Integrand size = 8, antiderivative size = 76 \[ \int x \text {arcsinh}(a+b x) \, dx=\frac {3 a \sqrt {1+(a+b x)^2}}{4 b^2}-\frac {x \sqrt {1+(a+b x)^2}}{4 b}+\frac {\left (1-2 a^2\right ) \text {arcsinh}(a+b x)}{4 b^2}+\frac {1}{2} x^2 \text {arcsinh}(a+b x) \]
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Time = 0.05 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.625, Rules used = {5859, 5828, 757, 655, 221} \[ \int x \text {arcsinh}(a+b x) \, dx=\frac {\left (1-2 a^2\right ) \text {arcsinh}(a+b x)}{4 b^2}+\frac {1}{2} x^2 \text {arcsinh}(a+b x)+\frac {3 a \sqrt {(a+b x)^2+1}}{4 b^2}-\frac {x \sqrt {(a+b x)^2+1}}{4 b} \]
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Rule 221
Rule 655
Rule 757
Rule 5828
Rule 5859
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \left (-\frac {a}{b}+\frac {x}{b}\right ) \text {arcsinh}(x) \, dx,x,a+b x\right )}{b} \\ & = \frac {1}{2} x^2 \text {arcsinh}(a+b x)-\frac {1}{2} \text {Subst}\left (\int \frac {\left (-\frac {a}{b}+\frac {x}{b}\right )^2}{\sqrt {1+x^2}} \, dx,x,a+b x\right ) \\ & = -\frac {x \sqrt {1+(a+b x)^2}}{4 b}+\frac {1}{2} x^2 \text {arcsinh}(a+b x)-\frac {1}{4} \text {Subst}\left (\int \frac {-\frac {1-2 a^2}{b^2}-\frac {3 a x}{b^2}}{\sqrt {1+x^2}} \, dx,x,a+b x\right ) \\ & = \frac {3 a \sqrt {1+(a+b x)^2}}{4 b^2}-\frac {x \sqrt {1+(a+b x)^2}}{4 b}+\frac {1}{2} x^2 \text {arcsinh}(a+b x)+\frac {\left (1-2 a^2\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1+x^2}} \, dx,x,a+b x\right )}{4 b^2} \\ & = \frac {3 a \sqrt {1+(a+b x)^2}}{4 b^2}-\frac {x \sqrt {1+(a+b x)^2}}{4 b}+\frac {\left (1-2 a^2\right ) \text {arcsinh}(a+b x)}{4 b^2}+\frac {1}{2} x^2 \text {arcsinh}(a+b x) \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.79 \[ \int x \text {arcsinh}(a+b x) \, dx=\frac {(3 a-b x) \sqrt {1+a^2+2 a b x+b^2 x^2}+\left (1-2 a^2+2 b^2 x^2\right ) \text {arcsinh}(a+b x)}{4 b^2} \]
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Time = 0.01 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.97
method | result | size |
derivativedivides | \(\frac {\frac {\operatorname {arcsinh}\left (b x +a \right ) \left (b x +a \right )^{2}}{2}-\operatorname {arcsinh}\left (b x +a \right ) a \left (b x +a \right )-\frac {\left (b x +a \right ) \sqrt {1+\left (b x +a \right )^{2}}}{4}+\frac {\operatorname {arcsinh}\left (b x +a \right )}{4}+a \sqrt {1+\left (b x +a \right )^{2}}}{b^{2}}\) | \(74\) |
default | \(\frac {\frac {\operatorname {arcsinh}\left (b x +a \right ) \left (b x +a \right )^{2}}{2}-\operatorname {arcsinh}\left (b x +a \right ) a \left (b x +a \right )-\frac {\left (b x +a \right ) \sqrt {1+\left (b x +a \right )^{2}}}{4}+\frac {\operatorname {arcsinh}\left (b x +a \right )}{4}+a \sqrt {1+\left (b x +a \right )^{2}}}{b^{2}}\) | \(74\) |
parts | \(\frac {x^{2} \operatorname {arcsinh}\left (b x +a \right )}{2}-\frac {b \left (\frac {x \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{2 b^{2}}-\frac {3 a \left (\frac {\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{b^{2}}-\frac {a \ln \left (\frac {b^{2} x +a b}{\sqrt {b^{2}}}+\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\right )}{b \sqrt {b^{2}}}\right )}{2 b}-\frac {\left (a^{2}+1\right ) \ln \left (\frac {b^{2} x +a b}{\sqrt {b^{2}}}+\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\right )}{2 b^{2} \sqrt {b^{2}}}\right )}{2}\) | \(170\) |
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Time = 0.25 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.99 \[ \int x \text {arcsinh}(a+b x) \, dx=\frac {{\left (2 \, b^{2} x^{2} - 2 \, a^{2} + 1\right )} \log \left (b x + a + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right ) - \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} {\left (b x - 3 \, a\right )}}{4 \, b^{2}} \]
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Time = 0.18 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.37 \[ \int x \text {arcsinh}(a+b x) \, dx=\begin {cases} - \frac {a^{2} \operatorname {asinh}{\left (a + b x \right )}}{2 b^{2}} + \frac {3 a \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1}}{4 b^{2}} + \frac {x^{2} \operatorname {asinh}{\left (a + b x \right )}}{2} - \frac {x \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1}}{4 b} + \frac {\operatorname {asinh}{\left (a + b x \right )}}{4 b^{2}} & \text {for}\: b \neq 0 \\\frac {x^{2} \operatorname {asinh}{\left (a \right )}}{2} & \text {otherwise} \end {cases} \]
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Leaf count of result is larger than twice the leaf count of optimal. 149 vs. \(2 (64) = 128\).
Time = 0.19 (sec) , antiderivative size = 149, normalized size of antiderivative = 1.96 \[ \int x \text {arcsinh}(a+b x) \, dx=\frac {1}{2} \, x^{2} \operatorname {arsinh}\left (b x + a\right ) - \frac {1}{4} \, b {\left (\frac {3 \, a^{2} \operatorname {arsinh}\left (\frac {2 \, {\left (b^{2} x + a b\right )}}{\sqrt {-4 \, a^{2} b^{2} + 4 \, {\left (a^{2} + 1\right )} b^{2}}}\right )}{b^{3}} + \frac {\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} x}{b^{2}} - \frac {{\left (a^{2} + 1\right )} \operatorname {arsinh}\left (\frac {2 \, {\left (b^{2} x + a b\right )}}{\sqrt {-4 \, a^{2} b^{2} + 4 \, {\left (a^{2} + 1\right )} b^{2}}}\right )}{b^{3}} - \frac {3 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} a}{b^{3}}\right )} \]
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Time = 0.29 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.46 \[ \int x \text {arcsinh}(a+b x) \, dx=\frac {1}{2} \, x^{2} \log \left (b x + a + \sqrt {{\left (b x + a\right )}^{2} + 1}\right ) - \frac {1}{4} \, {\left (\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} {\left (\frac {x}{b^{2}} - \frac {3 \, a}{b^{3}}\right )} - \frac {{\left (2 \, a^{2} - 1\right )} \log \left (-a b - {\left (x {\left | b \right |} - \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right )} {\left | b \right |}\right )}{b^{2} {\left | b \right |}}\right )} b \]
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Timed out. \[ \int x \text {arcsinh}(a+b x) \, dx=\int x\,\mathrm {asinh}\left (a+b\,x\right ) \,d x \]
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