Integrand size = 10, antiderivative size = 184 \[ \int x^3 \text {Shi}(a+b x) \, dx=\frac {a \cosh (a+b x)}{2 b^4}+\frac {a^3 \cosh (a+b x)}{4 b^4}-\frac {3 x \cosh (a+b x)}{2 b^3}-\frac {a^2 x \cosh (a+b x)}{4 b^3}+\frac {a x^2 \cosh (a+b x)}{4 b^2}-\frac {x^3 \cosh (a+b x)}{4 b}+\frac {3 \sinh (a+b x)}{2 b^4}+\frac {a^2 \sinh (a+b x)}{4 b^4}-\frac {a x \sinh (a+b x)}{2 b^3}+\frac {3 x^2 \sinh (a+b x)}{4 b^2}-\frac {a^4 \text {Shi}(a+b x)}{4 b^4}+\frac {1}{4} x^4 \text {Shi}(a+b x) \]
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Time = 0.31 (sec) , antiderivative size = 184, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {6667, 6874, 2718, 3377, 2717, 3379} \[ \int x^3 \text {Shi}(a+b x) \, dx=-\frac {a^4 \text {Shi}(a+b x)}{4 b^4}+\frac {a^3 \cosh (a+b x)}{4 b^4}+\frac {a^2 \sinh (a+b x)}{4 b^4}-\frac {a^2 x \cosh (a+b x)}{4 b^3}+\frac {3 \sinh (a+b x)}{2 b^4}+\frac {a \cosh (a+b x)}{2 b^4}-\frac {a x \sinh (a+b x)}{2 b^3}-\frac {3 x \cosh (a+b x)}{2 b^3}+\frac {3 x^2 \sinh (a+b x)}{4 b^2}+\frac {a x^2 \cosh (a+b x)}{4 b^2}+\frac {1}{4} x^4 \text {Shi}(a+b x)-\frac {x^3 \cosh (a+b x)}{4 b} \]
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Rule 2717
Rule 2718
Rule 3377
Rule 3379
Rule 6667
Rule 6874
Rubi steps \begin{align*} \text {integral}& = \frac {1}{4} x^4 \text {Shi}(a+b x)-\frac {1}{4} b \int \frac {x^4 \sinh (a+b x)}{a+b x} \, dx \\ & = \frac {1}{4} x^4 \text {Shi}(a+b x)-\frac {1}{4} b \int \left (-\frac {a^3 \sinh (a+b x)}{b^4}+\frac {a^2 x \sinh (a+b x)}{b^3}-\frac {a x^2 \sinh (a+b x)}{b^2}+\frac {x^3 \sinh (a+b x)}{b}+\frac {a^4 \sinh (a+b x)}{b^4 (a+b x)}\right ) \, dx \\ & = \frac {1}{4} x^4 \text {Shi}(a+b x)-\frac {1}{4} \int x^3 \sinh (a+b x) \, dx+\frac {a^3 \int \sinh (a+b x) \, dx}{4 b^3}-\frac {a^4 \int \frac {\sinh (a+b x)}{a+b x} \, dx}{4 b^3}-\frac {a^2 \int x \sinh (a+b x) \, dx}{4 b^2}+\frac {a \int x^2 \sinh (a+b x) \, dx}{4 b} \\ & = \frac {a^3 \cosh (a+b x)}{4 b^4}-\frac {a^2 x \cosh (a+b x)}{4 b^3}+\frac {a x^2 \cosh (a+b x)}{4 b^2}-\frac {x^3 \cosh (a+b x)}{4 b}-\frac {a^4 \text {Shi}(a+b x)}{4 b^4}+\frac {1}{4} x^4 \text {Shi}(a+b x)+\frac {a^2 \int \cosh (a+b x) \, dx}{4 b^3}-\frac {a \int x \cosh (a+b x) \, dx}{2 b^2}+\frac {3 \int x^2 \cosh (a+b x) \, dx}{4 b} \\ & = \frac {a^3 \cosh (a+b x)}{4 b^4}-\frac {a^2 x \cosh (a+b x)}{4 b^3}+\frac {a x^2 \cosh (a+b x)}{4 b^2}-\frac {x^3 \cosh (a+b x)}{4 b}+\frac {a^2 \sinh (a+b x)}{4 b^4}-\frac {a x \sinh (a+b x)}{2 b^3}+\frac {3 x^2 \sinh (a+b x)}{4 b^2}-\frac {a^4 \text {Shi}(a+b x)}{4 b^4}+\frac {1}{4} x^4 \text {Shi}(a+b x)+\frac {a \int \sinh (a+b x) \, dx}{2 b^3}-\frac {3 \int x \sinh (a+b x) \, dx}{2 b^2} \\ & = \frac {a \cosh (a+b x)}{2 b^4}+\frac {a^3 \cosh (a+b x)}{4 b^4}-\frac {3 x \cosh (a+b x)}{2 b^3}-\frac {a^2 x \cosh (a+b x)}{4 b^3}+\frac {a x^2 \cosh (a+b x)}{4 b^2}-\frac {x^3 \cosh (a+b x)}{4 b}+\frac {a^2 \sinh (a+b x)}{4 b^4}-\frac {a x \sinh (a+b x)}{2 b^3}+\frac {3 x^2 \sinh (a+b x)}{4 b^2}-\frac {a^4 \text {Shi}(a+b x)}{4 b^4}+\frac {1}{4} x^4 \text {Shi}(a+b x)+\frac {3 \int \cosh (a+b x) \, dx}{2 b^3} \\ & = \frac {a \cosh (a+b x)}{2 b^4}+\frac {a^3 \cosh (a+b x)}{4 b^4}-\frac {3 x \cosh (a+b x)}{2 b^3}-\frac {a^2 x \cosh (a+b x)}{4 b^3}+\frac {a x^2 \cosh (a+b x)}{4 b^2}-\frac {x^3 \cosh (a+b x)}{4 b}+\frac {3 \sinh (a+b x)}{2 b^4}+\frac {a^2 \sinh (a+b x)}{4 b^4}-\frac {a x \sinh (a+b x)}{2 b^3}+\frac {3 x^2 \sinh (a+b x)}{4 b^2}-\frac {a^4 \text {Shi}(a+b x)}{4 b^4}+\frac {1}{4} x^4 \text {Shi}(a+b x) \\ \end{align*}
