Optimal. Leaf size=28 \[ \frac {(a-b) \text {ArcTan}(\sinh (c+d x))}{d}+\frac {b \sinh (c+d x)}{d} \]
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Rubi [A]
time = 0.03, antiderivative size = 28, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {3269, 396, 209}
\begin {gather*} \frac {(a-b) \text {ArcTan}(\sinh (c+d x))}{d}+\frac {b \sinh (c+d x)}{d} \end {gather*}
Antiderivative was successfully verified.
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Rule 209
Rule 396
Rule 3269
Rubi steps
\begin {align*} \int \text {sech}(c+d x) \left (a+b \sinh ^2(c+d x)\right ) \, dx &=\frac {\text {Subst}\left (\int \frac {a+b x^2}{1+x^2} \, dx,x,\sinh (c+d x)\right )}{d}\\ &=\frac {b \sinh (c+d x)}{d}+\frac {(a-b) \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\sinh (c+d x)\right )}{d}\\ &=\frac {(a-b) \tan ^{-1}(\sinh (c+d x))}{d}+\frac {b \sinh (c+d x)}{d}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 37, normalized size = 1.32 \begin {gather*} \frac {a \text {ArcTan}(\sinh (c+d x))}{d}-\frac {b \text {ArcTan}(\sinh (c+d x))}{d}+\frac {b \sinh (c+d x)}{d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.83, size = 34, normalized size = 1.21
method | result | size |
derivativedivides | \(\frac {2 a \arctan \left ({\mathrm e}^{d x +c}\right )+b \left (\sinh \left (d x +c \right )-2 \arctan \left ({\mathrm e}^{d x +c}\right )\right )}{d}\) | \(34\) |
default | \(\frac {2 a \arctan \left ({\mathrm e}^{d x +c}\right )+b \left (\sinh \left (d x +c \right )-2 \arctan \left ({\mathrm e}^{d x +c}\right )\right )}{d}\) | \(34\) |
risch | \(\frac {b \,{\mathrm e}^{d x +c}}{2 d}-\frac {{\mathrm e}^{-d x -c} b}{2 d}+\frac {i \ln \left ({\mathrm e}^{d x +c}+i\right ) a}{d}-\frac {i \ln \left ({\mathrm e}^{d x +c}+i\right ) b}{d}-\frac {i \ln \left ({\mathrm e}^{d x +c}-i\right ) a}{d}+\frac {i \ln \left ({\mathrm e}^{d x +c}-i\right ) b}{d}\) | \(97\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.49, size = 56, normalized size = 2.00 \begin {gather*} \frac {1}{2} \, b {\left (\frac {4 \, \arctan \left (e^{\left (-d x - c\right )}\right )}{d} + \frac {e^{\left (d x + c\right )}}{d} - \frac {e^{\left (-d x - c\right )}}{d}\right )} + \frac {a \arctan \left (\sinh \left (d x + c\right )\right )}{d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 101 vs.
\(2 (28) = 56\).
time = 0.39, size = 101, normalized size = 3.61 \begin {gather*} \frac {b \cosh \left (d x + c\right )^{2} + 2 \, b \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + b \sinh \left (d x + c\right )^{2} + 4 \, {\left ({\left (a - b\right )} \cosh \left (d x + c\right ) + {\left (a - b\right )} \sinh \left (d x + c\right )\right )} \arctan \left (\cosh \left (d x + c\right ) + \sinh \left (d x + c\right )\right ) - b}{2 \, {\left (d \cosh \left (d x + c\right ) + d \sinh \left (d x + c\right )\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 \sinh ^{2}{\left (c + d x \right )}\right ) \operatorname {sech}{\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 = 0.43, size = 40, normalized size = 1.43 \begin {gather*} \frac {4 \, {\left (a - b\right )} \arctan \left (e^{\left (d x + c\right )}\right ) + b e^{\left (d x + c\right )} - b 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 = 1.78, size = 88, normalized size = 3.14 \begin {gather*} \frac {2\,\mathrm {atan}\left (\frac {{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,\left (a\,\sqrt {d^2}-b\,\sqrt {d^2}\right )}{d\,\sqrt {a^2-2\,a\,b+b^2}}\right )\,\sqrt {a^2-2\,a\,b+b^2}}{\sqrt {d^2}}-\frac {b\,{\mathrm {e}}^{-c-d\,x}}{2\,d}+\frac {b\,{\mathrm {e}}^{c+d\,x}}{2\,d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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