Optimal. Leaf size=47 \[ \frac {(a+b) \cosh ^3(c+d x)}{3 d}-\frac {(a+2 b) \cosh (c+d x)}{d}-\frac {b \text {sech}(c+d x)}{d} \]
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Rubi [A] time = 0.05, antiderivative size = 47, 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 = {3664, 448} \[ \frac {(a+b) \cosh ^3(c+d x)}{3 d}-\frac {(a+2 b) \cosh (c+d x)}{d}-\frac {b \text {sech}(c+d x)}{d} \]
Antiderivative was successfully verified.
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Rule 448
Rule 3664
Rubi steps
\begin {align*} \int \sinh ^3(c+d x) \left (a+b \tanh ^2(c+d x)\right ) \, dx &=\frac {\operatorname {Subst}\left (\int \frac {\left (-1+x^2\right ) \left (a+b-b x^2\right )}{x^4} \, dx,x,\text {sech}(c+d x)\right )}{d}\\ &=\frac {\operatorname {Subst}\left (\int \left (-b+\frac {-a-b}{x^4}+\frac {a+2 b}{x^2}\right ) \, dx,x,\text {sech}(c+d x)\right )}{d}\\ &=-\frac {(a+2 b) \cosh (c+d x)}{d}+\frac {(a+b) \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 = 73, normalized size = 1.55 \[ -\frac {3 a \cosh (c+d x)}{4 d}+\frac {a \cosh (3 (c+d x))}{12 d}-\frac {7 b \cosh (c+d x)}{4 d}+\frac {b \cosh (3 (c+d x))}{12 d}-\frac {b \text {sech}(c+d x)}{d} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.78, size = 91, normalized size = 1.94 \[ \frac {{\left (a + b\right )} \cosh \left (d x + c\right )^{4} + {\left (a + b\right )} \sinh \left (d x + c\right )^{4} - 4 \, {\left (2 \, a + 5 \, b\right )} \cosh \left (d x + c\right )^{2} + 2 \, {\left (3 \, {\left (a + b\right )} \cosh \left (d x + c\right )^{2} - 4 \, a - 10 \, b\right )} \sinh \left (d x + c\right )^{2} - 9 \, a - 45 \, b}{24 \, d \cosh \left (d x + c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.17, size = 120, normalized size = 2.55 \[ -\frac {{\left (9 \, a e^{\left (2 \, d x + 2 \, c\right )} + 21 \, b e^{\left (2 \, d x + 2 \, c\right )} - a - b\right )} e^{\left (-3 \, d x - 3 \, c\right )} - {\left (a e^{\left (3 \, d x + 24 \, c\right )} + b e^{\left (3 \, d x + 24 \, c\right )} - 9 \, a e^{\left (d x + 22 \, c\right )} - 21 \, b e^{\left (d x + 22 \, c\right )}\right )} e^{\left (-21 \, c\right )} + \frac {48 \, b e^{\left (d x + c\right )}}{e^{\left (2 \, d x + 2 \, c\right )} + 1}}{24 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.26, size = 75, normalized size = 1.60 \[ \frac {a \left (-\frac {2}{3}+\frac {\left (\sinh ^{2}\left (d x +c \right )\right )}{3}\right ) \cosh \left (d x +c \right )+b \left (\frac {\sinh ^{4}\left (d x +c \right )}{3 \cosh \left (d x +c \right )}-\frac {4 \left (\sinh ^{2}\left (d x +c \right )\right )}{3 \cosh \left (d x +c \right )}-\frac {8}{3 \cosh \left (d x +c \right )}\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.39, size = 136, normalized size = 2.89 \[ -\frac {1}{24} \, b {\left (\frac {21 \, e^{\left (-d x - c\right )} - e^{\left (-3 \, d x - 3 \, c\right )}}{d} + \frac {20 \, e^{\left (-2 \, d x - 2 \, c\right )} + 69 \, e^{\left (-4 \, d x - 4 \, c\right )} - 1}{d {\left (e^{\left (-3 \, d x - 3 \, c\right )} + e^{\left (-5 \, d x - 5 \, c\right )}\right )}}\right )} + \frac {1}{24} \, a {\left (\frac {e^{\left (3 \, d x + 3 \, c\right )}}{d} - \frac {9 \, e^{\left (d x + c\right )}}{d} - \frac {9 \, e^{\left (-d x - c\right )}}{d} + \frac {e^{\left (-3 \, d x - 3 \, c\right )}}{d}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.17, size = 99, normalized size = 2.11 \[ \frac {{\mathrm {e}}^{-3\,c-3\,d\,x}\,\left (a+b\right )}{24\,d}+\frac {{\mathrm {e}}^{3\,c+3\,d\,x}\,\left (a+b\right )}{24\,d}-\frac {{\mathrm {e}}^{c+d\,x}\,\left (3\,a+7\,b\right )}{8\,d}-\frac {{\mathrm {e}}^{-c-d\,x}\,\left (3\,a+7\,b\right )}{8\,d}-\frac {2\,b\,{\mathrm {e}}^{c+d\,x}}{d\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}+1\right )} \]
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 \[ \int \left (a + b \tanh ^{2}{\left (c + d x \right )}\right ) \sinh ^{3}{\left (c + d x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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