Integrand size = 14, antiderivative size = 52 \[ \int \frac {b+c+\cosh (x)}{a+b \sinh (x)} \, dx=-\frac {2 (b+c) \text {arctanh}\left (\frac {b-a \tanh \left (\frac {x}{2}\right )}{\sqrt {a^2+b^2}}\right )}{\sqrt {a^2+b^2}}+\frac {\log (a+b \sinh (x))}{b} \]
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Time = 0.10 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {4486, 2739, 632, 212, 2747, 31} \[ \int \frac {b+c+\cosh (x)}{a+b \sinh (x)} \, dx=\frac {\log (a+b \sinh (x))}{b}-\frac {2 (b+c) \text {arctanh}\left (\frac {b-a \tanh \left (\frac {x}{2}\right )}{\sqrt {a^2+b^2}}\right )}{\sqrt {a^2+b^2}} \]
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Rule 31
Rule 212
Rule 632
Rule 2739
Rule 2747
Rule 4486
Rubi steps \begin{align*} \text {integral}& = \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 {\text {Subst}\left (\int \frac {1}{a+x} \, dx,x,b \sinh (x)\right )}{b}+(2 (b+c)) \text {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)) \text {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) \text {arctanh}\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*}
Time = 0.16 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.15 \[ \int \frac {b+c+\cosh (x)}{a+b \sinh (x)} \, dx=\frac {2 (b+c) \arctan \left (\frac {b-a \tanh \left (\frac {x}{2}\right )}{\sqrt {-a^2-b^2}}\right )}{\sqrt {-a^2-b^2}}+\frac {\log (a+b \sinh (x))}{b} \]
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Time = 0.69 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.96
method | result | size |
parts | \(\frac {\ln \left (a +b \sinh \left (x \right )\right )}{b}+\frac {2 \left (b +c \right ) \operatorname {arctanh}\left (\frac {2 a \tanh \left (\frac {x}{2}\right )-2 b}{2 \sqrt {a^{2}+b^{2}}}\right )}{\sqrt {a^{2}+b^{2}}}\) | \(50\) |
default | \(-\frac {\ln \left (\tanh \left (\frac {x}{2}\right )+1\right )}{b}-\frac {\ln \left (\tanh \left (\frac {x}{2}\right )-1\right )}{b}+\frac {\ln \left (\tanh \left (\frac {x}{2}\right )^{2} a -2 b \tanh \left (\frac {x}{2}\right )-a \right )-\frac {2 \left (-b^{2}-c b \right ) \operatorname {arctanh}\left (\frac {2 a \tanh \left (\frac {x}{2}\right )-2 b}{2 \sqrt {a^{2}+b^{2}}}\right )}{\sqrt {a^{2}+b^{2}}}}{b}\) | \(98\) |
risch | \(\frac {x}{b}-\frac {2 x \,a^{2} b}{a^{2} b^{2}+b^{4}}-\frac {2 x \,b^{3}}{a^{2} b^{2}+b^{4}}+\frac {\ln \left ({\mathrm e}^{x}-\frac {-a \,b^{2}-c a b +\sqrt {a^{2} b^{4}+2 a^{2} b^{3} c +a^{2} b^{2} c^{2}+b^{6}+2 b^{5} c +b^{4} c^{2}}}{b^{2} \left (b +c \right )}\right ) a^{2}}{\left (a^{2}+b^{2}\right ) b}+\frac {b \ln \left ({\mathrm e}^{x}-\frac {-a \,b^{2}-c a b +\sqrt {a^{2} b^{4}+2 a^{2} b^{3} c +a^{2} b^{2} c^{2}+b^{6}+2 b^{5} c +b^{4} c^{2}}}{b^{2} \left (b +c \right )}\right )}{a^{2}+b^{2}}+\frac {\ln \left ({\mathrm e}^{x}-\frac {-a \,b^{2}-c a b +\sqrt {a^{2} b^{4}+2 a^{2} b^{3} c +a^{2} b^{2} c^{2}+b^{6}+2 b^{5} c +b^{4} c^{2}}}{b^{2} \left (b +c \right )}\right ) \sqrt {a^{2} b^{4}+2 a^{2} b^{3} c +a^{2} b^{2} c^{2}+b^{6}+2 b^{5} c +b^{4} c^{2}}}{\left (a^{2}+b^{2}\right ) b}+\frac {\ln \left ({\mathrm e}^{x}+\frac {a \,b^{2}+c a b +\sqrt {a^{2} b^{4}+2 a^{2} b^{3} c +a^{2} b^{2} c^{2}+b^{6}+2 b^{5} c +b^{4} c^{2}}}{b^{2} \left (b +c \right )}\right ) a^{2}}{\left (a^{2}+b^{2}\right ) b}+\frac {b \ln \left ({\mathrm e}^{x}+\frac {a \,b^{2}+c a b +\sqrt {a^{2} b^{4}+2 a^{2} b^{3} c +a^{2} b^{2} c^{2}+b^{6}+2 b^{5} c +b^{4} c^{2}}}{b^{2} \left (b +c \right )}\right )}{a^{2}+b^{2}}-\frac {\ln \left ({\mathrm e}^{x}+\frac {a \,b^{2}+c a b +\sqrt {a^{2} b^{4}+2 a^{2} b^{3} c +a^{2} b^{2} c^{2}+b^{6}+2 b^{5} c +b^{4} c^{2}}}{b^{2} \left (b +c \right )}\right ) \sqrt {a^{2} b^{4}+2 a^{2} b^{3} c +a^{2} b^{2} c^{2}+b^{6}+2 b^{5} c +b^{4} c^{2}}}{\left (a^{2}+b^{2}\right ) b}\) | \(634\) |
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Leaf count of result is larger than twice the leaf count of optimal. 167 vs. \(2 (48) = 96\).
