Optimal. Leaf size=15 \[ \frac {\tanh ^3(a+b x)}{3 b} \]
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Rubi [A] time = 0.03, antiderivative size = 15, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {2607, 30} \[ \frac {\tanh ^3(a+b x)}{3 b} \]
Antiderivative was successfully verified.
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Rule 30
Rule 2607
Rubi steps
\begin {align*} \int \text {sech}^2(a+b x) \tanh ^2(a+b x) \, dx &=\frac {i \operatorname {Subst}\left (\int x^2 \, dx,x,i \tanh (a+b x)\right )}{b}\\ &=\frac {\tanh ^3(a+b x)}{3 b}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 15, normalized size = 1.00 \[ \frac {\tanh ^3(a+b x)}{3 b} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.42, size = 138, normalized size = 9.20 \[ -\frac {8 \, {\left (\cosh \left (b x + a\right )^{2} + \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + \sinh \left (b x + a\right )^{2}\right )}}{3 \, {\left (b \cosh \left (b x + a\right )^{4} + 4 \, b \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{3} + b \sinh \left (b x + a\right )^{4} + 4 \, b \cosh \left (b x + a\right )^{2} + 2 \, {\left (3 \, b \cosh \left (b x + a\right )^{2} + 2 \, b\right )} \sinh \left (b x + a\right )^{2} + 4 \, {\left (b \cosh \left (b x + a\right )^{3} + b \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right ) + 3 \, b\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.13, size = 31, normalized size = 2.07 \[ -\frac {2 \, {\left (3 \, e^{\left (4 \, b x + 4 \, a\right )} + 1\right )}}{3 \, b {\left (e^{\left (2 \, b x + 2 \, a\right )} + 1\right )}^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.32, size = 42, normalized size = 2.80 \[ \frac {-\frac {\sinh \left (b x +a \right )}{2 \cosh \left (b x +a \right )^{3}}+\frac {\left (\frac {2}{3}+\frac {\mathrm {sech}\left (b x +a \right )^{2}}{3}\right ) \tanh \left (b x +a \right )}{2}}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.33, size = 13, normalized size = 0.87 \[ \frac {\tanh \left (b x + a\right )^{3}}{3 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.07, size = 31, normalized size = 2.07 \[ -\frac {2\,\left (3\,{\mathrm {e}}^{4\,a+4\,b\,x}+1\right )}{3\,b\,{\left ({\mathrm {e}}^{2\,a+2\,b\,x}+1\right )}^3} \]
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 \[ \int \tanh ^{2}{\left (a + b x \right )} \operatorname {sech}^{2}{\left (a + b x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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