Optimal. Leaf size=53 \[ \frac {2 (b+c) \tanh ^{-1}\left (\frac {a \tanh \left (\frac {x}{2}\right )+b}{\sqrt {a^2+b^2}}\right )}{\sqrt {a^2+b^2}}-\frac {\log (a-b \sinh (x))}{b} \]
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Rubi [A] time = 0.13, antiderivative size = 53, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {4401, 2660, 618, 206, 2668, 31} \[ \frac {2 (b+c) \tanh ^{-1}\left (\frac {a \tanh \left (\frac {x}{2}\right )+b}{\sqrt {a^2+b^2}}\right )}{\sqrt {a^2+b^2}}-\frac {\log (a-b \sinh (x))}{b} \]
Antiderivative was successfully verified.
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Rule 31
Rule 206
Rule 618
Rule 2660
Rule 2668
Rule 4401
Rubi steps
\begin {align*} \int \frac {b+c+\cosh (x)}{a-b \sinh (x)} \, dx &=\int \left (\frac {\left (1+\frac {b}{c}\right ) c}{a-b \sinh (x)}+\frac {\cosh (x)}{a-b \sinh (x)}\right ) \, dx\\ &=(b+c) \int \frac {1}{a-b \sinh (x)} \, dx+\int \frac {\cosh (x)}{a-b \sinh (x)} \, dx\\ &=-\frac {\operatorname {Subst}\left (\int \frac {1}{a+x} \, dx,x,-b \sinh (x)\right )}{b}+(2 (b+c)) \operatorname {Subst}\left (\int \frac {1}{a-2 b x-a x^2} \, dx,x,\tanh \left (\frac {x}{2}\right )\right )\\ &=-\frac {\log (a-b \sinh (x))}{b}-(4 (b+c)) \operatorname {Subst}\left (\int \frac {1}{4 \left (a^2+b^2\right )-x^2} \, dx,x,-2 b-2 a \tanh \left (\frac {x}{2}\right )\right )\\ &=\frac {2 (b+c) \tanh ^{-1}\left (\frac {b+a \tanh \left (\frac {x}{2}\right )}{\sqrt {a^2+b^2}}\right )}{\sqrt {a^2+b^2}}-\frac {\log (a-b \sinh (x))}{b}\\ \end {align*}
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Mathematica [A] time = 0.11, size = 62, normalized size = 1.17 \[ -\frac {2 (b+c) \tan ^{-1}\left (\frac {a \tanh \left (\frac {x}{2}\right )+b}{\sqrt {-a^2-b^2}}\right )}{\sqrt {-a^2-b^2}}-\frac {\log (b \sinh (x)-a)}{b} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.42, size = 174, normalized size = 3.28 \[ \frac {\sqrt {a^{2} + b^{2}} {\left (b^{2} + b c\right )} \log \left (\frac {b^{2} \cosh \relax (x)^{2} + b^{2} \sinh \relax (x)^{2} - 2 \, a b \cosh \relax (x) + 2 \, a^{2} + b^{2} + 2 \, {\left (b^{2} \cosh \relax (x) - a b\right )} \sinh \relax (x) + 2 \, \sqrt {a^{2} + b^{2}} {\left (b \cosh \relax (x) + b \sinh \relax (x) - a\right )}}{b \cosh \relax (x)^{2} + b \sinh \relax (x)^{2} - 2 \, a \cosh \relax (x) + 2 \, {\left (b \cosh \relax (x) - a\right )} \sinh \relax (x) - b}\right ) + {\left (a^{2} + b^{2}\right )} x - {\left (a^{2} + b^{2}\right )} \log \left (\frac {2 \, {\left (b \sinh \relax (x) - a\right )}}{\cosh \relax (x) - \sinh \relax (x)}\right )}{a^{2} b + b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 88, normalized size = 1.66 \[ -\frac {{\left (b + c\right )} \log \left (\frac {{\left | 2 \, b e^{x} - 2 \, a - 2 \, \sqrt {a^{2} + b^{2}} \right |}}{{\left | 2 \, b e^{x} - 2 \, a + 2 \, \sqrt {a^{2} + b^{2}} \right |}}\right )}{\sqrt {a^{2} + b^{2}}} + \frac {x}{b} - \frac {\log \left ({\left | b e^{\left (2 \, x\right )} - 2 \, a e^{x} - b \right |}\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.14, size = 119, normalized size = 2.25 \[ -\frac {\ln \left (a \left (\tanh ^{2}\left (\frac {x}{2}\right )\right )+2 \tanh \left (\frac {x}{2}\right ) b -a \right )}{b}+\frac {2 b \arctanh \left (\frac {2 a \tanh \left (\frac {x}{2}\right )+2 b}{2 \sqrt {a^{2}+b^{2}}}\right )}{\sqrt {a^{2}+b^{2}}}+\frac {2 \arctanh \left (\frac {2 a \tanh \left (\frac {x}{2}\right )+2 b}{2 \sqrt {a^{2}+b^{2}}}\right ) c}{\sqrt {a^{2}+b^{2}}}+\frac {\ln \left (\tanh \left (\frac {x}{2}\right )-1\right )}{b}+\frac {\ln \left (\tanh \left (\frac {x}{2}\right )+1\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.41, size = 119, normalized size = 2.25 \[ -\frac {b \log \left (\frac {b e^{\left (-x\right )} + a - \sqrt {a^{2} + b^{2}}}{b e^{\left (-x\right )} + a + \sqrt {a^{2} + b^{2}}}\right )}{\sqrt {a^{2} + b^{2}}} - \frac {c \log \left (\frac {b e^{\left (-x\right )} + a - \sqrt {a^{2} + b^{2}}}{b e^{\left (-x\right )} + a + \sqrt {a^{2} + b^{2}}}\right )}{\sqrt {a^{2} + b^{2}}} - \frac {\log \left (b \sinh \relax (x) - a\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.32, size = 177, normalized size = 3.34 \[ \frac {x}{b}+\frac {\ln \left (b\,\sqrt {a^2+b^2}+a^2\,{\mathrm {e}}^x+b^2\,{\mathrm {e}}^x+a\,{\mathrm {e}}^x\,\sqrt {a^2+b^2}\right )\,\left (b^2\,\sqrt {a^2+b^2}-a^2-b^2+b\,c\,\sqrt {a^2+b^2}\right )}{a^2\,b+b^3}-\frac {\ln \left (b\,\sqrt {a^2+b^2}-a^2\,{\mathrm {e}}^x-b^2\,{\mathrm {e}}^x+a\,{\mathrm {e}}^x\,\sqrt {a^2+b^2}\right )\,\left (b^2\,\sqrt {a^2+b^2}+a^2+b^2+b\,c\,\sqrt {a^2+b^2}\right )}{a^2\,b+b^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 89.17, size = 845, normalized size = 15.94 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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