Optimal. Leaf size=63 \[ \frac {1}{8} x \left (8 a^2-b^2\right )+\frac {a b \cosh (2 c+2 d x)}{2 d}+\frac {b^2 \sinh (2 c+2 d x) \cosh (2 c+2 d x)}{16 d} \]
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Rubi [A] time = 0.04, antiderivative size = 63, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {2666, 2644} \[ \frac {1}{8} x \left (8 a^2-b^2\right )+\frac {a b \cosh (2 c+2 d x)}{2 d}+\frac {b^2 \sinh (2 c+2 d x) \cosh (2 c+2 d x)}{16 d} \]
Antiderivative was successfully verified.
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Rule 2644
Rule 2666
Rubi steps
\begin {align*} \int (a+b \cosh (c+d x) \sinh (c+d x))^2 \, dx &=\int \left (a+\frac {1}{2} b \sinh (2 c+2 d x)\right )^2 \, dx\\ &=\frac {1}{8} \left (8 a^2-b^2\right ) x+\frac {a b \cosh (2 c+2 d x)}{2 d}+\frac {b^2 \cosh (2 c+2 d x) \sinh (2 c+2 d x)}{16 d}\\ \end {align*}
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Mathematica [A] time = 0.12, size = 50, normalized size = 0.79 \[ \frac {4 \left (8 a^2-b^2\right ) (c+d x)+16 a b \cosh (2 (c+d x))+b^2 \sinh (4 (c+d x))}{32 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.49, size = 80, normalized size = 1.27 \[ \frac {b^{2} \cosh \left (d x + c\right )^{3} \sinh \left (d x + c\right ) + b^{2} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3} + 4 \, a b \cosh \left (d x + c\right )^{2} + 4 \, a b \sinh \left (d x + c\right )^{2} + {\left (8 \, a^{2} - b^{2}\right )} d x}{8 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.12, size = 81, normalized size = 1.29 \[ \frac {1}{8} \, {\left (8 \, a^{2} - b^{2}\right )} x + \frac {b^{2} e^{\left (4 \, d x + 4 \, c\right )}}{64 \, d} + \frac {a b e^{\left (2 \, d x + 2 \, c\right )}}{4 \, d} + \frac {a b e^{\left (-2 \, d x - 2 \, c\right )}}{4 \, d} - \frac {b^{2} e^{\left (-4 \, d x - 4 \, c\right )}}{64 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.43, size = 68, normalized size = 1.08 \[ \frac {b^{2} \left (\frac {\sinh \left (d x +c \right ) \left (\cosh ^{3}\left (d x +c \right )\right )}{4}-\frac {\cosh \left (d x +c \right ) \sinh \left (d x +c \right )}{8}-\frac {d x}{8}-\frac {c}{8}\right )+a \left (\cosh ^{2}\left (d x +c \right )\right ) b +a^{2} \left (d x +c \right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.39, size = 63, normalized size = 1.00 \[ a^{2} x - \frac {1}{64} \, b^{2} {\left (\frac {8 \, {\left (d x + c\right )}}{d} - \frac {e^{\left (4 \, d x + 4 \, c\right )}}{d} + \frac {e^{\left (-4 \, d x - 4 \, c\right )}}{d}\right )} + \frac {a b \cosh \left (d x + c\right )^{2}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.14, size = 44, normalized size = 0.70 \[ \frac {\frac {\mathrm {sinh}\left (4\,c+4\,d\,x\right )\,b^2}{32}+\frac {a\,\mathrm {cosh}\left (2\,c+2\,d\,x\right )\,b}{2}}{d}+a^2\,x-\frac {b^2\,x}{8} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.93, size = 129, normalized size = 2.05 \[ \begin {cases} a^{2} x + \frac {a b \sinh ^{2}{\left (c + d x \right )}}{d} - \frac {b^{2} x \sinh ^{4}{\left (c + d x \right )}}{8} + \frac {b^{2} x \sinh ^{2}{\left (c + d x \right )} \cosh ^{2}{\left (c + d x \right )}}{4} - \frac {b^{2} x \cosh ^{4}{\left (c + d x \right )}}{8} + \frac {b^{2} \sinh ^{3}{\left (c + d x \right )} \cosh {\left (c + d x \right )}}{8 d} + \frac {b^{2} \sinh {\left (c + d x \right )} \cosh ^{3}{\left (c + d x \right )}}{8 d} & \text {for}\: d \neq 0 \\x \left (a + b \sinh {\relax (c )} \cosh {\relax (c )}\right )^{2} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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