Optimal. Leaf size=63 \[ \frac {3 \sinh (b x)}{2 b^4}-\frac {3 x \cosh (b x)}{2 b^3}+\frac {3 x^2 \sinh (b x)}{4 b^2}+\frac {1}{4} x^4 \text {Shi}(b x)-\frac {x^3 \cosh (b x)}{4 b} \]
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Rubi [A] time = 0.08, antiderivative size = 63, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6532, 12, 3296, 2637} \[ \frac {3 x^2 \sinh (b x)}{4 b^2}+\frac {3 \sinh (b x)}{2 b^4}-\frac {3 x \cosh (b x)}{2 b^3}+\frac {1}{4} x^4 \text {Shi}(b x)-\frac {x^3 \cosh (b x)}{4 b} \]
Antiderivative was successfully verified.
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Rule 12
Rule 2637
Rule 3296
Rule 6532
Rubi steps
\begin {align*} \int x^3 \text {Shi}(b x) \, dx &=\frac {1}{4} x^4 \text {Shi}(b x)-\frac {1}{4} b \int \frac {x^3 \sinh (b x)}{b} \, dx\\ &=\frac {1}{4} x^4 \text {Shi}(b x)-\frac {1}{4} \int x^3 \sinh (b x) \, dx\\ &=-\frac {x^3 \cosh (b x)}{4 b}+\frac {1}{4} x^4 \text {Shi}(b x)+\frac {3 \int x^2 \cosh (b x) \, dx}{4 b}\\ &=-\frac {x^3 \cosh (b x)}{4 b}+\frac {3 x^2 \sinh (b x)}{4 b^2}+\frac {1}{4} x^4 \text {Shi}(b x)-\frac {3 \int x \sinh (b x) \, dx}{2 b^2}\\ &=-\frac {3 x \cosh (b x)}{2 b^3}-\frac {x^3 \cosh (b x)}{4 b}+\frac {3 x^2 \sinh (b x)}{4 b^2}+\frac {1}{4} x^4 \text {Shi}(b x)+\frac {3 \int \cosh (b x) \, dx}{2 b^3}\\ &=-\frac {3 x \cosh (b x)}{2 b^3}-\frac {x^3 \cosh (b x)}{4 b}+\frac {3 \sinh (b x)}{2 b^4}+\frac {3 x^2 \sinh (b x)}{4 b^2}+\frac {1}{4} x^4 \text {Shi}(b x)\\ \end {align*}
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Mathematica [A] time = 0.03, size = 53, normalized size = 0.84 \[ \frac {3 \left (b^2 x^2+2\right ) \sinh (b x)}{4 b^4}-\frac {x \left (b^2 x^2+6\right ) \cosh (b x)}{4 b^3}+\frac {1}{4} x^4 \text {Shi}(b x) \]
Antiderivative was successfully verified.
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fricas [F] time = 1.41, size = 0, normalized size = 0.00 \[ {\rm integral}\left (x^{3} \operatorname {Shi}\left (b x\right ), x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{3} {\rm Shi}\left (b x\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 56, normalized size = 0.89 \[ \frac {\frac {b^{4} x^{4} \Shi \left (b x \right )}{4}-\frac {b^{3} x^{3} \cosh \left (b x \right )}{4}+\frac {3 b^{2} x^{2} \sinh \left (b x \right )}{4}-\frac {3 b x \cosh \left (b x \right )}{2}+\frac {3 \sinh \left (b x \right )}{2}}{b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{3} {\rm Shi}\left (b x\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int x^3\,\mathrm {sinhint}\left (b\,x\right ) \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.08, size = 61, normalized size = 0.97 \[ \frac {x^{4} \operatorname {Shi}{\left (b x \right )}}{4} - \frac {x^{3} \cosh {\left (b x \right )}}{4 b} + \frac {3 x^{2} \sinh {\left (b x \right )}}{4 b^{2}} - \frac {3 x \cosh {\left (b x \right )}}{2 b^{3}} + \frac {3 \sinh {\left (b x \right )}}{2 b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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