Integrand size = 12, antiderivative size = 117 \[ \int e^{\text {arccosh}(a+b x)^2} x \, dx=\frac {\sqrt {\pi } \text {erfi}(1-\text {arccosh}(a+b x))}{8 b^2 e}+\frac {\sqrt {\pi } \text {erfi}(1+\text {arccosh}(a+b x))}{8 b^2 e}+\frac {a \sqrt {\pi } \text {erfi}\left (\frac {1}{2} (-1+2 \text {arccosh}(a+b x))\right )}{4 b^2 \sqrt [4]{e}}-\frac {a \sqrt {\pi } \text {erfi}\left (\frac {1}{2} (1+2 \text {arccosh}(a+b x))\right )}{4 b^2 \sqrt [4]{e}} \]
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Time = 0.21 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.00, number of steps used = 17, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {6015, 6873, 12, 6874, 5623, 2266, 2235, 5625} \[ \int e^{\text {arccosh}(a+b x)^2} x \, dx=\frac {\sqrt {\pi } \text {erfi}(1-\text {arccosh}(a+b x))}{8 e b^2}+\frac {\sqrt {\pi } \text {erfi}(\text {arccosh}(a+b x)+1)}{8 e b^2}+\frac {\sqrt {\pi } a \text {erfi}\left (\frac {1}{2} (2 \text {arccosh}(a+b x)-1)\right )}{4 \sqrt [4]{e} b^2}-\frac {\sqrt {\pi } a \text {erfi}\left (\frac {1}{2} (2 \text {arccosh}(a+b x)+1)\right )}{4 \sqrt [4]{e} b^2} \]
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Rule 12
Rule 2235
Rule 2266
Rule 5623
Rule 5625
Rule 6015
Rule 6873
Rule 6874
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int e^{x^2} \left (-\frac {a}{b}+\frac {\cosh (x)}{b}\right ) \sinh (x) \, dx,x,\text {arccosh}(a+b x)\right )}{b} \\ & = \frac {\text {Subst}\left (\int \frac {e^{x^2} (-a+\cosh (x)) \sinh (x)}{b} \, dx,x,\text {arccosh}(a+b x)\right )}{b} \\ & = \frac {\text {Subst}\left (\int e^{x^2} (-a+\cosh (x)) \sinh (x) \, dx,x,\text {arccosh}(a+b x)\right )}{b^2} \\ & = \frac {\text {Subst}\left (\int \left (-a e^{x^2} \sinh (x)+e^{x^2} \cosh (x) \sinh (x)\right ) \, dx,x,\text {arccosh}(a+b x)\right )}{b^2} \\ & = \frac {\text {Subst}\left (\int e^{x^2} \cosh (x) \sinh (x) \, dx,x,\text {arccosh}(a+b x)\right )}{b^2}-\frac {a \text {Subst}\left (\int e^{x^2} \sinh (x) \, dx,x,\text {arccosh}(a+b x)\right )}{b^2} \\ & = \frac {\text {Subst}\left (\int \left (-\frac {1}{4} e^{-2 x+x^2}+\frac {1}{4} e^{2 x+x^2}\right ) \, dx,x,\text {arccosh}(a+b x)\right )}{b^2}-\frac {a \text {Subst}\left (\int \left (-\frac {1}{2} e^{-x+x^2}+\frac {e^{x+x^2}}{2}\right ) \, dx,x,\text {arccosh}(a+b x)\right )}{b^2} \\ & = -\frac {\text {Subst}\left (\int e^{-2 x+x^2} \, dx,x,\text {arccosh}(a+b x)\right )}{4 b^2}+\frac {\text {Subst}\left (\int e^{2 x+x^2} \, dx,x,\text {arccosh}(a+b x)\right )}{4 b^2}+\frac {a \text {Subst}\left (\int e^{-x+x^2} \, dx,x,\text {arccosh}(a+b x)\right )}{2 b^2}-\frac {a \text {Subst}\left (\int e^{x+x^2} \, dx,x,\text {arccosh}(a+b x)\right )}{2 b^2} \\ & = -\frac {\text {Subst}\left (\int e^{\frac {1}{4} (-2+2 x)^2} \, dx,x,\text {arccosh}(a+b x)\right )}{4 b^2 e}+\frac {\text {Subst}\left (\int e^{\frac {1}{4} (2+2 x)^2} \, dx,x,\text {arccosh}(a+b x)\right )}{4 b^2 e}+\frac {a \text {Subst}\left (\int e^{\frac {1}{4} (-1+2 x)^2} \, dx,x,\text {arccosh}(a+b x)\right )}{2 b^2 \sqrt [4]{e}}-\frac {a \text {Subst}\left (\int e^{\frac {1}{4} (1+2 x)^2} \, dx,x,\text {arccosh}(a+b x)\right )}{2 b^2 \sqrt [4]{e}} \\ & = \frac {\sqrt {\pi } \text {erfi}(1-\text {arccosh}(a+b x))}{8 b^2 e}+\frac {\sqrt {\pi } \text {erfi}(1+\text {arccosh}(a+b x))}{8 b^2 e}+\frac {a \sqrt {\pi } \text {erfi}\left (\frac {1}{2} (-1+2 \text {arccosh}(a+b x))\right )}{4 b^2 \sqrt [4]{e}}-\frac {a \sqrt {\pi } \text {erfi}\left (\frac {1}{2} (1+2 \text {arccosh}(a+b x))\right )}{4 b^2 \sqrt [4]{e}} \\ \end{align*}
Time = 0.13 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.65 \[ \int e^{\text {arccosh}(a+b x)^2} x \, dx=\frac {\sqrt {\pi } \left (-2 a e^{3/4} \text {erfi}\left (\frac {1}{2}-\text {arccosh}(a+b x)\right )+\text {erfi}(1-\text {arccosh}(a+b x))-2 a e^{3/4} \text {erfi}\left (\frac {1}{2}+\text {arccosh}(a+b x)\right )+\text {erfi}(1+\text {arccosh}(a+b x))\right )}{8 b^2 e} \]
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\[\int {\mathrm e}^{\operatorname {arccosh}\left (b x +a \right )^{2}} x d x\]
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\[ \int e^{\text {arccosh}(a+b x)^2} x \, dx=\int { x e^{\left (\operatorname {arcosh}\left (b x + a\right )^{2}\right )} \,d x } \]
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\[ \int e^{\text {arccosh}(a+b x)^2} x \, dx=\int x e^{\operatorname {acosh}^{2}{\left (a + b x \right )}}\, dx \]
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\[ \int e^{\text {arccosh}(a+b x)^2} x \, dx=\int { x e^{\left (\operatorname {arcosh}\left (b x + a\right )^{2}\right )} \,d x } \]
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\[ \int e^{\text {arccosh}(a+b x)^2} x \, dx=\int { x e^{\left (\operatorname {arcosh}\left (b x + a\right )^{2}\right )} \,d x } \]
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Timed out. \[ \int e^{\text {arccosh}(a+b x)^2} x \, dx=\int x\,{\mathrm {e}}^{{\mathrm {acosh}\left (a+b\,x\right )}^2} \,d x \]
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