Optimal. Leaf size=40 \[ \frac {\text {ArcSin}(\tanh (a+b x))}{2 b}+\frac {\sqrt {\text {sech}^2(a+b x)} \tanh (a+b x)}{2 b} \]
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Rubi [A]
time = 0.01, antiderivative size = 40, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {4207, 201, 222}
\begin {gather*} \frac {\text {ArcSin}(\tanh (a+b x))}{2 b}+\frac {\tanh (a+b x) \sqrt {\text {sech}^2(a+b x)}}{2 b} \end {gather*}
Antiderivative was successfully verified.
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Rule 201
Rule 222
Rule 4207
Rubi steps
\begin {align*} \int \text {sech}^2(a+b x)^{3/2} \, dx &=\frac {\text {Subst}\left (\int \sqrt {1-x^2} \, dx,x,\tanh (a+b x)\right )}{b}\\ &=\frac {\sqrt {\text {sech}^2(a+b x)} \tanh (a+b x)}{2 b}+\frac {\text {Subst}\left (\int \frac {1}{\sqrt {1-x^2}} \, dx,x,\tanh (a+b x)\right )}{2 b}\\ &=\frac {\sin ^{-1}(\tanh (a+b x))}{2 b}+\frac {\sqrt {\text {sech}^2(a+b x)} \tanh (a+b x)}{2 b}\\ \end {align*}
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Mathematica [A]
time = 0.04, size = 46, normalized size = 1.15 \begin {gather*} \frac {\text {sech}(a+b x) (\text {ArcTan}(\sinh (a+b x))+\text {sech}(a+b x) \tanh (a+b x))}{2 b \sqrt {\text {sech}^2(a+b x)}} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains complex when optimal does not.
time = 2.42, size = 183, normalized size = 4.58
method | result | size |
risch | \(\frac {\sqrt {\frac {{\mathrm e}^{2 b x +2 a}}{\left ({\mathrm e}^{2 b x +2 a}+1\right )^{2}}}\, \left ({\mathrm e}^{2 b x +2 a}-1\right )}{\left ({\mathrm e}^{2 b x +2 a}+1\right ) b}+\frac {i \ln \left ({\mathrm e}^{b x}+i {\mathrm e}^{-a}\right ) \sqrt {\frac {{\mathrm e}^{2 b x +2 a}}{\left ({\mathrm e}^{2 b x +2 a}+1\right )^{2}}}\, \left ({\mathrm e}^{2 b x +2 a}+1\right ) {\mathrm e}^{-b x -a}}{2 b}-\frac {i \ln \left ({\mathrm e}^{b x}-i {\mathrm e}^{-a}\right ) \sqrt {\frac {{\mathrm e}^{2 b x +2 a}}{\left ({\mathrm e}^{2 b x +2 a}+1\right )^{2}}}\, \left ({\mathrm e}^{2 b x +2 a}+1\right ) {\mathrm e}^{-b x -a}}{2 b}\) | \(183\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.48, size = 65, normalized size = 1.62 \begin {gather*} -\frac {\arctan \left (e^{\left (-b x - a\right )}\right )}{b} + \frac {e^{\left (-b x - a\right )} - e^{\left (-3 \, b x - 3 \, a\right )}}{b {\left (2 \, e^{\left (-2 \, b x - 2 \, a\right )} + e^{\left (-4 \, b x - 4 \, a\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. 267 vs.
\(2 (34) = 68\).
time = 0.37, size = 267, normalized size = 6.68 \begin {gather*} \frac {\cosh \left (b x + a\right )^{3} + 3 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{2} + \sinh \left (b x + a\right )^{3} + {\left (\cosh \left (b x + a\right )^{4} + 4 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{3} + \sinh \left (b x + a\right )^{4} + 2 \, {\left (3 \, \cosh \left (b x + a\right )^{2} + 1\right )} \sinh \left (b x + a\right )^{2} + 2 \, \cosh \left (b x + a\right )^{2} + 4 \, {\left (\cosh \left (b x + a\right )^{3} + \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right ) + 1\right )} \arctan \left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right ) + {\left (3 \, \cosh \left (b x + a\right )^{2} - 1\right )} \sinh \left (b x + a\right ) - \cosh \left (b x + a\right )}{b \cosh \left (b x + a\right )^{4} + 4 \, b \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{3} + b \sinh \left (b x + a\right )^{4} + 2 \, b \cosh \left (b x + a\right )^{2} + 2 \, {\left (3 \, b \cosh \left (b x + a\right )^{2} + b\right )} \sinh \left (b x + a\right )^{2} + 4 \, {\left (b \cosh \left (b x + a\right )^{3} + b \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right ) + b} \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 (\operatorname {sech}^{2}{\left (a + b x \right )}\right )^{\frac {3}{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 76 vs.
\(2 (34) = 68\).
time = 0.40, size = 76, normalized size = 1.90 \begin {gather*} \frac {\pi + \frac {4 \, {\left (e^{\left (b x + a\right )} - e^{\left (-b x - a\right )}\right )}}{{\left (e^{\left (b x + a\right )} - e^{\left (-b x - a\right )}\right )}^{2} + 4} + 2 \, \arctan \left (\frac {1}{2} \, {\left (e^{\left (2 \, b x + 2 \, a\right )} - 1\right )} e^{\left (-b x - a\right )}\right )}{4 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int {\left (\frac {1}{{\mathrm {cosh}\left (a+b\,x\right )}^2}\right )}^{3/2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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