Optimal. Leaf size=61 \[ \frac {(3 a+4 b) \sinh (c+d x) \cosh (c+d x)}{8 d}+\frac {1}{8} x (3 a+4 b)+\frac {a \sinh (c+d x) \cosh ^3(c+d x)}{4 d} \]
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Rubi [A] time = 0.05, antiderivative size = 61, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {4045, 2635, 8} \[ \frac {(3 a+4 b) \sinh (c+d x) \cosh (c+d x)}{8 d}+\frac {1}{8} x (3 a+4 b)+\frac {a \sinh (c+d x) \cosh ^3(c+d x)}{4 d} \]
Antiderivative was successfully verified.
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Rule 8
Rule 2635
Rule 4045
Rubi steps
\begin {align*} \int \cosh ^4(c+d x) \left (a+b \text {sech}^2(c+d x)\right ) \, dx &=\frac {a \cosh ^3(c+d x) \sinh (c+d x)}{4 d}+\frac {1}{4} (3 a+4 b) \int \cosh ^2(c+d x) \, dx\\ &=\frac {(3 a+4 b) \cosh (c+d x) \sinh (c+d x)}{8 d}+\frac {a \cosh ^3(c+d x) \sinh (c+d x)}{4 d}+\frac {1}{8} (3 a+4 b) \int 1 \, dx\\ &=\frac {1}{8} (3 a+4 b) x+\frac {(3 a+4 b) \cosh (c+d x) \sinh (c+d x)}{8 d}+\frac {a \cosh ^3(c+d x) \sinh (c+d x)}{4 d}\\ \end {align*}
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Mathematica [A] time = 0.09, size = 45, normalized size = 0.74 \[ \frac {4 (3 a+4 b) (c+d x)+8 (a+b) \sinh (2 (c+d x))+a \sinh (4 (c+d x))}{32 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.40, size = 61, normalized size = 1.00 \[ \frac {a \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3} + {\left (3 \, a + 4 \, b\right )} d x + {\left (a \cosh \left (d x + c\right )^{3} + 4 \, {\left (a + b\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )}{8 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.14, size = 116, normalized size = 1.90 \[ \frac {8 \, {\left (d x + c\right )} {\left (3 \, a + 4 \, b\right )} + a e^{\left (4 \, d x + 4 \, c\right )} + 8 \, a e^{\left (2 \, d x + 2 \, c\right )} + 8 \, b e^{\left (2 \, d x + 2 \, c\right )} - {\left (18 \, a e^{\left (4 \, d x + 4 \, c\right )} + 24 \, b e^{\left (4 \, d x + 4 \, c\right )} + 8 \, a e^{\left (2 \, d x + 2 \, c\right )} + 8 \, b e^{\left (2 \, d x + 2 \, c\right )} + a\right )} e^{\left (-4 \, d x - 4 \, c\right )}}{64 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.46, size = 66, normalized size = 1.08 \[ \frac {a \left (\left (\frac {\left (\cosh ^{3}\left (d x +c \right )\right )}{4}+\frac {3 \cosh \left (d x +c \right )}{8}\right ) \sinh \left (d x +c \right )+\frac {3 d x}{8}+\frac {3 c}{8}\right )+b \left (\frac {\cosh \left (d x +c \right ) \sinh \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 97, normalized size = 1.59 \[ \frac {1}{64} \, a {\left (24 \, x + \frac {e^{\left (4 \, d x + 4 \, c\right )}}{d} + \frac {8 \, e^{\left (2 \, d x + 2 \, c\right )}}{d} - \frac {8 \, e^{\left (-2 \, d x - 2 \, c\right )}}{d} - \frac {e^{\left (-4 \, d x - 4 \, c\right )}}{d}\right )} + \frac {1}{8} \, b {\left (4 \, x + \frac {e^{\left (2 \, d x + 2 \, c\right )}}{d} - \frac {e^{\left (-2 \, d x - 2 \, c\right )}}{d}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.13, size = 50, normalized size = 0.82 \[ \frac {\frac {a\,\mathrm {sinh}\left (2\,c+2\,d\,x\right )}{4}+\frac {a\,\mathrm {sinh}\left (4\,c+4\,d\,x\right )}{32}+\frac {b\,\mathrm {sinh}\left (2\,c+2\,d\,x\right )}{4}}{d}+\frac {3\,a\,x}{8}+\frac {b\,x}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (a + b \operatorname {sech}^{2}{\left (c + d x \right )}\right ) \cosh ^{4}{\left (c + d x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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