Optimal. Leaf size=46 \[ -\frac {x}{8}-\frac {\cosh (a+b x) \sinh (a+b x)}{8 b}+\frac {\cosh ^3(a+b x) \sinh (a+b x)}{4 b} \]
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Rubi [A]
time = 0.03, antiderivative size = 46, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {2648, 2715, 8}
\begin {gather*} \frac {\sinh (a+b x) \cosh ^3(a+b x)}{4 b}-\frac {\sinh (a+b x) \cosh (a+b x)}{8 b}-\frac {x}{8} \end {gather*}
Antiderivative was successfully verified.
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Rule 8
Rule 2648
Rule 2715
Rubi steps
\begin {align*} \int \cosh ^2(a+b x) \sinh ^2(a+b x) \, dx &=\frac {\cosh ^3(a+b x) \sinh (a+b x)}{4 b}-\frac {1}{4} \int \cosh ^2(a+b x) \, dx\\ &=-\frac {\cosh (a+b x) \sinh (a+b x)}{8 b}+\frac {\cosh ^3(a+b x) \sinh (a+b x)}{4 b}-\frac {\int 1 \, dx}{8}\\ &=-\frac {x}{8}-\frac {\cosh (a+b x) \sinh (a+b x)}{8 b}+\frac {\cosh ^3(a+b x) \sinh (a+b x)}{4 b}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 23, normalized size = 0.50 \begin {gather*} \frac {-4 (a+b x)+\sinh (4 (a+b x))}{32 b} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.60, size = 43, normalized size = 0.93
method | result | size |
risch | \(-\frac {x}{8}+\frac {{\mathrm e}^{4 b x +4 a}}{64 b}-\frac {{\mathrm e}^{-4 b x -4 a}}{64 b}\) | \(33\) |
derivativedivides | \(\frac {\frac {\left (\cosh ^{3}\left (b x +a \right )\right ) \sinh \left (b x +a \right )}{4}-\frac {\cosh \left (b x +a \right ) \sinh \left (b x +a \right )}{8}-\frac {b x}{8}-\frac {a}{8}}{b}\) | \(43\) |
default | \(\frac {\frac {\left (\cosh ^{3}\left (b x +a \right )\right ) \sinh \left (b x +a \right )}{4}-\frac {\cosh \left (b x +a \right ) \sinh \left (b x +a \right )}{8}-\frac {b x}{8}-\frac {a}{8}}{b}\) | \(43\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.26, size = 39, normalized size = 0.85 \begin {gather*} -\frac {b x + a}{8 \, b} + \frac {e^{\left (4 \, b x + 4 \, a\right )}}{64 \, b} - \frac {e^{\left (-4 \, b x - 4 \, a\right )}}{64 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.39, size = 40, normalized size = 0.87 \begin {gather*} \frac {\cosh \left (b x + a\right )^{3} \sinh \left (b x + a\right ) + \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{3} - b x}{8 \, b} \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. 92 vs.
\(2 (37) = 74\).
time = 0.18, size = 92, normalized size = 2.00 \begin {gather*} \begin {cases} - \frac {x \sinh ^{4}{\left (a + b x \right )}}{8} + \frac {x \sinh ^{2}{\left (a + b x \right )} \cosh ^{2}{\left (a + b x \right )}}{4} - \frac {x \cosh ^{4}{\left (a + b x \right )}}{8} + \frac {\sinh ^{3}{\left (a + b x \right )} \cosh {\left (a + b x \right )}}{8 b} + \frac {\sinh {\left (a + b x \right )} \cosh ^{3}{\left (a + b x \right )}}{8 b} & \text {for}\: b \neq 0 \\x \sinh ^{2}{\left (a \right )} \cosh ^{2}{\left (a \right )} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.39, size = 32, normalized size = 0.70 \begin {gather*} -\frac {1}{8} \, x + \frac {e^{\left (4 \, b x + 4 \, a\right )}}{64 \, b} - \frac {e^{\left (-4 \, b x - 4 \, a\right )}}{64 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.11, size = 18, normalized size = 0.39 \begin {gather*} \frac {\mathrm {sinh}\left (4\,a+4\,b\,x\right )}{32\,b}-\frac {x}{8} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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