Time = 0.14 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.51 \[ \int x^3 \text {Shi}(a+b x) \, dx=\frac {\left (2 a+a^3-6 b x-a^2 b x+a b^2 x^2-b^3 x^3\right ) \cosh (a+b x)+\left (6+a^2-2 a b x+3 b^2 x^2\right ) \sinh (a+b x)+\left (-a^4+b^4 x^4\right ) \text {Shi}(a+b x)}{4 b^4} \]
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Time = 0.45 (sec) , antiderivative size = 155, normalized size of antiderivative = 0.84
method | result | size |
parts | \(\frac {x^{4} \operatorname {Shi}\left (b x +a \right )}{4}-\frac {a^{4} \operatorname {Shi}\left (b x +a \right )-4 a^{3} \cosh \left (b x +a \right )+6 a^{2} \left (\left (b x +a \right ) \cosh \left (b x +a \right )-\sinh \left (b x +a \right )\right )-4 a \left (\left (b x +a \right )^{2} \cosh \left (b x +a \right )-2 \left (b x +a \right ) \sinh \left (b x +a \right )+2 \cosh \left (b x +a \right )\right )+\left (b x +a \right )^{3} \cosh \left (b x +a \right )-3 \left (b x +a \right )^{2} \sinh \left (b x +a \right )+6 \left (b x +a \right ) \cosh \left (b x +a \right )-6 \sinh \left (b x +a \right )}{4 b^{4}}\) | \(155\) |
derivativedivides | \(\frac {\frac {\operatorname {Shi}\left (b x +a \right ) b^{4} x^{4}}{4}-\frac {a^{4} \operatorname {Shi}\left (b x +a \right )}{4}+a^{3} \cosh \left (b x +a \right )-\frac {3 a^{2} \left (\left (b x +a \right ) \cosh \left (b x +a \right )-\sinh \left (b x +a \right )\right )}{2}+a \left (\left (b x +a \right )^{2} \cosh \left (b x +a \right )-2 \left (b x +a \right ) \sinh \left (b x +a \right )+2 \cosh \left (b x +a \right )\right )-\frac {\left (b x +a \right )^{3} \cosh \left (b x +a \right )}{4}+\frac {3 \left (b x +a \right )^{2} \sinh \left (b x +a \right )}{4}-\frac {3 \left (b x +a \right ) \cosh \left (b x +a \right )}{2}+\frac {3 \sinh \left (b x +a \right )}{2}}{b^{4}}\) | \(156\) |
default | \(\frac {\frac {\operatorname {Shi}\left (b x +a \right ) b^{4} x^{4}}{4}-\frac {a^{4} \operatorname {Shi}\left (b x +a \right )}{4}+a^{3} \cosh \left (b x +a \right )-\frac {3 a^{2} \left (\left (b x +a \right ) \cosh \left (b x +a \right )-\sinh \left (b x +a \right )\right )}{2}+a \left (\left (b x +a \right )^{2} \cosh \left (b x +a \right )-2 \left (b x +a \right ) \sinh \left (b x +a \right )+2 \cosh \left (b x +a \right )\right )-\frac {\left (b x +a \right )^{3} \cosh \left (b x +a \right )}{4}+\frac {3 \left (b x +a \right )^{2} \sinh \left (b x +a \right )}{4}-\frac {3 \left (b x +a \right ) \cosh \left (b x +a \right )}{2}+\frac {3 \sinh \left (b x +a \right )}{2}}{b^{4}}\) | \(156\) |
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\[ \int x^3 \text {Shi}(a+b x) \, dx=\int { x^{3} {\rm Shi}\left (b x + a\right ) \,d x } \]
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\[ \int x^3 \text {Shi}(a+b x) \, dx=\int x^{3} \operatorname {Shi}{\left (a + b x \right )}\, dx \]
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\[ \int x^3 \text {Shi}(a+b x) \, dx=\int { x^{3} {\rm Shi}\left (b x + a\right ) \,d x } \]
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\[ \int x^3 \text {Shi}(a+b x) \, dx=\int { x^{3} {\rm Shi}\left (b x + a\right ) \,d x } \]
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Timed out. \[ \int x^3 \text {Shi}(a+b x) \, dx=\int x^3\,\mathrm {sinhint}\left (a+b\,x\right ) \,d x \]
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