Optimal. Leaf size=30 \[ \frac {(a+b) \sinh ^3(c+d x)}{3 d}+\frac {a \sinh (c+d x)}{d} \]
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Rubi [A] time = 0.04, antiderivative size = 30, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.048, Rules used = {3676} \[ \frac {(a+b) \sinh ^3(c+d x)}{3 d}+\frac {a \sinh (c+d x)}{d} \]
Antiderivative was successfully verified.
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Rule 3676
Rubi steps
\begin {align*} \int \cosh ^3(c+d x) \left (a+b \tanh ^2(c+d x)\right ) \, dx &=\frac {\operatorname {Subst}\left (\int \left (a+(a+b) x^2\right ) \, dx,x,\sinh (c+d x)\right )}{d}\\ &=\frac {a \sinh (c+d x)}{d}+\frac {(a+b) \sinh ^3(c+d x)}{3 d}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 44, normalized size = 1.47 \[ \frac {a \sinh ^3(c+d x)}{3 d}+\frac {a \sinh (c+d x)}{d}+\frac {b \sinh ^3(c+d x)}{3 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.39, size = 45, normalized size = 1.50 \[ \frac {{\left (a + b\right )} \sinh \left (d x + c\right )^{3} + 3 \, {\left ({\left (a + b\right )} \cosh \left (d x + c\right )^{2} + 3 \, a - b\right )} \sinh \left (d x + c\right )}{12 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.16, size = 94, normalized size = 3.13 \[ -\frac {{\left (9 \, a e^{\left (2 \, d x + 2 \, c\right )} - 3 \, b e^{\left (2 \, d x + 2 \, c\right )} + a + b\right )} e^{\left (-3 \, d x - 3 \, c\right )} - {\left (a e^{\left (3 \, d x + 12 \, c\right )} + b e^{\left (3 \, d x + 12 \, c\right )} + 9 \, a e^{\left (d x + 10 \, c\right )} - 3 \, b e^{\left (d x + 10 \, c\right )}\right )} e^{\left (-9 \, c\right )}}{24 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.32, size = 37, normalized size = 1.23 \[ \frac {\frac {b \left (\sinh ^{3}\left (d x +c \right )\right )}{3}+a \left (\frac {2}{3}+\frac {\left (\cosh ^{2}\left (d x +c \right )\right )}{3}\right ) \sinh \left (d x +c \right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.34, size = 83, normalized size = 2.77 \[ \frac {b {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{3}}{24 \, d} + \frac {1}{24} \, a {\left (\frac {e^{\left (3 \, d x + 3 \, c\right )}}{d} + \frac {9 \, e^{\left (d x + c\right )}}{d} - \frac {9 \, e^{\left (-d x - c\right )}}{d} - \frac {e^{\left (-3 \, d x - 3 \, c\right )}}{d}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.21, size = 74, normalized size = 2.47 \[ \frac {{\mathrm {e}}^{3\,c+3\,d\,x}\,\left (a+b\right )}{24\,d}-\frac {{\mathrm {e}}^{-3\,c-3\,d\,x}\,\left (a+b\right )}{24\,d}+\frac {{\mathrm {e}}^{c+d\,x}\,\left (3\,a-b\right )}{8\,d}-\frac {{\mathrm {e}}^{-c-d\,x}\,\left (3\,a-b\right )}{8\,d} \]
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 \[ \int \left (a + b \tanh ^{2}{\left (c + d x \right )}\right ) \cosh ^{3}{\left (c + d x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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