Optimal. Leaf size=56 \[ \frac {4 e^{a+b x+2 (c+d x)} \, _2F_1\left (2,1+\frac {b}{2 d};2+\frac {b}{2 d};-e^{2 (c+d x)}\right )}{b+2 d} \]
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Rubi [A]
time = 0.02, antiderivative size = 56, normalized size of antiderivative = 1.00, number of steps
used = 1, number of rules used = 1, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {5600}
\begin {gather*} \frac {4 e^{a+b x+2 (c+d x)} \, _2F_1\left (2,\frac {b}{2 d}+1;\frac {b}{2 d}+2;-e^{2 (c+d x)}\right )}{b+2 d} \end {gather*}
Antiderivative was successfully verified.
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Rule 5600
Rubi steps
\begin {align*} \int e^{a+b x} \text {sech}^2(c+d x) \, dx &=\frac {4 e^{a+b x+2 (c+d x)} \, _2F_1\left (2,1+\frac {b}{2 d};2+\frac {b}{2 d};-e^{2 (c+d x)}\right )}{b+2 d}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 56, normalized size = 1.00 \begin {gather*} \frac {4 e^{a+b x+2 (c+d x)} \, _2F_1\left (2,1+\frac {b}{2 d};2+\frac {b}{2 d};-e^{2 (c+d x)}\right )}{b+2 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 1.19, size = 0, normalized size = 0.00 \[\int {\mathrm e}^{b x +a} \mathrm {sech}\left (d x +c \right )^{2}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} e^{a} \int e^{b x} \operatorname {sech}^{2}{\left (c + d x \right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {{\mathrm {e}}^{a+b\,x}}{{\mathrm {cosh}\left (c+d\,x\right )}^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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