Time = 0.27 (sec) , antiderivative size = 167, normalized size of antiderivative = 3.21 \[ \int \frac {b+c+\cosh (x)}{a+b \sinh (x)} \, dx=\frac {\sqrt {a^{2} + b^{2}} {\left (b^{2} + b c\right )} \log \left (\frac {b^{2} \cosh \left (x\right )^{2} + b^{2} \sinh \left (x\right )^{2} + 2 \, a b \cosh \left (x\right ) + 2 \, a^{2} + b^{2} + 2 \, {\left (b^{2} \cosh \left (x\right ) + a b\right )} \sinh \left (x\right ) - 2 \, \sqrt {a^{2} + b^{2}} {\left (b \cosh \left (x\right ) + b \sinh \left (x\right ) + a\right )}}{b \cosh \left (x\right )^{2} + b \sinh \left (x\right )^{2} + 2 \, a \cosh \left (x\right ) + 2 \, {\left (b \cosh \left (x\right ) + a\right )} \sinh \left (x\right ) - b}\right ) - {\left (a^{2} + b^{2}\right )} x + {\left (a^{2} + b^{2}\right )} \log \left (\frac {2 \, {\left (b \sinh \left (x\right ) + a\right )}}{\cosh \left (x\right ) - \sinh \left (x\right )}\right )}{a^{2} b + b^{3}} \]
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Result contains complex when optimal does not.
Time = 30.48 (sec) , antiderivative size = 585, normalized size of antiderivative = 11.25 \[ \int \frac {b+c+\cosh (x)}{a+b \sinh (x)} \, dx=\text {Too large to display} \]
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Leaf count of result is larger than twice the leaf count of optimal. 122 vs. \(2 (48) = 96\).
Time = 0.28 (sec) , antiderivative size = 122, normalized size of antiderivative = 2.35 \[ \int \frac {b+c+\cosh (x)}{a+b \sinh (x)} \, dx=\frac {b \log \left (\frac {b e^{\left (-x\right )} - a - \sqrt {a^{2} + b^{2}}}{b e^{\left (-x\right )} - a + \sqrt {a^{2} + b^{2}}}\right )}{\sqrt {a^{2} + b^{2}}} + \frac {c \log \left (\frac {b e^{\left (-x\right )} - a - \sqrt {a^{2} + b^{2}}}{b e^{\left (-x\right )} - a + \sqrt {a^{2} + b^{2}}}\right )}{\sqrt {a^{2} + b^{2}}} + \frac {\log \left (b \sinh \left (x\right ) + a\right )}{b} \]
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Time = 0.28 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.67 \[ \int \frac {b+c+\cosh (x)}{a+b \sinh (x)} \, dx=\frac {{\left (b + c\right )} \log \left (\frac {{\left | 2 \, b e^{x} + 2 \, a - 2 \, \sqrt {a^{2} + b^{2}} \right |}}{{\left | 2 \, b e^{x} + 2 \, a + 2 \, \sqrt {a^{2} + b^{2}} \right |}}\right )}{\sqrt {a^{2} + b^{2}}} - \frac {x}{b} + \frac {\log \left ({\left | b e^{\left (2 \, x\right )} + 2 \, a e^{x} - b \right |}\right )}{b} \]
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Time = 2.56 (sec) , antiderivative size = 178, normalized size of antiderivative = 3.42 \[ \int \frac {b+c+\cosh (x)}{a+b \sinh (x)} \, dx=\frac {\ln \left (a^2\,{\mathrm {e}}^x-b\,\sqrt {a^2+b^2}+b^2\,{\mathrm {e}}^x+a\,{\mathrm {e}}^x\,\sqrt {a^2+b^2}\right )\,\left (b^2\,\sqrt {a^2+b^2}+a^2+b^2+b\,c\,\sqrt {a^2+b^2}\right )}{a^2\,b+b^3}-\frac {\ln \left (b\,\sqrt {a^2+b^2}+a^2\,{\mathrm {e}}^x+b^2\,{\mathrm {e}}^x-a\,{\mathrm {e}}^x\,\sqrt {a^2+b^2}\right )\,\left (b^2\,\sqrt {a^2+b^2}-a^2-b^2+b\,c\,\sqrt {a^2+b^2}\right )}{a^2\,b+b^3}-\frac {x}{b} \]
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