Optimal. Leaf size=54 \[ \frac {2 \text {ArcTan}\left (\frac {b+(a-c) \tanh \left (\frac {x}{2}\right )}{\sqrt {a^2-b^2-c^2}}\right )}{\sqrt {a^2-b^2-c^2}} \]
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Rubi [A]
time = 0.06, 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 = {3244, 3203,
632, 210} \begin {gather*} \frac {2 \text {ArcTan}\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}} \end {gather*}
Antiderivative was successfully verified.
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Rule 210
Rule 632
Rule 3203
Rule 3244
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 \text {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 \text {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 = 0.03, size = 54, normalized size = 1.00 \begin {gather*} \frac {2 \text {ArcTan}\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 {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 1.41, size = 53, normalized size = 0.98
method | result | size |
default | \(\frac {2 \arctan \left (\frac {2 \left (a -c \right ) \tanh \left (\frac {x}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}-c^{2}}}\right )}{\sqrt {a^{2}-b^{2}-c^{2}}}\) | \(53\) |
risch | \(-\frac {\ln \left ({\mathrm e}^{x}+\frac {c \sqrt {-a^{2}+b^{2}+c^{2}}-a^{2}+b^{2}+c^{2}}{\left (a +b \right ) \sqrt {-a^{2}+b^{2}+c^{2}}}\right )}{\sqrt {-a^{2}+b^{2}+c^{2}}}+\frac {\ln \left ({\mathrm e}^{x}+\frac {c \sqrt {-a^{2}+b^{2}+c^{2}}+a^{2}-b^{2}-c^{2}}{\left (a +b \right ) \sqrt {-a^{2}+b^{2}+c^{2}}}\right )}{\sqrt {-a^{2}+b^{2}+c^{2}}}\) | \(139\) |
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 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.42, size = 234, normalized size = 4.33 \begin {gather*} \left [-\frac {\sqrt {-a^{2} + b^{2} + c^{2}} \log \left (\frac {2 \, {\left (a + b\right )} c \cosh \left (x\right ) + {\left (a^{2} + 2 \, a b + b^{2}\right )} \cosh \left (x\right )^{2} + {\left (a^{2} + 2 \, a b + b^{2}\right )} \sinh \left (x\right )^{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 \left (x\right )\right )} \sinh \left (x\right ) - 2 \, \sqrt {-a^{2} + b^{2} + c^{2}} {\left ({\left (a + b\right )} \cosh \left (x\right ) + {\left (a + b\right )} \sinh \left (x\right ) + c\right )}}{{\left (a + b\right )} \cosh \left (x\right )^{2} + {\left (a + b\right )} \sinh \left (x\right )^{2} + 2 \, c \cosh \left (x\right ) + 2 \, {\left ({\left (a + b\right )} \cosh \left (x\right ) + c\right )} \sinh \left (x\right ) + a - b}\right )}{a^{2} - b^{2} - c^{2}}, -\frac {2 \, \arctan \left (-\frac {{\left (a + b\right )} \cosh \left (x\right ) + {\left (a + b\right )} \sinh \left (x\right ) + c}{\sqrt {a^{2} - b^{2} - c^{2}}}\right )}{\sqrt {a^{2} - b^{2} - c^{2}}}\right ] \end {gather*}
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 \begin {gather*} \int \frac {\operatorname {sech}{\left (x \right )}}{a + b \tanh {\left (x \right )} + c \operatorname {sech}{\left (x \right )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.40, size = 46, normalized size = 0.85 \begin {gather*} \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}}} \end {gather*}
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 \begin {gather*} \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}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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