Optimal. Leaf size=100 \[ \frac {b \sinh (x)+c \cosh (x)}{3 \sqrt {b^2-c^2} \left (\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)\right )^2}-\frac {\sqrt {b^2-c^2} \sinh (x)+c}{3 c \sqrt {b^2-c^2} (b \sinh (x)+c \cosh (x))} \]
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Rubi [A] time = 0.08, antiderivative size = 100, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {3116, 3114} \[ \frac {b \sinh (x)+c \cosh (x)}{3 \sqrt {b^2-c^2} \left (\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)\right )^2}-\frac {\sqrt {b^2-c^2} \sinh (x)+c}{3 c \sqrt {b^2-c^2} (b \sinh (x)+c \cosh (x))} \]
Antiderivative was successfully verified.
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Rule 3114
Rule 3116
Rubi steps
\begin {align*} \int \frac {1}{\left (\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)\right )^2} \, dx &=\frac {c \cosh (x)+b \sinh (x)}{3 \sqrt {b^2-c^2} \left (\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)\right )^2}+\frac {\int \frac {1}{\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)} \, dx}{3 \sqrt {b^2-c^2}}\\ &=\frac {c \cosh (x)+b \sinh (x)}{3 \sqrt {b^2-c^2} \left (\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)\right )^2}-\frac {c+\sqrt {b^2-c^2} \sinh (x)}{3 c \sqrt {b^2-c^2} (c \cosh (x)+b \sinh (x))}\\ \end {align*}
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Mathematica [A] time = 0.17, size = 68, normalized size = 0.68 \[ -\frac {-2 c \sqrt {b^2-c^2}+b^2 \sinh ^3(x)+2 b c \cosh ^3(x)+2 c^2 \sinh (x)+c^2 \sinh (x) \cosh ^2(x)}{3 c (b \sinh (x)+c \cosh (x))^3} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.43, size = 660, normalized size = 6.60 \[ -\frac {2 \, {\left (3 \, {\left (b^{2} + 2 \, b c + c^{2}\right )} \cosh \relax (x)^{4} + 12 \, {\left (b^{2} + 2 \, b c + c^{2}\right )} \cosh \relax (x) \sinh \relax (x)^{3} + 3 \, {\left (b^{2} + 2 \, b c + c^{2}\right )} \sinh \relax (x)^{4} + 6 \, {\left (b^{2} - c^{2}\right )} \cosh \relax (x)^{2} + 6 \, {\left (3 \, {\left (b^{2} + 2 \, b c + c^{2}\right )} \cosh \relax (x)^{2} + b^{2} - c^{2}\right )} \sinh \relax (x)^{2} - b^{2} + 2 \, b c - c^{2} + 12 \, {\left ({\left (b^{2} + 2 \, b c + c^{2}\right )} \cosh \relax (x)^{3} + {\left (b^{2} - c^{2}\right )} \cosh \relax (x)\right )} \sinh \relax (x) - 8 \, {\left ({\left (b + c\right )} \cosh \relax (x)^{3} + 3 \, {\left (b + c\right )} \cosh \relax (x)^{2} \sinh \relax (x) + 3 \, {\left (b + c\right )} \cosh \relax (x) \sinh \relax (x)^{2} + {\left (b + c\right )} \sinh \relax (x)^{3}\right )} \sqrt {b^{2} - c^{2}}\right )}}{3 \, {\left ({\left (b^{4} + 4 \, b^{3} c + 6 \, b^{2} c^{2} + 4 \, b c^{3} + c^{4}\right )} \cosh \relax (x)^{6} + 6 \, {\left (b^{4} + 4 \, b^{3} c + 6 \, b^{2} c^{2} + 4 \, b c^{3} + c^{4}\right )} \cosh \relax (x) \sinh \relax (x)^{5} + {\left (b^{4} + 4 \, b^{3} c + 6 \, b^{2} c^{2} + 4 \, b c^{3} + c^{4}\right )} \sinh \relax (x)^{6} - 3 \, {\left (b^{4} + 2 \, b^{3} c - 2 \, b c^{3} - c^{4}\right )} \cosh \relax (x)^{4} - 3 \, {\left (b^{4} + 2 \, b^{3} c - 2 \, b c^{3} - c^{4} - 5 \, {\left (b^{4} + 4 \, b^{3} c + 6 \, b^{2} c^{2} + 4 \, b c^{3} + c^{4}\right )} \cosh \relax (x)^{2}\right )} \sinh \relax (x)^{4} - b^{4} + 2 \, b^{3} c - 2 \, b c^{3} + c^{4} + 4 \, {\left (5 \, {\left (b^{4} + 4 \, b^{3} c + 6 \, b^{2} c^{2} + 4 \, b c^{3} + c^{4}\right )} \cosh \relax (x)^{3} - 3 \, {\left (b^{4} + 2 \, b^{3} c - 2 \, b c^{3} - c^{4}\right )} \cosh \relax (x)\right )} \sinh \relax (x)^{3} + 3 \, {\left (b^{4} - 2 \, b^{2} c^{2} + c^{4}\right )} \cosh \relax (x)^{2} + 3 \, {\left (5 \, {\left (b^{4} + 4 \, b^{3} c + 6 \, b^{2} c^{2} + 4 \, b c^{3} + c^{4}\right )} \cosh \relax (x)^{4} + b^{4} - 2 \, b^{2} c^{2} + c^{4} - 6 \, {\left (b^{4} + 2 \, b^{3} c - 2 \, b c^{3} - c^{4}\right )} \cosh \relax (x)^{2}\right )} \sinh \relax (x)^{2} + 6 \, {\left ({\left (b^{4} + 4 \, b^{3} c + 6 \, b^{2} c^{2} + 4 \, b c^{3} + c^{4}\right )} \cosh \relax (x)^{5} - 2 \, {\left (b^{4} + 2 \, b^{3} c - 2 \, b c^{3} - c^{4}\right )} \cosh \relax (x)^{3} + {\left (b^{4} - 2 \, b^{2} c^{2} + c^{4}\right )} \cosh \relax (x)\right )} \sinh \relax (x)\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.36, size = 217, normalized size = 2.17 \[ \frac {2 \left (\sqrt {b^{2}-c^{2}}+b \right ) \left (\frac {\left (\sqrt {b^{2}-c^{2}}+b \right ) \left (\tanh ^{2}\left (\frac {x}{2}\right )\right )}{c^{2}}+\frac {\left (2 b^{2}-c^{2}+2 \sqrt {b^{2}-c^{2}}\, b \right ) \tanh \left (\frac {x}{2}\right )}{c^{3}}+\frac {\frac {4 \sqrt {b^{2}-c^{2}}\, b^{2}}{3}-\frac {2 \sqrt {b^{2}-c^{2}}\, c^{2}}{3}+\frac {4 b^{3}}{3}-\frac {4 b \,c^{2}}{3}}{c^{4}}\right )}{c^{2} \left (\tanh ^{2}\left (\frac {x}{2}\right )+\frac {2 \sqrt {\left (b -c \right ) \left (b +c \right )}\, \tanh \left (\frac {x}{2}\right )}{c}+\frac {2 \tanh \left (\frac {x}{2}\right ) b}{c}+\frac {2 \sqrt {\left (b -c \right ) \left (b +c \right )}\, b}{c^{2}}+\frac {2 b^{2}}{c^{2}}-1\right ) \left (\tanh \left (\frac {x}{2}\right )+\frac {\sqrt {\left (b -c \right ) \left (b +c \right )}}{c}+\frac {b}{c}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: RuntimeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {1}{{\left (b\,\mathrm {cosh}\relax (x)+\sqrt {b^2-c^2}+c\,\mathrm {sinh}\relax (x)\right )}^2} \,d x \]
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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