Optimal. Leaf size=71 \[ -\frac{a^2 \text{Chi}(a+b x)}{2 b^2}+\frac{a \sinh (a+b x)}{2 b^2}+\frac{\cosh (a+b x)}{2 b^2}+\frac{1}{2} x^2 \text{Chi}(a+b x)-\frac{x \sinh (a+b x)}{2 b} \]
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Rubi [A] time = 0.210045, antiderivative size = 71, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.75, Rules used = {6533, 6742, 2637, 3296, 2638, 3301} \[ -\frac{a^2 \text{Chi}(a+b x)}{2 b^2}+\frac{a \sinh (a+b x)}{2 b^2}+\frac{\cosh (a+b x)}{2 b^2}+\frac{1}{2} x^2 \text{Chi}(a+b x)-\frac{x \sinh (a+b x)}{2 b} \]
Antiderivative was successfully verified.
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Rule 6533
Rule 6742
Rule 2637
Rule 3296
Rule 2638
Rule 3301
Rubi steps
\begin{align*} \int x \text{Chi}(a+b x) \, dx &=\frac{1}{2} x^2 \text{Chi}(a+b x)-\frac{1}{2} b \int \frac{x^2 \cosh (a+b x)}{a+b x} \, dx\\ &=\frac{1}{2} x^2 \text{Chi}(a+b x)-\frac{1}{2} b \int \left (-\frac{a \cosh (a+b x)}{b^2}+\frac{x \cosh (a+b x)}{b}+\frac{a^2 \cosh (a+b x)}{b^2 (a+b x)}\right ) \, dx\\ &=\frac{1}{2} x^2 \text{Chi}(a+b x)-\frac{1}{2} \int x \cosh (a+b x) \, dx+\frac{a \int \cosh (a+b x) \, dx}{2 b}-\frac{a^2 \int \frac{\cosh (a+b x)}{a+b x} \, dx}{2 b}\\ &=-\frac{a^2 \text{Chi}(a+b x)}{2 b^2}+\frac{1}{2} x^2 \text{Chi}(a+b x)+\frac{a \sinh (a+b x)}{2 b^2}-\frac{x \sinh (a+b x)}{2 b}+\frac{\int \sinh (a+b x) \, dx}{2 b}\\ &=\frac{\cosh (a+b x)}{2 b^2}-\frac{a^2 \text{Chi}(a+b x)}{2 b^2}+\frac{1}{2} x^2 \text{Chi}(a+b x)+\frac{a \sinh (a+b x)}{2 b^2}-\frac{x \sinh (a+b x)}{2 b}\\ \end{align*}
Mathematica [A] time = 0.0976715, size = 47, normalized size = 0.66 \[ \frac{\left (b^2 x^2-a^2\right ) \text{Chi}(a+b x)+(a-b x) \sinh (a+b x)+\cosh (a+b x)}{2 b^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.047, size = 60, normalized size = 0.9 \begin{align*}{\frac{1}{{b}^{2}} \left ({\it Chi} \left ( bx+a \right ) \left ({\frac{ \left ( bx+a \right ) ^{2}}{2}}-a \left ( bx+a \right ) \right ) -{\frac{ \left ( bx+a \right ) \sinh \left ( bx+a \right ) }{2}}+{\frac{\cosh \left ( bx+a \right ) }{2}}+a\sinh \left ( bx+a \right ) \right ) } \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{\rm Chi}\left (b x + a\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 \operatorname{Chi}\left (b x + a\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 \operatorname{Chi}\left (a + 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{\rm Chi}\left (b x + a\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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