Optimal. Leaf size=46 \[ \frac {\sinh (a+b x) \cosh ^3(a+b x)}{4 b}+\frac {3 \sinh (a+b x) \cosh (a+b x)}{8 b}+\frac {3 x}{8} \]
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Rubi [A] time = 0.02, antiderivative size = 46, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {2635, 8} \[ \frac {\sinh (a+b x) \cosh ^3(a+b x)}{4 b}+\frac {3 \sinh (a+b x) \cosh (a+b x)}{8 b}+\frac {3 x}{8} \]
Antiderivative was successfully verified.
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Rule 8
Rule 2635
Rubi steps
\begin {align*} \int \cosh ^4(a+b x) \, dx &=\frac {\cosh ^3(a+b x) \sinh (a+b x)}{4 b}+\frac {3}{4} \int \cosh ^2(a+b x) \, dx\\ &=\frac {3 \cosh (a+b x) \sinh (a+b x)}{8 b}+\frac {\cosh ^3(a+b x) \sinh (a+b x)}{4 b}+\frac {3 \int 1 \, dx}{8}\\ &=\frac {3 x}{8}+\frac {3 \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 = 0.05, size = 33, normalized size = 0.72 \[ \frac {12 (a+b x)+8 \sinh (2 (a+b x))+\sinh (4 (a+b x))}{32 b} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.81, size = 49, normalized size = 1.07 \[ \frac {\cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{3} + 3 \, b x + {\left (\cosh \left (b x + a\right )^{3} + 4 \, \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )}{8 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.12, size = 60, normalized size = 1.30 \[ \frac {3}{8} \, x + \frac {e^{\left (4 \, b x + 4 \, a\right )}}{64 \, b} + \frac {e^{\left (2 \, b x + 2 \, a\right )}}{8 \, b} - \frac {e^{\left (-2 \, b x - 2 \, a\right )}}{8 \, b} - \frac {e^{\left (-4 \, b x - 4 \, a\right )}}{64 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.21, size = 39, normalized size = 0.85 \[ \frac {\left (\frac {\left (\cosh ^{3}\left (b x +a \right )\right )}{4}+\frac {3 \cosh \left (b x +a \right )}{8}\right ) \sinh \left (b x +a \right )+\frac {3 b x}{8}+\frac {3 a}{8}}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 60, normalized size = 1.30 \[ \frac {3}{8} \, x + \frac {e^{\left (4 \, b x + 4 \, a\right )}}{64 \, b} + \frac {e^{\left (2 \, b x + 2 \, a\right )}}{8 \, b} - \frac {e^{\left (-2 \, b x - 2 \, a\right )}}{8 \, b} - \frac {e^{\left (-4 \, b x - 4 \, a\right )}}{64 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.08, size = 31, normalized size = 0.67 \[ \frac {3\,x}{8}+\frac {\frac {\mathrm {sinh}\left (2\,a+2\,b\,x\right )}{4}+\frac {\mathrm {sinh}\left (4\,a+4\,b\,x\right )}{32}}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.85, size = 95, normalized size = 2.07 \[ \begin {cases} \frac {3 x \sinh ^{4}{\left (a + b x \right )}}{8} - \frac {3 x \sinh ^{2}{\left (a + b x \right )} \cosh ^{2}{\left (a + b x \right )}}{4} + \frac {3 x \cosh ^{4}{\left (a + b x \right )}}{8} - \frac {3 \sinh ^{3}{\left (a + b x \right )} \cosh {\left (a + b x \right )}}{8 b} + \frac {5 \sinh {\left (a + b x \right )} \cosh ^{3}{\left (a + b x \right )}}{8 b} & \text {for}\: b \neq 0 \\x \cosh ^{4}{\relax (a )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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