Integrand size = 30, antiderivative size = 54 \[ \int \frac {\left (1+a^2+2 a b x+b^2 x^2\right )^{3/2}}{\text {arcsinh}(a+b x)^2} \, dx=-\frac {\left (1+(a+b x)^2\right )^2}{b \text {arcsinh}(a+b x)}+\frac {\text {Shi}(2 \text {arcsinh}(a+b x))}{b}+\frac {\text {Shi}(4 \text {arcsinh}(a+b x))}{2 b} \]
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Time = 0.12 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {5860, 5790, 5819, 5556, 3379} \[ \int \frac {\left (1+a^2+2 a b x+b^2 x^2\right )^{3/2}}{\text {arcsinh}(a+b x)^2} \, dx=\frac {\text {Shi}(2 \text {arcsinh}(a+b x))}{b}+\frac {\text {Shi}(4 \text {arcsinh}(a+b x))}{2 b}-\frac {\left ((a+b x)^2+1\right )^2}{b \text {arcsinh}(a+b x)} \]
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Rule 3379
Rule 5556
Rule 5790
Rule 5819
Rule 5860
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {\left (1+x^2\right )^{3/2}}{\text {arcsinh}(x)^2} \, dx,x,a+b x\right )}{b} \\ & = -\frac {\left (1+(a+b x)^2\right )^2}{b \text {arcsinh}(a+b x)}+\frac {4 \text {Subst}\left (\int \frac {x \left (1+x^2\right )}{\text {arcsinh}(x)} \, dx,x,a+b x\right )}{b} \\ & = -\frac {\left (1+(a+b x)^2\right )^2}{b \text {arcsinh}(a+b x)}+\frac {4 \text {Subst}\left (\int \frac {\cosh ^3(x) \sinh (x)}{x} \, dx,x,\text {arcsinh}(a+b x)\right )}{b} \\ & = -\frac {\left (1+(a+b x)^2\right )^2}{b \text {arcsinh}(a+b x)}+\frac {4 \text {Subst}\left (\int \left (\frac {\sinh (2 x)}{4 x}+\frac {\sinh (4 x)}{8 x}\right ) \, dx,x,\text {arcsinh}(a+b x)\right )}{b} \\ & = -\frac {\left (1+(a+b x)^2\right )^2}{b \text {arcsinh}(a+b x)}+\frac {\text {Subst}\left (\int \frac {\sinh (4 x)}{x} \, dx,x,\text {arcsinh}(a+b x)\right )}{2 b}+\frac {\text {Subst}\left (\int \frac {\sinh (2 x)}{x} \, dx,x,\text {arcsinh}(a+b x)\right )}{b} \\ & = -\frac {\left (1+(a+b x)^2\right )^2}{b \text {arcsinh}(a+b x)}+\frac {\text {Shi}(2 \text {arcsinh}(a+b x))}{b}+\frac {\text {Shi}(4 \text {arcsinh}(a+b x))}{2 b} \\ \end{align*}
Time = 0.22 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.30 \[ \int \frac {\left (1+a^2+2 a b x+b^2 x^2\right )^{3/2}}{\text {arcsinh}(a+b x)^2} \, dx=\frac {-2 \left (1+a^2+2 a b x+b^2 x^2\right )^2+2 \text {arcsinh}(a+b x) \text {Shi}(2 \text {arcsinh}(a+b x))+\text {arcsinh}(a+b x) \text {Shi}(4 \text {arcsinh}(a+b x))}{2 b \text {arcsinh}(a+b x)} \]
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Time = 0.70 (sec) , antiderivative size = 72, normalized size of antiderivative = 1.33
method | result | size |
default | \(\frac {8 \,\operatorname {Shi}\left (2 \,\operatorname {arcsinh}\left (b x +a \right )\right ) \operatorname {arcsinh}\left (b x +a \right )+4 \,\operatorname {Shi}\left (4 \,\operatorname {arcsinh}\left (b x +a \right )\right ) \operatorname {arcsinh}\left (b x +a \right )-4 \cosh \left (2 \,\operatorname {arcsinh}\left (b x +a \right )\right )-\cosh \left (4 \,\operatorname {arcsinh}\left (b x +a \right )\right )-3}{8 b \,\operatorname {arcsinh}\left (b x +a \right )}\) | \(72\) |
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\[ \int \frac {\left (1+a^2+2 a b x+b^2 x^2\right )^{3/2}}{\text {arcsinh}(a+b x)^2} \, dx=\int { \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2} + 1\right )}^{\frac {3}{2}}}{\operatorname {arsinh}\left (b x + a\right )^{2}} \,d x } \]
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\[ \int \frac {\left (1+a^2+2 a b x+b^2 x^2\right )^{3/2}}{\text {arcsinh}(a+b x)^2} \, dx=\int \frac {\left (a^{2} + 2 a b x + b^{2} x^{2} + 1\right )^{\frac {3}{2}}}{\operatorname {asinh}^{2}{\left (a + b x \right )}}\, dx \]
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\[ \int \frac {\left (1+a^2+2 a b x+b^2 x^2\right )^{3/2}}{\text {arcsinh}(a+b x)^2} \, dx=\int { \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2} + 1\right )}^{\frac {3}{2}}}{\operatorname {arsinh}\left (b x + a\right )^{2}} \,d x } \]
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\[ \int \frac {\left (1+a^2+2 a b x+b^2 x^2\right )^{3/2}}{\text {arcsinh}(a+b x)^2} \, dx=\int { \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2} + 1\right )}^{\frac {3}{2}}}{\operatorname {arsinh}\left (b x + a\right )^{2}} \,d x } \]
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Timed out. \[ \int \frac {\left (1+a^2+2 a b x+b^2 x^2\right )^{3/2}}{\text {arcsinh}(a+b x)^2} \, dx=\int \frac {{\left (a^2+2\,a\,b\,x+b^2\,x^2+1\right )}^{3/2}}{{\mathrm {asinh}\left (a+b\,x\right )}^2} \,d x \]
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