Optimal. Leaf size=74 \[ -\frac {\text {Chi}(2 b x)}{2 b^2}+\frac {\text {Chi}(b x) \cosh (b x)}{b^2}-\frac {\log (x)}{2 b^2}+\frac {\sinh ^2(b x)}{2 b^2}+\frac {1}{2} x^2 \text {Chi}(b x)^2-\frac {x \text {Chi}(b x) \sinh (b x)}{b} \]
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Rubi [A] time = 0.09, antiderivative size = 74, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 8, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {6537, 6543, 12, 2564, 30, 6547, 3312, 3301} \[ -\frac {\text {Chi}(2 b x)}{2 b^2}+\frac {\text {Chi}(b x) \cosh (b x)}{b^2}-\frac {\log (x)}{2 b^2}+\frac {\sinh ^2(b x)}{2 b^2}+\frac {1}{2} x^2 \text {Chi}(b x)^2-\frac {x \text {Chi}(b x) \sinh (b x)}{b} \]
Antiderivative was successfully verified.
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Rule 12
Rule 30
Rule 2564
Rule 3301
Rule 3312
Rule 6537
Rule 6543
Rule 6547
Rubi steps
\begin {align*} \int x \text {Chi}(b x)^2 \, dx &=\frac {1}{2} x^2 \text {Chi}(b x)^2-\int x \cosh (b x) \text {Chi}(b x) \, dx\\ &=\frac {1}{2} x^2 \text {Chi}(b x)^2-\frac {x \text {Chi}(b x) \sinh (b x)}{b}+\frac {\int \text {Chi}(b x) \sinh (b x) \, dx}{b}+\int \frac {\cosh (b x) \sinh (b x)}{b} \, dx\\ &=\frac {\cosh (b x) \text {Chi}(b x)}{b^2}+\frac {1}{2} x^2 \text {Chi}(b x)^2-\frac {x \text {Chi}(b x) \sinh (b x)}{b}-\frac {\int \frac {\cosh ^2(b x)}{b x} \, dx}{b}+\frac {\int \cosh (b x) \sinh (b x) \, dx}{b}\\ &=\frac {\cosh (b x) \text {Chi}(b x)}{b^2}+\frac {1}{2} x^2 \text {Chi}(b x)^2-\frac {x \text {Chi}(b x) \sinh (b x)}{b}-\frac {\int \frac {\cosh ^2(b x)}{x} \, dx}{b^2}-\frac {\operatorname {Subst}(\int x \, dx,x,i \sinh (b x))}{b^2}\\ &=\frac {\cosh (b x) \text {Chi}(b x)}{b^2}+\frac {1}{2} x^2 \text {Chi}(b x)^2-\frac {x \text {Chi}(b x) \sinh (b x)}{b}+\frac {\sinh ^2(b x)}{2 b^2}-\frac {\int \left (\frac {1}{2 x}+\frac {\cosh (2 b x)}{2 x}\right ) \, dx}{b^2}\\ &=\frac {\cosh (b x) \text {Chi}(b x)}{b^2}+\frac {1}{2} x^2 \text {Chi}(b x)^2-\frac {\log (x)}{2 b^2}-\frac {x \text {Chi}(b x) \sinh (b x)}{b}+\frac {\sinh ^2(b x)}{2 b^2}-\frac {\int \frac {\cosh (2 b x)}{x} \, dx}{2 b^2}\\ &=\frac {\cosh (b x) \text {Chi}(b x)}{b^2}+\frac {1}{2} x^2 \text {Chi}(b x)^2-\frac {\text {Chi}(2 b x)}{2 b^2}-\frac {\log (x)}{2 b^2}-\frac {x \text {Chi}(b x) \sinh (b x)}{b}+\frac {\sinh ^2(b x)}{2 b^2}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 57, normalized size = 0.77 \[ \frac {2 b^2 x^2 \text {Chi}(b x)^2-2 \text {Chi}(2 b x)+4 \text {Chi}(b x) (\cosh (b x)-b x \sinh (b x))+\cosh (2 b x)-2 \log (x)}{4 b^2} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.45, size = 0, normalized size = 0.00 \[ {\rm integral}\left (x \operatorname {Chi}\left (b x\right )^{2}, x\right ) \]
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 \[ \int x {\rm Chi}\left (b x\right )^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 69, normalized size = 0.93 \[ \frac {x^{2} \Chi \left (b x \right )^{2}}{2}-\frac {x \Chi \left (b x \right ) \sinh \left (b x \right )}{b}+\frac {\Chi \left (b x \right ) \cosh \left (b x \right )}{b^{2}}+\frac {\cosh ^{2}\left (b x \right )}{2 b^{2}}-\frac {\ln \left (b x \right )}{2 b^{2}}-\frac {\Chi \left (2 b x \right )}{2 b^{2}} \]
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 \[ \int x {\rm Chi}\left (b x\right )^{2}\,{d x} \]
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 \[ \int x\,{\mathrm {coshint}\left (b\,x\right )}^2 \,d x \]
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 \[ \int x \operatorname {Chi}^{2}\left (b x\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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