Integrand size = 12, antiderivative size = 327 \[ \int x^2 \text {Chi}(a+b x)^2 \, dx=-\frac {2 x}{3 b^2}-\frac {a \cosh (2 a+2 b x)}{3 b^3}+\frac {x \cosh (2 a+2 b x)}{6 b^2}-\frac {2 a \cosh (a+b x) \text {Chi}(a+b x)}{3 b^3}+\frac {4 x \cosh (a+b x) \text {Chi}(a+b x)}{3 b^2}+\frac {a^2 (a+b x) \text {Chi}(a+b x)^2}{3 b^3}-\frac {a x (a+b x) \text {Chi}(a+b x)^2}{3 b^2}+\frac {x^2 (a+b x) \text {Chi}(a+b x)^2}{3 b}+\frac {a \text {Chi}(2 a+2 b x)}{b^3}+\frac {a \log (a+b x)}{b^3}-\frac {2 \cosh (a+b x) \sinh (a+b x)}{3 b^3}-\frac {4 \text {Chi}(a+b x) \sinh (a+b x)}{3 b^3}-\frac {2 a^2 \text {Chi}(a+b x) \sinh (a+b x)}{3 b^3}+\frac {2 a x \text {Chi}(a+b x) \sinh (a+b x)}{3 b^2}-\frac {2 x^2 \text {Chi}(a+b x) \sinh (a+b x)}{3 b}-\frac {\sinh (2 a+2 b x)}{12 b^3}+\frac {2 \text {Shi}(2 a+2 b x)}{3 b^3}+\frac {a^2 \text {Shi}(2 a+2 b x)}{b^3} \]
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Time = 0.99 (sec) , antiderivative size = 327, normalized size of antiderivative = 1.00, number of steps used = 39, number of rules used = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.583, Rules used = {6674, 6678, 5736, 6873, 6874, 2718, 3377, 2717, 3379, 6684, 2715, 8, 3393, 3382, 6676, 5556, 12, 6682, 6670} \[ \int x^2 \text {Chi}(a+b x)^2 \, dx=\frac {a^2 (a+b x) \text {Chi}(a+b x)^2}{3 b^3}-\frac {2 a^2 \text {Chi}(a+b x) \sinh (a+b x)}{3 b^3}+\frac {a^2 \text {Shi}(2 a+2 b x)}{b^3}+\frac {a \text {Chi}(2 a+2 b x)}{b^3}-\frac {4 \text {Chi}(a+b x) \sinh (a+b x)}{3 b^3}-\frac {2 a \text {Chi}(a+b x) \cosh (a+b x)}{3 b^3}+\frac {2 \text {Shi}(2 a+2 b x)}{3 b^3}+\frac {a \log (a+b x)}{b^3}-\frac {\sinh (2 a+2 b x)}{12 b^3}-\frac {a \cosh (2 a+2 b x)}{3 b^3}-\frac {2 \sinh (a+b x) \cosh (a+b x)}{3 b^3}-\frac {a x (a+b x) \text {Chi}(a+b x)^2}{3 b^2}+\frac {2 a x \text {Chi}(a+b x) \sinh (a+b x)}{3 b^2}+\frac {4 x \text {Chi}(a+b x) \cosh (a+b x)}{3 b^2}+\frac {x \cosh (2 a+2 b x)}{6 b^2}+\frac {x^2 (a+b x) \text {Chi}(a+b x)^2}{3 b}-\frac {2 x^2 \text {Chi}(a+b x) \sinh (a+b x)}{3 b}-\frac {2 x}{3 b^2} \]
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Rule 8
Rule 12
Rule 2715
Rule 2717
Rule 2718
Rule 3377
Rule 3379
Rule 3382
Rule 3393
Rule 5556
Rule 5736
Rule 6670
Rule 6674
Rule 6676
Rule 6678
Rule 6682
Rule 6684
Rule 6873
Rule 6874
