Integrand size = 8, antiderivative size = 31 \[ \int e^{\text {arccosh}(a+b x)} \, dx=\frac {e^{2 \text {arccosh}(a+b x)}}{4 b}-\frac {\text {arccosh}(a+b x)}{2 b} \]
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Time = 0.01 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6014, 2320, 12, 14} \[ \int e^{\text {arccosh}(a+b x)} \, dx=\frac {e^{2 \text {arccosh}(a+b x)}}{4 b}-\frac {\text {arccosh}(a+b x)}{2 b} \]
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Rule 12
Rule 14
Rule 2320
Rule 6014
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int e^x \sinh (x) \, dx,x,\text {arccosh}(a+b x)\right )}{b} \\ & = \frac {\text {Subst}\left (\int \frac {-1+x^2}{2 x} \, dx,x,e^{\text {arccosh}(a+b x)}\right )}{b} \\ & = \frac {\text {Subst}\left (\int \frac {-1+x^2}{x} \, dx,x,e^{\text {arccosh}(a+b x)}\right )}{2 b} \\ & = \frac {\text {Subst}\left (\int \left (-\frac {1}{x}+x\right ) \, dx,x,e^{\text {arccosh}(a+b x)}\right )}{2 b} \\ & = \frac {e^{2 \text {arccosh}(a+b x)}}{4 b}-\frac {\text {arccosh}(a+b x)}{2 b} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(69\) vs. \(2(31)=62\).
Time = 0.02 (sec) , antiderivative size = 69, normalized size of antiderivative = 2.23 \[ \int e^{\text {arccosh}(a+b x)} \, dx=\frac {(a+b x) \left (a+b x+\sqrt {-1+a+b x} \sqrt {1+a+b x}\right )-\log \left (a+b x+\sqrt {-1+a+b x} \sqrt {1+a+b x}\right )}{2 b} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(146\) vs. \(2(41)=82\).
Time = 0.72 (sec) , antiderivative size = 147, normalized size of antiderivative = 4.74
method | result | size |
default | \(a x +\frac {b \,x^{2}}{2}+\frac {\sqrt {b x +a -1}\, \left (b x +a +1\right )^{\frac {3}{2}}}{2 b}-\frac {\sqrt {b x +a -1}\, \sqrt {b x +a +1}}{2 b}-\frac {\sqrt {\left (b x +a -1\right ) \left (b x +a +1\right )}\, \ln \left (\frac {\frac {b \left (1+a \right )}{2}+\frac {\left (a -1\right ) b}{2}+b^{2} x}{\sqrt {b^{2}}}+\sqrt {b^{2} x^{2}+\left (b \left (1+a \right )+\left (a -1\right ) b \right ) x +\left (a -1\right ) \left (1+a \right )}\right )}{2 \sqrt {b x +a +1}\, \sqrt {b x +a -1}\, \sqrt {b^{2}}}\) | \(147\) |
parts | \(a x +\frac {b \,x^{2}}{2}+\frac {\sqrt {b x +a -1}\, \left (b x +a +1\right )^{\frac {3}{2}}}{2 b}-\frac {\sqrt {b x +a -1}\, \sqrt {b x +a +1}}{2 b}-\frac {\sqrt {\left (b x +a -1\right ) \left (b x +a +1\right )}\, \ln \left (\frac {\frac {b \left (1+a \right )}{2}+\frac {\left (a -1\right ) b}{2}+b^{2} x}{\sqrt {b^{2}}}+\sqrt {b^{2} x^{2}+\left (b \left (1+a \right )+\left (a -1\right ) b \right ) x +\left (a -1\right ) \left (1+a \right )}\right )}{2 \sqrt {b x +a +1}\, \sqrt {b x +a -1}\, \sqrt {b^{2}}}\) | \(147\) |
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Time = 0.26 (sec) , antiderivative size = 66, normalized size of antiderivative = 2.13 \[ \int e^{\text {arccosh}(a+b x)} \, dx=\frac {b^{2} x^{2} + 2 \, a b x + \sqrt {b x + a + 1} {\left (b x + a\right )} \sqrt {b x + a - 1} + \log \left (-b x + \sqrt {b x + a + 1} \sqrt {b x + a - 1} - a\right )}{2 \, b} \]
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Time = 1.42 (sec) , antiderivative size = 87, normalized size of antiderivative = 2.81 \[ \int e^{\text {arccosh}(a+b x)} \, dx=a x + \frac {b x^{2}}{2} + \begin {cases} \frac {2 \left (\frac {\left (a + b x + 1\right )^{\frac {3}{2}}}{4} - \frac {\sqrt {a + b x + 1}}{4}\right ) \sqrt {a + b x - 1} - \log {\left (2 \sqrt {a + b x - 1} + 2 \sqrt {a + b x + 1} \right )}}{b} & \text {for}\: b \neq 0 \\x \sqrt {a - 1} \sqrt {a + 1} & \text {otherwise} \end {cases} \]
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Leaf count of result is larger than twice the leaf count of optimal. 143 vs. \(2 (41) = 82\).
Time = 0.19 (sec) , antiderivative size = 143, normalized size of antiderivative = 4.61 \[ \int e^{\text {arccosh}(a+b x)} \, dx=\frac {1}{2} \, b x^{2} + a x - \frac {a^{2} \log \left (2 \, b^{2} x + 2 \, a b + 2 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} - 1} b\right )}{2 \, b} + \frac {1}{2} \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} - 1} x + \frac {{\left (a^{2} - 1\right )} \log \left (2 \, b^{2} x + 2 \, a b + 2 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} - 1} b\right )}{2 \, b} + \frac {\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} - 1} a}{2 \, b} \]
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Leaf count of result is larger than twice the leaf count of optimal. 151 vs. \(2 (41) = 82\).
Time = 0.33 (sec) , antiderivative size = 151, normalized size of antiderivative = 4.87 \[ \int e^{\text {arccosh}(a+b x)} \, dx=\frac {1}{2} \, b x^{2} + a x + \frac {\sqrt {b x + a + 1} \sqrt {b x + a - 1} {\left (b x - a - 2\right )} + 2 \, {\left (\sqrt {b x + a + 1} \sqrt {b x + a - 1} + 2 \, \log \left (\sqrt {b x + a + 1} - \sqrt {b x + a - 1}\right )\right )} a - 2 \, {\left (2 \, a + 1\right )} \log \left (\sqrt {b x + a + 1} - \sqrt {b x + a - 1}\right ) + 2 \, \sqrt {b x + a + 1} \sqrt {b x + a - 1} + 4 \, \log \left (\sqrt {b x + a + 1} - \sqrt {b x + a - 1}\right )}{2 \, b} \]
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Time = 0.18 (sec) , antiderivative size = 79, normalized size of antiderivative = 2.55 \[ \int e^{\text {arccosh}(a+b x)} \, dx=a\,x+\frac {b\,x^2}{2}-\frac {\ln \left (a+\sqrt {a+b\,x-1}\,\sqrt {a+b\,x+1}+b\,x\right )}{2\,b}+\frac {x\,\sqrt {a+b\,x-1}\,\sqrt {a+b\,x+1}}{2}+\frac {a\,\sqrt {a+b\,x-1}\,\sqrt {a+b\,x+1}}{2\,b} \]
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