Integrand size = 12, antiderivative size = 128 \[ \int x^3 \cosh (b x) \text {Shi}(b x) \, dx=\frac {4 x}{b^3}+\frac {x^3}{6 b}-\frac {4 \cosh (b x) \sinh (b x)}{b^4}-\frac {x^2 \cosh (b x) \sinh (b x)}{2 b^2}+\frac {2 x \sinh ^2(b x)}{b^3}-\frac {6 \cosh (b x) \text {Shi}(b x)}{b^4}-\frac {3 x^2 \cosh (b x) \text {Shi}(b x)}{b^2}+\frac {6 x \sinh (b x) \text {Shi}(b x)}{b^3}+\frac {x^3 \sinh (b x) \text {Shi}(b x)}{b}+\frac {3 \text {Shi}(2 b x)}{b^4} \]
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Time = 0.12 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.00, number of steps used = 20, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.917, Rules used = {6683, 12, 3392, 30, 2715, 8, 6677, 5480, 6675, 5556, 3379} \[ \int x^3 \cosh (b x) \text {Shi}(b x) \, dx=\frac {3 \text {Shi}(2 b x)}{b^4}-\frac {6 \text {Shi}(b x) \cosh (b x)}{b^4}-\frac {4 \sinh (b x) \cosh (b x)}{b^4}+\frac {6 x \text {Shi}(b x) \sinh (b x)}{b^3}+\frac {4 x}{b^3}+\frac {2 x \sinh ^2(b x)}{b^3}-\frac {3 x^2 \text {Shi}(b x) \cosh (b x)}{b^2}-\frac {x^2 \sinh (b x) \cosh (b x)}{2 b^2}+\frac {x^3 \text {Shi}(b x) \sinh (b x)}{b}+\frac {x^3}{6 b} \]
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Rule 8
Rule 12
Rule 30
Rule 2715
Rule 3379
Rule 3392
Rule 5480
Rule 5556
Rule 6675
Rule 6677
Rule 6683
Rubi steps \begin{align*} \text {integral}& = \frac {x^3 \sinh (b x) \text {Shi}(b x)}{b}-\frac {3 \int x^2 \sinh (b x) \text {Shi}(b x) \, dx}{b}-\int \frac {x^2 \sinh ^2(b x)}{b} \, dx \\ & = -\frac {3 x^2 \cosh (b x) \text {Shi}(b x)}{b^2}+\frac {x^3 \sinh (b x) \text {Shi}(b x)}{b}+\frac {6 \int x \cosh (b x) \text {Shi}(b x) \, dx}{b^2}-\frac {\int x^2 \sinh ^2(b x) \, dx}{b}+\frac {3 \int \frac {x \cosh (b x) \sinh (b x)}{b} \, dx}{b} \\ & = -\frac {x^2 \cosh (b x) \sinh (b x)}{2 b^2}+\frac {x \sinh ^2(b x)}{2 b^3}-\frac {3 x^2 \cosh (b x) \text {Shi}(b x)}{b^2}+\frac {6 x \sinh (b x) \text {Shi}(b x)}{b^3}+\frac {x^3 \sinh (b x) \text {Shi}(b x)}{b}-\frac {\int \sinh ^2(b x) \, dx}{2 b^3}-\frac {6 \int \sinh (b x) \text {Shi}(b x) \, dx}{b^3}+\frac {3 \int x \cosh (b x) \sinh (b x) \, dx}{b^2}-\frac {6 \int \frac {\sinh ^2(b x)}{b} \, dx}{b^2}+\frac {\int x^2 \, dx}{2 b} \\ & = \frac {x^3}{6 b}-\frac {\cosh (b x) \sinh (b x)}{4 b^4}-\frac {x^2 \cosh (b x) \sinh (b x)}{2 b^2}+\frac {2 x \sinh ^2(b x)}{b^3}-\frac {6 \cosh (b x) \text {Shi}(b x)}{b^4}-\frac {3 x^2 \cosh (b x) \text {Shi}(b x)}{b^2}+\frac {6 x \sinh (b x) \text {Shi}(b x)}{b^3}+\frac {x^3 \sinh (b x) \text {Shi}(b x)}{b}+\frac {\int 1 \, dx}{4 b^3}-\frac {3 \int \sinh ^2(b x) \, dx}{2 b^3}+\frac {6 \int \frac {\cosh (b x) \sinh (b x)}{b x} \, dx}{b^3}-\frac {6 \int \sinh ^2(b x) \, dx}{b^3} \\ & = \frac {x}{4 b^3}+\frac {x^3}{6 b}-\frac {4 \cosh (b x) \sinh (b x)}{b^4}-\frac {x^2 \cosh (b x) \sinh (b x)}{2 b^2}+\frac {2 x \sinh ^2(b x)}{b^3}-\frac {6 \cosh (b x) \text {Shi}(b x)}{b^4}-\frac {3 x^2 \cosh (b x) \text {Shi}(b x)}{b^2}+\frac {6 x \sinh (b x) \text {Shi}(b x)}{b^3}+\frac {x^3 \sinh (b x) \text {Shi}(b x)}{b}+\frac {6 \int \frac {\cosh (b x) \sinh (b x)}{x} \, dx}{b^4}+\frac {3 \int 1 \, dx}{4 b^3}+\frac {3 \int 1 \, dx}{b^3} \\ & = \frac {4 x}{b^3}+\frac {x^3}{6 b}-\frac {4 \cosh (b x) \sinh (b x)}{b^4}-\frac {x^2 \cosh (b x) \sinh (b x)}{2 b^2}+\frac {2 x \sinh ^2(b x)}{b^3}-\frac {6 \cosh (b x) \text {Shi}(b x)}{b^4}-\frac {3 x^2 \cosh (b