Integrand size = 16, antiderivative size = 186 \[ \int x^2 \cosh (a+b x) \text {Chi}(a+b x) \, dx=\frac {x}{b^2}+\frac {a \cosh (2 a+2 b x)}{4 b^3}-\frac {x \cosh (2 a+2 b x)}{4 b^2}-\frac {2 x \cosh (a+b x) \text {Chi}(a+b x)}{b^2}-\frac {a \text {Chi}(2 a+2 b x)}{b^3}-\frac {a \log (a+b x)}{b^3}+\frac {\cosh (a+b x) \sinh (a+b x)}{b^3}+\frac {2 \text {Chi}(a+b x) \sinh (a+b x)}{b^3}+\frac {x^2 \text {Chi}(a+b x) \sinh (a+b x)}{b}+\frac {\sinh (2 a+2 b x)}{8 b^3}-\frac {\text {Shi}(2 a+2 b x)}{b^3}-\frac {a^2 \text {Shi}(2 a+2 b x)}{2 b^3} \]
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Time = 0.43 (sec) , antiderivative size = 186, normalized size of antiderivative = 1.00, number of steps used = 21, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {6678, 5736, 6873, 6874, 2718, 3377, 2717, 3379, 6684, 2715, 8, 3393, 3382, 6676, 5556, 12} \[ \int x^2 \cosh (a+b x) \text {Chi}(a+b x) \, dx=-\frac {a^2 \text {Shi}(2 a+2 b x)}{2 b^3}-\frac {a \text {Chi}(2 a+2 b x)}{b^3}+\frac {2 \text {Chi}(a+b x) \sinh (a+b x)}{b^3}-\frac {\text {Shi}(2 a+2 b x)}{b^3}-\frac {a \log (a+b x)}{b^3}+\frac {\sinh (2 a+2 b x)}{8 b^3}+\frac {a \cosh (2 a+2 b x)}{4 b^3}+\frac {\sinh (a+b x) \cosh (a+b x)}{b^3}-\frac {2 x \text {Chi}(a+b x) \cosh (a+b x)}{b^2}-\frac {x \cosh (2 a+2 b x)}{4 b^2}+\frac {x^2 \text {Chi}(a+b x) \sinh (a+b x)}{b}+\frac {x}{b^2} \]
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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 6676
Rule 6678
Rule 6684
Rule 6873
Rule 6874
Rubi steps \begin{align*} \text {integral}& = \frac {x^2 \text {Chi}(a+b x) \sinh (a+b x)}{b}-\frac {2 \int x \text {Chi}(a+b x) \sinh (a+b x) \, dx}{b}-\int \frac {x^2 \cosh (a+b x) \sinh (a+b x)}{a+b x} \, dx \\ & = -\frac {2 x \cosh (a+b x) \text {Chi}(a+b x)}{b^2}+\frac {x^2 \text {Chi}(a+b x) \sinh (a+b x)}{b}-\frac {1}{2} \int \frac {x^2 \sinh (2 (a+b x))}{a+b x} \, dx+\frac {2 \int \cosh (a+b x) \text {Chi}(a+b x) \, dx}{b^2}+\frac {2 \int \frac {x \cosh ^2(a+b x)}{a+b x} \, dx}{b} \\ & = -\frac {2 x \cosh (a+b x) \text {Chi}(a+b x)}{b^2}+\frac {2 \text {Chi}(a+b x) \sinh (a+b x)}{b^3}+\frac {x^2 \text {Chi}(a+b x) \sinh (a+b x)}{b}-\frac {1}{2} \int \frac {x^2 \sinh (2 a+2 b x)}{a+b x} \, dx-\frac {2 \int \frac {\cosh (a+b x) \sinh (a+b x)}{a+b x} \, dx}{b^2}+\frac {2 \int \left (\frac {\cosh ^2(a+b x)}{b}-\frac {a \cosh ^2(a+b x)}{b (a+b x)}\right ) \, dx}{b} \\ & = -\frac {2 x \cosh (a+b x) \text {Chi}(a+b x)}{b^2}+\frac {2 \text {Chi}(a+b x) \sinh (a+b x)}{b^3}+\frac {x^2 \text {Chi}(a+b x) \sinh (a+b x)}{b}-\frac {1}{2} \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 {2 \int \cosh ^2(a+b x) \, dx}{b^2}-\frac {2 \int \frac {\sinh (2 a+2 b x)}{2 (a+b x)} \, dx}{b^2}-\frac {(2 a) \int \frac {\cosh ^2(a+b x)}{a+b x} \, dx}{b^2} \\ & = -\frac {2 x \cosh (a+b x) \text {Chi}(a+b x)}{b^2}+\frac {\cosh (a+b x) \sinh (a+b x)}{b^3}+\frac {2 \text {Chi}(a+b x) \sinh (a+b x)}{b^3}+\frac {x^2 \text {Chi}(a+b x) \sinh (a+b x)}{b}+\frac {\int 1 \, dx}{b^2}-\frac {\int \frac {\sinh (2 a+2 b x)}{a+b x} \, dx}{b^2}+\frac {a \int \sinh (2 a+2 b x) \, dx}{2 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}{b^2}-\frac {a^2 \int \frac {\sinh (2 a+2 b x)}{a+b x} \, dx}{2 b^2}-\frac {\int x \sinh (2 a+2 b x) \, dx}{2 b} \\ & = \frac {x}{b^2}+\frac {a \cosh (2 a+2 b x)}{4 b^3}-\frac {x \cosh (2 a+2 b x)}{4 b^2}-\frac {2 x \cosh (a+b x) \text {Chi}(a+b x)}{b^2}-\frac {a \log (a+b x)}{b^3}+\frac {\cosh (a+b x) \sinh (a+b x)}{b^3}+\frac {2 \text {Chi}(a+b x) \sinh (a+b x)}{b^3}+\frac {x^2 \text {Chi}(a+b x) \sinh (a+b x)}{b}-\frac {\text {Shi}(2 a+2 b x)}{b^3}-\frac {a^2 \text {Shi}(2 a+2 b x)}{2 b^3}+\frac {\int \cosh (2 a+2 b x) \, dx}{4 b^2}-\frac {a \int \frac {\cosh (2 a+2 b x)}{a+b x} \, dx}{b^2} \\ & = \frac {x}{b^2}+\frac {a \cosh (2 a+2 b x)}{4 b^3}-\frac {x \cosh (2 a+2 b x)}{4 b^2}-\frac {2 x \cosh (a+b x) \text {Chi}(a+b x)}{b^2}-\frac {a \text {Chi}(2 a+2 b x)}{b^3}-\frac {a \log (a+b x)}{b^3}+\frac {\cosh (a+b x) \sinh (a+b x)}{b^3}+\frac {2 \text {Chi}(a+b x) \sinh (a+b x)}{b^3}+\frac {x^2 \text {Chi}(a+b x) \sinh (a+b x)}{b}+\frac {\sinh (2 a+2 b x)}{8 b^3}-\frac {\text {Shi}(2 a+2 b x)}{b^3}-\frac {a^2 \text {Shi}(2 a+2 b x)}{2 b^3} \\ \end{align*}
