Optimal. Leaf size=327 \[ \frac{a^2 (a+b x) \text{Chi}(a+b x)^2}{3 b^3}-\frac{2 a^2 \text{Chi}(a+b x) \sinh (a+b x)}{3 b^3}+\frac{a^2 \text{Shi}(2 a+2 b x)}{b^3}-\frac{a x (a+b x) \text{Chi}(a+b x)^2}{3 b^2}+\frac{a \text{Chi}(2 a+2 b x)}{b^3}+\frac{2 a x \text{Chi}(a+b x) \sinh (a+b x)}{3 b^2}-\frac{4 \text{Chi}(a+b x) \sinh (a+b x)}{3 b^3}-\frac{2 a \text{Chi}(a+b x) \cosh (a+b x)}{3 b^3}+\frac{4 x \text{Chi}(a+b x) \cosh (a+b x)}{3 b^2}+\frac{2 \text{Shi}(2 a+2 b x)}{3 b^3}+\frac{a \log (a+b x)}{b^3}-\frac{\sinh (2 a+2 b x)}{12 b^3}-\frac{a \cosh (2 a+2 b x)}{3 b^3}+\frac{x \cosh (2 a+2 b x)}{6 b^2}-\frac{2 \sinh (a+b x) \cosh (a+b x)}{3 b^3}+\frac{x^2 (a+b x) \text{Chi}(a+b x)^2}{3 b}-\frac{2 x^2 \text{Chi}(a+b x) \sinh (a+b x)}{3 b}-\frac{2 x}{3 b^2} \]
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Rubi [A] time = 1.30589, antiderivative size = 327, normalized size of antiderivative = 1., number of steps used = 39, number of rules used = 19, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 1.583, Rules used = {6539, 6543, 5617, 6741, 6742, 2638, 3296, 2637, 3298, 6549, 2635, 8, 3312, 3301, 6541, 5448, 12, 6547, 6535} \[ \frac{a^2 (a+b x) \text{Chi}(a+b x)^2}{3 b^3}-\frac{2 a^2 \text{Chi}(a+b x) \sinh (a+b x)}{3 b^3}+\frac{a^2 \text{Shi}(2 a+2 b x)}{b^3}-\frac{a x (a+b x) \text{Chi}(a+b x)^2}{3 b^2}+\frac{a \text{Chi}(2 a+2 b x)}{b^3}+\frac{2 a x \text{Chi}(a+b x) \sinh (a+b x)}{3 b^2}-\frac{4 \text{Chi}(a+b x) \sinh (a+b x)}{3 b^3}-\frac{2 a \text{Chi}(a+b x) \cosh (a+b x)}{3 b^3}+\frac{4 x \text{Chi}(a+b x) \cosh (a+b x)}{3 b^2}+\frac{2 \text{Shi}(2 a+2 b x)}{3 b^3}+\frac{a \log (a+b x)}{b^3}-\frac{\sinh (2 a+2 b x)}{12 b^3}-\frac{a \cosh (2 a+2 b x)}{3 b^3}+\frac{x \cosh (2 a+2 b x)}{6 b^2}-\frac{2 \sinh (a+b x) \cosh (a+b x)}{3 b^3}+\frac{x^2 (a+b x) \text{Chi}(a+b x)^2}{3 b}-\frac{2 x^2 \text{Chi}(a+b x) \sinh (a+b x)}{3 b}-\frac{2 x}{3 b^2} \]
Antiderivative was successfully verified.
