Optimal. Leaf size=40 \[ \frac {(2 a+b) \tan ^{-1}(\sinh (c+d x))}{2 d}+\frac {b \tanh (c+d x) \text {sech}(c+d x)}{2 d} \]
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Rubi [A] time = 0.03, antiderivative size = 40, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {4046, 3770} \[ \frac {(2 a+b) \tan ^{-1}(\sinh (c+d x))}{2 d}+\frac {b \tanh (c+d x) \text {sech}(c+d x)}{2 d} \]
Antiderivative was successfully verified.
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Rule 3770
Rule 4046
Rubi steps
\begin {align*} \int \text {sech}(c+d x) \left (a+b \text {sech}^2(c+d x)\right ) \, dx &=\frac {b \text {sech}(c+d x) \tanh (c+d x)}{2 d}+\frac {1}{2} (2 a+b) \int \text {sech}(c+d x) \, dx\\ &=\frac {(2 a+b) \tan ^{-1}(\sinh (c+d x))}{2 d}+\frac {b \text {sech}(c+d x) \tanh (c+d x)}{2 d}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 48, normalized size = 1.20 \[ \frac {a \tan ^{-1}(\sinh (c+d x))}{d}+\frac {b \tan ^{-1}(\sinh (c+d x))}{2 d}+\frac {b \tanh (c+d x) \text {sech}(c+d x)}{2 d} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.41, size = 321, normalized size = 8.02 \[ \frac {b \cosh \left (d x + c\right )^{3} + 3 \, b \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{2} + b \sinh \left (d x + c\right )^{3} + {\left ({\left (2 \, a + b\right )} \cosh \left (d x + c\right )^{4} + 4 \, {\left (2 \, a + b\right )} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3} + {\left (2 \, a + b\right )} \sinh \left (d x + c\right )^{4} + 2 \, {\left (2 \, a + b\right )} \cosh \left (d x + c\right )^{2} + 2 \, {\left (3 \, {\left (2 \, a + b\right )} \cosh \left (d x + c\right )^{2} + 2 \, a + b\right )} \sinh \left (d x + c\right )^{2} + 4 \, {\left ({\left (2 \, a + b\right )} \cosh \left (d x + c\right )^{3} + {\left (2 \, a + b\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right ) + 2 \, a + b\right )} \arctan \left (\cosh \left (d x + c\right ) + \sinh \left (d x + c\right )\right ) - b \cosh \left (d x + c\right ) + {\left (3 \, b \cosh \left (d x + c\right )^{2} - b\right )} \sinh \left (d x + c\right )}{d \cosh \left (d x + c\right )^{4} + 4 \, d \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3} + d \sinh \left (d x + c\right )^{4} + 2 \, d \cosh \left (d x + c\right )^{2} + 2 \, {\left (3 \, d \cosh \left (d x + c\right )^{2} + d\right )} \sinh \left (d x + c\right )^{2} + 4 \, {\left (d \cosh \left (d x + c\right )^{3} + d \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right ) + d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.12, size = 84, normalized size = 2.10 \[ \frac {{\left (\pi + 2 \, \arctan \left (\frac {1}{2} \, {\left (e^{\left (2 \, d x + 2 \, c\right )} - 1\right )} e^{\left (-d x - c\right )}\right )\right )} {\left (2 \, a + b\right )} + \frac {4 \, b {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}}{{\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{2} + 4}}{4 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.30, size = 45, normalized size = 1.12 \[ \frac {2 a \arctan \left ({\mathrm e}^{d x +c}\right )}{d}+\frac {b \,\mathrm {sech}\left (d x +c \right ) \tanh \left (d x +c \right )}{2 d}+\frac {b \arctan \left ({\mathrm e}^{d x +c}\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.40, size = 81, normalized size = 2.02 \[ -b {\left (\frac {\arctan \left (e^{\left (-d x - c\right )}\right )}{d} - \frac {e^{\left (-d x - c\right )} - e^{\left (-3 \, d x - 3 \, c\right )}}{d {\left (2 \, e^{\left (-2 \, d x - 2 \, c\right )} + e^{\left (-4 \, d x - 4 \, c\right )} + 1\right )}}\right )} + \frac {a \arctan \left (\sinh \left (d x + c\right )\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.16, size = 124, normalized size = 3.10 \[ \frac {\mathrm {atan}\left (\frac {{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,\left (2\,a\,\sqrt {d^2}+b\,\sqrt {d^2}\right )}{d\,\sqrt {4\,a^2+4\,a\,b+b^2}}\right )\,\sqrt {4\,a^2+4\,a\,b+b^2}}{\sqrt {d^2}}+\frac {b\,{\mathrm {e}}^{c+d\,x}}{d\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}+1\right )}-\frac {2\,b\,{\mathrm {e}}^{c+d\,x}}{d\,\left (2\,{\mathrm {e}}^{2\,c+2\,d\,x}+{\mathrm {e}}^{4\,c+4\,d\,x}+1\right )} \]
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 \[ \int \left (a + b \operatorname {sech}^{2}{\left (c + d x \right )}\right ) \operatorname {sech}{\left (c + d x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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