Integrand size = 28, antiderivative size = 61 \[ \int \sqrt {1+a^2+2 a b x+b^2 x^2} \text {arcsinh}(a+b x) \, dx=-\frac {(a+b x)^2}{4 b}+\frac {(a+b x) \sqrt {1+(a+b x)^2} \text {arcsinh}(a+b x)}{2 b}+\frac {\text {arcsinh}(a+b x)^2}{4 b} \]
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Time = 0.05 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {5860, 5785, 5783, 30} \[ \int \sqrt {1+a^2+2 a b x+b^2 x^2} \text {arcsinh}(a+b x) \, dx=\frac {\sqrt {(a+b x)^2+1} (a+b x) \text {arcsinh}(a+b x)}{2 b}+\frac {\text {arcsinh}(a+b x)^2}{4 b}-\frac {(a+b x)^2}{4 b} \]
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Rule 30
Rule 5783
Rule 5785
Rule 5860
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \sqrt {1+x^2} \text {arcsinh}(x) \, dx,x,a+b x\right )}{b} \\ & = \frac {(a+b x) \sqrt {1+(a+b x)^2} \text {arcsinh}(a+b x)}{2 b}-\frac {\text {Subst}(\int x \, dx,x,a+b x)}{2 b}+\frac {\text {Subst}\left (\int \frac {\text {arcsinh}(x)}{\sqrt {1+x^2}} \, dx,x,a+b x\right )}{2 b} \\ & = -\frac {(a+b x)^2}{4 b}+\frac {(a+b x) \sqrt {1+(a+b x)^2} \text {arcsinh}(a+b x)}{2 b}+\frac {\text {arcsinh}(a+b x)^2}{4 b} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.00 \[ \int \sqrt {1+a^2+2 a b x+b^2 x^2} \text {arcsinh}(a+b x) \, dx=\frac {-b x (2 a+b x)+2 (a+b x) \sqrt {1+a^2+2 a b x+b^2 x^2} \text {arcsinh}(a+b x)+\text {arcsinh}(a+b x)^2}{4 b} \]
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Time = 0.86 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.49
method | result | size |
default | \(\frac {2 \,\operatorname {arcsinh}\left (b x +a \right ) \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\, b x -b^{2} x^{2}+2 \,\operatorname {arcsinh}\left (b x +a \right ) \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\, a -2 a b x +\operatorname {arcsinh}\left (b x +a \right )^{2}-a^{2}-1}{4 b}\) | \(91\) |
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Time = 0.26 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.61 \[ \int \sqrt {1+a^2+2 a b x+b^2 x^2} \text {arcsinh}(a+b x) \, dx=-\frac {b^{2} x^{2} + 2 \, a b x - 2 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} {\left (b x + a\right )} \log \left (b x + a + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right ) - \log \left (b x + a + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right )^{2}}{4 \, b} \]
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\[ \int \sqrt {1+a^2+2 a b x+b^2 x^2} \text {arcsinh}(a+b x) \, dx=\int \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} \operatorname {asinh}{\left (a + b x \right )}\, dx \]
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Leaf count of result is larger than twice the leaf count of optimal. 238 vs. \(2 (53) = 106\).
Time = 0.25 (sec) , antiderivative size = 238, normalized size of antiderivative = 3.90 \[ \int \sqrt {1+a^2+2 a b x+b^2 x^2} \text {arcsinh}(a+b x) \, dx=-\frac {1}{4} \, {\left (x^{2} + \frac {2 \, a x}{b} + \frac {2 \, \operatorname {arsinh}\left (b x + a\right ) \operatorname {arsinh}\left (\frac {2 \, {\left (b^{2} x + a b\right )}}{\sqrt {-4 \, a^{2} b^{2} + 4 \, {\left (a^{2} + 1\right )} b^{2}}}\right )}{b^{2}} - \frac {\operatorname {arsinh}\left (\frac {2 \, {\left (b^{2} x + a b\right )}}{\sqrt {-4 \, a^{2} b^{2} + 4 \, {\left (a^{2} + 1\right )} b^{2}}}\right )^{2}}{b^{2}}\right )} b - \frac {1}{2} \, {\left (\frac {a^{2} \operatorname {arsinh}\left (\frac {2 \, {\left (b^{2} x + a b\right )}}{\sqrt {-4 \, a^{2} b^{2} + 4 \, {\left (a^{2} + 1\right )} b^{2}}}\right )}{b} - \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} x - \frac {{\left (a^{2} + 1\right )} \operatorname {arsinh}\left (\frac {2 \, {\left (b^{2} x + a b\right )}}{\sqrt {-4 \, a^{2} b^{2} + 4 \, {\left (a^{2} + 1\right )} b^{2}}}\right )}{b} - \frac {\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} a}{b}\right )} \operatorname {arsinh}\left (b x + a\right ) \]
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\[ \int \sqrt {1+a^2+2 a b x+b^2 x^2} \text {arcsinh}(a+b x) \, dx=\int { \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} \operatorname {arsinh}\left (b x + a\right ) \,d x } \]
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Timed out. \[ \int \sqrt {1+a^2+2 a b x+b^2 x^2} \text {arcsinh}(a+b x) \, dx=\int \mathrm {asinh}\left (a+b\,x\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2+1} \,d x \]
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