Optimal. Leaf size=94 \[ \frac{64 a^3 (7 A+5 B) \sinh (x)}{105 \sqrt{a \cosh (x)+a}}+\frac{16}{105} a^2 (7 A+5 B) \sinh (x) \sqrt{a \cosh (x)+a}+\frac{2}{35} a (7 A+5 B) \sinh (x) (a \cosh (x)+a)^{3/2}+\frac{2}{7} B \sinh (x) (a \cosh (x)+a)^{5/2} \]
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Rubi [A] time = 0.0922257, antiderivative size = 94, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176, Rules used = {2751, 2647, 2646} \[ \frac{64 a^3 (7 A+5 B) \sinh (x)}{105 \sqrt{a \cosh (x)+a}}+\frac{16}{105} a^2 (7 A+5 B) \sinh (x) \sqrt{a \cosh (x)+a}+\frac{2}{35} a (7 A+5 B) \sinh (x) (a \cosh (x)+a)^{3/2}+\frac{2}{7} B \sinh (x) (a \cosh (x)+a)^{5/2} \]
Antiderivative was successfully verified.
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Rule 2751
Rule 2647
Rule 2646
Rubi steps
\begin{align*} \int (a+a \cosh (x))^{5/2} (A+B \cosh (x)) \, dx &=\frac{2}{7} B (a+a \cosh (x))^{5/2} \sinh (x)+\frac{1}{7} (7 A+5 B) \int (a+a \cosh (x))^{5/2} \, dx\\ &=\frac{2}{35} a (7 A+5 B) (a+a \cosh (x))^{3/2} \sinh (x)+\frac{2}{7} B (a+a \cosh (x))^{5/2} \sinh (x)+\frac{1}{35} (8 a (7 A+5 B)) \int (a+a \cosh (x))^{3/2} \, dx\\ &=\frac{16}{105} a^2 (7 A+5 B) \sqrt{a+a \cosh (x)} \sinh (x)+\frac{2}{35} a (7 A+5 B) (a+a \cosh (x))^{3/2} \sinh (x)+\frac{2}{7} B (a+a \cosh (x))^{5/2} \sinh (x)+\frac{1}{105} \left (32 a^2 (7 A+5 B)\right ) \int \sqrt{a+a \cosh (x)} \, dx\\ &=\frac{64 a^3 (7 A+5 B) \sinh (x)}{105 \sqrt{a+a \cosh (x)}}+\frac{16}{105} a^2 (7 A+5 B) \sqrt{a+a \cosh (x)} \sinh (x)+\frac{2}{35} a (7 A+5 B) (a+a \cosh (x))^{3/2} \sinh (x)+\frac{2}{7} B (a+a \cosh (x))^{5/2} \sinh (x)\\ \end{align*}
Mathematica [A] time = 0.119413, size = 60, normalized size = 0.64 \[ \frac{1}{210} a^2 \tanh \left (\frac{x}{2}\right ) \sqrt{a (\cosh (x)+1)} ((392 A+505 B) \cosh (x)+6 (7 A+20 B) \cosh (2 x)+1246 A+15 B \cosh (3 x)+1040 B) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.046, size = 71, normalized size = 0.8 \begin{align*}{\frac{8\,{a}^{3}\sqrt{2}}{105}\cosh \left ({\frac{x}{2}} \right ) \sinh \left ({\frac{x}{2}} \right ) \left ( 30\,B \left ( \sinh \left ( x/2 \right ) \right ) ^{6}+ \left ( 21\,A+105\,B \right ) \left ( \sinh \left ({\frac{x}{2}} \right ) \right ) ^{4}+ \left ( 70\,A+140\,B \right ) \left ( \sinh \left ({\frac{x}{2}} \right ) \right ) ^{2}+105\,A+105\,B \right ){\frac{1}{\sqrt{a \left ( \cosh \left ({\frac{x}{2}} \right ) \right ) ^{2}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.6449, size = 320, normalized size = 3.4 \begin{align*} \frac{1}{60} \,{\left (3 \, \sqrt{2} a^{\frac{5}{2}} e^{\left (\frac{5}{2} \, x\right )} + 25 \, \sqrt{2} a^{\frac{5}{2}} e^{\left (\frac{3}{2} \, x\right )} + 150 \, \sqrt{2} a^{\frac{5}{2}} e^{\left (\frac{1}{2} \, x\right )} - 150 \, \sqrt{2} a^{\frac{5}{2}} e^{\left (-\frac{1}{2} \, x\right )} - 25 \, \sqrt{2} a^{\frac{5}{2}} e^{\left (-\frac{3}{2} \, x\right )} - 3 \, \sqrt{2} a^{\frac{5}{2}} e^{\left (-\frac{5}{2} \, x\right )}\right )} A + \frac{1}{168} \,{\left ({\left (3 \, \sqrt{2} a^{\frac{5}{2}} e^{\left (-x\right )} + 21 \, \sqrt{2} a^{\frac{5}{2}} e^{\left (-2 \, x\right )} + 70 \, \sqrt{2} a^{\frac{5}{2}} e^{\left (-3 \, x\right )} + 210 \, \sqrt{2} a^{\frac{5}{2}} e^{\left (-4 \, x\right )} - 105 \, \sqrt{2} a^{\frac{5}{2}} e^{\left (-5 \, x\right )} - 7 \, \sqrt{2} a^{\frac{5}{2}} e^{\left (-6 \, x\right )}\right )} e^{\left (\frac{9}{2} \, x\right )} +{\left (7 \, \sqrt{2} a^{\frac{5}{2}} e^{\left (-x\right )} + 105 \, \sqrt{2} a^{\frac{5}{2}} e^{\left (-2 \, x\right )} - 210 \, \sqrt{2} a^{\frac{5}{2}} e^{\left (-3 \, x\right )} - 70 \, \sqrt{2} a^{\frac{5}{2}} e^{\left (-4 \, x\right )} - 21 \, \sqrt{2} a^{\frac{5}{2}} e^{\left (-5 \, x\right )} - 3 \, \sqrt{2} a^{\frac{5}{2}} e^{\left (-6 \, x\right )}\right )} e^{\left (\frac{5}{2} \, x\right )}\right )} B \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.26143, size = 1553, normalized size = 16.52 \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 = 1.24837, size = 207, normalized size = 2.2 \begin{align*} -\frac{1}{840} \, \sqrt{2}{\left (\frac{{\left (2100 \, A a^{6} e^{\left (3 \, x\right )} + 1575 \, B a^{6} e^{\left (3 \, x\right )} + 350 \, A a^{6} e^{\left (2 \, x\right )} + 385 \, B a^{6} e^{\left (2 \, x\right )} + 42 \, A a^{6} e^{x} + 105 \, B a^{6} e^{x} + 15 \, B a^{6}\right )} e^{\left (-\frac{7}{2} \, x\right )}}{a^{\frac{7}{2}}} - \frac{15 \, B a^{\frac{19}{2}} e^{\left (\frac{7}{2} \, x\right )} + 42 \, A a^{\frac{19}{2}} e^{\left (\frac{5}{2} \, x\right )} + 105 \, B a^{\frac{19}{2}} e^{\left (\frac{5}{2} \, x\right )} + 350 \, A a^{\frac{19}{2}} e^{\left (\frac{3}{2} \, x\right )} + 385 \, B a^{\frac{19}{2}} e^{\left (\frac{3}{2} \, x\right )} + 2100 \, A a^{\frac{19}{2}} e^{\left (\frac{1}{2} \, x\right )} + 1575 \, B a^{\frac{19}{2}} e^{\left (\frac{1}{2} \, x\right )}}{a^{7}}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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