Optimal. Leaf size=96 \[ b^2 \text{CosIntegral}(2 b x)-\frac{1}{4} b^2 \text{Si}(b x)^2-\frac{\text{Si}(b x) \sin (b x)}{2 x^2}-\frac{b \text{Si}(b x) \cos (b x)}{2 x}-\frac{\sin ^2(b x)}{4 x^2}-\frac{b \sin (2 b x)}{4 x}-\frac{b \sin (b x) \cos (b x)}{2 x} \]
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Rubi [A] time = 0.203468, antiderivative size = 96, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 10, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.833, Rules used = {6515, 6521, 6686, 12, 4406, 3297, 3302, 3314, 29, 3312} \[ b^2 \text{CosIntegral}(2 b x)-\frac{1}{4} b^2 \text{Si}(b x)^2-\frac{\text{Si}(b x) \sin (b x)}{2 x^2}-\frac{b \text{Si}(b x) \cos (b x)}{2 x}-\frac{\sin ^2(b x)}{4 x^2}-\frac{b \sin (2 b x)}{4 x}-\frac{b \sin (b x) \cos (b x)}{2 x} \]
Antiderivative was successfully verified.
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Rule 6515
Rule 6521
Rule 6686
Rule 12
Rule 4406
Rule 3297
Rule 3302
Rule 3314
Rule 29
Rule 3312
Rubi steps
\begin{align*} \int \frac{\sin (b x) \text{Si}(b x)}{x^3} \, dx &=-\frac{\sin (b x) \text{Si}(b x)}{2 x^2}+\frac{1}{2} b \int \frac{\sin ^2(b x)}{b x^3} \, dx+\frac{1}{2} b \int \frac{\cos (b x) \text{Si}(b x)}{x^2} \, dx\\ &=-\frac{b \cos (b x) \text{Si}(b x)}{2 x}-\frac{\sin (b x) \text{Si}(b x)}{2 x^2}+\frac{1}{2} \int \frac{\sin ^2(b x)}{x^3} \, dx+\frac{1}{2} b^2 \int \frac{\cos (b x) \sin (b x)}{b x^2} \, dx-\frac{1}{2} b^2 \int \frac{\sin (b x) \text{Si}(b x)}{x} \, dx\\ &=-\frac{b \cos (b x) \sin (b x)}{2 x}-\frac{\sin ^2(b x)}{4 x^2}-\frac{b \cos (b x) \text{Si}(b x)}{2 x}-\frac{\sin (b x) \text{Si}(b x)}{2 x^2}-\frac{1}{4} b^2 \text{Si}(b x)^2+\frac{1}{2} b \int \frac{\cos (b x) \sin (b x)}{x^2} \, dx+\frac{1}{2} b^2 \int \frac{1}{x} \, dx-b^2 \int \frac{\sin ^2(b x)}{x} \, dx\\ &=\frac{1}{2} b^2 \log (x)-\frac{b \cos (b x) \sin (b x)}{2 x}-\frac{\sin ^2(b x)}{4 x^2}-\frac{b \cos (b x) \text{Si}(b x)}{2 x}-\frac{\sin (b x) \text{Si}(b x)}{2 x^2}-\frac{1}{4} b^2 \text{Si}(b x)^2+\frac{1}{2} b \int \frac{\sin (2 b x)}{2 x^2} \, dx-b^2 \int \left (\frac{1}{2 x}-\frac{\cos (2 b x)}{2 x}\right ) \, dx\\ &=-\frac{b \cos (b x) \sin (b x)}{2 x}-\frac{\sin ^2(b x)}{4 x^2}-\frac{b \cos (b x) \text{Si}(b x)}{2 x}-\frac{\sin (b x) \text{Si}(b x)}{2 x^2}-\frac{1}{4} b^2 \text{Si}(b x)^2+\frac{1}{4} b \int \frac{\sin (2 b x)}{x^2} \, dx+\frac{1}{2} b^2 \int \frac{\cos (2 b x)}{x} \, dx\\ &=\frac{1}{2} b^2 \text{Ci}(2 b x)-\frac{b \cos (b x) \sin (b x)}{2 x}-\frac{\sin ^2(b x)}{4 x^2}-\frac{b \sin (2 b x)}{4 x}-\frac{b \cos (b x) \text{Si}(b x)}{2 x}-\frac{\sin (b x) \text{Si}(b x)}{2 x^2}-\frac{1}{4} b^2 \text{Si}(b x)^2+\frac{1}{2} b^2 \int \frac{\cos (2 b x)}{x} \, dx\\ &=b^2 \text{Ci}(2 b x)-\frac{b \cos (b x) \sin (b x)}{2 x}-\frac{\sin ^2(b x)}{4 x^2}-\frac{b \sin (2 b x)}{4 x}-\frac{b \cos (b x) \text{Si}(b x)}{2 x}-\frac{\sin (b x) \text{Si}(b x)}{2 x^2}-\frac{1}{4} b^2 \text{Si}(b x)^2\\ \end{align*}
Mathematica [F] time = 1.00412, size = 0, normalized size = 0. \[ \int \frac{\sin (b x) \text{Si}(b x)}{x^3} \, dx \]
Verification is Not applicable to the result.
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Maple [F] time = 0.056, size = 0, normalized size = 0. \begin{align*} \int{\frac{{\it Si} \left ( bx \right ) \sin \left ( bx \right ) }{{x}^{3}}}\, 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 \frac{{\rm Si}\left (b x\right ) \sin \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{\sin \left (b x\right ) \operatorname{Si}\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 [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sin{\left (b x \right )} \operatorname{Si}{\left (b x \right )}}{x^{3}}\, 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 \frac{{\rm Si}\left (b x\right ) \sin \left (b x\right )}{x^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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