Optimal. Leaf size=24 \[ \frac {b \cosh (c+d x)}{d}-\frac {a \coth (c+d x)}{d} \]
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Rubi [A] time = 0.05, antiderivative size = 24, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {3220, 3767, 8, 2638} \[ \frac {b \cosh (c+d x)}{d}-\frac {a \coth (c+d x)}{d} \]
Antiderivative was successfully verified.
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Rule 8
Rule 2638
Rule 3220
Rule 3767
Rubi steps
\begin {align*} \int \text {csch}^2(c+d x) \left (a+b \sinh ^3(c+d x)\right ) \, dx &=-\int \left (-a \text {csch}^2(c+d x)-b \sinh (c+d x)\right ) \, dx\\ &=a \int \text {csch}^2(c+d x) \, dx+b \int \sinh (c+d x) \, dx\\ &=\frac {b \cosh (c+d x)}{d}-\frac {(i a) \operatorname {Subst}(\int 1 \, dx,x,-i \coth (c+d x))}{d}\\ &=\frac {b \cosh (c+d x)}{d}-\frac {a \coth (c+d x)}{d}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 35, normalized size = 1.46 \[ -\frac {a \coth (c+d x)}{d}+\frac {b \sinh (c) \sinh (d x)}{d}+\frac {b \cosh (c) \cosh (d x)}{d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.93, size = 40, normalized size = 1.67 \[ -\frac {a \cosh \left (d x + c\right ) - {\left (b \cosh \left (d x + c\right ) + a\right )} \sinh \left (d x + c\right )}{d \sinh \left (d x + c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.14, size = 59, normalized size = 2.46 \[ \frac {b e^{\left (d x + c\right )} + \frac {b e^{\left (2 \, d x + 2 \, c\right )} - 4 \, a e^{\left (d x + c\right )} - b}{e^{\left (3 \, d x + 3 \, c\right )} - e^{\left (d x + c\right )}}}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.08, size = 23, normalized size = 0.96 \[ \frac {-\coth \left (d x +c \right ) a +b \cosh \left (d x +c \right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.32, size = 47, normalized size = 1.96 \[ \frac {1}{2} \, b {\left (\frac {e^{\left (d x + c\right )}}{d} + \frac {e^{\left (-d x - c\right )}}{d}\right )} + \frac {2 \, a}{d {\left (e^{\left (-2 \, d x - 2 \, c\right )} - 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.09, size = 47, normalized size = 1.96 \[ \frac {b\,{\mathrm {e}}^{-c-d\,x}}{2\,d}-\frac {2\,a}{d\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}-1\right )}+\frac {b\,{\mathrm {e}}^{c+d\,x}}{2\,d} \]
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 \[ \int \left (a + b \sinh ^{3}{\left (c + d x \right )}\right ) \operatorname {csch}^{2}{\left (c + d x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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