Optimal. Leaf size=67 \[ \frac{2 \sinh (x)}{3 a \left (a^2-b^2\right ) (a \cosh (x)+b \sinh (x))}+\frac{a \sinh (x)+b \cosh (x)}{3 \left (a^2-b^2\right ) (a \cosh (x)+b \sinh (x))^3} \]
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Rubi [A] time = 0.0315738, antiderivative size = 67, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {3076, 3075} \[ \frac{2 \sinh (x)}{3 a \left (a^2-b^2\right ) (a \cosh (x)+b \sinh (x))}+\frac{a \sinh (x)+b \cosh (x)}{3 \left (a^2-b^2\right ) (a \cosh (x)+b \sinh (x))^3} \]
Antiderivative was successfully verified.
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Rule 3076
Rule 3075
Rubi steps
\begin{align*} \int \frac{1}{(a \cosh (x)+b \sinh (x))^4} \, dx &=\frac{b \cosh (x)+a \sinh (x)}{3 \left (a^2-b^2\right ) (a \cosh (x)+b \sinh (x))^3}+\frac{2 \int \frac{1}{(a \cosh (x)+b \sinh (x))^2} \, dx}{3 \left (a^2-b^2\right )}\\ &=\frac{b \cosh (x)+a \sinh (x)}{3 \left (a^2-b^2\right ) (a \cosh (x)+b \sinh (x))^3}+\frac{2 \sinh (x)}{3 a \left (a^2-b^2\right ) (a \cosh (x)+b \sinh (x))}\\ \end{align*}
Mathematica [A] time = 0.132782, size = 64, normalized size = 0.96 \[ \frac{\sinh (x) \left (\left (a^2+b^2\right ) \cosh (2 x)+2 a^2-b^2\right )+a b \cosh (3 x)}{3 a (a-b) (a+b) (a \cosh (x)+b \sinh (x))^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.081, size = 87, normalized size = 1.3 \begin{align*} -2\,{\frac{1}{ \left ( a+2\,\tanh \left ( x/2 \right ) b+a \left ( \tanh \left ( x/2 \right ) \right ) ^{2} \right ) ^{3}} \left ( -{\frac{ \left ( \tanh \left ( x/2 \right ) \right ) ^{5}}{a}}-2\,{\frac{b \left ( \tanh \left ( x/2 \right ) \right ) ^{4}}{{a}^{2}}}-2/3\,{\frac{ \left ({a}^{2}+2\,{b}^{2} \right ) \left ( \tanh \left ( x/2 \right ) \right ) ^{3}}{{a}^{3}}}-2\,{\frac{ \left ( \tanh \left ( x/2 \right ) \right ) ^{2}b}{{a}^{2}}}-{\frac{\tanh \left ( x/2 \right ) }{a}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.12521, size = 672, normalized size = 10.03 \begin{align*} \frac{4 \,{\left (a - b\right )} e^{\left (-2 \, x\right )}}{a^{5} + a^{4} b - 2 \, a^{3} b^{2} - 2 \, a^{2} b^{3} + a b^{4} + b^{5} + 3 \,{\left (a^{5} - a^{4} b - 2 \, a^{3} b^{2} + 2 \, a^{2} b^{3} + a b^{4} - b^{5}\right )} e^{\left (-2 \, x\right )} + 3 \,{\left (a^{5} - 3 \, a^{4} b + 2 \, a^{3} b^{2} + 2 \, a^{2} b^{3} - 3 \, a b^{4} + b^{5}\right )} e^{\left (-4 \, x\right )} +{\left (a^{5} - 5 \, a^{4} b + 10 \, a^{3} b^{2} - 10 \, a^{2} b^{3} + 5 \, a b^{4} - b^{5}\right )} e^{\left (-6 \, x\right )}} + \frac{4 \, a}{3 \,{\left (a^{5} + a^{4} b - 2 \, a^{3} b^{2} - 2 \, a^{2} b^{3} + a b^{4} + b^{5} + 3 \,{\left (a^{5} - a^{4} b - 2 \, a^{3} b^{2} + 2 \, a^{2} b^{3} + a b^{4} - b^{5}\right )} e^{\left (-2 \, x\right )} + 3 \,{\left (a^{5} - 3 \, a^{4} b + 2 \, a^{3} b^{2} + 2 \, a^{2} b^{3} - 3 \, a b^{4} + b^{5}\right )} e^{\left (-4 \, x\right )} +{\left (a^{5} - 5 \, a^{4} b + 10 \, a^{3} b^{2} - 10 \, a^{2} b^{3} + 5 \, a