Integrand size = 12, antiderivative size = 125 \[ \int x^3 \sinh (b x) \text {Shi}(b x) \, dx=-\frac {x^2}{b^2}+\frac {3 \text {Chi}(2 b x)}{b^4}-\frac {3 \log (x)}{b^4}+\frac {2 x \cosh (b x) \sinh (b x)}{b^3}-\frac {4 \sinh ^2(b x)}{b^4}-\frac {x^2 \sinh ^2(b x)}{2 b^2}+\frac {6 x \cosh (b x) \text {Shi}(b x)}{b^3}+\frac {x^3 \cosh (b x) \text {Shi}(b x)}{b}-\frac {6 \sinh (b x) \text {Shi}(b x)}{b^4}-\frac {3 x^2 \sinh (b x) \text {Shi}(b x)}{b^2} \]
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Time = 0.14 (sec) , antiderivative size = 125, normalized size of antiderivative = 1.00, number of steps used = 18, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.833, Rules used = {6677, 12, 5480, 3391, 30, 6683, 2644, 6681, 3393, 3382} \[ \int x^3 \sinh (b x) \text {Shi}(b x) \, dx=\frac {3 \text {Chi}(2 b x)}{b^4}-\frac {6 \text {Shi}(b x) \sinh (b x)}{b^4}-\frac {3 \log (x)}{b^4}-\frac {4 \sinh ^2(b x)}{b^4}+\frac {6 x \text {Shi}(b x) \cosh (b x)}{b^3}+\frac {2 x \sinh (b x) \cosh (b x)}{b^3}-\frac {3 x^2 \text {Shi}(b x) \sinh (b x)}{b^2}-\frac {x^2}{b^2}-\frac {x^2 \sinh ^2(b x)}{2 b^2}+\frac {x^3 \text {Shi}(b x) \cosh (b x)}{b} \]
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Rule 12
Rule 30
Rule 2644
Rule 3382
Rule 3391
Rule 3393
Rule 5480
Rule 6677
Rule 6681
Rule 6683
Rubi steps \begin{align*} \text {integral}& = \frac {x^3 \cosh (b x) \text {Shi}(b x)}{b}-\frac {3 \int x^2 \cosh (b x) \text {Shi}(b x) \, dx}{b}-\int \frac {x^2 \cosh (b x) \sinh (b x)}{b} \, dx \\ & = \frac {x^3 \cosh (b x) \text {Shi}(b x)}{b}-\frac {3 x^2 \sinh (b x) \text {Shi}(b x)}{b^2}+\frac {6 \int x \sinh (b x) \text {Shi}(b x) \, dx}{b^2}-\frac {\int x^2 \cosh (b x) \sinh (b x) \, dx}{b}+\frac {3 \int \frac {x \sinh ^2(b x)}{b} \, dx}{b} \\ & = -\frac {x^2 \sinh ^2(b x)}{2 b^2}+\frac {6 x \cosh (b x) \text {Shi}(b x)}{b^3}+\frac {x^3 \cosh (b x) \text {Shi}(b x)}{b}-\frac {3 x^2 \sinh (b x) \text {Shi}(b x)}{b^2}-\frac {6 \int \cosh (b x) \text {Shi}(b x) \, dx}{b^3}+\frac {\int x \sinh ^2(b x) \, dx}{b^2}+\frac {3 \int x \sinh ^2(b x) \, dx}{b^2}-\frac {6 \int \frac {\cosh (b x) \sinh (b x)}{b} \, dx}{b^2} \\ & = \frac {2 x \cosh (b x) \sinh (b x)}{b^3}-\frac {\sinh ^2(b x)}{b^4}-\frac {x^2 \sinh ^2(b x)}{2 b^2}+\frac {6 x \cosh (b x) \text {Shi}(b x)}{b^3}+\frac {x^3 \cosh (b x) \text {Shi}(b x)}{b}-\frac {6 \sinh (b x) \text {Shi}(b x)}{b^4}-\frac {3 x^2 \sinh (b x) \text {Shi}(b x)}{b^2}-\frac {6 \int \cosh (b x) \sinh (b x) \, dx}{b^3}+\frac {6 \int \frac {\sinh ^2(b x)}{b x} \, dx}{b^3}-\frac {\int x \, dx}{2 b^2}-\frac {3 \int x \, dx}{2 b^2} \\ & = -\frac {x^2}{b^2}+\frac {2 x \cosh (b x) \sinh (b x)}{b^3}-\frac {\sinh ^2(b x)}{b^4}-\frac {x^2 \sinh ^2(b x)}{2 b^2}+\frac {6 x \cosh (b x) \text {Shi}(b x)}{b^3}+\frac {x^3 \cosh (b x) \text {Shi}(b x)}{b}-\frac {6 \sinh (b x) \text {Shi}(b x)}{b^4}-\frac {3 x^2 \sinh (b x) \text {Shi}(b x)}{b^2}+\frac {6 \int \frac {\sinh ^2(b x)}{x} \, dx}{b^4}+\frac {6 \text {Subst}(\int x \, dx,x,i \sinh (b x))}{b^4} \\ & = -\frac {x^2}{b^2}+\frac {2 x \cosh (b x) \sinh (b x)}{b^3}-\frac {4 \sinh ^2(b x)}{b^4}-\frac {x^2 \sinh ^2(b x)}{2 b^2}+\frac {6 x \cosh (b x) \text {Shi}(b x)}{b^3}+\frac {x^3 \cosh (b x) \text {Shi}(b x)}{b}-\frac {6 \sinh (b x) \text {Shi}(b x)}{b^4}-\frac {3 x^2 \sinh (b