Optimal. Leaf size=15 \[ a x+\frac {b \tanh (c+d x)}{d} \]
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Rubi [A] time = 0.01, antiderivative size = 15, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {3767, 8} \[ a x+\frac {b \tanh (c+d x)}{d} \]
Antiderivative was successfully verified.
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Rule 8
Rule 3767
Rubi steps
\begin {align*} \int \left (a+b \text {sech}^2(c+d x)\right ) \, dx &=a x+b \int \text {sech}^2(c+d x) \, dx\\ &=a x+\frac {(i b) \operatorname {Subst}(\int 1 \, dx,x,-i \tanh (c+d x))}{d}\\ &=a x+\frac {b \tanh (c+d x)}{d}\\ \end {align*}
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Mathematica [A] time = 0.00, size = 15, normalized size = 1.00 \[ a x+\frac {b \tanh (c+d x)}{d} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.40, size = 36, normalized size = 2.40 \[ \frac {{\left (a d x - b\right )} \cosh \left (d x + c\right ) + b \sinh \left (d x + c\right )}{d \cosh \left (d x + c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.12, size = 23, normalized size = 1.53 \[ a x - \frac {2 \, b}{d {\left (e^{\left (2 \, d x + 2 \, c\right )} + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.27, size = 16, normalized size = 1.07 \[ a x +\frac {b \tanh \left (d x +c \right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.37, size = 23, normalized size = 1.53 \[ a x + \frac {2 \, b}{d {\left (e^{\left (-2 \, d x - 2 \, c\right )} + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.39, size = 23, normalized size = 1.53 \[ a\,x-\frac {2\,b}{d\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}+1\right )} \]
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 \[ \int \left (a + b \operatorname {sech}^{2}{\left (c + d x \right )}\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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