Optimal. Leaf size=54 \[ \frac {2 \tan ^{-1}\left (\frac {(a-c) \tanh \left (\frac {x}{2}\right )+b}{\sqrt {a^2-b^2-c^2}}\right )}{\sqrt {a^2-b^2-c^2}} \]
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Rubi [A] time = 0.08, antiderivative size = 54, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {3165, 3124, 618, 204} \[ \frac {2 \tan ^{-1}\left (\frac {(a-c) \tanh \left (\frac {x}{2}\right )+b}{\sqrt {a^2-b^2-c^2}}\right )}{\sqrt {a^2-b^2-c^2}} \]
Antiderivative was successfully verified.
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Rule 204
Rule 618
Rule 3124
Rule 3165
Rubi steps
\begin {align*} \int \frac {\text {sech}(x)}{a+c \text {sech}(x)+b \tanh (x)} \, dx &=\int \frac {1}{c+a \cosh (x)+b \sinh (x)} \, dx\\ &=2 \operatorname {Subst}\left (\int \frac {1}{a+c+2 b x-(-a+c) x^2} \, dx,x,\tanh \left (\frac {x}{2}\right )\right )\\ &=-\left (4 \operatorname {Subst}\left (\int \frac {1}{-4 \left (a^2-b^2-c^2\right )-x^2} \, dx,x,2 b+2 (a-c) \tanh \left (\frac {x}{2}\right )\right )\right )\\ &=\frac {2 \tan ^{-1}\left (\frac {b+(a-c) \tanh \left (\frac {x}{2}\right )}{\sqrt {a^2-b^2-c^2}}\right )}{\sqrt {a^2-b^2-c^2}}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 54, normalized size = 1.00 \[ \frac {2 \tan ^{-1}\left (\frac {(a-c) \tanh \left (\frac {x}{2}\right )+b}{\sqrt {a^2-b^2-c^2}}\right )}{\sqrt {a^2-b^2-c^2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.44, size = 234, normalized size = 4.33 \[ \left [-\frac {\sqrt {-a^{2} + b^{2} + c^{2}} \log \left (\frac {2 \, {\left (a + b\right )} c \cosh \relax (x) + {\left (a^{2} + 2 \, a b + b^{2}\right )} \cosh \relax (x)^{2} + {\left (a^{2} + 2 \, a b + b^{2}\right )} \sinh \relax (x)^{2} - a^{2} + b^{2} + 2 \, c^{2} + 2 \, {\left ({\left (a + b\right )} c + {\left (a^{2} + 2 \, a b + b^{2}\right )} \cosh \relax (x)\right )} \sinh \relax (x) - 2 \, \sqrt {-a^{2} + b^{2} + c^{2}} {\left ({\left (a + b\right )} \cosh \relax (x) + {\left (a + b\right )} \sinh \relax (x) + c\right )}}{{\left (a + b\right )} \cosh \relax (x)^{2} + {\left (a + b\right )} \sinh \relax (x)^{2} + 2 \, c \cosh \relax (x) + 2 \, {\left ({\left (a + b\right )} \cosh \relax (x) + c\right )} \sinh \relax (x) + a - b}\right )}{a^{2} - b^{2} - c^{2}}, -\frac {2 \, \arctan \left (-\frac {{\left (a + b\right )} \cosh \relax (x) + {\left (a + b\right )} \sinh \relax (x) + c}{\sqrt {a^{2} - b^{2} - c^{2}}}\right )}{\sqrt {a^{2} - b^{2} - c^{2}}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.11, size = 46, normalized size = 0.85 \[ \frac {2 \, \arctan \left (\frac {a e^{x} + b e^{x} + c}{\sqrt {a^{2} - b^{2} - c^{2}}}\right )}{\sqrt {a^{2} - b^{2} - c^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.24, size = 53, normalized size = 0.98 \[ \frac {2 \arctan \left (\frac {2 \left (a -c \right ) \tanh \left (\frac {x}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}-c^{2}}}\right )}{\sqrt {a^{2}-b^{2}-c^{2}}} \]
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 \[ \text {Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.82, size = 78, normalized size = 1.44 \[ \frac {2\,\mathrm {atan}\left (\frac {c}{\sqrt {a^2-b^2-c^2}}+\frac {a\,{\mathrm {e}}^x}{\sqrt {a^2-b^2-c^2}}+\frac {b\,{\mathrm {e}}^x}{\sqrt {a^2-b^2-c^2}}\right )}{\sqrt {a^2-b^2-c^2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {sech}{\relax (x )}}{a + b \tanh {\relax (x )} + c \operatorname {sech}{\relax (x )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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