Optimal. Leaf size=30 \[ \frac {(a+b) \sinh (c+d x)}{d}+\frac {a \sinh ^3(c+d x)}{3 d} \]
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Rubi [A] time = 0.05, antiderivative size = 30, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {4044, 3013} \[ \frac {(a+b) \sinh (c+d x)}{d}+\frac {a \sinh ^3(c+d x)}{3 d} \]
Antiderivative was successfully verified.
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Rule 3013
Rule 4044
Rubi steps
\begin {align*} \int \cosh ^3(c+d x) \left (a+b \text {sech}^2(c+d x)\right ) \, dx &=\int \cosh (c+d x) \left (b+a \cosh ^2(c+d x)\right ) \, dx\\ &=\frac {i \operatorname {Subst}\left (\int \left (a+b-a x^2\right ) \, dx,x,-i \sinh (c+d x)\right )}{d}\\ &=\frac {(a+b) \sinh (c+d x)}{d}+\frac {a \sinh ^3(c+d x)}{3 d}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 50, normalized size = 1.67 \[ \frac {a \sinh ^3(c+d x)}{3 d}+\frac {a \sinh (c+d x)}{d}+\frac {b \sinh (c) \cosh (d x)}{d}+\frac {b \cosh (c) \sinh (d x)}{d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.39, size = 41, normalized size = 1.37 \[ \frac {a \sinh \left (d x + c\right )^{3} + 3 \, {\left (a \cosh \left (d x + c\right )^{2} + 3 \, a + 4 \, b\right )} \sinh \left (d x + c\right )}{12 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.16, size = 72, normalized size = 2.40 \[ \frac {a e^{\left (3 \, d x + 3 \, c\right )} + 9 \, a e^{\left (d x + c\right )} + 12 \, b e^{\left (d x + c\right )} - {\left (9 \, a e^{\left (2 \, d x + 2 \, c\right )} + 12 \, b e^{\left (2 \, d x + 2 \, c\right )} + a\right )} e^{\left (-3 \, d x - 3 \, c\right )}}{24 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.43, size = 34, normalized size = 1.13 \[ \frac {a \left (\frac {2}{3}+\frac {\left (\cosh ^{2}\left (d x +c \right )\right )}{3}\right ) \sinh \left (d x +c \right )+b \sinh \left (d x +c \right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.31, size = 85, normalized size = 2.83 \[ \frac {1}{24} \, a {\left (\frac {e^{\left (3 \, d x + 3 \, c\right )}}{d} + \frac {9 \, e^{\left (d x + c\right )}}{d} - \frac {9 \, e^{\left (-d x - c\right )}}{d} - \frac {e^{\left (-3 \, d x - 3 \, c\right )}}{d}\right )} + \frac {1}{2} \, b {\left (\frac {e^{\left (d x + c\right )}}{d} - \frac {e^{\left (-d x - c\right )}}{d}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.09, size = 34, normalized size = 1.13 \[ \frac {3\,a\,\mathrm {sinh}\left (c+d\,x\right )+3\,b\,\mathrm {sinh}\left (c+d\,x\right )+a\,{\mathrm {sinh}\left (c+d\,x\right )}^3}{3\,d} \]
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 \[ \int \left (a + b \operatorname {sech}^{2}{\left (c + d x \right )}\right ) \cosh ^{3}{\left (c + d x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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