Optimal. Leaf size=142 \[ -\frac{3 x^2 \text{Chi}(b x) \cosh (b x)}{b^2}+\frac{3 \text{Chi}(2 b x)}{b^4}+\frac{6 x \text{Chi}(b x) \sinh (b x)}{b^3}-\frac{6 \text{Chi}(b x) \cosh (b x)}{b^4}+\frac{x^2}{2 b^2}-\frac{x^2 \sinh ^2(b x)}{2 b^2}+\frac{3 \log (x)}{b^4}-\frac{13 \sinh ^2(b x)}{4 b^4}-\frac{3 \cosh ^2(b x)}{4 b^4}+\frac{2 x \sinh (b x) \cosh (b x)}{b^3}+\frac{x^3 \text{Chi}(b x) \sinh (b x)}{b} \]
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Rubi [A] time = 0.212921, antiderivative size = 142, normalized size of antiderivative = 1., number of steps used = 18, number of rules used = 10, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.833, Rules used = {6543, 12, 5372, 3310, 30, 6549, 2564, 6547, 3312, 3301} \[ -\frac{3 x^2 \text{Chi}(b x) \cosh (b x)}{b^2}+\frac{3 \text{Chi}(2 b x)}{b^4}+\frac{6 x \text{Chi}(b x) \sinh (b x)}{b^3}-\frac{6 \text{Chi}(b x) \cosh (b x)}{b^4}+\frac{x^2}{2 b^2}-\frac{x^2 \sinh ^2(b x)}{2 b^2}+\frac{3 \log (x)}{b^4}-\frac{13 \sinh ^2(b x)}{4 b^4}-\frac{3 \cosh ^2(b x)}{4 b^4}+\frac{2 x \sinh (b x) \cosh (b x)}{b^3}+\frac{x^3 \text{Chi}(b x) \sinh (b x)}{b} \]
Antiderivative was successfully verified.
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Rule 6543
Rule 12
Rule 5372
Rule 3310
Rule 30
Rule 6549
Rule 2564
Rule 6547
Rule 3312
Rule 3301
Rubi steps
\begin{align*} \int x^3 \cosh (b x) \text{Chi}(b x) \, dx &=\frac{x^3 \text{Chi}(b x) \sinh (b x)}{b}-\frac{3 \int x^2 \text{Chi}(b x) \sinh (b x) \, dx}{b}-\int \frac{x^2 \cosh (b x) \sinh (b x)}{b} \, dx\\ &=-\frac{3 x^2 \cosh (b x) \text{Chi}(b x)}{b^2}+\frac{x^3 \text{Chi}(b x) \sinh (b x)}{b}+\frac{6 \int x \cosh (b x) \text{Chi}(b x) \, dx}{b^2}-\frac{\int x^2 \cosh (b x) \sinh (b x) \, dx}{b}+\frac{3 \int \frac{x \cosh ^2(b x)}{b} \, dx}{b}\\ &=-\frac{3 x^2 \cosh (b x) \text{Chi}(b x)}{b^2}+\frac{6 x \text{Chi}(b x) \sinh (b x)}{b^3}+\frac{x^3 \text{Chi}(b x) \sinh (b x)}{b}-\frac{x^2 \sinh ^2(b x)}{2 b^2}-\frac{6 \int \text{Chi}(b x) \sinh (b x) \, dx}{b^3}+\frac{\int x \sinh ^2(b x) \, dx}{b^2}+\frac{3 \int x \cosh ^2(b x) \, dx}{b^2}-\frac{6 \int \frac{\cosh (b x) \sinh (b x)}{b} \, dx}{b^2}\\ &=-\frac{3 \cosh ^2(b x)}{4 b^4}-\frac{6 \cosh (b x) \text{Chi}(b x)}{b^4}-\frac{3 x^2 \cosh (b x) \text{Chi}(b x)}{b^2}+\frac{2 x \cosh (b x) \sinh (b x)}{b^3}+\frac{6 x \text{Chi}(b x) \sinh (b x)}{b^3}+\frac{x^3 \text{Chi}(b x) \sinh (b x)}{b}-\frac{\sinh ^2(b x)}{4 b^4}-\frac{x^2 \sinh ^2(b x)}{2 b^2}+\frac{6 \int \frac{\cosh ^2(b x)}{b x} \, dx}{b^3}-\frac{6 \int \cosh (b x) \sinh (b x) \, dx}{b^3}-\frac{\int x \, dx}{2 b^2}+\frac{3 \int x \, dx}{2 b^2}\\ &=\frac{x^2}{2 b^2}-\frac{3 \cosh ^2(b x)}{4 b^4}-\frac{6 \cosh (b x) \text{Chi}(b x)}{b^4}-\frac{3 x^2 \cosh (b x) \text{Chi}(b x)}{b^2}+\frac{2 x \cosh (b x) \sinh (b x)}{b^3}+\frac{6 x \text{Chi}(b x) \sinh (b x)}{b^3}+\frac{x^3 \text{Chi}(b x) \sinh (b