Optimal. Leaf size=86 \[ \frac{5 x \text{sech}^2(x)}{16 a \sqrt{a \text{sech}^4(x)}}+\frac{5 \tanh (x)}{16 a \sqrt{a \text{sech}^4(x)}}+\frac{\sinh (x) \cosh ^3(x)}{6 a \sqrt{a \text{sech}^4(x)}}+\frac{5 \sinh (x) \cosh (x)}{24 a \sqrt{a \text{sech}^4(x)}} \]
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Rubi [A] time = 0.035756, antiderivative size = 86, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {4123, 2635, 8} \[ \frac{5 x \text{sech}^2(x)}{16 a \sqrt{a \text{sech}^4(x)}}+\frac{5 \tanh (x)}{16 a \sqrt{a \text{sech}^4(x)}}+\frac{\sinh (x) \cosh ^3(x)}{6 a \sqrt{a \text{sech}^4(x)}}+\frac{5 \sinh (x) \cosh (x)}{24 a \sqrt{a \text{sech}^4(x)}} \]
Antiderivative was successfully verified.
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Rule 4123
Rule 2635
Rule 8
Rubi steps
\begin{align*} \int \frac{1}{\left (a \text{sech}^4(x)\right )^{3/2}} \, dx &=\frac{\text{sech}^2(x) \int \cosh ^6(x) \, dx}{a \sqrt{a \text{sech}^4(x)}}\\ &=\frac{\cosh ^3(x) \sinh (x)}{6 a \sqrt{a \text{sech}^4(x)}}+\frac{\left (5 \text{sech}^2(x)\right ) \int \cosh ^4(x) \, dx}{6 a \sqrt{a \text{sech}^4(x)}}\\ &=\frac{5 \cosh (x) \sinh (x)}{24 a \sqrt{a \text{sech}^4(x)}}+\frac{\cosh ^3(x) \sinh (x)}{6 a \sqrt{a \text{sech}^4(x)}}+\frac{\left (5 \text{sech}^2(x)\right ) \int \cosh ^2(x) \, dx}{8 a \sqrt{a \text{sech}^4(x)}}\\ &=\frac{5 \cosh (x) \sinh (x)}{24 a \sqrt{a \text{sech}^4(x)}}+\frac{\cosh ^3(x) \sinh (x)}{6 a \sqrt{a \text{sech}^4(x)}}+\frac{5 \tanh (x)}{16 a \sqrt{a \text{sech}^4(x)}}+\frac{\left (5 \text{sech}^2(x)\right ) \int 1 \, dx}{16 a \sqrt{a \text{sech}^4(x)}}\\ &=\frac{5 x \text{sech}^2(x)}{16 a \sqrt{a \text{sech}^4(x)}}+\frac{5 \cosh (x) \sinh (x)}{24 a \sqrt{a \text{sech}^4(x)}}+\frac{\cosh ^3(x) \sinh (x)}{6 a \sqrt{a \text{sech}^4(x)}}+\frac{5 \tanh (x)}{16 a \sqrt{a \text{sech}^4(x)}}\\ \end{align*}
Mathematica [A] time = 0.0360809, size = 38, normalized size = 0.44 \[ \frac{(60 x+45 \sinh (2 x)+9 \sinh (4 x)+\sinh (6 x)) \text{sech}^6(x)}{192 \left (a \text{sech}^4(x)\right )^{3/2}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.058, size = 230, normalized size = 2.7 \begin{align*}{\frac{5\,{{\rm e}^{2\,x}}x}{16\,a \left ({{\rm e}^{2\,x}}+1 \right ) ^{2}}{\frac{1}{\sqrt{{\frac{a{{\rm e}^{4\,x}}}{ \left ({{\rm e}^{2\,x}}+1 \right ) ^{4}}}}}}}+{\frac{{{\rm e}^{8\,x}}}{384\,a \left ({{\rm e}^{2\,x}}+1 \right ) ^{2}}{\frac{1}{\sqrt{{\frac{a{{\rm e}^{4\,x}}}{ \left ({{\rm e}^{2\,x}}+1 \right ) ^{4}}}}}}}+{\frac{3\,{{\rm e}^{6\,x}}}{128\,a \left ({{\rm e}^{2\,x}}+1 \right ) ^{2}}{\frac{1}{\sqrt{{\frac{a{{\rm e}^{4\,x}}}{ \left ({{\rm e}^{2\,x}}+1 \right ) ^{4}}}}}}}+{\frac{15\,{{\rm e}^{4\,x}}}{128\,a \left ({{\rm e}^{2\,x}}+1 \right ) ^{2}}{\frac{1}{\sqrt{{\frac{a{{\rm e}^{4\,x}}}{ \left ({{\rm e}^{2\,x}}+1 \right ) ^{4}}}}}}}-{\frac{15}{128\,a \left ({{\rm e}^{2\,x}}+1 \right ) ^{2}}{\frac{1}{\sqrt{{\frac{a{{\rm e}^{4\,x}}}{ \left ({{\rm e}^{2\,x}}+1 \right ) ^{4}}}}}}}-{\frac{3\,{{\rm e}^{-2\,x}}}{128\,a \left ({{\rm e}^{2\,x}}+1 \right ) ^{2}}{\frac{1}{\sqrt{{\frac{a{{\rm e}^{4\,x}}}{ \left ({{\rm e}^{2\,x}}+1 \right ) ^{4}}}}}}}-{\frac{{{\rm e}^{-4\,x}}}{384\,a \left ({{\rm e}^{2\,x}}+1 \right ) ^{2}}{\frac{1}{\sqrt{{\frac{a{{\rm e}^{4\,x}}}{ \left ({{\rm e}^{2\,x}}+1 \right ) ^{4}}}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.68946, size = 88, normalized size = 1.02 \begin{align*} \frac{{\left (9 \, \sqrt{a} e^{\left (-2 \, x\right )} + 45 \, \sqrt{a} e^{\left (-4 \, x\right )} - 45 \, \sqrt{a} e^{\left (-8 \, x\right )} - 9 \, \sqrt{a} e^{\left (-10 \, x\right )} - \sqrt{a} e^{\left (-12 \, x\right )} + \sqrt{a}\right )} e^{\left (6 \, x\right )}}{384 \, a^{2}} + \frac{5 \, x}{16 \, a^{\frac{3}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.64698, size = 3687, normalized size = 42.87 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (a \operatorname{sech}^{4}{\left (x \right )}\right )^{\frac{3}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.13889, size = 70, normalized size = 0.81 \begin{align*} -\frac{{\left (110 \, e^{\left (6 \, x\right )} + 45 \, e^{\left (4 \, x\right )} + 9 \, e^{\left (2 \, x\right )} + 1\right )} e^{\left (-6 \, x\right )} - 120 \, x - e^{\left (6 \, x\right )} - 9 \, e^{\left (4 \, x\right )} - 45 \, e^{\left (2 \, x\right )}}{384 \, a^{\frac{3}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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