Optimal. Leaf size=121 \[ -\frac{e^{-a-b x} \, _2F_1\left (1,-\frac{b}{2 d};1-\frac{b}{2 d};-e^{2 (c+d x)}\right )}{b}-\frac{e^{a+b x} \, _2F_1\left (1,\frac{b}{2 d};\frac{b}{2 d}+1;-e^{2 (c+d x)}\right )}{b}+\frac{e^{-a-b x}}{2 b}+\frac{e^{a+b x}}{2 b} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.106408, antiderivative size = 121, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {5601, 2194, 2251} \[ -\frac{e^{-a-b x} \, _2F_1\left (1,-\frac{b}{2 d};1-\frac{b}{2 d};-e^{2 (c+d x)}\right )}{b}-\frac{e^{a+b x} \, _2F_1\left (1,\frac{b}{2 d};\frac{b}{2 d}+1;-e^{2 (c+d x)}\right )}{b}+\frac{e^{-a-b x}}{2 b}+\frac{e^{a+b x}}{2 b} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 5601
Rule 2194
Rule 2251
Rubi steps
\begin{align*} \int \sinh (a+b x) \tanh (c+d x) \, dx &=\int \left (-\frac{1}{2} e^{-a-b x}+\frac{1}{2} e^{a+b x}+\frac{e^{-a-b x}}{1+e^{2 (c+d x)}}-\frac{e^{a+b x}}{1+e^{2 (c+d x)}}\right ) \, dx\\ &=-\left (\frac{1}{2} \int e^{-a-b x} \, dx\right )+\frac{1}{2} \int e^{a+b x} \, dx+\int \frac{e^{-a-b x}}{1+e^{2 (c+d x)}} \, dx-\int \frac{e^{a+b x}}{1+e^{2 (c+d x)}} \, dx\\ &=\frac{e^{-a-b x}}{2 b}+\frac{e^{a+b x}}{2 b}-\frac{e^{-a-b x} \, _2F_1\left (1,-\frac{b}{2 d};1-\frac{b}{2 d};-e^{2 (c+d x)}\right )}{b}-\frac{e^{a+b x} \, _2F_1\left (1,\frac{b}{2 d};1+\frac{b}{2 d};-e^{2 (c+d x)}\right )}{b}\\ \end{align*}
Mathematica [B] time = 14.1913, size = 278, normalized size = 2.3 \[ \frac{e^{-a-b x-c} \left ((b-2 d) \left (2 b \text{sech}(c) e^{2 (a+x (b+d)+c)} \, _2F_1\left (1,\frac{b}{2 d}+1;\frac{b}{2 d}+2;-e^{2 (c+d x)}\right )-(b+2 d) \left (\left (e^{2 a}+2 e^{2 c}+1\right ) \text{sech}(c) \, _2F_1\left (1,-\frac{b}{2 d};1-\frac{b}{2 d};-e^{2 (c+d x)}\right )+2 \text{sech}(c) e^{2 (a+b x+c)} \, _2F_1\left (1,\frac{b}{2 d};\frac{b}{2 d}+1;-e^{2 (c+d x)}\right )-4 \tanh (c) e^{a+b x+c} \cosh (a+b x)-e^{2 a} \text{sech}(c)-\text{sech}(c)\right )\right )-\left (e^{2 a}-1\right ) b (b+2 d) \text{sech}(c) e^{2 (c+d x)} \, _2F_1\left (1,1-\frac{b}{2 d};2-\frac{b}{2 d};-e^{2 (c+d x)}\right )\right )}{4 \left (b^3-4 b d^2\right )} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.069, size = 0, normalized size = 0. \begin{align*} \int \sinh \left ( bx+a \right ) \tanh \left ( dx+c \right ) \, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \frac{{\left (e^{\left (2 \, b x + 2 \, a\right )} + 1\right )} e^{\left (-b x - a\right )}}{2 \, b} - \frac{1}{2} \, \int \frac{2 \,{\left (e^{\left (2 \, b x + 2 \, a\right )} - 1\right )}}{e^{\left (b x + 2 \, d x + a + 2 \, c\right )} + e^{\left (b x + a\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\sinh \left (b x + a\right ) \tanh \left (d x + c\right ), x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \sinh{\left (a + b x \right )} \tanh{\left (c + d x \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \sinh \left (b x + a\right ) \tanh \left (d x + c\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]