x)}{b}-\frac{\sinh ^2(b x)}{4 b^4}-\frac{x^2 \sinh ^2(b x)}{2 b^2}+\frac{6 \int \frac{\cosh ^2(b x)}{x} \, dx}{b^4}+\frac{6 \operatorname{Subst}(\int x \, dx,x,i \sinh (b x))}{b^4}\\ &=\frac{x^2}{2 b^2}-\frac{3 \cosh ^2(b x)}{4 b^4}-\frac{6 \cosh (b x) \text{Chi}(b x)}{b^4}-\frac{3 x^2 \cosh (b x) \text{Chi}(b x)}{b^2}+\frac{2 x \cosh (b x) \sinh (b x)}{b^3}+\frac{6 x \text{Chi}(b x) \sinh (b x)}{b^3}+\frac{x^3 \text{Chi}(b x) \sinh (b x)}{b}-\frac{13 \sinh ^2(b x)}{4 b^4}-\frac{x^2 \sinh ^2(b x)}{2 b^2}+\frac{6 \int \left (\frac{1}{2 x}+\frac{\cosh (2 b x)}{2 x}\right ) \, dx}{b^4}\\ &=\frac{x^2}{2 b^2}-\frac{3 \cosh ^2(b x)}{4 b^4}-\frac{6 \cosh (b x) \text{Chi}(b x)}{b^4}-\frac{3 x^2 \cosh (b x) \text{Chi}(b x)}{b^2}+\frac{3 \log (x)}{b^4}+\frac{2 x \cosh (b x) \sinh (b x)}{b^3}+\frac{6 x \text{Chi}(b x) \sinh (b x)}{b^3}+\frac{x^3 \text{Chi}(b x) \sinh (b x)}{b}-\frac{13 \sinh ^2(b x)}{4 b^4}-\frac{x^2 \sinh ^2(b x)}{2 b^2}+\frac{3 \int \frac{\cosh (2 b x)}{x} \, dx}{b^4}\\ &=\frac{x^2}{2 b^2}-\frac{3 \cosh ^2(b x)}{4 b^4}-\frac{6 \cosh (b x) \text{Chi}(b x)}{b^4}-\frac{3 x^2 \cosh (b x) \text{Chi}(b x)}{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{6 x \text{Chi}(b x) \sinh (b x)}{b^3}+\frac{x^3 \text{Chi}(b x) \sinh (b x)}{b}-\frac{13 \sinh ^2(b x)}{4 b^4}-\frac{x^2 \sinh ^2(b x)}{2 b^2}\\ \end{align*}
Mathematica [A] time = 0.101638, size = 94, normalized size = 0.66 \[ \frac{4 \text{Chi}(b x) \left (b x \left (b^2 x^2+6\right ) \sinh (b x)-3 \left (b^2 x^2+2\right ) \cosh (b x)\right )+3 b^2 x^2-b^2 x^2 \cosh (2 b x)+12 \text{Chi}(2 b x)+4 b x \sinh (2 b x)-8 \cosh (2 b x)+12 \log (x)}{4 b^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.062, size = 125, normalized size = 0.9 \begin{align*}{\frac{{x}^{3}{\it Chi} \left ( bx \right ) \sinh \left ( bx \right ) }{b}}-3\,{\frac{{x}^{2}{\it Chi} \left ( bx \right ) \cosh \left ( bx \right ) }{{b}^{2}}}+6\,{\frac{x{\it Chi} \left ( bx \right ) \sinh \left ( bx \right ) }{{b}^{3}}}-6\,{\frac{{\it Chi} \left ( bx \right ) \cosh \left ( bx \right ) }{{b}^{4}}}-{\frac{{x}^{2} \left ( \cosh \left ( bx \right ) \right ) ^{2}}{2\,{b}^{2}}}+2\,{\frac{x\cosh \left ( bx \right ) \sinh \left ( bx \right ) }{{b}^{3}}}+{\frac{{x}^{2}}{{b}^{2}}}-4\,{\frac{ \left ( \cosh \left ( bx \right ) \right ) ^{2}}{{b}^{4}}}+3\,{\frac{\ln \left ( bx \right ) }{{b}^{4}}}+3\,{\frac{{\it Chi} \left ( 2\,bx \right ) }{{b}^{4}}} \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^{3}{\rm Chi}\left (b x\right ) \cosh \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^{3} \cosh \left (b x\right ) \operatorname{Chi}\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^{3} \cosh{\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^{3}{\rm Chi}\left (b x\right ) \cosh \left (b x\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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