Optimal. Leaf size=90 \[ -\frac {\text {Shi}(2 b x)}{b^3}+\frac {2 \text {Shi}(b x) \cosh (b x)}{b^3}+\frac {5 \sinh (b x) \cosh (b x)}{4 b^3}-\frac {2 x \text {Shi}(b x) \sinh (b x)}{b^2}-\frac {5 x}{4 b^2}-\frac {x \sinh ^2(b x)}{2 b^2}+\frac {x^2 \text {Shi}(b x) \cosh (b x)}{b} \]
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Rubi [A] time = 0.13, antiderivative size = 90, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 9, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.750, Rules used = {6542, 12, 5372, 2635, 8, 6548, 6540, 5448, 3298} \[ -\frac {\text {Shi}(2 b x)}{b^3}-\frac {2 x \text {Shi}(b x) \sinh (b x)}{b^2}+\frac {2 \text {Shi}(b x) \cosh (b x)}{b^3}-\frac {5 x}{4 b^2}-\frac {x \sinh ^2(b x)}{2 b^2}+\frac {5 \sinh (b x) \cosh (b x)}{4 b^3}+\frac {x^2 \text {Shi}(b x) \cosh (b x)}{b} \]
Antiderivative was successfully verified.
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Rule 8
Rule 12
Rule 2635
Rule 3298
Rule 5372
Rule 5448
Rule 6540
Rule 6542
Rule 6548
Rubi steps
\begin {align*} \int x^2 \sinh (b x) \text {Shi}(b x) \, dx &=\frac {x^2 \cosh (b x) \text {Shi}(b x)}{b}-\frac {2 \int x \cosh (b x) \text {Shi}(b x) \, dx}{b}-\int \frac {x \cosh (b x) \sinh (b x)}{b} \, dx\\ &=\frac {x^2 \cosh (b x) \text {Shi}(b x)}{b}-\frac {2 x \sinh (b x) \text {Shi}(b x)}{b^2}+\frac {2 \int \sinh (b x) \text {Shi}(b x) \, dx}{b^2}-\frac {\int x \cosh (b x) \sinh (b x) \, dx}{b}+\frac {2 \int \frac {\sinh ^2(b x)}{b} \, dx}{b}\\ &=-\frac {x \sinh ^2(b x)}{2 b^2}+\frac {2 \cosh (b x) \text {Shi}(b x)}{b^3}+\frac {x^2 \cosh (b x) \text {Shi}(b x)}{b}-\frac {2 x \sinh (b x) \text {Shi}(b x)}{b^2}+\frac {\int \sinh ^2(b x) \, dx}{2 b^2}-\frac {2 \int \frac {\cosh (b x) \sinh (b x)}{b x} \, dx}{b^2}+\frac {2 \int \sinh ^2(b x) \, dx}{b^2}\\ &=\frac {5 \cosh (b x) \sinh (b x)}{4 b^3}-\frac {x \sinh ^2(b x)}{2 b^2}+\frac {2 \cosh (b x) \text {Shi}(b x)}{b^3}+\frac {x^2 \cosh (b x) \text {Shi}(b x)}{b}-\frac {2 x \sinh (b x) \text {Shi}(b x)}{b^2}-\frac {2 \int \frac {\cosh (b x) \sinh (b x)}{x} \, dx}{b^3}-\frac {\int 1 \, dx}{4 b^2}-\frac {\int 1 \, dx}{b^2}\\ &=-\frac {5 x}{4 b^2}+\frac {5 \cosh (b x) \sinh (b x)}{4 b^3}-\frac {x \sinh ^2(b x)}{2 b^2}+\frac {2 \cosh (b x) \text {Shi}(b x)}{b^3}+\frac {x^2 \cosh (b x) \text {Shi}(b x)}{b}-\frac {2 x \sinh (b x) \text {Shi}(b x)}{b^2}-\frac {2 \int \frac {\sinh (2 b x)}{2 x} \, dx}{b^3}\\ &=-\frac {5 x}{4 b^2}+\frac {5 \cosh (b x) \sinh (b x)}{4 b^3}-\frac {x \sinh ^2(b x)}{2 b^2}+\frac {2 \cosh (b x) \text {Shi}(b x)}{b^3}+\frac {x^2 \cosh (b x) \text {Shi}(b x)}{b}-\frac {2 x \sinh (b x) \text {Shi}(b x)}{b^2}-\frac {\int \frac {\sinh (2 b x)}{x} \, dx}{b^3}\\ &=-\frac {5 x}{4 b^2}+\frac {5 \cosh (b x) \sinh (b x)}{4 b^3}-\frac {x \sinh ^2(b x)}{2 b^2}+\frac {2 \cosh (b x) \text {Shi}(b x)}{b^3}+\frac {x^2 \cosh (b x) \text {Shi}(b x)}{b}-\frac {2 x \sinh (b x) \text {Shi}(b x)}{b^2}-\frac {\text {Shi}(2 b x)}{b^3}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 64, normalized size = 0.71 \[ \frac {8 \text {Shi}(b x) \left (\left (b^2 x^2+2\right ) \cosh (b x)-2 b x \sinh (b x)\right )-8 \text {Shi}(2 b x)-8 b x+5 \sinh (2 b x)-2 b x \cosh (2 b x)}{8 b^3} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.34, size = 0, normalized size = 0.00 \[ {\rm integral}\left (x^{2} \sinh \left (b x\right ) \operatorname {Shi}\left (b x\right ), x\right ) \]
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 \[ \int x^{2} {\rm Shi}\left (b x\right ) \sinh \left (b x\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 68, normalized size = 0.76 \[ \frac {\Shi \left (b x \right ) \left (b^{2} x^{2} \cosh \left (b x \right )-2 b x \sinh \left (b x \right )+2 \cosh \left (b x \right )\right )-\frac {b x \left (\cosh ^{2}\left (b x \right )\right )}{2}+\frac {5 \sinh \left (b x \right ) \cosh \left (b x \right )}{4}-\frac {3 b x}{4}-\Shi \left (2 b x \right )}{b^{3}} \]
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 \[ \int x^{2} {\rm Shi}\left (b x\right ) \sinh \left (b x\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int x^2\,\mathrm {sinhint}\left (b\,x\right )\,\mathrm {sinh}\left (b\,x\right ) \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{2} \sinh {\left (b x \right )} \operatorname {Shi}{\left (b x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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