Optimal. Leaf size=34 \[ -\frac {\sqrt {b^2-c^2} \sinh (x)+c}{c (b \sinh (x)+c \cosh (x))} \]
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Rubi [A] time = 0.04, antiderivative size = 34, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.042, Rules used = {3114} \[ -\frac {\sqrt {b^2-c^2} \sinh (x)+c}{c (b \sinh (x)+c \cosh (x))} \]
Antiderivative was successfully verified.
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Rule 3114
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)} \, dx &=-\frac {c+\sqrt {b^2-c^2} \sinh (x)}{c (c \cosh (x)+b \sinh (x))}\\ \end {align*}
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Mathematica [A] time = 0.08, size = 36, normalized size = 1.06 \[ \frac {-\sqrt {b^2-c^2} \sinh (x)-c}{c (b \sinh (x)+c \cosh (x))} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.40, size = 88, normalized size = 2.59 \[ -\frac {2 \, {\left ({\left (b + c\right )} \cosh \relax (x) + {\left (b + c\right )} \sinh \relax (x) - \sqrt {b^{2} - c^{2}}\right )}}{{\left (b^{2} + 2 \, b c + c^{2}\right )} \cosh \relax (x)^{2} + 2 \, {\left (b^{2} + 2 \, b c + c^{2}\right )} \cosh \relax (x) \sinh \relax (x) + {\left (b^{2} + 2 \, b c + c^{2}\right )} \sinh \relax (x)^{2} - b^{2} + c^{2}} \]
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 [C] time = 0.62, size = 596, normalized size = 17.53 \[ \frac {\left (\munderset {\textit {\_R} =\RootOf \left (\left ({| \left (b -c \right ) \left (b +c \right )|} \mathrm {signum}\left (\left (b -c \right ) \left (b +c \right )\right )^{2}-2 \sqrt {{| \left (b -c \right ) \left (b +c \right )|}}\, \mathrm {signum}\left (\left (b -c \right ) \left (b +c \right )\right ) b +{| \left (b -c \right ) \left (b +c \right )|}-2 \sqrt {{| \left (b -c \right ) \left (b +c \right )|}}\, b +2 b^{2}\right ) \textit {\_Z}^{4}+\left (-4 \sqrt {{| \left (b -c \right ) \left (b +c \right )|}}\, \mathrm {signum}\left (\left (b -c \right ) \left (b +c \right )\right ) c -4 \sqrt {{| \left (b -c \right ) \left (b +c \right )|}}\, c +8 c b \right ) \textit {\_Z}^{3}+\left (-2 {| \left (b -c \right ) \left (b +c \right )|} \mathrm {signum}\left (\left (b -c \right ) \left (b +c \right )\right )^{2}-2 {| \left (b -c \right ) \left (b +c \right )|}+4 b^{2}+8 c^{2}\right ) \textit {\_Z}^{2}+\left (4 \sqrt {{| \left (b -c \right ) \left (b +c \right )|}}\, \mathrm {signum}\left (\left (b -c \right ) \left (b +c \right )\right ) c +4 \sqrt {{| \left (b -c \right ) \left (b +c \right )|}}\, c +8 c b \right ) \textit {\_Z} +{| \left (b -c \right ) \left (b +c \right )|} \mathrm {signum}\left (\left (b -c \right ) \left (b +c \right )\right )^{2}+2 \sqrt {{| \left (b -c \right ) \left (b +c \right )|}}\, \mathrm {signum}\left (\left (b -c \right ) \left (b +c \right )\right ) b +{| \left (b -c \right ) \left (b +c \right )|}+2 \sqrt {{| \left (b -c \right ) \left (b +c \right )|}}\, b +2 b^{2}\right )}{\sum }\frac {\left (2 b \,\textit {\_R}^{2}+4 \textit {\_R} c +2 b +\sqrt {{| b^{2}-c^{2}|}}\, \left (1-i+\textit {\_R}^{2} \left (-1+i-i \mathrm {signum}\left (b^{2}-c^{2}\right )-\mathrm {signum}\left (b^{2}-c^{2}\right )\right )+i \mathrm {signum}\left (b^{2}-c^{2}\right )+\mathrm {signum}\left (b^{2}-c^{2}\right )\right )\right ) \ln \left (\tanh \left (\frac {x}{2}\right )-\textit {\_R} \right )}{{| \left (b -c \right ) \left (b +c \right )|} \textit {\_R}^{3} \mathrm {signum}\left (\left (b -c \right ) \left (b +c \right )\right )^{2}-{| \left (b -c \right ) \left (b +c \right )|} \textit {\_R} \mathrm {signum}\left (\left (b -c \right ) \left (b +c \right )\right )^{2}+{| \left (b -c \right ) \left (b +c \right )|} \textit {\_R}^{3}+2 b^{2} \textit {\_R}^{3}+6 \textit {\_R}^{2} b c -{| \left (b -c \right ) \left (b +c \right )|} \textit {\_R} +2 b^{2} \textit {\_R} +4 c^{2} \textit {\_R} +2 c b +\sqrt {{| \left (b -c \right ) \left (b +c \right )|}}\, \left (-2 \,\mathrm {signum}\left (\left (b -c \right ) \left (b +c \right )\right ) b \,\textit {\_R}^{3}-3 \,\mathrm {signum}\left (\left (b -c \right ) \left (b +c \right )\right ) c \,\textit {\_R}^{2}-2 b \,\textit {\_R}^{3}-3 c \,\textit {\_R}^{2}+\mathrm {signum}\left (\left (b -c \right ) \left (b +c \right )\right ) c +c \right )}\right )}{2} \]
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 = -0.03 \[ \int \frac {1}{b\,\mathrm {cosh}\relax (x)+\sqrt {b^2-c^2}+c\,\mathrm {sinh}\relax (x)} \,d x \]
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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