Optimal. Leaf size=24 \[ \frac {a \cosh (c+d x)}{d}-\frac {b \text {sech}(c+d x)}{d} \]
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Rubi [A]
time = 0.02, 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 = {4218, 14}
\begin {gather*} \frac {a \cosh (c+d x)}{d}-\frac {b \text {sech}(c+d x)}{d} \end {gather*}
Antiderivative was successfully verified.
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Rule 14
Rule 4218
Rubi steps
\begin {align*} \int \left (a+b \text {sech}^2(c+d x)\right ) \sinh (c+d x) \, dx &=\frac {\text {Subst}\left (\int \frac {b+a x^2}{x^2} \, dx,x,\cosh (c+d x)\right )}{d}\\ &=\frac {\text {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*}
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Mathematica [A]
time = 0.02, size = 35, normalized size = 1.46 \begin {gather*} \frac {a \cosh (c) \cosh (d x)}{d}-\frac {b \text {sech}(c+d x)}{d}+\frac {a \sinh (c) \sinh (d x)}{d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.66, size = 26, normalized size = 1.08
method | result | size |
derivativedivides | \(-\frac {b \,\mathrm {sech}\left (d x +c \right )-\frac {a}{\mathrm {sech}\left (d x +c \right )}}{d}\) | \(26\) |
default | \(-\frac {b \,\mathrm {sech}\left (d x +c \right )-\frac {a}{\mathrm {sech}\left (d x +c \right )}}{d}\) | \(26\) |
risch | \(\frac {a \,{\mathrm e}^{d x +c}}{2 d}+\frac {{\mathrm e}^{-d x -c} a}{2 d}-\frac {2 b \,{\mathrm e}^{d x +c}}{d \left (1+{\mathrm e}^{2 d x +2 c}\right )}\) | \(54\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 36, normalized size = 1.50 \begin {gather*} \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 )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 38, normalized size = 1.58 \begin {gather*} \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 )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (a + b \operatorname {sech}^{2}{\left (c + d x \right )}\right ) \sinh {\left (c + d x \right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.41, size = 45, normalized size = 1.88 \begin {gather*} \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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.08, size = 26, normalized size = 1.08 \begin {gather*} -\frac {b-a\,{\mathrm {cosh}\left (c+d\,x\right )}^2}{d\,\mathrm {cosh}\left (c+d\,x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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