Integrand size = 30, antiderivative size = 31 \[ \int \frac {\sqrt {1+a^2+2 a b x+b^2 x^2}}{\text {arcsinh}(a+b x)} \, dx=\frac {\text {Chi}(2 \text {arcsinh}(a+b x))}{2 b}+\frac {\log (\text {arcsinh}(a+b x))}{2 b} \]
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Time = 0.09 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {5860, 5791, 3393, 3382} \[ \int \frac {\sqrt {1+a^2+2 a b x+b^2 x^2}}{\text {arcsinh}(a+b x)} \, dx=\frac {\text {Chi}(2 \text {arcsinh}(a+b x))}{2 b}+\frac {\log (\text {arcsinh}(a+b x))}{2 b} \]
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Rule 3382
Rule 3393
Rule 5791
Rule 5860
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {\sqrt {1+x^2}}{\text {arcsinh}(x)} \, dx,x,a+b x\right )}{b} \\ & = \frac {\text {Subst}\left (\int \frac {\cosh ^2(x)}{x} \, dx,x,\text {arcsinh}(a+b x)\right )}{b} \\ & = \frac {\text {Subst}\left (\int \left (\frac {1}{2 x}+\frac {\cosh (2 x)}{2 x}\right ) \, dx,x,\text {arcsinh}(a+b x)\right )}{b} \\ & = \frac {\log (\text {arcsinh}(a+b x))}{2 b}+\frac {\text {Subst}\left (\int \frac {\cosh (2 x)}{x} \, dx,x,\text {arcsinh}(a+b x)\right )}{2 b} \\ & = \frac {\text {Chi}(2 \text {arcsinh}(a+b x))}{2 b}+\frac {\log (\text {arcsinh}(a+b x))}{2 b} \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.77 \[ \int \frac {\sqrt {1+a^2+2 a b x+b^2 x^2}}{\text {arcsinh}(a+b x)} \, dx=\frac {\text {Chi}(2 \text {arcsinh}(a+b x))+\log (\text {arcsinh}(a+b x))}{2 b} \]
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Time = 0.58 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.74
method | result | size |
default | \(\frac {\ln \left (\operatorname {arcsinh}\left (b x +a \right )\right )+\operatorname {Chi}\left (2 \,\operatorname {arcsinh}\left (b x +a \right )\right )}{2 b}\) | \(23\) |
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\[ \int \frac {\sqrt {1+a^2+2 a b x+b^2 x^2}}{\text {arcsinh}(a+b x)} \, dx=\int { \frac {\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}}{\operatorname {arsinh}\left (b x + a\right )} \,d x } \]
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\[ \int \frac {\sqrt {1+a^2+2 a b x+b^2 x^2}}{\text {arcsinh}(a+b x)} \, dx=\int \frac {\sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1}}{\operatorname {asinh}{\left (a + b x \right )}}\, dx \]
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\[ \int \frac {\sqrt {1+a^2+2 a b x+b^2 x^2}}{\text {arcsinh}(a+b x)} \, dx=\int { \frac {\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}}{\operatorname {arsinh}\left (b x + a\right )} \,d x } \]
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\[ \int \frac {\sqrt {1+a^2+2 a b x+b^2 x^2}}{\text {arcsinh}(a+b x)} \, dx=\int { \frac {\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}}{\operatorname {arsinh}\left (b x + a\right )} \,d x } \]
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Timed out. \[ \int \frac {\sqrt {1+a^2+2 a b x+b^2 x^2}}{\text {arcsinh}(a+b x)} \, dx=\int \frac {\sqrt {a^2+2\,a\,b\,x+b^2\,x^2+1}}{\mathrm {asinh}\left (a+b\,x\right )} \,d x \]
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