Optimal. Leaf size=78 \[ \frac{1}{6} a \sinh (x) \cosh ^3(x) \sqrt{a \cosh ^4(x)}+\frac{5}{24} a \sinh (x) \cosh (x) \sqrt{a \cosh ^4(x)}+\frac{5}{16} a \tanh (x) \sqrt{a \cosh ^4(x)}+\frac{5}{16} a x \text{sech}^2(x) \sqrt{a \cosh ^4(x)} \]
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Rubi [A] time = 0.0352139, antiderivative size = 78, 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 = {3207, 2635, 8} \[ \frac{1}{6} a \sinh (x) \cosh ^3(x) \sqrt{a \cosh ^4(x)}+\frac{5}{24} a \sinh (x) \cosh (x) \sqrt{a \cosh ^4(x)}+\frac{5}{16} a \tanh (x) \sqrt{a \cosh ^4(x)}+\frac{5}{16} a x \text{sech}^2(x) \sqrt{a \cosh ^4(x)} \]
Antiderivative was successfully verified.
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Rule 3207
Rule 2635
Rule 8
Rubi steps
\begin{align*} \int \left (a \cosh ^4(x)\right )^{3/2} \, dx &=\left (a \sqrt{a \cosh ^4(x)} \text{sech}^2(x)\right ) \int \cosh ^6(x) \, dx\\ &=\frac{1}{6} a \cosh ^3(x) \sqrt{a \cosh ^4(x)} \sinh (x)+\frac{1}{6} \left (5 a \sqrt{a \cosh ^4(x)} \text{sech}^2(x)\right ) \int \cosh ^4(x) \, dx\\ &=\frac{5}{24} a \cosh (x) \sqrt{a \cosh ^4(x)} \sinh (x)+\frac{1}{6} a \cosh ^3(x) \sqrt{a \cosh ^4(x)} \sinh (x)+\frac{1}{8} \left (5 a \sqrt{a \cosh ^4(x)} \text{sech}^2(x)\right ) \int \cosh ^2(x) \, dx\\ &=\frac{5}{24} a \cosh (x) \sqrt{a \cosh ^4(x)} \sinh (x)+\frac{1}{6} a \cosh ^3(x) \sqrt{a \cosh ^4(x)} \sinh (x)+\frac{5}{16} a \sqrt{a \cosh ^4(x)} \tanh (x)+\frac{1}{16} \left (5 a \sqrt{a \cosh ^4(x)} \text{sech}^2(x)\right ) \int 1 \, dx\\ &=\frac{5}{16} a x \sqrt{a \cosh ^4(x)} \text{sech}^2(x)+\frac{5}{24} a \cosh (x) \sqrt{a \cosh ^4(x)} \sinh (x)+\frac{1}{6} a \cosh ^3(x) \sqrt{a \cosh ^4(x)} \sinh (x)+\frac{5}{16} a \sqrt{a \cosh ^4(x)} \tanh (x)\\ \end{align*}
Mathematica [A] time = 0.0636691, size = 38, normalized size = 0.49 \[ \frac{1}{192} (60 x+45 \sinh (2 x)+9 \sinh (4 x)+\sinh (6 x)) \text{sech}^6(x) \left (a \cosh ^4(x)\right )^{3/2} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.111, size = 131, normalized size = 1.7 \begin{align*}{\frac{\sqrt{8} \left ( \cosh \left ( 2\,x \right ) +1 \right ) \sqrt{2}}{384\,\sinh \left ( 2\,x \right ) }\sqrt{a \left ( -1+\cosh \left ( 2\,x \right ) \right ) \left ( \cosh \left ( 2\,x \right ) +1 \right ) }\sqrt{a} \left ( 2\,\sqrt{a \left ( \sinh \left ( 2\,x \right ) \right ) ^{2}}\sqrt{a} \left ( \sinh \left ( 2\,x \right ) \right ) ^{2}+9\,\cosh \left ( 2\,x \right ) \sqrt{a \left ( \sinh \left ( 2\,x \right ) \right ) ^{2}}\sqrt{a}+24\,\sqrt{a \left ( \sinh \left ( 2\,x \right ) \right ) ^{2}}\sqrt{a}+15\,\ln \left ( \sqrt{a}\cosh \left ( 2\,x \right ) +\sqrt{a \left ( \sinh \left ( 2\,x \right ) \right ) ^{2}} \right ) a \right ){\frac{1}{\sqrt{a \left ( \cosh \left ( 2\,x \right ) +1 \right ) ^{2}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.65225, size = 84, normalized size = 1.08 \begin{align*} \frac{5}{16} \, a^{\frac{3}{2}} x + \frac{1}{384} \,{\left (9 \, a^{\frac{3}{2}} e^{\left (-2 \, x\right )} + 45 \, a^{\frac{3}{2}} e^{\left (-4 \, x\right )} - 45 \, a^{\frac{3}{2}} e^{\left (-8 \, x\right )} - 9 \, a^{\frac{3}{2}} e^{\left (-10 \, x\right )} - a^{\frac{3}{2}} e^{\left (-12 \, x\right )} + a^{\frac{3}{2}}\right )} e^{\left (6 \, x\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.1175, size = 2068, normalized size = 26.51 \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(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.1516, size = 70, normalized size = 0.9 \begin{align*} -\frac{1}{384} \,{\left ({\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 )}\right )} a^{\frac{3}{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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