Time = 0.19 (sec) , antiderivative size = 123, normalized size of antiderivative = 0.66 \[ \int x^2 \cosh (a+b x) \text {Chi}(a+b x) \, dx=\frac {8 b x+2 a \cosh (2 (a+b x))-2 b x \cosh (2 (a+b x))-8 a \text {Chi}(2 (a+b x))-8 a \log (a+b x)+8 \text {Chi}(a+b x) \left (-2 b x \cosh (a+b x)+\left (2+b^2 x^2\right ) \sinh (a+b x)\right )+5 \sinh (2 (a+b x))-8 \text {Shi}(2 (a+b x))-4 a^2 \text {Shi}(2 (a+b x))}{8 b^3} \]
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Time = 2.55 (sec) , antiderivative size = 175, normalized size of antiderivative = 0.94
method | result | size |
derivativedivides | \(\frac {\operatorname {Chi}\left (b x +a \right ) \left (a^{2} \sinh \left (b x +a \right )-2 a \left (\left (b x +a \right ) \sinh \left (b x +a \right )-\cosh \left (b x +a \right )\right )+\left (b x +a \right )^{2} \sinh \left (b x +a \right )-2 \left (b x +a \right ) \cosh \left (b x +a \right )+2 \sinh \left (b x +a \right )\right )-\frac {a^{2} \operatorname {Shi}\left (2 b x +2 a \right )}{2}+a \cosh \left (b x +a \right )^{2}-a \ln \left (b x +a \right )-a \,\operatorname {Chi}\left (2 b x +2 a \right )-\frac {\left (b x +a \right ) \cosh \left (b x +a \right )^{2}}{2}+\frac {5 \cosh \left (b x +a \right ) \sinh \left (b x +a \right )}{4}+\frac {5 b x}{4}+\frac {5 a}{4}-\operatorname {Shi}\left (2 b x +2 a \right )}{b^{3}}\) | \(175\) |
default | \(\frac {\operatorname {Chi}\left (b x +a \right ) \left (a^{2} \sinh \left (b x +a \right )-2 a \left (\left (b x +a \right ) \sinh \left (b x +a \right )-\cosh \left (b x +a \right )\right )+\left (b x +a \right )^{2} \sinh \left (b x +a \right )-2 \left (b x +a \right ) \cosh \left (b x +a \right )+2 \sinh \left (b x +a \right )\right )-\frac {a^{2} \operatorname {Shi}\left (2 b x +2 a \right )}{2}+a \cosh \left (b x +a \right )^{2}-a \ln \left (b x +a \right )-a \,\operatorname {Chi}\left (2 b x +2 a \right )-\frac {\left (b x +a \right ) \cosh \left (b x +a \right )^{2}}{2}+\frac {5 \cosh \left (b x +a \right ) \sinh \left (b x +a \right )}{4}+\frac {5 b x}{4}+\frac {5 a}{4}-\operatorname {Shi}\left (2 b x +2 a \right )}{b^{3}}\) | \(175\) |
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\[ \int x^2 \cosh (a+b x) \text {Chi}(a+b x) \, dx=\int { x^{2} {\rm Chi}\left (b x + a\right ) \cosh \left (b x + a\right ) \,d x } \]
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\[ \int x^2 \cosh (a+b x) \text {Chi}(a+b x) \, dx=\int x^{2} \cosh {\left (a + b x \right )} \operatorname {Chi}\left (a + b x\right )\, dx \]
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\[ \int x^2 \cosh (a+b x) \text {Chi}(a+b x) \, dx=\int { x^{2} {\rm Chi}\left (b x + a\right ) \cosh \left (b x + a\right ) \,d x } \]
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\[ \int x^2 \cosh (a+b x) \text {Chi}(a+b x) \, dx=\int { x^{2} {\rm Chi}\left (b x + a\right ) \cosh \left (b x + a\right ) \,d x } \]
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Timed out. \[ \int x^2 \cosh (a+b x) \text {Chi}(a+b x) \, dx=\int x^2\,\mathrm {coshint}\left (a+b\,x\right )\,\mathrm {cosh}\left (a+b\,x\right ) \,d x \]
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