Optimal. Leaf size=44 \[ -\frac {(a-b) \cosh (c+d x)}{d}+\frac {a \cosh ^3(c+d x)}{3 d}+\frac {b \text {sech}(c+d x)}{d} \]
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Rubi [A] time = 0.06, antiderivative size = 44, 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 = {4133, 448} \[ -\frac {(a-b) \cosh (c+d x)}{d}+\frac {a \cosh ^3(c+d x)}{3 d}+\frac {b \text {sech}(c+d x)}{d} \]
Antiderivative was successfully verified.
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Rule 448
Rule 4133
Rubi steps
\begin {align*} \int \left (a+b \text {sech}^2(c+d x)\right ) \sinh ^3(c+d x) \, dx &=-\frac {\operatorname {Subst}\left (\int \frac {\left (1-x^2\right ) \left (b+a x^2\right )}{x^2} \, dx,x,\cosh (c+d x)\right )}{d}\\ &=-\frac {\operatorname {Subst}\left (\int \left (a \left (1-\frac {b}{a}\right )+\frac {b}{x^2}-a x^2\right ) \, dx,x,\cosh (c+d x)\right )}{d}\\ &=-\frac {(a-b) \cosh (c+d x)}{d}+\frac {a \cosh ^3(c+d x)}{3 d}+\frac {b \text {sech}(c+d x)}{d}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 53, normalized size = 1.20 \[ -\frac {3 a \cosh (c+d x)}{4 d}+\frac {a \cosh (3 (c+d x))}{12 d}+\frac {b \cosh (c+d x)}{d}+\frac {b \text {sech}(c+d x)}{d} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.44, size = 85, normalized size = 1.93 \[ \frac {a \cosh \left (d x + c\right )^{4} + a \sinh \left (d x + c\right )^{4} - 4 \, {\left (2 \, a - 3 \, b\right )} \cosh \left (d x + c\right )^{2} + 2 \, {\left (3 \, a \cosh \left (d x + c\right )^{2} - 4 \, a + 6 \, b\right )} \sinh \left (d x + c\right )^{2} - 9 \, a + 36 \, b}{24 \, d \cosh \left (d x + c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.13, size = 85, normalized size = 1.93 \[ \frac {a {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{3} - 12 \, a {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )} + 12 \, b {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )} + \frac {48 \, b}{e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}}}{24 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.31, size = 56, normalized size = 1.27 \[ \frac {a \left (-\frac {2}{3}+\frac {\left (\sinh ^{2}\left (d x +c \right )\right )}{3}\right ) \cosh \left (d x +c \right )+b \left (\frac {\sinh ^{2}\left (d x +c \right )}{\cosh \left (d x +c \right )}+\frac {2}{\cosh \left (d x +c \right )}\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.34, size = 111, normalized size = 2.52 \[ \frac {1}{24} \, a {\left (\frac {e^{\left (3 \, d x + 3 \, c\right )}}{d} - \frac {9 \, e^{\left (d x + c\right )}}{d} - \frac {9 \, e^{\left (-d x - c\right )}}{d} + \frac {e^{\left (-3 \, d x - 3 \, c\right )}}{d}\right )} + \frac {1}{2} \, b {\left (\frac {e^{\left (-d x - c\right )}}{d} + \frac {5 \, e^{\left (-2 \, d x - 2 \, c\right )} + 1}{d {\left (e^{\left (-d x - c\right )} + e^{\left (-3 \, d x - 3 \, c\right )}\right )}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.18, size = 44, normalized size = 1.00 \[ \frac {a\,{\mathrm {cosh}\left (c+d\,x\right )}^3}{3\,d}-\frac {\mathrm {cosh}\left (c+d\,x\right )\,\left (a-b\right )}{d}+\frac {b}{d\,\mathrm {cosh}\left (c+d\,x\right )} \]
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 \[ \int \left (a + b \operatorname {sech}^{2}{\left (c + d x \right )}\right ) \sinh ^{3}{\left (c + d x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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