Optimal. Leaf size=71 \[ -\frac{8 a^2 (5 A-3 B) \sinh (x)}{15 \sqrt{a-a \cosh (x)}}-\frac{2}{15} a (5 A-3 B) \sinh (x) \sqrt{a-a \cosh (x)}+\frac{2}{5} B \sinh (x) (a-a \cosh (x))^{3/2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0802921, antiderivative size = 71, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {2751, 2647, 2646} \[ -\frac{8 a^2 (5 A-3 B) \sinh (x)}{15 \sqrt{a-a \cosh (x)}}-\frac{2}{15} a (5 A-3 B) \sinh (x) \sqrt{a-a \cosh (x)}+\frac{2}{5} B \sinh (x) (a-a \cosh (x))^{3/2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2751
Rule 2647
Rule 2646
Rubi steps
\begin{align*} \int (a-a \cosh (x))^{3/2} (A+B \cosh (x)) \, dx &=\frac{2}{5} B (a-a \cosh (x))^{3/2} \sinh (x)-\frac{1}{5} (-5 A+3 B) \int (a-a \cosh (x))^{3/2} \, dx\\ &=-\frac{2}{15} a (5 A-3 B) \sqrt{a-a \cosh (x)} \sinh (x)+\frac{2}{5} B (a-a \cosh (x))^{3/2} \sinh (x)+\frac{1}{15} (4 a (5 A-3 B)) \int \sqrt{a-a \cosh (x)} \, dx\\ &=-\frac{8 a^2 (5 A-3 B) \sinh (x)}{15 \sqrt{a-a \cosh (x)}}-\frac{2}{15} a (5 A-3 B) \sqrt{a-a \cosh (x)} \sinh (x)+\frac{2}{5} B (a-a \cosh (x))^{3/2} \sinh (x)\\ \end{align*}
Mathematica [A] time = 0.0971045, size = 47, normalized size = 0.66 \[ -\frac{1}{15} a \coth \left (\frac{x}{2}\right ) \sqrt{a-a \cosh (x)} (2 (5 A-9 B) \cosh (x)-50 A+3 B \cosh (2 x)+39 B) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.049, size = 55, normalized size = 0.8 \begin{align*}{\frac{8\,{a}^{2}}{15}\sinh \left ({\frac{x}{2}} \right ) \cosh \left ({\frac{x}{2}} \right ) \left ( 6\,B \left ( \sinh \left ( x/2 \right ) \right ) ^{4}+ \left ( 5\,A-3\,B \right ) \left ( \sinh \left ({\frac{x}{2}} \right ) \right ) ^{2}-10\,A+6\,B \right ){\frac{1}{\sqrt{-2\, \left ( \sinh \left ( x/2 \right ) \right ) ^{2}a}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [B] time = 1.6047, size = 269, normalized size = 3.79 \begin{align*} \frac{1}{6} \,{\left (\frac{9 \, \sqrt{2} a^{\frac{3}{2}} e^{\left (-x\right )}}{\left (-e^{\left (-x\right )}\right )^{\frac{3}{2}}} + \frac{9 \, \sqrt{2} a^{\frac{3}{2}} e^{\left (-2 \, x\right )}}{\left (-e^{\left (-x\right )}\right )^{\frac{3}{2}}} - \frac{\sqrt{2} a^{\frac{3}{2}} e^{\left (-3 \, x\right )}}{\left (-e^{\left (-x\right )}\right )^{\frac{3}{2}}} - \frac{\sqrt{2} a^{\frac{3}{2}}}{\left (-e^{\left (-x\right )}\right )^{\frac{3}{2}}}\right )} A + \frac{1}{20} \, B{\left (\frac{{\left (5 \, \sqrt{2} a^{\frac{3}{2}} e^{\left (-x\right )} - 15 \, \sqrt{2} a^{\frac{3}{2}} e^{\left (-2 \, x\right )} - 5 \, \sqrt{2} a^{\frac{3}{2}} e^{\left (-3 \, x\right )} - \sqrt{2} a^{\frac{3}{2}}\right )} e^{x}}{\left (-e^{\left (-x\right )}\right )^{\frac{3}{2}}} - \frac{5 \, \sqrt{2} a^{\frac{3}{2}} e^{\left (-x\right )} + 15 \, \sqrt{2} a^{\frac{3}{2}} e^{\left (-2 \, x\right )} - 5 \, \sqrt{2} a^{\frac{3}{2}} e^{\left (-3 \, x\right )} + \sqrt{2} a^{\frac{3}{2}} e^{\left (-4 \, x\right )}}{\left (-e^{\left (-x\right )}\right )^{\frac{3}{2}}}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] time = 2.15376, size = 807, normalized size = 11.37 \begin{align*} -\frac{\sqrt{\frac{1}{2}}{\left (3 \, B a \cosh \left (x\right )^{5} + 3 \, B a \sinh \left (x\right )^{5} + 5 \,{\left (2 \, A - 3 \, B\right )} a \cosh \left (x\right )^{4} - 30 \,{\left (3 \, A - 2 \, B\right )} a \cosh \left (x\right )^{3} + 5 \,{\left (3 \, B a \cosh \left (x\right ) +{\left (2 \, A - 3 \, B\right )} a\right )} \sinh \left (x\right )^{4} - 30 \,{\left (3 \, A - 2 \, B\right )} a \cosh \left (x\right )^{2} + 10 \,{\left (3 \, B a \cosh \left (x\right )^{2} + 2 \,{\left (2 \, A - 3 \, B\right )} a \cosh \left (x\right ) - 3 \,{\left (3 \, A - 2 \, B\right )} a\right )} \sinh \left (x\right )^{3} + 5 \,{\left (2 \, A - 3 \, B\right )} a \cosh \left (x\right ) + 30 \,{\left (B a \cosh \left (x\right )^{3} +{\left (2 \, A - 3 \, B\right )} a \cosh \left (x\right )^{2} - 3 \,{\left (3 \, A - 2 \, B\right )} a \cosh \left (x\right ) -{\left (3 \, A - 2 \, B\right )} a\right )} \sinh \left (x\right )^{2} + 3 \, B a + 5 \,{\left (3 \, B a \cosh \left (x\right )^{4} + 4 \,{\left (2 \, A - 3 \, B\right )} a \cosh \left (x\right )^{3} - 18 \,{\left (3 \, A - 2 \, B\right )} a \cosh \left (x\right )^{2} - 12 \,{\left (3 \, A - 2 \, B\right )} a \cosh \left (x\right ) +{\left (2 \, A - 3 \, B\right )} a\right )} \sinh \left (x\right )\right )} \sqrt{-\frac{a}{\cosh \left (x\right ) + \sinh \left (x\right )}}}{30 \,{\left (\cosh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Giac [B] time = 1.25391, size = 286, normalized size = 4.03 \begin{align*} \frac{1}{60} \, \sqrt{2}{\left (\frac{{\left (90 \, A a^{4} e^{\left (2 \, x\right )} \mathrm{sgn}\left (-e^{x} + 1\right ) - 60 \, B a^{4} e^{\left (2 \, x\right )} \mathrm{sgn}\left (-e^{x} + 1\right ) - 10 \, A a^{4} e^{x} \mathrm{sgn}\left (-e^{x} + 1\right ) + 15 \, B a^{4} e^{x} \mathrm{sgn}\left (-e^{x} + 1\right ) - 3 \, B a^{4} \mathrm{sgn}\left (-e^{x} + 1\right )\right )} e^{\left (-2 \, x\right )}}{\sqrt{-a e^{x}} a^{2}} + \frac{3 \, \sqrt{-a e^{x}} B a^{6} e^{\left (2 \, x\right )} \mathrm{sgn}\left (-e^{x} + 1\right ) + 10 \, \sqrt{-a e^{x}} A a^{6} e^{x} \mathrm{sgn}\left (-e^{x} + 1\right ) - 15 \, \sqrt{-a e^{x}} B a^{6} e^{x} \mathrm{sgn}\left (-e^{x} + 1\right ) - 90 \, \sqrt{-a e^{x}} A a^{6} \mathrm{sgn}\left (-e^{x} + 1\right ) + 60 \, \sqrt{-a e^{x}} B a^{6} \mathrm{sgn}\left (-e^{x} + 1\right )}{a^{5}}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]