Optimal. Leaf size=63 \[ \frac {3 \cos (b x)}{2 b^4}+\frac {3 x \sin (b x)}{2 b^3}-\frac {3 x^2 \cos (b x)}{4 b^2}+\frac {1}{4} x^4 \text {Ci}(b x)-\frac {x^3 \sin (b x)}{4 b} \]
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Rubi [A] time = 0.07, antiderivative size = 63, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6504, 12, 3296, 2638} \[ -\frac {3 x^2 \cos (b x)}{4 b^2}+\frac {3 x \sin (b x)}{2 b^3}+\frac {3 \cos (b x)}{2 b^4}+\frac {1}{4} x^4 \text {CosIntegral}(b x)-\frac {x^3 \sin (b x)}{4 b} \]
Antiderivative was successfully verified.
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Rule 12
Rule 2638
Rule 3296
Rule 6504
Rubi steps
\begin {align*} \int x^3 \text {Ci}(b x) \, dx &=\frac {1}{4} x^4 \text {Ci}(b x)-\frac {1}{4} b \int \frac {x^3 \cos (b x)}{b} \, dx\\ &=\frac {1}{4} x^4 \text {Ci}(b x)-\frac {1}{4} \int x^3 \cos (b x) \, dx\\ &=\frac {1}{4} x^4 \text {Ci}(b x)-\frac {x^3 \sin (b x)}{4 b}+\frac {3 \int x^2 \sin (b x) \, dx}{4 b}\\ &=-\frac {3 x^2 \cos (b x)}{4 b^2}+\frac {1}{4} x^4 \text {Ci}(b x)-\frac {x^3 \sin (b x)}{4 b}+\frac {3 \int x \cos (b x) \, dx}{2 b^2}\\ &=-\frac {3 x^2 \cos (b x)}{4 b^2}+\frac {1}{4} x^4 \text {Ci}(b x)+\frac {3 x \sin (b x)}{2 b^3}-\frac {x^3 \sin (b x)}{4 b}-\frac {3 \int \sin (b x) \, dx}{2 b^3}\\ &=\frac {3 \cos (b x)}{2 b^4}-\frac {3 x^2 \cos (b x)}{4 b^2}+\frac {1}{4} x^4 \text {Ci}(b x)+\frac {3 x \sin (b x)}{2 b^3}-\frac {x^3 \sin (b x)}{4 b}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 53, normalized size = 0.84 \[ -\frac {3 \left (b^2 x^2-2\right ) \cos (b x)}{4 b^4}-\frac {x \left (b^2 x^2-6\right ) \sin (b x)}{4 b^3}+\frac {1}{4} x^4 \text {Ci}(b x) \]
Antiderivative was successfully verified.
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fricas [F] time = 1.02, size = 0, normalized size = 0.00 \[ {\rm integral}\left (x^{3} \operatorname {Ci}\left (b x\right ), x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.52, size = 49, normalized size = 0.78 \[ \frac {1}{4} \, x^{4} \operatorname {Ci}\left (b x\right ) - \frac {3 \, {\left (b^{2} x^{2} - 2\right )} \cos \left (b x\right )}{4 \, b^{4}} - \frac {{\left (b^{3} x^{3} - 6 \, b x\right )} \sin \left (b x\right )}{4 \, b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 56, normalized size = 0.89 \[ \frac {\frac {b^{4} x^{4} \Ci \left (b x \right )}{4}-\frac {\sin \left (b x \right ) b^{3} x^{3}}{4}-\frac {3 b^{2} x^{2} \cos \left (b x \right )}{4}+\frac {3 \cos \left (b x \right )}{2}+\frac {3 b x \sin \left (b x \right )}{2}}{b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{3} {\rm Ci}\left (b x\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \frac {6\,\cos \left (b\,x\right )-3\,b^2\,x^2\,\cos \left (b\,x\right )-b^3\,x^3\,\sin \left (b\,x\right )+6\,b\,x\,\sin \left (b\,x\right )}{4\,b^4}+\frac {x^4\,\mathrm {cosint}\left (b\,x\right )}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 2.13, size = 85, normalized size = 1.35 \[ - \frac {x^{4} \log {\left (b x \right )}}{4} + \frac {x^{4} \log {\left (b^{2} x^{2} \right )}}{8} + \frac {x^{4} \operatorname {Ci}{\left (b x \right )}}{4} - \frac {x^{3} \sin {\left (b x \right )}}{4 b} - \frac {3 x^{2} \cos {\left (b x \right )}}{4 b^{2}} + \frac {3 x \sin {\left (b x \right )}}{2 b^{3}} + \frac {3 \cos {\left (b x \right )}}{2 b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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