b^{4} - b^{5}\right )} e^{\left (-6 \, x\right )}\right )}} + \frac{4 \, b}{3 \,{\left (a^{5} + a^{4} b - 2 \, a^{3} b^{2} - 2 \, a^{2} b^{3} + a b^{4} + b^{5} + 3 \,{\left (a^{5} - a^{4} b - 2 \, a^{3} b^{2} + 2 \, a^{2} b^{3} + a b^{4} - b^{5}\right )} e^{\left (-2 \, x\right )} + 3 \,{\left (a^{5} - 3 \, a^{4} b + 2 \, a^{3} b^{2} + 2 \, a^{2} b^{3} - 3 \, a b^{4} + b^{5}\right )} e^{\left (-4 \, x\right )} +{\left (a^{5} - 5 \, a^{4} b + 10 \, a^{3} b^{2} - 10 \, a^{2} b^{3} + 5 \, a b^{4} - b^{5}\right )} e^{\left (-6 \, x\right )}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.30457, size = 1234, normalized size = 18.42 \begin{align*} -\frac{8 \,{\left ({\left (2 \, a + b\right )} \cosh \left (x\right ) +{\left (a + 2 \, b\right )} \sinh \left (x\right )\right )}}{3 \,{\left ({\left (a^{5} + 5 \, a^{4} b + 10 \, a^{3} b^{2} + 10 \, a^{2} b^{3} + 5 \, a b^{4} + b^{5}\right )} \cosh \left (x\right )^{5} + 5 \,{\left (a^{5} + 5 \, a^{4} b + 10 \, a^{3} b^{2} + 10 \, a^{2} b^{3} + 5 \, a b^{4} + b^{5}\right )} \cosh \left (x\right ) \sinh \left (x\right )^{4} +{\left (a^{5} + 5 \, a^{4} b + 10 \, a^{3} b^{2} + 10 \, a^{2} b^{3} + 5 \, a b^{4} + b^{5}\right )} \sinh \left (x\right )^{5} + 3 \,{\left (a^{5} + 3 \, a^{4} b + 2 \, a^{3} b^{2} - 2 \, a^{2} b^{3} - 3 \, a b^{4} - b^{5}\right )} \cosh \left (x\right )^{3} +{\left (3 \, a^{5} + 9 \, a^{4} b + 6 \, a^{3} b^{2} - 6 \, a^{2} b^{3} - 9 \, a b^{4} - 3 \, b^{5} + 10 \,{\left (a^{5} + 5 \, a^{4} b + 10 \, a^{3} b^{2} + 10 \, a^{2} b^{3} + 5 \, a b^{4} + b^{5}\right )} \cosh \left (x\right )^{2}\right )} \sinh \left (x\right )^{3} +{\left (10 \,{\left (a^{5} + 5 \, a^{4} b + 10 \, a^{3} b^{2} + 10 \, a^{2} b^{3} + 5 \, a b^{4} + b^{5}\right )} \cosh \left (x\right )^{3} + 9 \,{\left (a^{5} + 3 \, a^{4} b + 2 \, a^{3} b^{2} - 2 \, a^{2} b^{3} - 3 \, a b^{4} - b^{5}\right )} \cosh \left (x\right )\right )} \sinh \left (x\right )^{2} + 2 \,{\left (2 \, a^{5} + a^{4} b - 4 \, a^{3} b^{2} - 2 \, a^{2} b^{3} + 2 \, a b^{4} + b^{5}\right )} \cosh \left (x\right ) +{\left (2 \, a^{5} + 4 \, a^{4} b - 4 \, a^{3} b^{2} - 8 \, a^{2} b^{3} + 2 \, a b^{4} + 4 \, b^{5} + 5 \,{\left (a^{5} + 5 \, a^{4} b + 10 \, a^{3} b^{2} + 10 \, a^{2} b^{3} + 5 \, a b^{4} + b^{5}\right )} \cosh \left (x\right )^{4} + 9 \,{\left (a^{5} + 3 \, a^{4} b + 2 \, a^{3} b^{2} - 2 \, a^{2} b^{3} - 3 \, a b^{4} - b^{5}\right )} \cosh \left (x\right )^{2}\right )} \sinh \left (x\right )\right )}} \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 = 1.14393, size = 72, normalized size = 1.07 \begin{align*} -\frac{4 \,{\left (3 \, a e^{\left (2 \, x\right )} + 3 \, b e^{\left (2 \, x\right )} + a - b\right )}}{3 \,{\left (a^{2} + 2 \, a b + b^{2}\right )}{\left (a e^{\left (2 \, x\right )} + b e^{\left (2 \, x\right )} + a - b\right )}^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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