Integrand size = 12, antiderivative size = 257 \[ \int \frac {x^2}{\text {arcsinh}(a+b x)^3} \, dx=-\frac {a^2 \sqrt {1+(a+b x)^2}}{2 b^3 \text {arcsinh}(a+b x)^2}+\frac {a (a+b x) \sqrt {1+(a+b x)^2}}{b^3 \text {arcsinh}(a+b x)^2}-\frac {(a+b x)^2 \sqrt {1+(a+b x)^2}}{2 b^3 \text {arcsinh}(a+b x)^2}+\frac {a}{b^3 \text {arcsinh}(a+b x)}-\frac {a+b x}{b^3 \text {arcsinh}(a+b x)}-\frac {a^2 (a+b x)}{2 b^3 \text {arcsinh}(a+b x)}+\frac {2 a (a+b x)^2}{b^3 \text {arcsinh}(a+b x)}-\frac {3 (a+b x)^3}{2 b^3 \text {arcsinh}(a+b x)}-\frac {\text {Chi}(\text {arcsinh}(a+b x))}{8 b^3}+\frac {a^2 \text {Chi}(\text {arcsinh}(a+b x))}{2 b^3}+\frac {9 \text {Chi}(3 \text {arcsinh}(a+b x))}{8 b^3}-\frac {2 a \text {Shi}(2 \text {arcsinh}(a+b x))}{b^3} \]
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Time = 0.38 (sec) , antiderivative size = 257, normalized size of antiderivative = 1.00, number of steps used = 24, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {5859, 5829, 5773, 5818, 5774, 3382, 5779, 5780, 5556, 12, 3379, 5783} \[ \int \frac {x^2}{\text {arcsinh}(a+b x)^3} \, dx=\frac {a^2 \text {Chi}(\text {arcsinh}(a+b x))}{2 b^3}-\frac {a^2 (a+b x)}{2 b^3 \text {arcsinh}(a+b x)}-\frac {a^2 \sqrt {(a+b x)^2+1}}{2 b^3 \text {arcsinh}(a+b x)^2}-\frac {\text {Chi}(\text {arcsinh}(a+b x))}{8 b^3}+\frac {9 \text {Chi}(3 \text {arcsinh}(a+b x))}{8 b^3}-\frac {2 a \text {Shi}(2 \text {arcsinh}(a+b x))}{b^3}-\frac {3 (a+b x)^3}{2 b^3 \text {arcsinh}(a+b x)}+\frac {2 a (a+b x)^2}{b^3 \text {arcsinh}(a+b x)}-\frac {\sqrt {(a+b x)^2+1} (a+b x)^2}{2 b^3 \text {arcsinh}(a+b x)^2}-\frac {a+b x}{b^3 \text {arcsinh}(a+b x)}+\frac {a \sqrt {(a+b x)^2+1} (a+b x)}{b^3 \text {arcsinh}(a+b x)^2}+\frac {a}{b^3 \text {arcsinh}(a+b x)} \]
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Rule 12
Rule 3379
Rule 3382
Rule 5556
Rule 5773
Rule 5774
Rule 5779
Rule 5780
Rule 5783
Rule 5818
Rule 5829
Rule 5859
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {\left (-\frac {a}{b}+\frac {x}{b}\right )^2}{\text {arcsinh}(x)^3} \, dx,x,a+b x\right )}{b} \\ & = \frac {\text {Subst}\left (\int \left (\frac {a^2}{b^2 \text {arcsinh}(x)^3}-\frac {2 a x}{b^2 \text {arcsinh}(x)^3}+\frac {x^2}{b^2 \text {arcsinh}(x)^3}\right ) \, dx,x,a+b x\right )}{b} \\ & = \frac {\text {Subst}\left (\int \frac {x^2}{\text {arcsinh}(x)^3} \, dx,x,a+b x\right )}{b^3}-\frac {(2 a) \text {Subst}\left (\int \frac {x}{\text {arcsinh}(x)^3} \, dx,x,a+b x\right )}{b^3}+\frac {a^2 \text {Subst}\left (\int \frac {1}{\text {arcsinh}(x)^3} \, dx,x,a+b x\right )}{b^3} \\ & = -\frac {a^2 \sqrt {1+(a+b x)^2}}{2 b^3 \text {arcsinh}(a+b x)^2}+\frac {a (a+b x) \sqrt {1+(a+b x)^2}}{b^3 \text {arcsinh}(a+b x)^2}-\frac {(a+b x)^2 \sqrt {1+(a+b x)^2}}{2 b^3 \text {arcsinh}(a+b x)^2}+\frac {\text {Subst}\left (\int \frac {x}{\sqrt {1+x^2} \text {arcsinh}(x)^2} \, dx,x,a+b x\right )}{b^3}+\frac {3 \text {Subst}\left (\int \frac {x^3}{\sqrt {1+x^2} \text {arcsinh}(x)^2} \, dx,x,a+b x\right )}{2 b^3}-\frac {a \text {Subst}\left (\int \frac {1}{\sqrt {1+x^2} \text {arcsinh}(x)^2} \, dx,x,a+b x\right )}{b^3}-\frac {(2 a) \text {Subst}\left (\int \frac {x^2}{\sqrt {1+x^2} \text {arcsinh}(x)^2} \, dx,x,a+b x\right )}{b^3}+\frac {a^2 \text {Subst}\left (\int \frac {x}{\sqrt {1+x^2} \text {arcsinh}(x)^2} \, dx,x,a+b x\right )}{2 b^3} \\ & = -\frac {a^2 \sqrt {1+(a+b x)^2}}{2 b^3 \text {arcsinh}(a+b x)^2}+\frac {a (a+b x) \sqrt {1+(a+b x)^2}}{b^3 \text {arcsinh}(a+b x)^2}-\frac {(a+b x)^2 \sqrt {1+(a+b x)^2}}{2 b^3 \text {arcsinh}(a+b x)^2}+\frac {a}{b^3 \text {arcsinh}(a+b x)}-\frac {a+b x}{b^3 \text {arcsinh}(a+b x)}-\frac {a^2 (a+b x)}{2 b^3 \text {arcsinh}(a+b x)}+\frac {2 a (a+b x)^2}{b^3 \text {arcsinh}(a+b x)}-\frac {3 (a+b x)^3}{2 b^3 \text {arcsinh}(a+b x)}+\frac {\text {Subst}\left (\int \frac {1}{\text {arcsinh}(x)} \, dx,x,a+b x\right )}{b^3}+\frac {9 \text {Subst}\left (\int \frac {x^2}{\text {arcsinh}(x)} \, dx,x,a+b x\right )}{2 b^3}-\frac {(4 a) \text {Subst}\left (\int \frac {x}{\text {arcsinh}(x)} \, dx,x,a+b x\right )}{b^3}+\frac {a^2 \text {Subst}\left (\int \frac {1}{\text {arcsinh}(x)} \, dx,x,a+b x\right )}{2 b^3} \\ & = -\frac {a^2 \sqrt {1+(a+b x)^2}}{2 b^3 \text {arcsinh}(a+b x)^2}+\frac {a (a+b x) \sqrt {1+(a+b x)^2}}{b^3 \text {arcsinh}(a+b x)^2}-\frac {(a+b x)^2 \sqrt {1+(a+b x)^2}}{2 b^3 \text {arcsinh}(a+b x)^2}+\frac {a}{b^3 \text {arcsinh}(a+b x)}-\frac {a+b x}{b^3 \text {arcsinh}(a+b x)}-\frac {a^2 (a+b x)}{2 b^3 \text {arcsinh}(a+b x)}+\frac {2 a (a+b x)^2}{b^3 \text {arcsinh}(a+b x)}-\frac {3 (a+b x)^3}{2 b^3 \text {arcsinh}(a+b x)}+\frac {\text {Subst}\left (\int \frac {\cosh (x)}{x} \, dx,x,\text {arcsinh}(a+b x)\right )}{b^3}+\frac {9 \text {Subst}\left (\int \frac {\cosh (x) \sinh ^2(x)}{x} \, dx,x,\text {arcsinh}(a+b x)\right )}{2 b^3}-\frac {(4 a) \text {Subst}\left (\int \frac {\cosh (x) \sinh (x)}{x} \, dx,x,\text {arcsinh}(a+b x)\right )}{b^3}+\frac {a^2 \text {Subst}\left (\int \frac {\cosh (x)}{x} \, dx,x,\text {arcsinh}(a+b x)\right )}{2 b^3} \\ & = -\frac {a^2 \sqrt {1+(a+b x)^2}}{2 b^3 \text {arcsinh}(a+b x)^2}+\frac {a (a+b x) \sqrt {1+(a+b x)^2}}{b^3 \text {arcsinh}(a+b x)^2}-\frac {(a+b x)^2 \sqrt {1+(a+b x)^2}}{2 b^3 \text {arcsinh}(a+b x)^2}+\frac {a}{b^3 \text {arcsinh}(a+b x)}-\frac {a+b x}{b^3 \text {arcsinh}(a+b x)}-\frac {a^2 (a+b x)}{2 b^3 \text {arcsinh}(a+b x)}+\frac {2 a (a+b x)^2}{b^3 \text {arcsinh}(a+b x)}-\frac {3 (a+b x)^3}{2 b^3 \text {arcsinh}(a+b x)}+\frac {\text {Chi}(\text {arcsinh}(a+b x))}{b^3}+\frac {a^2 \text {Chi}(\text {arcsinh}(a+b x))}{2 b^3}+\frac {9 \text {Subst}\left (\int \left (-\frac {\cosh (x)}{4 x}+\frac {\cosh (3 x)}{4 x}\right ) \, dx,x,\text {arcsinh}(a+b x)\right )}{2 b^3}-\frac {(4 a) \text {Subst}\left (\int \frac {\sinh (2 x)}{2 x} \, dx,x,\text {arcsinh}(a+b x)\right )}{b^3} \\ & = -\frac {a^2 \sqrt {1+(a+b x)^2}}{2 