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Rule 6539
Rule 6543
Rule 5617
Rule 6741
Rule 6742
Rule 2638
Rule 3296
Rule 2637
Rule 3298
Rule 6549
Rule 2635
Rule 8
Rule 3312
Rule 3301
Rule 6541
Rule 5448
Rule 12
Rule 6547
Rule 6535
Rubi steps
\begin{align*} \int x^2 \text{Chi}(a+b x)^2 \, dx &=\frac{x^2 (a+b x) \text{Chi}(a+b x)^2}{3 b}-\frac{2}{3} \int x^2 \cosh (a+b x) \text{Chi}(a+b x) \, dx-\frac{(2 a) \int x \text{Chi}(a+b x)^2 \, dx}{3 b}\\ &=-\frac{a x (a+b x) \text{Chi}(a+b x)^2}{3 b^2}+\frac{x^2 (a+b x) \text{Chi}(a+b x)^2}{3 b}-\frac{2 x^2 \text{Chi}(a+b x) \sinh (a+b x)}{3 b}+\frac{2}{3} \int \frac{x^2 \cosh (a+b x) \sinh (a+b x)}{a+b x} \, dx+\frac{a^2 \int \text{Chi}(a+b x)^2 \, dx}{3 b^2}+\frac{4 \int x \text{Chi}(a+b x) \sinh (a+b x) \, dx}{3 b}+\frac{(2 a) \int x \cosh (a+b x) \text{Chi}(a+b x) \, dx}{3 b}\\ &=\frac{4 x \cosh (a+b x) \text{Chi}(a+b x)}{3 b^2}+\frac{a^2 (a+b x) \text{Chi}(a+b x)^2}{3 b^3}-\frac{a x (a+b x) \text{Chi}(a+b x)^2}{3 b^2}+\frac{x^2 (a+b x) \text{Chi}(a+b x)^2}{3 b}+\frac{2 a x \text{Chi}(a+b x) \sinh (a+b x)}{3 b^2}-\frac{2 x^2 \text{Chi}(a+b x) \sinh (a+b x)}{3 b}+\frac{1}{3} \int \frac{x^2 \sinh (2 (a+b x))}{a+b x} \, dx-\frac{4 \int \cosh (a+b x) \text{Chi}(a+b x) \, dx}{3 b^2}-\frac{(2 a) \int \text{Chi}(a+b x) \sinh (a+b x) \, dx}{3 b^2}-\frac{\left (2 a^2\right ) \int \cosh (a+b x) \text{Chi}(a+b x) \, dx}{3 b^2}-\frac{4 \int \frac{x \cosh ^2(a+b x)}{a+b x} \, dx}{3 b}-\frac{(2 a) \int \frac{x \cosh (a+b x) \sinh (a+b x)}{a+b x} \, dx}{3 b}\\ &=-\frac{2 a \cosh (a+b x) \text{Chi}(a+b x)}{3 b^3}+\frac{4 x \cosh (a+b x) \text{Chi}(a+b x)}{3 b^2}+\frac{a^2 (a+b x) \text{Chi}(a+b x)^2}{3 b^3}-\frac{a x (a+b x) \text{Chi}(a+b x)^2}{3 b^2}+\frac{x^2 (a+b x) \text{Chi}(a+b x)^2}{3 b}-\frac{4 \text{Chi}(a+b x) \sinh (a+b x)}{3 b^3}-\frac{2 a^2 \text{Chi}(a+b x) \sinh (a+b x)}{3 b^3}+\frac{2 a x \text{Chi}(a+b x) \sinh (a+b x)}{3 b^2}-\frac{2 x^2 \text{Chi}(a+b x) \sinh (a+b x)}{3 b}+\frac{1}{3} \int \frac{x^2 \sinh (2 a+2 b x)}{a+b x} \, dx+\frac{4 \int \frac{\cosh (a+b x) \sinh (a+b x)}{a+b x} \, dx}{3 b^2}+\frac{(2 a) \int \frac{\cosh ^2(a+b x)}{a+b x} \, dx}{3 b^2}+\frac{\left (2 a^2\right ) \int \frac{\cosh (a+b x) \sinh (a+b x)}{a+b x} \, dx}{3 b^2}-\frac{4 \int \left (\frac{\cosh ^2(a+b x)}{b}-\frac{a \cosh ^2(a+b x)}{b (a+b x)}\right ) \, dx}{3 b}-\frac{a \int \frac{x \sinh (2 (a+b x))}{a+b x} \, dx}{3 b}\\ &=-\frac{2 a \cosh (a+b x) \text{Chi}(a+b x)}{3 b^3}+\frac{4 x \cosh (a+b x) \text{Chi}(a+b x)}{3 b^2}+\frac{a^2 (a+b x) \text{Chi}(a+b x)^2}{3 b^3}-\frac{a x (a+b x) \text{Chi}(a+b x)^2}{3 b^2}+\frac{x^2 (a+b x) \text{Chi}(a+b x)^2}{3 b}-\frac{4 \text{Chi}(a+b x) \sinh (a+b x)}{3 b^3}-\frac{2 a^2 \text{Chi}(a+b x) \sinh (a+b x)}{3 b^3}+\frac{2 a x \text{Chi}(a+b x) \sinh (a+b x)}{3 b^2}-\frac{2 x^2 \text{Chi}(a+b x) \sinh (a+b x)}{3 b}+\frac{1}{3} \int \left (-\frac{a \sinh (2 a+2 b x)}{b^2}+\frac{x \sinh (2 a+2 b x)}{b}+\frac{a^2 \sinh (2 a+2 b x)}{b^2 (a+b x)}\right ) \, dx-\frac{4 \int \cosh ^2(a+b x) \, dx}{3 b^2}+\frac{4 \int \frac{\sinh (2 a+2 b x)}{2 (a+b x)} \, dx}{3 b^2}+\frac{(2 a) \int \left (\frac{1}{2 (a+b x)}+\frac{\cosh (2 a+2 b x)}{2 (a+b x)}\right ) \, dx}{3 b^2}+\frac{(4 a) \int \frac{\cosh ^2(a+b x)}{a+b x} \, dx}{3 b^2}+\frac{\left (2 a^2\right ) \int \frac{\sinh (2 a+2 b x)}{2 (a+b x)} \, dx}{3 b^2}-\frac{a \int \frac{x \sinh (2 a+2 b x)}{a+b x} \, dx}{3 b}\\ &=-\frac{2 a \cosh (a+b x) \text{Chi}(a+b x)}{3 b^3}+\frac{4 x \cosh (a+b x) \text{Chi}(a+b x)}{3 b^2}+\frac{a^2 (a+b x) \text{Chi}(a+b x)^2}{3 b^3}-\frac{a x (a+b x) \text{Chi}(a+b x)^2}{3 b^2}+\frac{x^2 (a+b x) \text{Chi}(a+b x)^2}{3 b}+\frac{a \log (a+b x)}{3 b^3}-\frac{2 \cosh (a+b x) \sinh (a+b x)}{3 b^3}-\frac{4 \text{Chi}(a+b x) \sinh (a+b x)}{3 b^3}-\frac{2 a^2 \text{Chi}(a+b x) \sinh (a+b x)}{3 b^3}+\frac{2 a x \text{Chi}(a+b x) \sinh (a+b x)}{3 b^2}-\frac{2 x^2 \text{Chi}(a+b x) \sinh (a+b x)}{3 b}-\frac{2 \int 1 \, dx}{3 b^2}+\frac{2 \int \frac{\sinh (2 a+2 b x)}{a+b x} \, dx}{3 b^2}+\frac{a \int \frac{\cosh (2 a+2 b x)}{a+b x} \, dx}{3 b^2}-\frac{a \int \sinh (2 a+2 b x) \, dx}{3 b^2}+\frac{(4 a) \int \left (\frac{1}{2 (a+b x)}+\frac{\cosh (2 a+2 b x)}{2 (a+b x)}\right ) \, dx}{3 b^2}+2 \frac{a^2 \int \frac{\sinh (2 a+2 b x)}{a+b x} \, dx}{3 b^2}+\frac{\int x \sinh (2 a+2 b x) \, dx}{3 b}-\frac{a \int \left (\frac{\sinh (2 a+2 b x)}{b}+\frac{a \sinh (2 a+2 b x)}{b (-a-b x)}\right ) \, dx}{3 b}\\ &=-\frac{2 x}{3 b^2}-\frac{a \cosh (2 a+2 b x)}{6 b^3}+\frac{x \cosh (2 a+2 b x)}{6 b^2}-\frac{2 a \cosh (a+b x) \text{Chi}(a+b x)}{3 b^3}+\frac{4 x \cosh (a+b x) \text{Chi}(a+b x)}{3 b^2}+\frac{a^2 (a+b x) \text{Chi}(a+b x)^2}{3 b^3}-\frac{a x (a+b x) \text{Chi}(a+b x)^2}{3 