Optimal. Leaf size=33 \[ a^2 x+\frac {2 a b \text {ArcTan}(\sinh (c+d x))}{d}+\frac {b^2 \tanh (c+d x)}{d} \]
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Rubi [A]
time = 0.02, antiderivative size = 33, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {3858, 3855,
3852, 8} \begin {gather*} a^2 x+\frac {2 a b \text {ArcTan}(\sinh (c+d x))}{d}+\frac {b^2 \tanh (c+d x)}{d} \end {gather*}
Antiderivative was successfully verified.
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Rule 8
Rule 3852
Rule 3855
Rule 3858
Rubi steps
\begin {align*} \int (a+b \text {sech}(c+d x))^2 \, dx &=a^2 x+(2 a b) \int \text {sech}(c+d x) \, dx+b^2 \int \text {sech}^2(c+d x) \, dx\\ &=a^2 x+\frac {2 a b \tan ^{-1}(\sinh (c+d x))}{d}+\frac {\left (i b^2\right ) \text {Subst}(\int 1 \, dx,x,-i \tanh (c+d x))}{d}\\ &=a^2 x+\frac {2 a b \tan ^{-1}(\sinh (c+d x))}{d}+\frac {b^2 \tanh (c+d x)}{d}\\ \end {align*}
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Mathematica [A]
time = 0.04, size = 32, normalized size = 0.97 \begin {gather*} \frac {a (a d x+2 b \text {ArcTan}(\sinh (c+d x)))+b^2 \tanh (c+d x)}{d} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains complex when optimal does not.
time = 1.50, size = 64, normalized size = 1.94
method | result | size |
risch | \(a^{2} x -\frac {2 b^{2}}{d \left (1+{\mathrm e}^{2 d x +2 c}\right )}+\frac {2 i b a \ln \left ({\mathrm e}^{d x +c}+i\right )}{d}-\frac {2 i b a \ln \left ({\mathrm e}^{d x +c}-i\right )}{d}\) | \(64\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 41, normalized size = 1.24 \begin {gather*} a^{2} x + \frac {2 \, a b \arctan \left (\sinh \left (d x + c\right )\right )}{d} + \frac {2 \, b^{2}}{d {\left (e^{\left (-2 \, d x - 2 \, c\right )} + 1\right )}} \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. 157 vs.
\(2 (33) = 66\).
time = 0.42, size = 157, normalized size = 4.76 \begin {gather*} \frac {a^{2} d x \cosh \left (d x + c\right )^{2} + 2 \, a^{2} d x \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + a^{2} d x \sinh \left (d x + c\right )^{2} + a^{2} d x - 2 \, b^{2} + 4 \, {\left (a b \cosh \left (d x + c\right )^{2} + 2 \, a b \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + a b \sinh \left (d x + c\right )^{2} + a b\right )} \arctan \left (\cosh \left (d x + c\right ) + \sinh \left (d x + c\right )\right )}{d \cosh \left (d x + c\right )^{2} + 2 \, d \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + d \sinh \left (d x + c\right )^{2} + d} \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}{\left (c + d x \right )}\right )^{2}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.39, size = 43, normalized size = 1.30 \begin {gather*} \frac {{\left (d x + c\right )} a^{2} + 4 \, a b \arctan \left (e^{\left (d x + c\right )}\right ) - \frac {2 \, b^{2}}{e^{\left (2 \, d x + 2 \, c\right )} + 1}}{d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.11, size = 70, normalized size = 2.12 \begin {gather*} a^2\,x-\frac {2\,b^2}{d\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}+1\right )}+\frac {4\,\mathrm {atan}\left (\frac {a\,b\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,\sqrt {d^2}}{d\,\sqrt {a^2\,b^2}}\right )\,\sqrt {a^2\,b^2}}{\sqrt {d^2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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