Rubi steps \begin{align*} \text {integral}& = \frac {x^2 (a+b x) \text {Chi}(a+b x)^2}{3 b}-\frac {2}{3} \int x^2 \cosh (a+b x) \text {Chi}(a+b x) \, dx-\frac {(2 a) \int x \text {Chi}(a+b x)^2 \, dx}{3 b} \\ & = -\frac {a x (a+b x) \text {Chi}(a+b x)^2}{3 b^2}+\frac {x^2 (a+b x) \text {Chi}(a+b x)^2}{3 b}-\frac {2 x^2 \text {Chi}(a+b x) \sinh (a+b x)}{3 b}+\frac {2}{3} \int \frac {x^2 \cosh (a+b x) \sinh (a+b x)}{a+b x} \, dx+\frac {a^2 \int \text {Chi}(a+b x)^2 \, dx}{3 b^2}+\frac {4 \int x \text {Chi}(a+b x) \sinh (a+b x) \, dx}{3 b}+\frac {(2 a) \int x \cosh (a+b x) \text {Chi}(a+b x) \, dx}{3 b} \\ & = \frac {4 x \cosh (a+b x) \text {Chi}(a+b x)}{3 b^2}+\frac {a^2 (a+b x) \text {Chi}(a+b x)^2}{3 b^3}-\frac {a x (a+b x) \text {Chi}(a+b x)^2}{3 b^2}+\frac {x^2 (a+b x) \text {Chi}(a+b x)^2}{3 b}+\frac {2 a x \text {Chi}(a+b x) \sinh (a+b x)}{3 b^2}-\frac {2 x^2 \text {Chi}(a+b x) \sinh (a+b x)}{3 b}+\frac {1}{3} \int \frac {x^2 \sinh (2 (a+b x))}{a+b x} \, dx-\frac {4 \int \cosh (a+b x) \text {Chi}(a+b x) \, dx}{3 b^2}-\frac {(2 a) \int \text {Chi}(a+b x) \sinh (a+b x) \, dx}{3 b^2}-\frac {\left (2 a^2\right ) \int \cosh (a+b x) \text {Chi}(a+b x) \, dx}{3 b^2}-\frac {4 \int \frac {x \cosh ^2(a+b x)}{a+b x} \, dx}{3 b}-\frac {(2 a) \int \frac {x \cosh (a+b x) \sinh (a+b x)}{a+b x} \, dx}{3 b} \\ & = -\frac {2 a \cosh (a+b x) \text {Chi}(a+b x)}{3 b^3}+\frac {4 x \cosh (a+b x) \text {Chi}(a+b x)}{3 b^2}+\frac {a^2 (a+b x) \text {Chi}(a+b x)^2}{3 b^3}-\frac {a x (a+b x) \text {Chi}(a+b x)^2}{3 b^2}+\frac {x^2 (a+b x) \text {Chi}(a+b x)^2}{3 b}-\frac {4 \text {Chi}(a+b x) \sinh (a+b x)}{3 b^3}-\frac {2 a^2 \text {Chi}(a+b x) \sinh (a+b x)}{3 b^3}+\frac {2 a x \text {Chi}(a+b x) \sinh (a+b x)}{3 b^2}-\frac {2 x^2 \text {Chi}(a+b x) \sinh (a+b x)}{3 b}+\frac {1}{3} \int \frac {x^2 \sinh (2 a+2 b x)}{a+b x} \, dx+\frac {4 \int \frac {\cosh (a+b x) \sinh (a+b x)}{a+b x} \, dx}{3 b^2}+\frac {(2 a) \int \frac {\cosh ^2(a+b x)}{a+b x} \, dx}{3 b^2}+\frac {\left (2 a^2\right ) \int \frac {\cosh (a+b x) \sinh (a+b x)}{a+b x} \, dx}{3 b^2}-\frac {4 \int \left (\frac {\cosh ^2(a+b x)}{b}-\frac {a \cosh ^2(a+b x)}{b (a+b x)}\right ) \, dx}{3 b}-\frac {a \int \frac {x \sinh (2 (a+b x))}{a+b x} \, dx}{3 b} \\ & = -\frac {2 a \cosh (a+b x) \text {Chi}(a+b x)}{3 b^3}+\frac {4 x \cosh (a+b x) \text {Chi}(a+b x)}{3 b^2}+\frac {a^2 (a+b x) \text {Chi}(a+b x)^2}{3 b^3}-\frac {a x (a+b x) \text {Chi}(a+b x)^2}{3 b^2}+\frac {x^2 (a+b x) \text {Chi}(a+b x)^2}{3 b}-\frac {4 \text {Chi}(a+b x) \sinh (a+b x)}{3 b^3}-\frac {2 a^2 \text {Chi}(a+b x) \sinh (a+b x)}{3 b^3}+\frac {2 a x \text {Chi}(a+b