Integrand size = 28, antiderivative size = 46 \[ \int \frac {\text {arcsinh}(a+b x)}{\left (1+a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx=\frac {(a+b x) \text {arcsinh}(a+b x)}{b \sqrt {1+(a+b x)^2}}-\frac {\log \left (1+(a+b x)^2\right )}{2 b} \]
[Out]
Time = 0.04 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.107, Rules used = {5860, 5787, 266} \[ \int \frac {\text {arcsinh}(a+b x)}{\left (1+a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx=\frac {(a+b x) \text {arcsinh}(a+b x)}{b \sqrt {(a+b x)^2+1}}-\frac {\log \left ((a+b x)^2+1\right )}{2 b} \]
[In]
[Out]
Rule 266
Rule 5787
Rule 5860
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {\text {arcsinh}(x)}{\left (1+x^2\right )^{3/2}} \, dx,x,a+b x\right )}{b} \\ & = \frac {(a+b x) \text {arcsinh}(a+b x)}{b \sqrt {1+(a+b x)^2}}-\frac {\text {Subst}\left (\int \frac {x}{1+x^2} \, dx,x,a+b x\right )}{b} \\ & = \frac {(a+b x) \text {arcsinh}(a+b x)}{b \sqrt {1+(a+b x)^2}}-\frac {\log \left (1+(a+b x)^2\right )}{2 b} \\ \end{align*}
Time = 0.07 (sec) , antiderivative size = 62, normalized size of antiderivative = 1.35 \[ \int \frac {\text {arcsinh}(a+b x)}{\left (1+a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx=\frac {(a+b x) \text {arcsinh}(a+b x)}{b \sqrt {1+a^2+2 a b x+b^2 x^2}}-\frac {\log \left (1+a^2+2 a b x+b^2 x^2\right )}{2 b} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. \(130\) vs. \(2(42)=84\).
Time = 0.61 (sec) , antiderivative size = 131, normalized size of antiderivative = 2.85
method | result | size |
default | \(\frac {2 \,\operatorname {arcsinh}\left (b x +a \right )}{b}-\frac {\left (b^{2} x^{2}-\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\, b x +2 a b x -a \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}+a^{2}+1\right ) \operatorname {arcsinh}\left (b x +a \right )}{b \left (b^{2} x^{2}+2 a b x +a^{2}+1\right )}-\frac {\ln \left (1+\left (b x +a +\sqrt {1+\left (b x +a \right )^{2}}\right )^{2}\right )}{b}\) | \(131\) |
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 115 vs. \(2 (42) = 84\).
Time = 0.26 (sec) , antiderivative size = 115, normalized size of antiderivative = 2.50 \[ \int \frac {\text {arcsinh}(a+b x)}{\left (1+a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx=\frac {2 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} {\left (b x + a\right )} \log \left (b x + a + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right ) - {\left (b^{2} x^{2} + 2 \, a b x + a^{2} + 1\right )} \log \left (b^{2} x^{2} + 2 \, a b x + a^{2} + 1\right )}{2 \, {\left (b^{3} x^{2} + 2 \, a b^{2} x + {\left (a^{2} + 1\right )} b\right )}} \]
[In]
[Out]
\[ \int \frac {\text {arcsinh}(a+b x)}{\left (1+a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx=\int \frac {\operatorname {asinh}{\left (a + b x \right )}}{\left (a^{2} + 2 a b x + b^{2} x^{2} + 1\right )^{\frac {3}{2}}}\, dx \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 119 vs. \(2 (42) = 84\).
Time = 0.34 (sec) , antiderivative size = 119, normalized size of antiderivative = 2.59 \[ \int \frac {\text {arcsinh}(a+b x)}{\left (1+a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx=-{\left (\frac {b^{2} x}{{\left (a^{2} b^{2} - {\left (a^{2} + 1\right )} b^{2}\right )} \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}} + \frac {a b}{{\left (a^{2} b^{2} - {\left (a^{2} + 1\right )} b^{2}\right )} \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}}\right )} \operatorname {arsinh}\left (b x + a\right ) - \frac {\log \left (b^{2} x^{2} + 2 \, a b x + a^{2} + 1\right )}{2 \, b} \]
[In]
[Out]
none
Time = 0.31 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.65 \[ \int \frac {\text {arcsinh}(a+b x)}{\left (1+a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx=\frac {{\left (x + \frac {a}{b}\right )} \log \left (b x + a + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right )}{\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}} - \frac {\log \left (b^{2} x^{2} + 2 \, a b x + a^{2} + 1\right )}{2 \, b} \]
[In]
[Out]
Timed out. \[ \int \frac {\text {arcsinh}(a+b x)}{\left (1+a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx=\int \frac {\mathrm {asinh}\left (a+b\,x\right )}{{\left (a^2+2\,a\,b\,x+b^2\,x^2+1\right )}^{3/2}} \,d x \]
[In]
[Out]