b^2}+\frac{x^2 (a+b x) \text{Chi}(a+b x)^2}{3 b}+\frac{a \text{Chi}(2 a+2 b x)}{3 b^3}+\frac{a \log (a+b x)}{b^3}-\frac{2 \cosh (a+b x) \sinh (a+b x)}{3 b^3}-\frac{4 \text{Chi}(a+b x) \sinh (a+b x)}{3 b^3}-\frac{2 a^2 \text{Chi}(a+b x) \sinh (a+b x)}{3 b^3}+\frac{2 a x \text{Chi}(a+b x) \sinh (a+b x)}{3 b^2}-\frac{2 x^2 \text{Chi}(a+b x) \sinh (a+b x)}{3 b}+\frac{2 \text{Shi}(2 a+2 b x)}{3 b^3}+\frac{2 a^2 \text{Shi}(2 a+2 b x)}{3 b^3}-\frac{\int \cosh (2 a+2 b x) \, dx}{6 b^2}-\frac{a \int \sinh (2 a+2 b x) \, dx}{3 b^2}+\frac{(2 a) \int \frac{\cosh (2 a+2 b x)}{a+b x} \, dx}{3 b^2}-\frac{a^2 \int \frac{\sinh (2 a+2 b x)}{-a-b x} \, dx}{3 b^2}\\ &=-\frac{2 x}{3 b^2}-\frac{a \cosh (2 a+2 b x)}{3 b^3}+\frac{x \cosh (2 a+2 b x)}{6 b^2}-\frac{2 a \cosh (a+b x) \text{Chi}(a+b x)}{3 b^3}+\frac{4 x \cosh (a+b x) \text{Chi}(a+b x)}{3 b^2}+\frac{a^2 (a+b x) \text{Chi}(a+b x)^2}{3 b^3}-\frac{a x (a+b x) \text{Chi}(a+b x)^2}{3 b^2}+\frac{x^2 (a+b x) \text{Chi}(a+b x)^2}{3 b}+\frac{a \text{Chi}(2 a+2 b x)}{b^3}+\frac{a \log (a+b x)}{b^3}-\frac{2 \cosh (a+b x) \sinh (a+b x)}{3 b^3}-\frac{4 \text{Chi}(a+b x) \sinh (a+b x)}{3 b^3}-\frac{2 a^2 \text{Chi}(a+b x) \sinh (a+b x)}{3 b^3}+\frac{2 a x \text{Chi}(a+b x) \sinh (a+b x)}{3 b^2}-\frac{2 x^2 \text{Chi}(a+b x) \sinh (a+b x)}{3 b}-\frac{\sinh (2 a+2 b x)}{12 b^3}+\frac{2 \text{Shi}(2 a+2 b x)}{3 b^3}+\frac{a^2 \text{Shi}(2 a+2 b x)}{b^3}\\ \end{align*}
Mathematica [A] time = 1.11654, size = 158, normalized size = 0.48 \[ \frac{4 \left (a^3+b^3 x^3\right ) \text{Chi}(a+b x)^2-8 \text{Chi}(a+b x) \left (\left (a^2-a b x+b^2 x^2+2\right ) \sinh (a+b x)+(a-2 b x) \cosh (a+b x)\right )+12 a^2 \text{Shi}(2 (a+b x))+12 a \text{Chi}(2 (a+b x))+8 \text{Shi}(2 (a+b x))+12 a \log (a+b x)-5 \sinh (2 (a+b x))-4 a \cosh (2 (a+b x))+2 b x \cosh (2 (a+b x))-8 a-8 b x}{12 b^3} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.051, size = 0, normalized size = 0. \begin{align*} \int{x}^{2} \left ({\it Chi} \left ( bx+a \right ) \right ) ^{2}\, dx \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 )^{2}\,{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 )^{2}, 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}^{2}\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 )^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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