x) \sinh (a+b x)}{3 b^2}-\frac {2 x^2 \text {Chi}(a+b x) \sinh (a+b x)}{3 b}+\frac {1}{3} \int \left (-\frac {a \sinh (2 a+2 b x)}{b^2}+\frac {x \sinh (2 a+2 b x)}{b}+\frac {a^2 \sinh (2 a+2 b x)}{b^2 (a+b x)}\right ) \, dx-\frac {4 \int \cosh ^2(a+b x) \, dx}{3 b^2}+\frac {4 \int \frac {\sinh (2 a+2 b x)}{2 (a+b x)} \, dx}{3 b^2}+\frac {(2 a) \int \left (\frac {1}{2 (a+b x)}+\frac {\cosh (2 a+2 b x)}{2 (a+b x)}\right ) \, dx}{3 b^2}+\frac {(4 a) \int \frac {\cosh ^2(a+b x)}{a+b x} \, dx}{3 b^2}+\frac {\left (2 a^2\right ) \int \frac {\sinh (2 a+2 b x)}{2 (a+b x)} \, dx}{3 b^2}-\frac {a \int \frac {x \sinh (2 a+2 b x)}{a+b x} \, dx}{3 b} \\ & = -\frac {2 a \cosh (a+b x) \text {Chi}(a+b x)}{3 b^3}+\frac {4 x \cosh (a+b x) \text {Chi}(a+b x)}{3 b^2}+\frac {a^2 (a+b x) \text {Chi}(a+b x)^2}{3 b^3}-\frac {a x (a+b x) \text {Chi}(a+b x)^2}{3 b^2}+\frac {x^2 (a+b x) \text {Chi}(a+b x)^2}{3 b}+\frac {a \log (a+b x)}{3 b^3}-\frac {2 \cosh (a+b x) \sinh (a+b x)}{3 b^3}-\frac {4 \text {Chi}(a+b x) \sinh (a+b x)}{3 b^3}-\frac {2 a^2 \text {Chi}(a+b x) \sinh (a+b x)}{3 b^3}+\frac {2 a x \text {Chi}(a+b x) \sinh (a+b x)}{3 b^2}-\frac {2 x^2 \text {Chi}(a+b x) \sinh (a+b x)}{3 b}-\frac {2 \int 1 \, dx}{3 b^2}+\frac {2 \int \frac {\sinh (2 a+2 b x)}{a+b x} \, dx}{3 b^2}+\frac {a \int \frac {\cosh (2 a+2 b x)}{a+b x} \, dx}{3 b^2}-\frac {a \int \sinh (2 a+2 b x) \, dx}{3 b^2}+\frac {(4 a) \int \left (\frac {1}{2 (a+b x)}+\frac {\cosh (2 a+2 b x)}{2 (a+b x)}\right ) \, dx}{3 b^2}+2 \frac {a^2 \int \frac {\sinh (2 a+2 b x)}{a+b x} \, dx}{3 b^2}+\frac {\int x \sinh (2 a+2 b x) \, dx}{3 b}-\frac {a \int \left (\frac {\sinh (2 a+2 b x)}{b}+\frac {a \sinh (2 a+2 b x)}{b (-a-b x)}\right ) \, dx}{3 b} \\ & = -\frac {2 x}{3 b^2}-\frac {a \cosh (2 a+2 b x)}{6 b^3}+\frac {x \cosh (2 a+2 b x)}{6 b^2}-\frac {2 a \cosh (a+b x) \text {Chi}(a+b x)}{3 b^3}+\frac {4 x \cosh (a+b x) \text {Chi}(a+b x)}{3 b^2}+\frac {a^2 (a+b x) \text {Chi}(a+b x)^2}{3 b^3}-\frac {a x (a+b x) \text {Chi}(a+b x)^2}{3 b^2}+\frac {x^2 (a+b x) \text {Chi}(a+b x)^2}{3 b}+\frac {a \text {Chi}(2 a+2 b x)}{3 b^3}+\frac {a \log (a+b x)}{b^3}-\frac {2 \cosh (a+b x) \sinh (a+b x)}{3 b^3}-\frac {4 \text {Chi}(a+b x) \sinh (a+b x)}{3 b^3}-\frac {2 a^2 \text {Chi}(a+b x) \sinh (a+b x)}{3 b^3}+\frac {2 a x \text {Chi}(a+b x) \sinh (a+b x)}{3 b^2}-\frac {2 x^2 \text {Chi}(a+b x) \sinh (a+b x)}{3 b}+\frac {2 \text {Shi}(2 a+2 b x)}{3 b^3}+\frac {2 a^2 \text {Shi}(2 a+2 b x)}{3 