Optimal. Leaf size=50 \[ -\frac {2 \tanh ^{-1}\left (\frac {a+(b-c) \tanh \left (\frac {x}{2}\right )}{\sqrt {a^2-b^2+c^2}}\right )}{\sqrt {a^2-b^2+c^2}} \]
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Rubi [A]
time = 0.06, antiderivative size = 50, 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 = {3245, 3203,
632, 210} \begin {gather*} -\frac {2 \tanh ^{-1}\left (\frac {a+(b-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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Rule 210
Rule 632
Rule 3203
Rule 3245
Rubi steps
\begin {align*} \int \frac {\text {csch}(x)}{a+b \coth (x)+c \text {csch}(x)} \, dx &=i \int \frac {1}{i c+i b \cosh (x)+i a \sinh (x)} \, dx\\ &=2 i \text {Subst}\left (\int \frac {1}{i b+i c+2 i a x-(-i b+i c) x^2} \, dx,x,\tanh \left (\frac {x}{2}\right )\right )\\ &=-\left (4 i \text {Subst}\left (\int \frac {1}{-4 \left (a^2-b^2+c^2\right )-x^2} \, dx,x,2 i a+2 (i b-i c) \tanh \left (\frac {x}{2}\right )\right )\right )\\ &=-\frac {2 \tanh ^{-1}\left (\frac {a+(b-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 = 1.08 \begin {gather*} \frac {2 \text {ArcTan}\left (\frac {a+(b-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 = 1.09, size = 53, normalized size = 1.06
method | result | size |
default | \(\frac {2 \arctan \left (\frac {2 \left (b -c \right ) \tanh \left (\frac {x}{2}\right )+2 a}{2 \sqrt {-a^{2}+b^{2}-c^{2}}}\right )}{\sqrt {-a^{2}+b^{2}-c^{2}}}\) | \(53\) |
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 = 0.41, size = 244, normalized size = 4.88 \begin {gather*} \left [\frac {\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 )}{\sqrt {a^{2} - b^{2} + c^{2}}}, \frac {2 \, \sqrt {-a^{2} + b^{2} - c^{2}} \arctan \left (\frac {\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 )}}{a^{2} - b^{2} + c^{2}}\right )}{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 {csch}{\left (x \right )}}{a + b \coth {\left (x \right )} + c \operatorname {csch}{\left (x \right )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.41, size = 46, normalized size = 0.92 \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 = 0.21, size = 78, normalized size = 1.56 \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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