Integrand size = 28, antiderivative size = 15 \[ \int \frac {\text {arcsinh}(a+b x)}{\sqrt {1+a^2+2 a b x+b^2 x^2}} \, dx=\frac {\text {arcsinh}(a+b x)^2}{2 b} \]
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Time = 0.03 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {5860, 5783} \[ \int \frac {\text {arcsinh}(a+b x)}{\sqrt {1+a^2+2 a b x+b^2 x^2}} \, dx=\frac {\text {arcsinh}(a+b x)^2}{2 b} \]
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Rule 5783
Rule 5860
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {\text {arcsinh}(x)}{\sqrt {1+x^2}} \, dx,x,a+b x\right )}{b} \\ & = \frac {\text {arcsinh}(a+b x)^2}{2 b} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.00 \[ \int \frac {\text {arcsinh}(a+b x)}{\sqrt {1+a^2+2 a b x+b^2 x^2}} \, dx=\frac {\text {arcsinh}(a+b x)^2}{2 b} \]
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Time = 0.70 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.93
method | result | size |
default | \(\frac {\operatorname {arcsinh}\left (b x +a \right )^{2}}{2 b}\) | \(14\) |
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Leaf count of result is larger than twice the leaf count of optimal. 32 vs. \(2 (13) = 26\).
Time = 0.26 (sec) , antiderivative size = 32, normalized size of antiderivative = 2.13 \[ \int \frac {\text {arcsinh}(a+b x)}{\sqrt {1+a^2+2 a b x+b^2 x^2}} \, dx=\frac {\log \left (b x + a + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right )^{2}}{2 \, b} \]
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Leaf count of result is larger than twice the leaf count of optimal. 24 vs. \(2 (10) = 20\).
Time = 0.21 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.60 \[ \int \frac {\text {arcsinh}(a+b x)}{\sqrt {1+a^2+2 a b x+b^2 x^2}} \, dx=\begin {cases} \frac {\operatorname {asinh}^{2}{\left (a + b x \right )}}{2 b} & \text {for}\: b \neq 0 \\\frac {x \operatorname {asinh}{\left (a \right )}}{\sqrt {a^{2} + 1}} & \text {otherwise} \end {cases} \]
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Leaf count of result is larger than twice the leaf count of optimal. 84 vs. \(2 (13) = 26\).
Time = 0.28 (sec) , antiderivative size = 84, normalized size of antiderivative = 5.60 \[ \int \frac {\text {arcsinh}(a+b x)}{\sqrt {1+a^2+2 a b x+b^2 x^2}} \, dx=\frac {\operatorname {arsinh}\left (b x + a\right ) \operatorname {arsinh}\left (\frac {2 \, {\left (b^{2} x + a b\right )}}{\sqrt {-4 \, a^{2} b^{2} + 4 \, {\left (a^{2} + 1\right )} b^{2}}}\right )}{b} - \frac {\operatorname {arsinh}\left (\frac {2 \, {\left (b^{2} x + a b\right )}}{\sqrt {-4 \, a^{2} b^{2} + 4 \, {\left (a^{2} + 1\right )} b^{2}}}\right )^{2}}{2 \, b} \]
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\[ \int \frac {\text {arcsinh}(a+b x)}{\sqrt {1+a^2+2 a b x+b^2 x^2}} \, dx=\int { \frac {\operatorname {arsinh}\left (b x + a\right )}{\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}} \,d x } \]
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Time = 2.85 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.87 \[ \int \frac {\text {arcsinh}(a+b x)}{\sqrt {1+a^2+2 a b x+b^2 x^2}} \, dx=\frac {{\mathrm {asinh}\left (a+b\,x\right )}^2}{2\,b} \]
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