right )}{4\,b}-\frac {x\,\mathrm {cosh}\left (a-c\right )}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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