Optimal. Leaf size=50 \[ -\frac {2 e^{a+b x+c+d x} \, _2F_1\left (1,\frac {b+d}{2 d};\frac {1}{2} \left (\frac {b}{d}+3\right );e^{2 (c+d x)}\right )}{b+d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.02, antiderivative size = 50, 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 = {5493} \[ -\frac {2 e^{a+b x+c+d x} \, _2F_1\left (1,\frac {b+d}{2 d};\frac {1}{2} \left (\frac {b}{d}+3\right );e^{2 (c+d x)}\right )}{b+d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 5493
Rubi steps
\begin {align*} \int e^{a+b x} \text {csch}(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 = 0.13, size = 59, normalized size = 1.18 \[ -\frac {2 (\sinh (c)+\cosh (c)) e^{a+x (b+d)} \, _2F_1\left (1,\frac {b+d}{2 d};\frac {b+3 d}{2 d};e^{2 d x} (\cosh (c)+\sinh (c))^2\right )}{b+d} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.47, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\operatorname {csch}\left (d x + c\right ) e^{\left (b x + a\right )}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \operatorname {csch}\left (d x + c\right ) e^{\left (b x + a\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.27, size = 0, normalized size = 0.00 \[ \int {\mathrm e}^{b x +a} \mathrm {csch}\left (d x +c \right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \operatorname {csch}\left (d x + c\right ) e^{\left (b x + a\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {{\mathrm {e}}^{a+b\,x}}{\mathrm {sinh}\left (c+d\,x\right )} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ e^{a} \int e^{b x} \operatorname {csch}{\left (c + d x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________