Optimal. Leaf size=109 \[ -\frac {\text {Chi}(2 b x)}{b^3}+\frac {2 \text {Chi}(b x) \cosh (b x)}{b^3}-\frac {\log (x)}{b^3}+\frac {\sinh ^2(b x)}{b^3}+\frac {\cosh ^2(b x)}{4 b^3}-\frac {2 x \text {Chi}(b x) \sinh (b x)}{b^2}-\frac {x \sinh (b x) \cosh (b x)}{2 b^2}+\frac {x^2 \text {Chi}(b x) \cosh (b x)}{b}-\frac {x^2}{4 b} \]
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Rubi [A] time = 0.13, antiderivative size = 109, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 9, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.750, 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 12
Rule 30
Rule 2564
Rule 3301
Rule 3310
Rule 3312
Rule 6543
Rule 6547
Rule 6549
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*}
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Mathematica [A] time = 0.11, 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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fricas [F] time = 1.35, size = 0, normalized size = 0.00 \[ {\rm integral}\left (x^{2} \operatorname {Chi}\left (b x\right ) \sinh \left (b x\right ), x\right ) \]
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 \[ \int x^{2} {\rm Chi}\left (b x\right ) \sinh \left (b x\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 96, normalized size = 0.88 \[ \frac {x^{2} \Chi \left (b x \right ) \cosh \left (b x \right )}{b}-\frac {2 x \Chi \left (b x \right ) \sinh \left (b x \right )}{b^{2}}+\frac {2 \Chi \left (b x \right ) \cosh \left (b x \right )}{b^{3}}-\frac {x \cosh \left (b x \right ) \sinh \left (b x \right )}{2 b^{2}}-\frac {x^{2}}{4 b}+\frac {5 \left (\cosh ^{2}\left (b x \right )\right )}{4 b^{3}}-\frac {\ln \left (b x \right )}{b^{3}}-\frac {\Chi \left (2 b x \right )}{b^{3}} \]
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 \[ \int x^{2} {\rm Chi}\left (b x\right ) \sinh \left (b x\right )\,{d x} \]
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 \[ \int x^2\,\mathrm {coshint}\left (b\,x\right )\,\mathrm {sinh}\left (b\,x\right ) \,d x \]
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 \[ \int x^{2} \sinh {\left (b x \right )} \operatorname {Chi}\left (b x\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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