Optimal. Leaf size=113 \[ \frac {2 a c \tanh ^{-1}\left (\frac {a+(b-c) \tanh \left (\frac {x}{2}\right )}{\sqrt {a^2-b^2+c^2}}\right )}{\left (a^2-b^2\right ) \sqrt {a^2-b^2+c^2}}-\frac {b \log (i a \sinh (x)+i b \cosh (x)+i c)}{a^2-b^2}+\frac {a x}{a^2-b^2} \]
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Rubi [A] time = 0.16, antiderivative size = 113, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {3160, 3137, 3124, 618, 204} \[ \frac {2 a c \tanh ^{-1}\left (\frac {a+(b-c) \tanh \left (\frac {x}{2}\right )}{\sqrt {a^2-b^2+c^2}}\right )}{\left (a^2-b^2\right ) \sqrt {a^2-b^2+c^2}}-\frac {b \log (i a \sinh (x)+i b \cosh (x)+i c)}{a^2-b^2}+\frac {a x}{a^2-b^2} \]
Antiderivative was successfully verified.
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Rule 204
Rule 618
Rule 3124
Rule 3137
Rule 3160
Rubi steps
\begin {align*} \int \frac {1}{a+b \coth (x)+c \text {csch}(x)} \, dx &=i \int \frac {\sinh (x)}{i c+i b \cosh (x)+i a \sinh (x)} \, dx\\ &=\frac {a x}{a^2-b^2}-\frac {b \log (i c+i b \cosh (x)+i a \sinh (x))}{a^2-b^2}-\frac {(i a c) \int \frac {1}{i c+i b \cosh (x)+i a \sinh (x)} \, dx}{a^2-b^2}\\ &=\frac {a x}{a^2-b^2}-\frac {b \log (i c+i b \cosh (x)+i a \sinh (x))}{a^2-b^2}-\frac {(2 i a c) \operatorname {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 )}{a^2-b^2}\\ &=\frac {a x}{a^2-b^2}-\frac {b \log (i c+i b \cosh (x)+i a \sinh (x))}{a^2-b^2}+\frac {(4 i a c) \operatorname {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 )}{a^2-b^2}\\ &=\frac {a x}{a^2-b^2}+\frac {2 a c \tanh ^{-1}\left (\frac {a+(b-c) \tanh \left (\frac {x}{2}\right )}{\sqrt {a^2-b^2+c^2}}\right )}{\left (a^2-b^2\right ) \sqrt {a^2-b^2+c^2}}-\frac {b \log (i c+i b \cosh (x)+i a \sinh (x))}{a^2-b^2}\\ \end {align*}
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Mathematica [A] time = 0.22, size = 86, normalized size = 0.76 \[ \frac {-\frac {2 a c \tan ^{-1}\left (\frac {a+(b-c) \tanh \left (\frac {x}{2}\right )}{\sqrt {-a^2+b^2-c^2}}\right )}{\sqrt {-a^2+b^2-c^2}}-b \log (a \sinh (x)+b \cosh (x)+c)+a x}{a^2-b^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.46, size = 438, normalized size = 3.88 \[ \left [-\frac {\sqrt {a^{2} - b^{2} + c^{2}} a c \log \left (\frac {2 \, {\left (a + b\right )} c \cosh \relax (x) + {\left (a^{2} + 2 \, a b + b^{2}\right )} \cosh \relax (x)^{2} + {\left (a^{2} + 2 \, a b + b^{2}\right )} \sinh \relax (x)^{2} + a^{2} - b^{2} + 2 \, c^{2} + 2 \, {\left ({\left (a + b\right )} c + {\left (a^{2} + 2 \, a b + b^{2}\right )} \cosh \relax (x)\right )} \sinh \relax (x) - 2 \, \sqrt {a^{2} - b^{2} + c^{2}} {\left ({\left (a + b\right )} \cosh \relax (x) + {\left (a + b\right )} \sinh \relax (x) + c\right )}}{{\left (a + b\right )} \cosh \relax (x)^{2} + {\left (a + b\right )} \sinh \relax (x)^{2} + 2 \, c \cosh \relax (x) + 2 \, {\left ({\left (a + b\right )} \cosh \relax (x) + c\right )} \sinh \relax (x) - a + b}\right ) - {\left (a^{3} + a^{2} b - a b^{2} - b^{3} + {\left (a + b\right )} c^{2}\right )} x + {\left (a^{2} b - b^{3} + b c^{2}\right )} \log \left (\frac {2 \, {\left (b \cosh \relax (x) + a \sinh \relax (x) + c\right )}}{\cosh \relax (x) - \sinh \relax (x)}\right )}{a^{4} - 2 \, a^{2} b^{2} + b^{4} + {\left (a^{2} - b^{2}\right )} c^{2}}, -\frac {2 \, \sqrt {-a^{2} + b^{2} - c^{2}} a c \arctan \left (\frac {\sqrt {-a^{2} + b^{2} - c^{2}} {\left ({\left (a + b\right )} \cosh \relax (x) + {\left (a + b\right )} \sinh \relax (x) + c\right )}}{a^{2} - b^{2} + c^{2}}\right ) - {\left (a^{3} + a^{2} b - a