x) \text {Shi}(b x)}{b^2}+\frac {6 x \sinh (b x) \text {Shi}(b x)}{b^3}+\frac {x^3 \sinh (b x) \text {Shi}(b x)}{b}+\frac {6 \int \frac {\sinh (2 b x)}{2 x} \, dx}{b^4} \\ & = \frac {4 x}{b^3}+\frac {x^3}{6 b}-\frac {4 \cosh (b x) \sinh (b x)}{b^4}-\frac {x^2 \cosh (b x) \sinh (b x)}{2 b^2}+\frac {2 x \sinh ^2(b x)}{b^3}-\frac {6 \cosh (b x) \text {Shi}(b x)}{b^4}-\frac {3 x^2 \cosh (b x) \text {Shi}(b x)}{b^2}+\frac {6 x \sinh (b x) \text {Shi}(b x)}{b^3}+\frac {x^3 \sinh (b x) \text {Shi}(b x)}{b}+\frac {3 \int \frac {\sinh (2 b x)}{x} \, dx}{b^4} \\ & = \frac {4 x}{b^3}+\frac {x^3}{6 b}-\frac {4 \cosh (b x) \sinh (b x)}{b^4}-\frac {x^2 \cosh (b x) \sinh (b x)}{2 b^2}+\frac {2 x \sinh ^2(b x)}{b^3}-\frac {6 \cosh (b x) \text {Shi}(b x)}{b^4}-\frac {3 x^2 \cosh (b x) \text {Shi}(b x)}{b^2}+\frac {6 x \sinh (b x) \text {Shi}(b x)}{b^3}+\frac {x^3 \sinh (b x) \text {Shi}(b x)}{b}+\frac {3 \text {Shi}(2 b x)}{b^4} \\ \end{align*}
Time = 0.07 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.73 \[ \int x^3 \cosh (b x) \text {Shi}(b x) \, dx=\frac {36 b x+2 b^3 x^3+12 b x \cosh (2 b x)-24 \sinh (2 b x)-3 b^2 x^2 \sinh (2 b x)+12 \left (-3 \left (2+b^2 x^2\right ) \cosh (b x)+b x \left (6+b^2 x^2\right ) \sinh (b x)\right ) \text {Shi}(b x)+36 \text {Shi}(2 b x)}{12 b^4} \]
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Time = 1.34 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.81
method | result | size |
derivativedivides | \(\frac {\operatorname {Shi}\left (b x \right ) \left (b^{3} x^{3} \sinh \left (b x \right )-3 b^{2} x^{2} \cosh \left (b x \right )+6 b x \sinh \left (b x \right )-6 \cosh \left (b x \right )\right )-\frac {b^{2} x^{2} \cosh \left (b x \right ) \sinh \left (b x \right )}{2}+\frac {b^{3} x^{3}}{6}+2 b x \cosh \left (b x \right )^{2}-4 \cosh \left (b x \right ) \sinh \left (b x \right )+2 b x +3 \,\operatorname {Shi}\left (2 b x \right )}{b^{4}}\) | \(104\) |
default | \(\frac {\operatorname {Shi}\left (b x \right ) \left (b^{3} x^{3} \sinh \left (b x \right )-3 b^{2} x^{2} \cosh \left (b x \right )+6 b x \sinh \left (b x \right )-6 \cosh \left (b x \right )\right )-\frac {b^{2} x^{2} \cosh \left (b x \right ) \sinh \left (b x \right )}{2}+\frac {b^{3} x^{3}}{6}+2 b x \cosh \left (b x \right )^{2}-4 \cosh \left (b x \right ) \sinh \left (b x \right )+2 b x +3 \,\operatorname {Shi}\left (2 b x \right )}{b^{4}}\) | \(104\) |
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\[ \int x^3 \cosh (b x) \text {Shi}(b x) \, dx=\int { x^{3} {\rm Shi}\left (b x\right ) \cosh \left (b x\right ) \,d x } \]
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\[ \int x^3 \cosh (b x) \text {Shi}(b x) \, dx=\int x^{3} \cosh {\left (b x \right )} \operatorname {Shi}{\left (b x \right )}\, dx \]
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\[ \int x^3 \cosh (b x) \text {Shi}(b x) \, dx=\int { x^{3} {\rm Shi}\left (b x\right ) \cosh \left (b x\right ) \,d x } \]
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\[ \int x^3 \cosh (b x) \text {Shi}(b x) \, dx=\int { x^{3} {\rm Shi}\left (b x\right ) \cosh \left (b x\right ) \,d x } \]
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Timed out. \[ \int x^3 \cosh (b x) \text {Shi}(b x) \, dx=\int x^3\,\mathrm {sinhint}\left (b\,x\right )\,\mathrm {cosh}\left (b\,x\right ) \,d x \]
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