Optimal. Leaf size=109 \[ -\frac{\text{Chi}(2 b x)}{b^3}-\frac{2 x \text{Chi}(b x) \sinh (b x)}{b^2}+\frac{2 \text{Chi}(b x) \cosh (b x)}{b^3}-\frac{\log (x)}{b^3}+\frac{\sinh ^2(b x)}{b^3}+\frac{\cosh ^2(b x)}{4 b^3}-\frac{x \sinh (b x) \cosh (b x)}{2 b^2}+\frac{x^2 \text{Chi}(b x) \cosh (b x)}{b}-\frac{x^2}{4 b} \]
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Rubi [A] time = 0.127257, antiderivative size = 109, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 9, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.75, Rules used = {6549, 12, 3310, 30, 6543, 2564, 6547, 3312, 3301} \[ -\frac{\text{Chi}(2 b x)}{b^3}-\frac{2 x \text{Chi}(b x) \sinh (b x)}{b^2}+\frac{2 \text{Chi}(b x) \cosh (b x)}{b^3}-\frac{\log (x)}{b^3}+\frac{\sinh ^2(b x)}{b^3}+\frac{\cosh ^2(b x)}{4 b^3}-\frac{x \sinh (b x) \cosh (b x)}{2 b^2}+\frac{x^2 \text{Chi}(b x) \cosh (b x)}{b}-\frac{x^2}{4 b} \]
Antiderivative was successfully verified.
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Rule 6549
Rule 12
Rule 3310
Rule 30
Rule 6543
Rule 2564
Rule 6547
Rule 3312
Rule 3301
Rubi steps
\begin{align*} \int x^2 \text{Chi}(b x) \sinh (b x) \, dx &=\frac{x^2 \cosh (b x) \text{Chi}(b x)}{b}-\frac{2 \int x \cosh (b x) \text{Chi}(b x) \, dx}{b}-\int \frac{x \cosh ^2(b x)}{b} \, dx\\ &=\frac{x^2 \cosh (b x) \text{Chi}(b x)}{b}-\frac{2 x \text{Chi}(b x) \sinh (b x)}{b^2}+\frac{2 \int \text{Chi}(b x) \sinh (b x) \, dx}{b^2}-\frac{\int x \cosh ^2(b x) \, dx}{b}+\frac{2 \int \frac{\cosh (b x) \sinh (b x)}{b} \, dx}{b}\\ &=\frac{\cosh ^2(b x)}{4 b^3}+\frac{2 \cosh (b x) \text{Chi}(b x)}{b^3}+\frac{x^2 \cosh (b x) \text{Chi}(b x)}{b}-\frac{x \cosh (b x) \sinh (b x)}{2 b^2}-\frac{2 x \text{Chi}(b x) \sinh (b x)}{b^2}-\frac{2 \int \frac{\cosh ^2(b x)}{b x} \, dx}{b^2}+\frac{2 \int \cosh (b x) \sinh (b x) \, dx}{b^2}-\frac{\int x \, dx}{2 b}\\ &=-\frac{x^2}{4 b}+\frac{\cosh ^2(b x)}{4 b^3}+\frac{2 \cosh (b x) \text{Chi}(b x)}{b^3}+\frac{x^2 \cosh (b x) \text{Chi}(b x)}{b}-\frac{x \cosh (b x) \sinh (b x)}{2 b^2}-\frac{2 x \text{Chi}(b x) \sinh (b x)}{b^2}-\frac{2 \int \frac{\cosh ^2(b x)}{x} \, dx}{b^3}-\frac{2 \operatorname{Subst}(\int x \, dx,x,i \sinh (b x))}{b^3}\\ &=-\frac{x^2}{4 b}+\frac{\cosh ^2(b x)}{4 b^3}+\frac{2 \cosh (b x) \text{Chi}(b x)}{b^3}+\frac{x^2 \cosh (b x) \text{Chi}(b x)}{b}-\frac{x \cosh (b x) \sinh (b x)}{2 b^2}-\frac{2 x \text{Chi}(b x) \sinh (b x)}{b^2}+\frac{\sinh ^2(b x)}{b^3}-\frac{2 \int \left (\frac{1}{2 x}+\frac{\cosh (2 b x)}{2 x}\right ) \, dx}{b^3}\\ &=-\frac{x^2}{4 b}+\frac{\cosh ^2(b x)}{4 b^3}+\frac{2 \cosh (b x) \text{Chi}(b x)}{b^3}+\frac{x^2 \cosh (b x) \text{Chi}(b x)}{b}-\frac{\log (x)}{b^3}-\frac{x \cosh (b x) \sinh (b x)}{2 b^2}-\frac{2 x \text{Chi}(b x) \sinh (b x)}{b^2}+\frac{\sinh ^2(b x)}{b^3}-\frac{\int \frac{\cosh (2 b x)}{x} \, dx}{b^3}\\ &=-\frac{x^2}{4 b}+\frac{\cosh ^2(b x)}{4 b^3}+\frac{2 \cosh (b x) \text{Chi}(b x)}{b^3}+\frac{x^2 \cosh (b x) \text{Chi}(b x)}{b}-\frac{\text{Chi}(2 b x)}{b^3}-\frac{\log (x)}{b^3}-\frac{x \cosh (b x) \sinh (b x)}{2 b^2}-\frac{2 x \text{Chi}(b x) \sinh (b x)}{b^2}+\frac{\sinh ^2(b x)}{b^3}\\ \end{align*}
Mathematica [A] time = 0.109465, size = 72, normalized size = 0.66 \[ -\frac{-8 \text{Chi}(b x) \left (\left (b^2 x^2+2\right ) \cosh (b x)-2 b x \sinh (b x)\right )+2 b^2 x^2+8 \text{Chi}(2 b x)+2 b x \sinh (2 b x)-5 \cosh (2 b x)+8 \log (x)}{8 b^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.062, size = 96, normalized size = 0.9 \begin{align*}{\frac{{x}^{2}{\it Chi} \left ( bx \right ) \cosh \left ( bx \right ) }{b}}-2\,{\frac{x{\it Chi} \left ( bx \right ) \sinh \left ( bx \right ) }{{b}^{2}}}+2\,{\frac{{\it Chi} \left ( bx \right ) \cosh \left ( bx \right ) }{{b}^{3}}}-{\frac{x\cosh \left ( bx \right ) \sinh \left ( bx \right ) }{2\,{b}^{2}}}-{\frac{{x}^{2}}{4\,b}}+{\frac{5\, \left ( \cosh \left ( bx \right ) \right ) ^{2}}{4\,{b}^{3}}}-{\frac{\ln \left ( bx \right ) }{{b}^{3}}}-{\frac{{\it Chi} \left ( 2\,bx \right ) }{{b}^{3}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{2}{\rm Chi}\left (b x\right ) \sinh \left (b x\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (x^{2} \operatorname{Chi}\left (b x\right ) \sinh \left (b x\right ), x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{2} \sinh{\left (b x \right )} \operatorname{Chi}\left (b x\right )\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{2}{\rm Chi}\left (b x\right ) \sinh \left (b x\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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