x) \text {Shi}(b x)}{b^2}-\frac {6 \int \left (\frac {1}{2 x}-\frac {\cosh (2 b x)}{2 x}\right ) \, dx}{b^4} \\ & = -\frac {x^2}{b^2}-\frac {3 \log (x)}{b^4}+\frac {2 x \cosh (b x) \sinh (b x)}{b^3}-\frac {4 \sinh ^2(b x)}{b^4}-\frac {x^2 \sinh ^2(b x)}{2 b^2}+\frac {6 x \cosh (b x) \text {Shi}(b x)}{b^3}+\frac {x^3 \cosh (b x) \text {Shi}(b x)}{b}-\frac {6 \sinh (b x) \text {Shi}(b x)}{b^4}-\frac {3 x^2 \sinh (b x) \text {Shi}(b x)}{b^2}+\frac {3 \int \frac {\cosh (2 b x)}{x} \, dx}{b^4} \\ & = -\frac {x^2}{b^2}+\frac {3 \text {Chi}(2 b x)}{b^4}-\frac {3 \log (x)}{b^4}+\frac {2 x \cosh (b x) \sinh (b x)}{b^3}-\frac {4 \sinh ^2(b x)}{b^4}-\frac {x^2 \sinh ^2(b x)}{2 b^2}+\frac {6 x \cosh (b x) \text {Shi}(b x)}{b^3}+\frac {x^3 \cosh (b x) \text {Shi}(b x)}{b}-\frac {6 \sinh (b x) \text {Shi}(b x)}{b^4}-\frac {3 x^2 \sinh (b x) \text {Shi}(b x)}{b^2} \\ \end{align*}
Time = 0.12 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.74 \[ \int x^3 \sinh (b x) \text {Shi}(b x) \, dx=-\frac {3 b^2 x^2+8 \cosh (2 b x)+b^2 x^2 \cosh (2 b x)-12 \text {Chi}(2 b x)+12 \log (x)-4 b x \sinh (2 b x)-4 \left (b x \left (6+b^2 x^2\right ) \cosh (b x)-3 \left (2+b^2 x^2\right ) \sinh (b x)\right ) \text {Shi}(b x)}{4 b^4} \]
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Time = 0.96 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.83
method | result | size |
derivativedivides | \(\frac {\operatorname {Shi}\left (b x \right ) \left (b^{3} x^{3} \cosh \left (b x \right )-3 b^{2} x^{2} \sinh \left (b x \right )+6 b x \cosh \left (b x \right )-6 \sinh \left (b x \right )\right )-\frac {b^{2} x^{2} \cosh \left (b x \right )^{2}}{2}+2 b x \cosh \left (b x \right ) \sinh \left (b x \right )-\frac {b^{2} x^{2}}{2}-4 \cosh \left (b x \right )^{2}-3 \ln \left (b x \right )+3 \,\operatorname {Chi}\left (2 b x \right )}{b^{4}}\) | \(104\) |
default | \(\frac {\operatorname {Shi}\left (b x \right ) \left (b^{3} x^{3} \cosh \left (b x \right )-3 b^{2} x^{2} \sinh \left (b x \right )+6 b x \cosh \left (b x \right )-6 \sinh \left (b x \right )\right )-\frac {b^{2} x^{2} \cosh \left (b x \right )^{2}}{2}+2 b x \cosh \left (b x \right ) \sinh \left (b x \right )-\frac {b^{2} x^{2}}{2}-4 \cosh \left (b x \right )^{2}-3 \ln \left (b x \right )+3 \,\operatorname {Chi}\left (2 b x \right )}{b^{4}}\) | \(104\) |
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\[ \int x^3 \sinh (b x) \text {Shi}(b x) \, dx=\int { x^{3} {\rm Shi}\left (b x\right ) \sinh \left (b x\right ) \,d x } \]
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\[ \int x^3 \sinh (b x) \text {Shi}(b x) \, dx=\int x^{3} \sinh {\left (b x \right )} \operatorname {Shi}{\left (b x \right )}\, dx \]
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\[ \int x^3 \sinh (b x) \text {Shi}(b x) \, dx=\int { x^{3} {\rm Shi}\left (b x\right ) \sinh \left (b x\right ) \,d x } \]
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\[ \int x^3 \sinh (b x) \text {Shi}(b x) \, dx=\int { x^{3} {\rm Shi}\left (b x\right ) \sinh \left (b x\right ) \,d x } \]
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Timed out. \[ \int x^3 \sinh (b x) \text {Shi}(b x) \, dx=\int x^3\,\mathrm {sinhint}\left (b\,x\right )\,\mathrm {sinh}\left (b\,x\right ) \,d x \]
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