Optimal. Leaf size=26 \[ \frac {\tanh (a+b x)}{b}-\frac {\tanh ^3(a+b x)}{3 b} \]
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Rubi [A]
time = 0.01, antiderivative size = 26, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 1, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {3852}
\begin {gather*} \frac {\tanh (a+b x)}{b}-\frac {\tanh ^3(a+b x)}{3 b} \end {gather*}
Antiderivative was successfully verified.
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Rule 3852
Rubi steps
\begin {align*} \int \text {sech}^4(a+b x) \, dx &=\frac {i \text {Subst}\left (\int \left (1+x^2\right ) \, dx,x,-i \tanh (a+b x)\right )}{b}\\ &=\frac {\tanh (a+b x)}{b}-\frac {\tanh ^3(a+b x)}{3 b}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 26, normalized size = 1.00 \begin {gather*} \frac {\tanh (a+b x)}{b}-\frac {\tanh ^3(a+b x)}{3 b} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 1.08, size = 32, normalized size = 1.23
method | result | size |
risch | \(-\frac {4 \left (3 \,{\mathrm e}^{2 b x +2 a}+1\right )}{3 b \left ({\mathrm e}^{2 b x +2 a}+1\right )^{3}}\) | \(32\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 90 vs.
\(2 (24) = 48\).
time = 0.27, size = 90, normalized size = 3.46 \begin {gather*} \frac {4 \, e^{\left (-2 \, b x - 2 \, a\right )}}{b {\left (3 \, e^{\left (-2 \, b x - 2 \, a\right )} + 3 \, e^{\left (-4 \, b x - 4 \, a\right )} + e^{\left (-6 \, b x - 6 \, a\right )} + 1\right )}} + \frac {4}{3 \, b {\left (3 \, e^{\left (-2 \, b x - 2 \, a\right )} + 3 \, e^{\left (-4 \, b x - 4 \, a\right )} + e^{\left (-6 \, b x - 6 \, 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. 164 vs.
\(2 (24) = 48\).
time = 0.35, size = 164, normalized size = 6.31 \begin {gather*} -\frac {8 \, {\left (2 \, \cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right )}}{3 \, {\left (b \cosh \left (b x + a\right )^{5} + 5 \, b \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{4} + b \sinh \left (b x + a\right )^{5} + 3 \, b \cosh \left (b x + a\right )^{3} + {\left (10 \, b \cosh \left (b x + a\right )^{2} + 3 \, b\right )} \sinh \left (b x + a\right )^{3} + {\left (10 \, b \cosh \left (b x + a\right )^{3} + 9 \, b \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )^{2} + 4 \, b \cosh \left (b x + a\right ) + {\left (5 \, b \cosh \left (b x + a\right )^{4} + 9 \, b \cosh \left (b x + a\right )^{2} + 2 \, b\right )} \sinh \left (b x + a\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 \operatorname {sech}^{4}{\left (a + b x \right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.39, size = 31, normalized size = 1.19 \begin {gather*} -\frac {4 \, {\left (3 \, e^{\left (2 \, b x + 2 \, a\right )} + 1\right )}}{3 \, b {\left (e^{\left (2 \, b x + 2 \, a\right )} + 1\right )}^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.06, size = 31, normalized size = 1.19 \begin {gather*} -\frac {4\,\left (3\,{\mathrm {e}}^{2\,a+2\,b\,x}+1\right )}{3\,b\,{\left ({\mathrm {e}}^{2\,a+2\,b\,x}+1\right )}^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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