Optimal. Leaf size=46 \[ \frac{1}{4} b^2 \text{Chi}(b x)-\frac{\text{Chi}(b x)}{2 x^2}-\frac{\cosh (b x)}{4 x^2}-\frac{b \sinh (b x)}{4 x} \]
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Rubi [A] time = 0.0734483, antiderivative size = 46, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {6533, 12, 3297, 3301} \[ \frac{1}{4} b^2 \text{Chi}(b x)-\frac{\text{Chi}(b x)}{2 x^2}-\frac{\cosh (b x)}{4 x^2}-\frac{b \sinh (b x)}{4 x} \]
Antiderivative was successfully verified.
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Rule 6533
Rule 12
Rule 3297
Rule 3301
Rubi steps
\begin{align*} \int \frac{\text{Chi}(b x)}{x^3} \, dx &=-\frac{\text{Chi}(b x)}{2 x^2}+\frac{1}{2} b \int \frac{\cosh (b x)}{b x^3} \, dx\\ &=-\frac{\text{Chi}(b x)}{2 x^2}+\frac{1}{2} \int \frac{\cosh (b x)}{x^3} \, dx\\ &=-\frac{\cosh (b x)}{4 x^2}-\frac{\text{Chi}(b x)}{2 x^2}+\frac{1}{4} b \int \frac{\sinh (b x)}{x^2} \, dx\\ &=-\frac{\cosh (b x)}{4 x^2}-\frac{\text{Chi}(b x)}{2 x^2}-\frac{b \sinh (b x)}{4 x}+\frac{1}{4} b^2 \int \frac{\cosh (b x)}{x} \, dx\\ &=-\frac{\cosh (b x)}{4 x^2}+\frac{1}{4} b^2 \text{Chi}(b x)-\frac{\text{Chi}(b x)}{2 x^2}-\frac{b \sinh (b x)}{4 x}\\ \end{align*}
Mathematica [A] time = 0.0128408, size = 46, normalized size = 1. \[ \frac{1}{4} b^2 \text{Chi}(b x)-\frac{\text{Chi}(b x)}{2 x^2}-\frac{\cosh (b x)}{4 x^2}-\frac{b \sinh (b x)}{4 x} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.045, size = 48, normalized size = 1. \begin{align*}{b}^{2} \left ( -{\frac{{\it Chi} \left ( bx \right ) }{2\,{b}^{2}{x}^{2}}}-{\frac{\cosh \left ( bx \right ) }{4\,{b}^{2}{x}^{2}}}-{\frac{\sinh \left ( bx \right ) }{4\,bx}}+{\frac{{\it Chi} \left ( bx \right ) }{4}} \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 \frac{{\rm Chi}\left (b x\right )}{x^{3}}\,{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 (\frac{\operatorname{Chi}\left (b x\right )}{x^{3}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 3.10077, size = 87, normalized size = 1.89 \begin{align*} - \frac{b^{2} \log{\left (b x \right )}}{4} + \frac{b^{2} \log{\left (b^{2} x^{2} \right )}}{8} + \frac{b^{2} \operatorname{Chi}\left (b x\right )}{4} - \frac{b \sinh{\left (b x \right )}}{4 x} + \frac{\log{\left (b x \right )}}{2 x^{2}} - \frac{\log{\left (b^{2} x^{2} \right )}}{4 x^{2}} - \frac{\cosh{\left (b x \right )}}{4 x^{2}} - \frac{\operatorname{Chi}\left (b x\right )}{2 x^{2}} \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 \frac{{\rm Chi}\left (b x\right )}{x^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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