Optimal. Leaf size=24 \[ \frac {a \cosh (c+d x)}{d}-\frac {b \text {sech}(c+d x)}{d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.03, antiderivative size = 24, normalized size of antiderivative = 1.00, 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 = {4133, 14} \[ \frac {a \cosh (c+d x)}{d}-\frac {b \text {sech}(c+d x)}{d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 14
Rule 4133
Rubi steps
\begin {align*} \int \left (a+b \text {sech}^2(c+d x)\right ) \sinh (c+d x) \, dx &=\frac {\operatorname {Subst}\left (\int \frac {b+a x^2}{x^2} \, dx,x,\cosh (c+d x)\right )}{d}\\ &=\frac {\operatorname {Subst}\left (\int \left (a+\frac {b}{x^2}\right ) \, dx,x,\cosh (c+d x)\right )}{d}\\ &=\frac {a \cosh (c+d x)}{d}-\frac {b \text {sech}(c+d x)}{d}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.02, size = 35, normalized size = 1.46 \[ \frac {a \sinh (c) \sinh (d x)}{d}+\frac {a \cosh (c) \cosh (d x)}{d}-\frac {b \text {sech}(c+d x)}{d} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.43, size = 38, normalized size = 1.58 \[ \frac {a \cosh \left (d x + c\right )^{2} + a \sinh \left (d x + c\right )^{2} + a - 2 \, b}{2 \, d \cosh \left (d x + c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.12, size = 45, normalized size = 1.88 \[ \frac {a {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )} - \frac {4 \, b}{e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}}}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.09, size = 26, normalized size = 1.08 \[ -\frac {b \,\mathrm {sech}\left (d x +c \right )-\frac {a}{\mathrm {sech}\left (d x +c \right )}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.32, size = 36, normalized size = 1.50 \[ \frac {a \cosh \left (d x + c\right )}{d} - \frac {2 \, b}{d {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.08, size = 26, normalized size = 1.08 \[ -\frac {b-a\,{\mathrm {cosh}\left (c+d\,x\right )}^2}{d\,\mathrm {cosh}\left (c+d\,x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (a + b \operatorname {sech}^{2}{\left (c + d x \right )}\right ) \sinh {\left (c + d x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________