Optimal. Leaf size=94 \[ -\frac{(3 A-5 B) \tan ^{-1}\left (\frac{\sqrt{a} \sinh (x)}{\sqrt{2} \sqrt{a-a \cosh (x)}}\right )}{16 \sqrt{2} a^{5/2}}-\frac{(3 A-5 B) \sinh (x)}{16 a (a-a \cosh (x))^{3/2}}-\frac{(A+B) \sinh (x)}{4 (a-a \cosh (x))^{5/2}} \]
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Rubi [A] time = 0.095537, antiderivative size = 94, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {2750, 2650, 2649, 206} \[ -\frac{(3 A-5 B) \tan ^{-1}\left (\frac{\sqrt{a} \sinh (x)}{\sqrt{2} \sqrt{a-a \cosh (x)}}\right )}{16 \sqrt{2} a^{5/2}}-\frac{(3 A-5 B) \sinh (x)}{16 a (a-a \cosh (x))^{3/2}}-\frac{(A+B) \sinh (x)}{4 (a-a \cosh (x))^{5/2}} \]
Antiderivative was successfully verified.
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Rule 2750
Rule 2650
Rule 2649
Rule 206
Rubi steps
\begin{align*} \int \frac{A+B \cosh (x)}{(a-a \cosh (x))^{5/2}} \, dx &=-\frac{(A+B) \sinh (x)}{4 (a-a \cosh (x))^{5/2}}+\frac{(3 A-5 B) \int \frac{1}{(a-a \cosh (x))^{3/2}} \, dx}{8 a}\\ &=-\frac{(A+B) \sinh (x)}{4 (a-a \cosh (x))^{5/2}}-\frac{(3 A-5 B) \sinh (x)}{16 a (a-a \cosh (x))^{3/2}}+\frac{(3 A-5 B) \int \frac{1}{\sqrt{a-a \cosh (x)}} \, dx}{32 a^2}\\ &=-\frac{(A+B) \sinh (x)}{4 (a-a \cosh (x))^{5/2}}-\frac{(3 A-5 B) \sinh (x)}{16 a (a-a \cosh (x))^{3/2}}+\frac{(i (3 A-5 B)) \operatorname{Subst}\left (\int \frac{1}{2 a-x^2} \, dx,x,\frac{i a \sinh (x)}{\sqrt{a-a \cosh (x)}}\right )}{16 a^2}\\ &=-\frac{(3 A-5 B) \tan ^{-1}\left (\frac{\sqrt{a} \sinh (x)}{\sqrt{2} \sqrt{a-a \cosh (x)}}\right )}{16 \sqrt{2} a^{5/2}}-\frac{(A+B) \sinh (x)}{4 (a-a \cosh (x))^{5/2}}-\frac{(3 A-5 B) \sinh (x)}{16 a (a-a \cosh (x))^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.360171, size = 108, normalized size = 1.15 \[ \frac{\sinh ^5\left (\frac{x}{2}\right ) \left (-(A+B) \text{csch}^4\left (\frac{x}{4}\right )+2 (3 A-5 B) \text{csch}^2\left (\frac{x}{4}\right )+(A+B) \text{sech}^4\left (\frac{x}{4}\right )+2 (3 A-5 B) \text{sech}^2\left (\frac{x}{4}\right )+8 (3 A-5 B) \log \left (\tanh \left (\frac{x}{4}\right )\right )\right )}{32 a^2 (\cosh (x)-1)^2 \sqrt{a-a \cosh (x)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.07, size = 118, normalized size = 1.3 \begin{align*} -{\frac{1}{32\,{a}^{2}} \left ( \left ( -6\,A+10\,B \right ) \cosh \left ({\frac{x}{2}} \right ) \left ( \sinh \left ({\frac{x}{2}} \right ) \right ) ^{2}+ \left ( 4\,A+4\,B \right ) \cosh \left ({\frac{x}{2}} \right ) + \left ( 3\,\ln \left ( 1+\cosh \left ( x/2 \right ) \right ) A-3\,\ln \left ( -1+\cosh \left ( x/2 \right ) \right ) A-5\,B\ln \left ( 1+\cosh \left ( x/2 \right ) \right ) +5\,B\ln \left ( -1+\cosh \left ( x/2 \right ) \right ) \right ) \left ( \sinh \left ({\frac{x}{2}} \right ) \right ) ^{4} \right ) \left ( 1+\cosh \left ({\frac{x}{2}} \right ) \right ) ^{-1} \left ( -1+\cosh \left ({\frac{x}{2}} \right ) \right ) ^{-1} \left ( \sinh \left ({\frac{x}{2}} \right ) \right ) ^{-1}{\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.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{B \cosh \left (x\right ) + A}{{\left (-a \cosh \left (x\right ) + a\right )}^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.24865, 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 [B] time = 1.33991, size = 255, normalized size = 2.71 \begin{align*} -\frac{\sqrt{2}{\left (3 \, A - 5 \, B\right )} \arctan \left (\frac{\sqrt{-a e^{x}}}{\sqrt{a}}\right )}{16 \, a^{\frac{5}{2}} \mathrm{sgn}\left (-e^{x} + 1\right )} + \frac{\sqrt{2}{\left (3 \, \sqrt{-a e^{x}} A a^{3} e^{\left (3 \, x\right )} - 5 \, \sqrt{-a e^{x}} B a^{3} e^{\left (3 \, x\right )} - 11 \, \sqrt{-a e^{x}} A a^{3} e^{\left (2 \, x\right )} - 3 \, \sqrt{-a e^{x}} B a^{3} e^{\left (2 \, x\right )} - 11 \, \sqrt{-a e^{x}} A a^{3} e^{x} - 3 \, \sqrt{-a e^{x}} B a^{3} e^{x} + 3 \, \sqrt{-a e^{x}} A a^{3} - 5 \, \sqrt{-a e^{x}} B a^{3}\right )}}{16 \,{\left (a e^{x} - a\right )}^{4} a^{2} \mathrm{sgn}\left (-e^{x} + 1\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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