Optimal. Leaf size=118 \[ \frac{a^3 \text{Chi}(a+b x)}{3 b^3}-\frac{a^2 \sinh (a+b x)}{3 b^3}+\frac{a x \sinh (a+b x)}{3 b^2}-\frac{2 \sinh (a+b x)}{3 b^3}-\frac{a \cosh (a+b x)}{3 b^3}+\frac{2 x \cosh (a+b x)}{3 b^2}+\frac{1}{3} x^3 \text{Chi}(a+b x)-\frac{x^2 \sinh (a+b x)}{3 b} \]
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Rubi [A] time = 0.280358, antiderivative size = 118, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 6, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.6, Rules used = {6533, 6742, 2637, 3296, 2638, 3301} \[ \frac{a^3 \text{Chi}(a+b x)}{3 b^3}-\frac{a^2 \sinh (a+b x)}{3 b^3}+\frac{a x \sinh (a+b x)}{3 b^2}-\frac{2 \sinh (a+b x)}{3 b^3}-\frac{a \cosh (a+b x)}{3 b^3}+\frac{2 x \cosh (a+b x)}{3 b^2}+\frac{1}{3} x^3 \text{Chi}(a+b x)-\frac{x^2 \sinh (a+b x)}{3 b} \]
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^2 \text{Chi}(a+b x) \, dx &=\frac{1}{3} x^3 \text{Chi}(a+b x)-\frac{1}{3} b \int \frac{x^3 \cosh (a+b x)}{a+b x} \, dx\\ &=\frac{1}{3} x^3 \text{Chi}(a+b x)-\frac{1}{3} b \int \left (\frac{a^2 \cosh (a+b x)}{b^3}-\frac{a x \cosh (a+b x)}{b^2}+\frac{x^2 \cosh (a+b x)}{b}-\frac{a^3 \cosh (a+b x)}{b^3 (a+b x)}\right ) \, dx\\ &=\frac{1}{3} x^3 \text{Chi}(a+b x)-\frac{1}{3} \int x^2 \cosh (a+b x) \, dx-\frac{a^2 \int \cosh (a+b x) \, dx}{3 b^2}+\frac{a^3 \int \frac{\cosh (a+b x)}{a+b x} \, dx}{3 b^2}+\frac{a \int x \cosh (a+b x) \, dx}{3 b}\\ &=\frac{a^3 \text{Chi}(a+b x)}{3 b^3}+\frac{1}{3} x^3 \text{Chi}(a+b x)-\frac{a^2 \sinh (a+b x)}{3 b^3}+\frac{a x \sinh (a+b x)}{3 b^2}-\frac{x^2 \sinh (a+b x)}{3 b}-\frac{a \int \sinh (a+b x) \, dx}{3 b^2}+\frac{2 \int x \sinh (a+b x) \, dx}{3 b}\\ &=-\frac{a \cosh (a+b x)}{3 b^3}+\frac{2 x \cosh (a+b x)}{3 b^2}+\frac{a^3 \text{Chi}(a+b x)}{3 b^3}+\frac{1}{3} x^3 \text{Chi}(a+b x)-\frac{a^2 \sinh (a+b x)}{3 b^3}+\frac{a x \sinh (a+b x)}{3 b^2}-\frac{x^2 \sinh (a+b x)}{3 b}-\frac{2 \int \cosh (a+b x) \, dx}{3 b^2}\\ &=-\frac{a \cosh (a+b x)}{3 b^3}+\frac{2 x \cosh (a+b x)}{3 b^2}+\frac{a^3 \text{Chi}(a+b x)}{3 b^3}+\frac{1}{3} x^3 \text{Chi}(a+b x)-\frac{2 \sinh (a+b x)}{3 b^3}-\frac{a^2 \sinh (a+b x)}{3 b^3}+\frac{a x \sinh (a+b x)}{3 b^2}-\frac{x^2 \sinh (a+b x)}{3 b}\\ \end{align*}
Mathematica [A] time = 0.177024, size = 64, normalized size = 0.54 \[ -\frac{-\left (a^3+b^3 x^3\right ) \text{Chi}(a+b x)+\left (a^2-a b x+b^2 x^2+2\right ) \sinh (a+b x)+(a-2 b x) \cosh (a+b x)}{3 b^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.048, size = 101, normalized size = 0.9 \begin{align*}{\frac{1}{{b}^{3}} \left ({\frac{{b}^{3}{x}^{3}{\it Chi} \left ( bx+a \right ) }{3}}-{\frac{ \left ( bx+a \right ) ^{2}\sinh \left ( bx+a \right ) }{3}}+{\frac{ \left ( 2\,bx+2\,a \right ) \cosh \left ( bx+a \right ) }{3}}-{\frac{2\,\sinh \left ( bx+a \right ) }{3}}+a \left ( \left ( bx+a \right ) \sinh \left ( bx+a \right ) -\cosh \left ( bx+a \right ) \right ) -{a}^{2}\sinh \left ( bx+a \right ) +{\frac{{a}^{3}{\it Chi} \left ( bx+a \right ) }{3}} \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^{2}{\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^{2} \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^{2} \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^{2}{\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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