Optimal. Leaf size=113 \[ \frac {a x}{a^2-b^2}+\frac {2 a c \tanh ^{-1}\left (\frac {a+(b-c) \tanh \left (\frac {x}{2}\right )}{\sqrt {a^2-b^2+c^2}}\right )}{\left (a^2-b^2\right ) \sqrt {a^2-b^2+c^2}}-\frac {b \log (i c+i b \cosh (x)+i a \sinh (x))}{a^2-b^2} \]
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Rubi [A]
time = 0.13, 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 = {3239, 3216,
3203, 632, 210} \begin {gather*} \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} \end {gather*}
Antiderivative was successfully verified.
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Rule 210
Rule 632
Rule 3203
Rule 3216
Rule 3239
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) \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 )}{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) \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 )}{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 = 0.16, size = 86, normalized size = 0.76 \begin {gather*} \frac {a x-\frac {2 a c \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}}-b \log (c+b \cosh (x)+a \sinh (x))}{a^2-b^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 1.25, size = 178, normalized size = 1.58
method | result | size |
default | \(-\frac {4 \ln \left (\tanh \left (\frac {x}{2}\right )-1\right )}{4 a +4 b}+\frac {\frac {2 \left (-b^{2}+b c \right ) \ln \left (b \left (\tanh ^{2}\left (\frac {x}{2}\right )\right )-c \left (\tanh ^{2}\left (\frac {x}{2}\right )\right )+2 a \tanh \left (\frac {x}{2}\right )+b +c \right )}{2 b -2 c}+\frac {2 \left (-a b -a c -\frac {\left (-b^{2}+b c \right ) a}{b -c}\right ) \arctan \left (\frac {2 \left (b -c \right ) \tanh \left (\frac {x}{2}\right )+2 a}{2 \sqrt {-a^{2}+b^{2}-c^{2}}}\right )}{\sqrt {-a^{2}+b^{2}-c^{2}}}}{\left (a +b \right ) \left (a -b \right )}+\frac {4 \ln \left (\tanh \left (\frac {x}{2}\right )+1\right )}{4 a -4 b}\) | \(178\) |
risch | \(\frac {x}{a +b}+\frac {2 x \,a^{2} b}{a^{4}-2 a^{2} b^{2}+a^{2} c^{2}+b^{4}-b^{2} c^{2}}-\frac {2 x \,b^{3}}{a^{4}-2 a^{2} b^{2}+a^{2} c^{2}+b^{4}-b^{2} c^{2}}+\frac {2 x b \,c^{2}}{a^{4}-2 a^{2} b^{2}+a^{2} c^{2}+b^{4}-b^{2} c^{2}}-\frac {\ln \left ({\mathrm e}^{x}+\frac {a \,c^{2}+\sqrt {a^{4} c^{2}-a^{2} b^{2} c^{2}+a^{2} c^{4}}}{\left (a +b \right ) a c}\right ) a^{2} b}{a^{4}-2 a^{2} b^{2}+a^{2} c^{2}+b^{4}-b^{2} c^{2}}+\frac {\ln \left ({\mathrm e}^{x}+\frac {a \,c^{2}+\sqrt {a^{4} c^{2}-a^{2} b^{2} c^{2}+a^{2} c^{4}}}{\left (a +b \right ) a c}\right ) b^{3}}{a^{4}-2 a^{2} b^{2}+a^{2} c^{2}+b^{4}-b^{2} c^{2}}-\frac {\ln \left ({\mathrm e}^{x}+\frac {a \,c^{2}+\sqrt {a^{4} c^{2}-a^{2} b^{2} c^{2}+a^{2} c^{4}}}{\left (a +b \right ) a c}\right ) b \,c^{2}}{a^{4}-2 a^{2} b^{2}+a^{2} c^{2}+b^{4}-b^{2} c^{2}}+\frac {\ln \left ({\mathrm e}^{x}+\frac {a \,c^{2}+\sqrt {a^{4} c^{2}-a^{2} b^{2} c^{2}+a^{2} c^{4}}}{\left (a +b \right ) a c}\right ) \sqrt {a^{4} c^{2}-a^{2} b^{2} c^{2}+a^{2} c^{4}}}{a^{4}-2 a^{2} b^{2}+a^{2} c^{2}+b^{4}-b^{2} c^{2}}-\frac {\ln \left ({\mathrm e}^{x}-\frac {-a \,c^{2}+\sqrt {a^{4} c^{2}-a^{2} b^{2} c^{2}+a^{2} c^{4}}}{\left (a +b \right ) a c}\right ) a^{2} b}{a^{4}-2 a^{2} b^{2}+a^{2} c^{2}+b^{4}-b^{2} c^{2}}+\frac {\ln \left ({\mathrm e}^{x}-\frac {-a \,c^{2}+\sqrt {a^{4} c^{2}-a^{2} b^{2} c^{2}+a^{2} c^{4}}}{\left (a +b \right ) a c}\right ) b^{3}}{a^{4}-2 a^{2} b^{2}+a^{2} c^{2}+b^{4}-b^{2} c^{2}}-\frac {\ln \left ({\mathrm e}^{x}-\frac {-a \,c^{2}+\sqrt {a^{4} c^{2}-a^{2} b^{2} c^{2}+a^{2} c^{4}}}{\left (a +b \right ) a c}\right ) b \,c^{2}}{a^{4}-2 a^{2} b^{2}+a^{2} c^{2}+b^{4}-b^{2} c^{2}}-\frac {\ln \left ({\mathrm e}^{x}-\frac {-a \,c^{2}+\sqrt {a^{4} c^{2}-a^{2} b^{2} c^{2}+a^{2} c^{4}}}{\left (a +b \right ) a c}\right ) \sqrt {a^{4} c^{2}-a^{2} b^{2} c^{2}+a^{2} c^{4}}}{a^{4}-2 a^{2} b^{2}+a^{2} c^{2}+b^{4}-b^{2} c^{2}}\) | \(880\) |
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 = 0.44, size = 438, normalized size = 3.88 \begin {gather*} \left [-\frac {\sqrt {a^{2} - b^{2} + c^{2}} a c \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 ) - {\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 \left (x\right ) + a \sinh \left (x\right ) + c\right )}}{\cosh \left (x\right ) - \sinh \left (x\right )}\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 \left (x\right ) + {\left (a + b\right )} \sinh \left (x\right ) + 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 \left (x\right ) + a \sinh \left (x\right ) + c\right )}}{\cosh \left (x\right ) - \sinh \left (x\right )}\right )}{a^{4} - 2 \, a^{2} b^{2} + b^{4} + {\left (a^{2} - b^{2}\right )} 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 {1}{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 = 106, normalized size = 0.94 \begin {gather*} -\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} \end {gather*}
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 \begin {gather*} \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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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