Integrand size = 8, antiderivative size = 31 \[ \int e^{\text {arcsinh}(a+b x)} \, dx=\frac {e^{2 \text {arcsinh}(a+b x)}}{4 b}+\frac {\text {arcsinh}(a+b x)}{2 b} \]
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Time = 0.02 (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 = {5872, 2320, 12, 14} \[ \int e^{\text {arcsinh}(a+b x)} \, dx=\frac {\text {arcsinh}(a+b x)}{2 b}+\frac {e^{2 \text {arcsinh}(a+b x)}}{4 b} \]
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Rule 12
Rule 14
Rule 2320
Rule 5872
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int e^x \cosh (x) \, dx,x,\text {arcsinh}(a+b x)\right )}{b} \\ & = \frac {\text {Subst}\left (\int \frac {1+x^2}{2 x} \, dx,x,e^{\text {arcsinh}(a+b x)}\right )}{b} \\ & = \frac {\text {Subst}\left (\int \frac {1+x^2}{x} \, dx,x,e^{\text {arcsinh}(a+b x)}\right )}{2 b} \\ & = \frac {\text {Subst}\left (\int \left (\frac {1}{x}+x\right ) \, dx,x,e^{\text {arcsinh}(a+b x)}\right )}{2 b} \\ & = \frac {e^{2 \text {arcsinh}(a+b x)}}{4 b}+\frac {\text {arcsinh}(a+b x)}{2 b} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.48 \[ \int e^{\text {arcsinh}(a+b x)} \, dx=\frac {(a+b x) \left (a+b x+\sqrt {1+a^2+2 a b x+b^2 x^2}\right )+\text {arcsinh}(a+b x)}{2 b} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(88\) vs. \(2(35)=70\).
Time = 0.76 (sec) , antiderivative size = 89, normalized size of antiderivative = 2.87
method | result | size |
default | \(a x +\frac {\left (2 b^{2} x +2 a b \right ) \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{4 b^{2}}+\frac {\ln \left (\frac {b^{2} x +a b}{\sqrt {b^{2}}}+\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\right )}{2 \sqrt {b^{2}}}+\frac {b \,x^{2}}{2}\) | \(89\) |
parts | \(a x +\frac {\left (2 b^{2} x +2 a b \right ) \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{4 b^{2}}+\frac {\ln \left (\frac {b^{2} x +a b}{\sqrt {b^{2}}}+\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\right )}{2 \sqrt {b^{2}}}+\frac {b \,x^{2}}{2}\) | \(89\) |
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Leaf count of result is larger than twice the leaf count of optimal. 73 vs. \(2 (35) = 70\).
Time = 0.25 (sec) , antiderivative size = 73, normalized size of antiderivative = 2.35 \[ \int e^{\text {arcsinh}(a+b x)} \, dx=\frac {b^{2} x^{2} + 2 \, a b x + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} {\left (b x + a\right )} - \log \left (-b x - a + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right )}{2 \, b} \]
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Time = 0.39 (sec) , antiderivative size = 124, normalized size of antiderivative = 4.00 \[ \int e^{\text {arcsinh}(a+b x)} \, dx=a x + \frac {b x^{2}}{2} + \begin {cases} \left (\frac {a}{2 b} + \frac {x}{2}\right ) \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + \frac {\log {\left (2 a b + 2 b^{2} x + 2 \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} \sqrt {b^{2}} \right )}}{2 \sqrt {b^{2}}} & \text {for}\: b^{2} \neq 0 \\\frac {\left (a^{2} + 2 a b x + 1\right )^{\frac {3}{2}}}{3 a b} & \text {for}\: a b \neq 0 \\x \sqrt {a^{2} + 1} & \text {otherwise} \end {cases} \]
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Leaf count of result is larger than twice the leaf count of optimal. 141 vs. \(2 (35) = 70\).
Time = 0.26 (sec) , antiderivative size = 141, normalized size of antiderivative = 4.55 \[ \int e^{\text {arcsinh}(a+b x)} \, dx=\frac {1}{2} \, b x^{2} + a x - \frac {a^{2} \operatorname {arsinh}\left (\frac {2 \, {\left (b^{2} x + a b\right )}}{\sqrt {-4 \, a^{2} b^{2} + 4 \, {\left (a^{2} + 1\right )} b^{2}}}\right )}{2 \, b} + \frac {1}{2} \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} x + \frac {{\left (a^{2} + 1\right )} \operatorname {arsinh}\left (\frac {2 \, {\left (b^{2} x + a b\right )}}{\sqrt {-4 \, a^{2} b^{2} + 4 \, {\left (a^{2} + 1\right )} b^{2}}}\right )}{2 \, b} + \frac {\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} a}{2 \, b} \]
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Leaf count of result is larger than twice the leaf count of optimal. 80 vs. \(2 (35) = 70\).
Time = 0.29 (sec) , antiderivative size = 80, normalized size of antiderivative = 2.58 \[ \int e^{\text {arcsinh}(a+b x)} \, dx=\frac {1}{2} \, b x^{2} + a x + \frac {1}{2} \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} {\left (x + \frac {a}{b}\right )} - \frac {\log \left (-a b - {\left (x {\left | b \right |} - \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right )} {\left | b \right |}\right )}{2 \, {\left | b \right |}} \]
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Timed out. \[ \int e^{\text {arcsinh}(a+b x)} \, dx=\int a+\sqrt {{\left (a+b\,x\right )}^2+1}+b\,x \,d x \]
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