b^{2} - b^{3} + {\left (a + b\right )} c^{2}\right )} x + {\left (a^{2} b - b^{3} + b c^{2}\right )} \log \left (\frac {2 \, {\left (b \cosh \relax (x) + a \sinh \relax (x) + c\right )}}{\cosh \relax (x) - \sinh \relax (x)}\right )}{a^{4} - 2 \, a^{2} b^{2} + b^{4} + {\left (a^{2} - b^{2}\right )} c^{2}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.14, size = 106, normalized size = 0.94 \[ -\frac {2 \, a c \arctan \left (\frac {a e^{x} + b e^{x} + c}{\sqrt {-a^{2} + b^{2} - c^{2}}}\right )}{{\left (a^{2} - b^{2}\right )} \sqrt {-a^{2} + b^{2} - c^{2}}} - \frac {b \log \left (a e^{\left (2 \, x\right )} + b e^{\left (2 \, x\right )} + 2 \, c e^{x} - a + b\right )}{a^{2} - b^{2}} + \frac {x}{a - b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.22, size = 421, normalized size = 3.73 \[ -\frac {4 \ln \left (\tanh \left (\frac {x}{2}\right )-1\right )}{4 a +4 b}+\frac {4 \ln \left (\tanh \left (\frac {x}{2}\right )+1\right )}{4 a -4 b}-\frac {\ln \left (\left (\tanh ^{2}\left (\frac {x}{2}\right )\right ) b -\left (\tanh ^{2}\left (\frac {x}{2}\right )\right ) c +2 a \tanh \left (\frac {x}{2}\right )+b +c \right ) b^{2}}{\left (a +b \right ) \left (a -b \right ) \left (b -c \right )}+\frac {\ln \left (\left (\tanh ^{2}\left (\frac {x}{2}\right )\right ) b -\left (\tanh ^{2}\left (\frac {x}{2}\right )\right ) c +2 a \tanh \left (\frac {x}{2}\right )+b +c \right ) c b}{\left (a +b \right ) \left (a -b \right ) \left (b -c \right )}-\frac {2 \arctan \left (\frac {2 \left (b -c \right ) \tanh \left (\frac {x}{2}\right )+2 a}{2 \sqrt {-a^{2}+b^{2}-c^{2}}}\right ) a b}{\left (a +b \right ) \left (a -b \right ) \sqrt {-a^{2}+b^{2}-c^{2}}}-\frac {2 \arctan \left (\frac {2 \left (b -c \right ) \tanh \left (\frac {x}{2}\right )+2 a}{2 \sqrt {-a^{2}+b^{2}-c^{2}}}\right ) a c}{\left (a +b \right ) \left (a -b \right ) \sqrt {-a^{2}+b^{2}-c^{2}}}+\frac {2 \arctan \left (\frac {2 \left (b -c \right ) \tanh \left (\frac {x}{2}\right )+2 a}{2 \sqrt {-a^{2}+b^{2}-c^{2}}}\right ) a \,b^{2}}{\left (a +b \right ) \left (a -b \right ) \sqrt {-a^{2}+b^{2}-c^{2}}\, \left (b -c \right )}-\frac {2 \arctan \left (\frac {2 \left (b -c \right ) \tanh \left (\frac {x}{2}\right )+2 a}{2 \sqrt {-a^{2}+b^{2}-c^{2}}}\right ) a c b}{\left (a +b \right ) \left (a -b \right ) \sqrt {-a^{2}+b^{2}-c^{2}}\, \left (b -c \right )} \]
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 \[ \text {Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.70, size = 324, normalized size = 2.87 \[ \frac {x}{a-b}-\frac {\ln \left (\frac {2\,\left (b+c\,{\mathrm {e}}^x\right )}{{\left (a+b\right )}^2}+\frac {2\,\left (b-a+c\,{\mathrm {e}}^x\right )\,\left (a^2\,b+b\,c^2-b^3+a\,c\,\sqrt {a^2-b^2+c^2}\right )}{\left (a+b\right )\,\left (a^2-b^2\right )\,\left (a^2-b^2+c^2\right )}\right )\,\left (a^2\,b+b\,c^2-b^3+a\,c\,\sqrt {a^2-b^2+c^2}\right )}{a^4-2\,a^2\,b^2+a^2\,c^2+b^4-b^2\,c^2}-\frac {\ln \left (\frac {2\,\left (b+c\,{\mathrm {e}}^x\right )}{{\left (a+b\right )}^2}+\frac {2\,\left (b-a+c\,{\mathrm {e}}^x\right )\,\left (a^2\,b+b\,c^2-b^3-a\,c\,\sqrt {a^2-b^2+c^2}\right )}{\left (a+b\right )\,\left (a^2-b^2\right )\,\left (a^2-b^2+c^2\right )}\right )\,\left (a^2\,b+b\,c^2-b^3-a\,c\,\sqrt {a^2-b^2+c^2}\right )}{a^4-2\,a^2\,b^2+a^2\,c^2+b^4-b^2\,c^2} \]
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 \[ \int \frac {1}{a + b \coth {\relax (x )} + c \operatorname {csch}{\relax (x )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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