Optimal. Leaf size=19 \[ -\frac {\coth ^2(x)}{2 b (a \coth (x)+b)^2} \]
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Rubi [A] time = 0.03, antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {3088, 37} \[ -\frac {\coth ^2(x)}{2 b (a \coth (x)+b)^2} \]
Antiderivative was successfully verified.
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Rule 37
Rule 3088
Rubi steps
\begin {align*} \int \frac {\cosh (x)}{(a \cosh (x)+b \sinh (x))^3} \, dx &=i \operatorname {Subst}\left (\int \frac {x}{(-i b+a x)^3} \, dx,x,-i \coth (x)\right )\\ &=-\frac {\coth ^2(x)}{2 b (b+a \coth (x))^2}\\ \end {align*}
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Mathematica [B] time = 0.06, size = 40, normalized size = 2.11 \[ \frac {a \sinh (2 x)+b \cosh (2 x)}{2 (a-b) (a+b) (a \cosh (x)+b \sinh (x))^2} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.42, size = 216, normalized size = 11.37 \[ -\frac {2 \, {\left ({\left (2 \, a + b\right )} \cosh \relax (x) + b \sinh \relax (x)\right )}}{{\left (a^{4} + 4 \, a^{3} b + 6 \, a^{2} b^{2} + 4 \, a b^{3} + b^{4}\right )} \cosh \relax (x)^{3} + 3 \, {\left (a^{4} + 4 \, a^{3} b + 6 \, a^{2} b^{2} + 4 \, a b^{3} + b^{4}\right )} \cosh \relax (x) \sinh \relax (x)^{2} + {\left (a^{4} + 4 \, a^{3} b + 6 \, a^{2} b^{2} + 4 \, a b^{3} + b^{4}\right )} \sinh \relax (x)^{3} + {\left (3 \, a^{4} + 4 \, a^{3} b - 2 \, a^{2} b^{2} - 4 \, a b^{3} - b^{4}\right )} \cosh \relax (x) + {\left (a^{4} + 4 \, a^{3} b + 2 \, a^{2} b^{2} - 4 \, a b^{3} - 3 \, b^{4} + 3 \, {\left (a^{4} + 4 \, a^{3} b + 6 \, a^{2} b^{2} + 4 \, a b^{3} + b^{4}\right )} \cosh \relax (x)^{2}\right )} \sinh \relax (x)} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.13, size = 48, normalized size = 2.53 \[ -\frac {2 \, {\left (a e^{\left (2 \, x\right )} + b e^{\left (2 \, x\right )} + a\right )}}{{\left (a^{2} + 2 \, a b + b^{2}\right )} {\left (a e^{\left (2 \, x\right )} + b e^{\left (2 \, x\right )} + a - b\right )}^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.25, size = 55, normalized size = 2.89 \[ -\frac {2 \left (-\frac {\tanh ^{3}\left (\frac {x}{2}\right )}{a}-\frac {b \left (\tanh ^{2}\left (\frac {x}{2}\right )\right )}{a^{2}}-\frac {\tanh \left (\frac {x}{2}\right )}{a}\right )}{\left (a +2 \tanh \left (\frac {x}{2}\right ) b +a \left (\tanh ^{2}\left (\frac {x}{2}\right )\right )\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.33, size = 167, normalized size = 8.79 \[ \frac {2 \, {\left (a - b\right )} e^{\left (-2 \, x\right )}}{a^{4} - 2 \, a^{2} b^{2} + b^{4} + 2 \, {\left (a^{4} - 2 \, a^{3} b + 2 \, a b^{3} - b^{4}\right )} e^{\left (-2 \, x\right )} + {\left (a^{4} - 4 \, a^{3} b + 6 \, a^{2} b^{2} - 4 \, a b^{3} + b^{4}\right )} e^{\left (-4 \, x\right )}} + \frac {2 \, a}{a^{4} - 2 \, a^{2} b^{2} + b^{4} + 2 \, {\left (a^{4} - 2 \, a^{3} b + 2 \, a b^{3} - b^{4}\right )} e^{\left (-2 \, x\right )} + {\left (a^{4} - 4 \, a^{3} b + 6 \, a^{2} b^{2} - 4 \, a b^{3} + b^{4}\right )} e^{\left (-4 \, x\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.52, size = 42, normalized size = 2.21 \[ -\frac {2\,a+{\mathrm {e}}^{2\,x}\,\left (2\,a+2\,b\right )}{{\left (a+b\right )}^2\,{\left (a-b+a\,{\mathrm {e}}^{2\,x}+b\,{\mathrm {e}}^{2\,x}\right )}^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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