b^3}-\frac {\int \cosh (2 a+2 b x) \, dx}{6 b^2}-\frac {a \int \sinh (2 a+2 b x) \, dx}{3 b^2}+\frac {(2 a) \int \frac {\cosh (2 a+2 b x)}{a+b x} \, dx}{3 b^2}-\frac {a^2 \int \frac {\sinh (2 a+2 b x)}{-a-b x} \, dx}{3 b^2} \\ & = -\frac {2 x}{3 b^2}-\frac {a \cosh (2 a+2 b x)}{3 b^3}+\frac {x \cosh (2 a+2 b x)}{6 b^2}-\frac {2 a \cosh (a+b x) \text {Chi}(a+b x)}{3 b^3}+\frac {4 x \cosh (a+b x) \text {Chi}(a+b x)}{3 b^2}+\frac {a^2 (a+b x) \text {Chi}(a+b x)^2}{3 b^3}-\frac {a x (a+b x) \text {Chi}(a+b x)^2}{3 b^2}+\frac {x^2 (a+b x) \text {Chi}(a+b x)^2}{3 b}+\frac {a \text {Chi}(2 a+2 b x)}{b^3}+\frac {a \log (a+b x)}{b^3}-\frac {2 \cosh (a+b x) \sinh (a+b x)}{3 b^3}-\frac {4 \text {Chi}(a+b x) \sinh (a+b x)}{3 b^3}-\frac {2 a^2 \text {Chi}(a+b x) \sinh (a+b x)}{3 b^3}+\frac {2 a x \text {Chi}(a+b x) \sinh (a+b x)}{3 b^2}-\frac {2 x^2 \text {Chi}(a+b x) \sinh (a+b x)}{3 b}-\frac {\sinh (2 a+2 b x)}{12 b^3}+\frac {2 \text {Shi}(2 a+2 b x)}{3 b^3}+\frac {a^2 \text {Shi}(2 a+2 b x)}{b^3} \\ \end{align*}
Time = 0.79 (sec) , antiderivative size = 158, normalized size of antiderivative = 0.48 \[ \int x^2 \text {Chi}(a+b x)^2 \, dx=\frac {-8 a-8 b x-4 a \cosh (2 (a+b x))+2 b x \cosh (2 (a+b x))+4 \left (a^3+b^3 x^3\right ) \text {Chi}(a+b x)^2+12 a \text {Chi}(2 (a+b x))+12 a \log (a+b x)-8 \text {Chi}(a+b x) \left ((a-2 b x) \cosh (a+b x)+\left (2+a^2-a b x+b^2 x^2\right ) \sinh (a+b x)\right )-5 \sinh (2 (a+b x))+8 \text {Shi}(2 (a+b x))+12 a^2 \text {Shi}(2 (a+b x))}{12 b^3} \]
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\[\int x^{2} \operatorname {Chi}\left (b x +a \right )^{2}d x\]
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\[ \int x^2 \text {Chi}(a+b x)^2 \, dx=\int { x^{2} {\rm Chi}\left (b x + a\right )^{2} \,d x } \]
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\[ \int x^2 \text {Chi}(a+b x)^2 \, dx=\int x^{2} \operatorname {Chi}^{2}\left (a + b x\right )\, dx \]
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\[ \int x^2 \text {Chi}(a+b x)^2 \, dx=\int { x^{2} {\rm Chi}\left (b x + a\right )^{2} \,d x } \]
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\[ \int x^2 \text {Chi}(a+b x)^2 \, dx=\int { x^{2} {\rm Chi}\left (b x + a\right )^{2} \,d x } \]
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Timed out. \[ \int x^2 \text {Chi}(a+b x)^2 \, dx=\int x^2\,{\mathrm {coshint}\left (a+b\,x\right )}^2 \,d x \]
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