Integrand size = 10, antiderivative size = 118 \[ \int x^2 \text {Shi}(a+b x) \, dx=-\frac {2 \cosh (a+b x)}{3 b^3}-\frac {a^2 \cosh (a+b x)}{3 b^3}+\frac {a x \cosh (a+b x)}{3 b^2}-\frac {x^2 \cosh (a+b x)}{3 b}-\frac {a \sinh (a+b x)}{3 b^3}+\frac {2 x \sinh (a+b x)}{3 b^2}+\frac {a^3 \text {Shi}(a+b x)}{3 b^3}+\frac {1}{3} x^3 \text {Shi}(a+b x) \]
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Time = 0.23 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {6667, 6874, 2718, 3377, 2717, 3379} \[ \int x^2 \text {Shi}(a+b x) \, dx=\frac {a^3 \text {Shi}(a+b x)}{3 b^3}-\frac {a^2 \cosh (a+b x)}{3 b^3}-\frac {a \sinh (a+b x)}{3 b^3}-\frac {2 \cosh (a+b x)}{3 b^3}+\frac {2 x \sinh (a+b x)}{3 b^2}+\frac {a x \cosh (a+b x)}{3 b^2}+\frac {1}{3} x^3 \text {Shi}(a+b x)-\frac {x^2 \cosh (a+b x)}{3 b} \]
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Rule 2717
Rule 2718
Rule 3377
Rule 3379
Rule 6667
Rule 6874
Rubi steps \begin{align*} \text {integral}& = \frac {1}{3} x^3 \text {Shi}(a+b x)-\frac {1}{3} b \int \frac {x^3 \sinh (a+b x)}{a+b x} \, dx \\ & = \frac {1}{3} x^3 \text {Shi}(a+b x)-\frac {1}{3} b \int \left (\frac {a^2 \sinh (a+b x)}{b^3}-\frac {a x \sinh (a+b x)}{b^2}+\frac {x^2 \sinh (a+b x)}{b}-\frac {a^3 \sinh (a+b x)}{b^3 (a+b x)}\right ) \, dx \\ & = \frac {1}{3} x^3 \text {Shi}(a+b x)-\frac {1}{3} \int x^2 \sinh (a+b x) \, dx-\frac {a^2 \int \sinh (a+b x) \, dx}{3 b^2}+\frac {a^3 \int \frac {\sinh (a+b x)}{a+b x} \, dx}{3 b^2}+\frac {a \int x \sinh (a+b x) \, dx}{3 b} \\ & = -\frac {a^2 \cosh (a+b x)}{3 b^3}+\frac {a x \cosh (a+b x)}{3 b^2}-\frac {x^2 \cosh (a+b x)}{3 b}+\frac {a^3 \text {Shi}(a+b x)}{3 b^3}+\frac {1}{3} x^3 \text {Shi}(a+b x)-\frac {a \int \cosh (a+b x) \, dx}{3 b^2}+\frac {2 \int x \cosh (a+b x) \, dx}{3 b} \\ & = -\frac {a^2 \cosh (a+b x)}{3 b^3}+\frac {a x \cosh (a+b x)}{3 b^2}-\frac {x^2 \cosh (a+b x)}{3 b}-\frac {a \sinh (a+b x)}{3 b^3}+\frac {2 x \sinh (a+b x)}{3 b^2}+\frac {a^3 \text {Shi}(a+b x)}{3 b^3}+\frac {1}{3} x^3 \text {Shi}(a+b x)-\frac {2 \int \sinh (a+b x) \, dx}{3 b^2} \\ & = -\frac {2 \cosh (a+b x)}{3 b^3}-\frac {a^2 \cosh (a+b x)}{3 b^3}+\frac {a x \cosh (a+b x)}{3 b^2}-\frac {x^2 \cosh (a+b x)}{3 b}-\frac {a \sinh (a+b x)}{3 b^3}+\frac {2 x \sinh (a+b x)}{3 b^2}+\frac {a^3 \text {Shi}(a+b x)}{3 b^3}+\frac {1}{3} x^3 \text {Shi}(a+b x) \\ \end{align*}
Time = 0.12 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.54 \[ \int x^2 \text {Shi}(a+b x) \, dx=-\frac {\left (2+a^2-a b x+b^2 x^2\right ) \cosh (a+b x)+(a-2 b x) \sinh (a+b x)-\left (a^3+b^3 x^3\right ) \text {Shi}(a+b x)}{3 b^3} \]
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Time = 0.46 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.85
method | result | size |
parts | \(\frac {x^{3} \operatorname {Shi}\left (b x +a \right )}{3}-\frac {-a^{3} \operatorname {Shi}\left (b x +a \right )+3 a^{2} \cosh \left (b x +a \right )-3 a \left (\left (b x +a \right ) \cosh \left (b x +a \right )-\sinh \left (b x +a \right )\right )+\left (b x +a \right )^{2} \cosh \left (b x +a \right )-2 \left (b x +a \right ) \sinh \left (b x +a \right )+2 \cosh \left (b x +a \right )}{3 b^{3}}\) | \(100\) |
derivativedivides | \(\frac {\frac {\operatorname {Shi}\left (b x +a \right ) b^{3} x^{3}}{3}+\frac {a^{3} \operatorname {Shi}\left (b x +a \right )}{3}-a^{2} \cosh \left (b x +a \right )+a \left (\left (b x +a \right ) \cosh \left (b x +a \right )-\sinh \left (b x +a \right )\right )-\frac {\left (b x +a \right )^{2} \cosh \left (b x +a \right )}{3}+\frac {2 \left (b x +a \right ) \sinh \left (b x +a \right )}{3}-\frac {2 \cosh \left (b x +a \right )}{3}}{b^{3}}\) | \(101\) |
default | \(\frac {\frac {\operatorname {Shi}\left (b x +a \right ) b^{3} x^{3}}{3}+\frac {a^{3} \operatorname {Shi}\left (b x +a \right )}{3}-a^{2} \cosh \left (b x +a \right )+a \left (\left (b x +a \right ) \cosh \left (b x +a \right )-\sinh \left (b x +a \right )\right )-\frac {\left (b x +a \right )^{2} \cosh \left (b x +a \right )}{3}+\frac {2 \left (b x +a \right ) \sinh \left (b x +a \right )}{3}-\frac {2 \cosh \left (b x +a \right )}{3}}{b^{3}}\) | \(101\) |
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\[ \int x^2 \text {Shi}(a+b x) \, dx=\int { x^{2} {\rm Shi}\left (b x + a\right ) \,d x } \]
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\[ \int x^2 \text {Shi}(a+b x) \, dx=\int x^{2} \operatorname {Shi}{\left (a + b x \right )}\, dx \]
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\[ \int x^2 \text {Shi}(a+b x) \, dx=\int { x^{2} {\rm Shi}\left (b x + a\right ) \,d x } \]
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\[ \int x^2 \text {Shi}(a+b x) \, dx=\int { x^{2} {\rm Shi}\left (b x + a\right ) \,d x } \]
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Timed out. \[ \int x^2 \text {Shi}(a+b x) \, dx=\int x^2\,\mathrm {sinhint}\left (a+b\,x\right ) \,d x \]
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