Optimal. Leaf size=25 \[ \frac{(a+b) \cosh (c+d x)}{d}+\frac{b \text{sech}(c+d x)}{d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0320672, antiderivative size = 25, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {3664, 14} \[ \frac{(a+b) \cosh (c+d x)}{d}+\frac{b \text{sech}(c+d x)}{d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3664
Rule 14
Rubi steps
\begin{align*} \int \sinh (c+d x) \left (a+b \tanh ^2(c+d x)\right ) \, dx &=-\frac{\operatorname{Subst}\left (\int \frac{a+b-b x^2}{x^2} \, dx,x,\text{sech}(c+d x)\right )}{d}\\ &=-\frac{\operatorname{Subst}\left (\int \left (-b+\frac{a+b}{x^2}\right ) \, dx,x,\text{sech}(c+d x)\right )}{d}\\ &=\frac{(a+b) \cosh (c+d x)}{d}+\frac{b \text{sech}(c+d x)}{d}\\ \end{align*}
Mathematica [A] time = 0.04502, size = 45, normalized size = 1.8 \[ \frac{a \sinh (c) \sinh (d x)}{d}+\frac{a \cosh (c) \cosh (d x)}{d}+\frac{b \cosh (c+d x)}{d}+\frac{b \text{sech}(c+d x)}{d} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.033, size = 43, normalized size = 1.7 \begin{align*}{\frac{1}{d} \left ( a\cosh \left ( dx+c \right ) +b \left ( -{\frac{ \left ( \sinh \left ( dx+c \right ) \right ) ^{2}}{\cosh \left ( dx+c \right ) }}+2\,\cosh \left ( dx+c \right ) \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [B] time = 1.1077, size = 90, normalized size = 3.6 \begin{align*} \frac{1}{2} \, b{\left (\frac{e^{\left (-d x - c\right )}}{d} + \frac{5 \, e^{\left (-2 \, d x - 2 \, c\right )} + 1}{d{\left (e^{\left (-d x - c\right )} + e^{\left (-3 \, d x - 3 \, c\right )}\right )}}\right )} + \frac{a \cosh \left (d x + c\right )}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.90525, size = 115, normalized size = 4.6 \begin{align*} \frac{{\left (a + b\right )} \cosh \left (d x + c\right )^{2} +{\left (a + b\right )} \sinh \left (d x + c\right )^{2} + a + 3 \, b}{2 \, d \cosh \left (d x + c\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 \left (a + b \tanh ^{2}{\left (c + d x \right )}\right ) \sinh{\left (c + d x \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [B] time = 1.20609, size = 107, normalized size = 4.28 \begin{align*} \frac{{\left (a e^{\left (d x + 6 \, c\right )} + b e^{\left (d x + 6 \, c\right )}\right )} e^{\left (-5 \, c\right )} + \frac{{\left (a e^{\left (2 \, d x + 2 \, c\right )} + 5 \, b e^{\left (2 \, d x + 2 \, c\right )} + a + b\right )} e^{\left (-c\right )}}{e^{\left (3 \, d x + 2 \, c\right )} + e^{\left (d x\right )}}}{2 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]