Optimal. Leaf size=43 \[ -\frac {1}{2} x (a-2 b)+\frac {a \sinh (c+d x) \cosh (c+d x)}{2 d}-\frac {b \tanh (c+d x)}{d} \]
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Rubi [A] time = 0.06, antiderivative size = 43, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {4132, 455, 388, 206} \[ -\frac {1}{2} x (a-2 b)+\frac {a \sinh (c+d x) \cosh (c+d x)}{2 d}-\frac {b \tanh (c+d x)}{d} \]
Antiderivative was successfully verified.
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Rule 206
Rule 388
Rule 455
Rule 4132
Rubi steps
\begin {align*} \int \left (a+b \text {sech}^2(c+d x)\right ) \sinh ^2(c+d x) \, dx &=\frac {\operatorname {Subst}\left (\int \frac {x^2 \left (a+b-b x^2\right )}{\left (1-x^2\right )^2} \, dx,x,\tanh (c+d x)\right )}{d}\\ &=\frac {a \cosh (c+d x) \sinh (c+d x)}{2 d}-\frac {\operatorname {Subst}\left (\int \frac {a-2 b x^2}{1-x^2} \, dx,x,\tanh (c+d x)\right )}{2 d}\\ &=\frac {a \cosh (c+d x) \sinh (c+d x)}{2 d}-\frac {b \tanh (c+d x)}{d}-\frac {(a-2 b) \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\tanh (c+d x)\right )}{2 d}\\ &=-\frac {1}{2} (a-2 b) x+\frac {a \cosh (c+d x) \sinh (c+d x)}{2 d}-\frac {b \tanh (c+d x)}{d}\\ \end {align*}
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Mathematica [A] time = 0.19, size = 57, normalized size = 1.33 \[ \frac {a (-c-d x)}{2 d}+\frac {a \sinh (2 (c+d x))}{4 d}+\frac {b \tanh ^{-1}(\tanh (c+d x))}{d}-\frac {b \tanh (c+d x)}{d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.41, size = 67, normalized size = 1.56 \[ \frac {a \sinh \left (d x + c\right )^{3} - 4 \, {\left ({\left (a - 2 \, b\right )} d x - 2 \, b\right )} \cosh \left (d x + c\right ) + {\left (3 \, a \cosh \left (d x + c\right )^{2} + a - 8 \, b\right )} \sinh \left (d x + c\right )}{8 \, d \cosh \left (d x + c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.16, size = 92, normalized size = 2.14 \[ -\frac {4 \, {\left (d x + c\right )} {\left (a - 2 \, b\right )} - a e^{\left (2 \, d x + 2 \, c\right )} - \frac {a e^{\left (4 \, d x + 4 \, c\right )} - 2 \, b e^{\left (4 \, d x + 4 \, c\right )} + 14 \, b e^{\left (2 \, d x + 2 \, c\right )} - a}{e^{\left (4 \, d x + 4 \, c\right )} + e^{\left (2 \, d x + 2 \, c\right )}}}{8 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.18, size = 45, normalized size = 1.05 \[ \frac {a \left (\frac {\cosh \left (d x +c \right ) \sinh \left (d x +c \right )}{2}-\frac {d x}{2}-\frac {c}{2}\right )+b \left (d x +c -\tanh \left (d x +c \right )\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.33, size = 62, normalized size = 1.44 \[ -\frac {1}{8} \, a {\left (4 \, x - \frac {e^{\left (2 \, d x + 2 \, c\right )}}{d} + \frac {e^{\left (-2 \, d x - 2 \, c\right )}}{d}\right )} + b {\left (x + \frac {c}{d} - \frac {2}{d {\left (e^{\left (-2 \, d x - 2 \, c\right )} + 1\right )}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.43, size = 55, normalized size = 1.28 \[ \frac {a\,\mathrm {cosh}\left (c+d\,x\right )\,\mathrm {sinh}\left (c+d\,x\right )}{2\,d}-\frac {\frac {a\,d\,x}{2}-b\,d\,x}{d}-\frac {b\,\mathrm {sinh}\left (c+d\,x\right )}{d\,\mathrm {cosh}\left (c+d\,x\right )} \]
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 \[ \int \left (a + b \operatorname {sech}^{2}{\left (c + d x \right )}\right ) \sinh ^{2}{\left (c + d x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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