Optimal. Leaf size=52 \[ \frac {2 e^{a+c+b x+d x} \, _2F_1\left (1,\frac {b+d}{2 d};\frac {1}{2} \left (3+\frac {b}{d}\right );-e^{2 (c+d x)}\right )}{b+d} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.01, antiderivative size = 52, normalized size of antiderivative = 1.00, number of steps
used = 1, number of rules used = 1, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {5600}
\begin {gather*} \frac {2 e^{a+b x+c+d x} \, _2F_1\left (1,\frac {b+d}{2 d};\frac {1}{2} \left (\frac {b}{d}+3\right );-e^{2 (c+d x)}\right )}{b+d} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 5600
Rubi steps
\begin {align*} \int e^{a+b x} \text {sech}(c+d x) \, dx &=\frac {2 e^{a+c+b x+d x} \, _2F_1\left (1,\frac {b+d}{2 d};\frac {1}{2} \left (3+\frac {b}{d}\right );-e^{2 (c+d x)}\right )}{b+d}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.02, size = 51, normalized size = 0.98 \begin {gather*} \frac {2 e^{a+c+(b+d) x} \, _2F_1\left (1,\frac {b+d}{2 d};\frac {1}{2} \left (3+\frac {b}{d}\right );-e^{2 (c+d x)}\right )}{b+d} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F]
time = 1.25, size = 0, normalized size = 0.00 \[\int {\mathrm e}^{b x +a} \mathrm {sech}\left (d x +c \right )\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} e^{a} \int e^{b x} \operatorname {sech}{\left (c + d x \right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
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 )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________