b^3 \text {arcsinh}(a+b x)^2}+\frac {a (a+b x) \sqrt {1+(a+b x)^2}}{b^3 \text {arcsinh}(a+b x)^2}-\frac {(a+b x)^2 \sqrt {1+(a+b x)^2}}{2 b^3 \text {arcsinh}(a+b x)^2}+\frac {a}{b^3 \text {arcsinh}(a+b x)}-\frac {a+b x}{b^3 \text {arcsinh}(a+b x)}-\frac {a^2 (a+b x)}{2 b^3 \text {arcsinh}(a+b x)}+\frac {2 a (a+b x)^2}{b^3 \text {arcsinh}(a+b x)}-\frac {3 (a+b x)^3}{2 b^3 \text {arcsinh}(a+b x)}+\frac {\text {Chi}(\text {arcsinh}(a+b x))}{b^3}+\frac {a^2 \text {Chi}(\text {arcsinh}(a+b x))}{2 b^3}-\frac {9 \text {Subst}\left (\int \frac {\cosh (x)}{x} \, dx,x,\text {arcsinh}(a+b x)\right )}{8 b^3}+\frac {9 \text {Subst}\left (\int \frac {\cosh (3 x)}{x} \, dx,x,\text {arcsinh}(a+b x)\right )}{8 b^3}-\frac {(2 a) \text {Subst}\left (\int \frac {\sinh (2 x)}{x} \, dx,x,\text {arcsinh}(a+b x)\right )}{b^3} \\ & = -\frac {a^2 \sqrt {1+(a+b x)^2}}{2 b^3 \text {arcsinh}(a+b x)^2}+\frac {a (a+b x) \sqrt {1+(a+b x)^2}}{b^3 \text {arcsinh}(a+b x)^2}-\frac {(a+b x)^2 \sqrt {1+(a+b x)^2}}{2 b^3 \text {arcsinh}(a+b x)^2}+\frac {a}{b^3 \text {arcsinh}(a+b x)}-\frac {a+b x}{b^3 \text {arcsinh}(a+b x)}-\frac {a^2 (a+b x)}{2 b^3 \text {arcsinh}(a+b x)}+\frac {2 a (a+b x)^2}{b^3 \text {arcsinh}(a+b x)}-\frac {3 (a+b x)^3}{2 b^3 \text {arcsinh}(a+b x)}-\frac {\text {Chi}(\text {arcsinh}(a+b x))}{8 b^3}+\frac {a^2 \text {Chi}(\text {arcsinh}(a+b x))}{2 b^3}+\frac {9 \text {Chi}(3 \text {arcsinh}(a+b x))}{8 b^3}-\frac {2 a \text {Shi}(2 \text {arcsinh}(a+b x))}{b^3} \\ \end{align*}
Time = 0.33 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.43 \[ \int \frac {x^2}{\text {arcsinh}(a+b x)^3} \, dx=\frac {-\frac {4 b x \left (b x \sqrt {1+a^2+2 a b x+b^2 x^2}+\left (2+2 a^2+5 a b x+3 b^2 x^2\right ) \text {arcsinh}(a+b x)\right )}{\text {arcsinh}(a+b x)^2}+\left (-1+4 a^2\right ) \text {Chi}(\text {arcsinh}(a+b x))+9 \text {Chi}(3 \text {arcsinh}(a+b x))-16 a \text {Shi}(2 \text {arcsinh}(a+b x))}{8 b^3} \]
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Time = 0.15 (sec) , antiderivative size = 215, normalized size of antiderivative = 0.84
method | result | size |
derivativedivides | \(\frac {\frac {\sqrt {1+\left (b x +a \right )^{2}}}{8 \operatorname {arcsinh}\left (b x +a \right )^{2}}+\frac {b x +a}{8 \,\operatorname {arcsinh}\left (b x +a \right )}-\frac {\operatorname {Chi}\left (\operatorname {arcsinh}\left (b x +a \right )\right )}{8}-\frac {\cosh \left (3 \,\operatorname {arcsinh}\left (b x +a \right )\right )}{8 \operatorname {arcsinh}\left (b x +a \right )^{2}}-\frac {3 \sinh \left (3 \,\operatorname {arcsinh}\left (b x +a \right )\right )}{8 \,\operatorname {arcsinh}\left (b x +a \right )}+\frac {9 \,\operatorname {Chi}\left (3 \,\operatorname {arcsinh}\left (b x +a \right )\right )}{8}-\frac {a \left (4 \,\operatorname {Shi}\left (2 \,\operatorname {arcsinh}\left (b x +a \right )\right ) \operatorname {arcsinh}\left (b x +a \right )^{2}-2 \cosh \left (2 \,\operatorname {arcsinh}\left (b x +a \right )\right ) \operatorname {arcsinh}\left (b x +a \right )-\sinh \left (2 \,\operatorname {arcsinh}\left (b x +a \right )\right )\right )}{2 \operatorname {arcsinh}\left (b x +a \right )^{2}}+\frac {a^{2} \left (\operatorname {Chi}\left (\operatorname {arcsinh}\left (b x +a \right )\right ) \operatorname {arcsinh}\left (b x +a \right )^{2}-\left (b x +a \right ) \operatorname {arcsinh}\left (b x +a \right )-\sqrt {1+\left (b x +a \right )^{2}}\right )}{2 \operatorname {arcsinh}\left (b x +a \right )^{2}}}{b^{3}}\) | \(215\) |
default | \(\frac {\frac {\sqrt {1+\left (b x +a \right )^{2}}}{8 \operatorname {arcsinh}\left (b x +a \right )^{2}}+\frac {b x +a}{8 \,\operatorname {arcsinh}\left (b x +a \right )}-\frac {\operatorname {Chi}\left (\operatorname {arcsinh}\left (b x +a \right )\right )}{8}-\frac {\cosh \left (3 \,\operatorname {arcsinh}\left (b x +a \right )\right )}{8 \operatorname {arcsinh}\left (b x +a \right )^{2}}-\frac {3 \sinh \left (3 \,\operatorname {arcsinh}\left (b x +a \right )\right )}{8 \,\operatorname {arcsinh}\left (b x +a \right )}+\frac {9 \,\operatorname {Chi}\left (3 \,\operatorname {arcsinh}\left (b x +a \right )\right )}{8}-\frac {a \left (4 \,\operatorname {Shi}\left (2 \,\operatorname {arcsinh}\left (b x +a \right )\right ) \operatorname {arcsinh}\left (b x +a \right )^{2}-2 \cosh \left (2 \,\operatorname {arcsinh}\left (b x +a \right )\right ) \operatorname {arcsinh}\left (b x +a \right )-\sinh \left (2 \,\operatorname {arcsinh}\left (b x +a \right )\right )\right )}{2 \operatorname {arcsinh}\left (b x +a \right )^{2}}+\frac {a^{2} \left (\operatorname {Chi}\left (\operatorname {arcsinh}\left (b x +a \right )\right ) \operatorname {arcsinh}\left (b x +a \right )^{2}-\left (b x +a \right ) \operatorname {arcsinh}\left (b x +a \right )-\sqrt {1+\left (b x +a \right )^{2}}\right )}{2 \operatorname {arcsinh}\left (b x +a \right )^{2}}}{b^{3}}\) | \(215\) |
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\[ \int \frac {x^2}{\text {arcsinh}(a+b x)^3} \, dx=\int { \frac {x^{2}}{\operatorname {arsinh}\left (b x + a\right )^{3}} \,d x } \]
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\[ \int \frac {x^2}{\text {arcsinh}(a+b x)^3} \, dx=\int \frac {x^{2}}{\operatorname {asinh}^{3}{\left (a + b x \right )}}\, dx \]
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\[ \int \frac {x^2}{\text {arcsinh}(a+b x)^3} \, dx=\int { \frac {x^{2}}{\operatorname {arsinh}\left (b x + a\right )^{3}} \,d x } \]
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\[ \int \frac {x^2}{\text {arcsinh}(a+b x)^3} \, dx=\int { \frac {x^{2}}{\operatorname {arsinh}\left (b x + a\right )^{3}} \,d x } \]
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Timed out. \[ \int \frac {x^2}{\text {arcsinh}(a+b x)^3} \, dx=\int \frac {x^2}{{\mathrm {asinh}\left (a+b\,x\right )}^3} \,d x \]
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