Optimal. Leaf size=46 \[ -\frac {1}{4} b^2 \text {Ci}(b x)-\frac {\text {Ci}(b x)}{2 x^2}-\frac {\cos (b x)}{4 x^2}+\frac {b \sin (b x)}{4 x} \]
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Rubi [A] time = 0.07, antiderivative size = 46, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6504, 12, 3297, 3302} \[ -\frac {1}{4} b^2 \text {CosIntegral}(b x)-\frac {\text {CosIntegral}(b x)}{2 x^2}-\frac {\cos (b x)}{4 x^2}+\frac {b \sin (b x)}{4 x} \]
Antiderivative was successfully verified.
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Rule 12
Rule 3297
Rule 3302
Rule 6504
Rubi steps
\begin {align*} \int \frac {\text {Ci}(b x)}{x^3} \, dx &=-\frac {\text {Ci}(b x)}{2 x^2}+\frac {1}{2} b \int \frac {\cos (b x)}{b x^3} \, dx\\ &=-\frac {\text {Ci}(b x)}{2 x^2}+\frac {1}{2} \int \frac {\cos (b x)}{x^3} \, dx\\ &=-\frac {\cos (b x)}{4 x^2}-\frac {\text {Ci}(b x)}{2 x^2}-\frac {1}{4} b \int \frac {\sin (b x)}{x^2} \, dx\\ &=-\frac {\cos (b x)}{4 x^2}-\frac {\text {Ci}(b x)}{2 x^2}+\frac {b \sin (b x)}{4 x}-\frac {1}{4} b^2 \int \frac {\cos (b x)}{x} \, dx\\ &=-\frac {\cos (b x)}{4 x^2}-\frac {1}{4} b^2 \text {Ci}(b x)-\frac {\text {Ci}(b x)}{2 x^2}+\frac {b \sin (b x)}{4 x}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 46, normalized size = 1.00 \[ -\frac {1}{4} b^2 \text {Ci}(b x)-\frac {\text {Ci}(b x)}{2 x^2}-\frac {\cos (b x)}{4 x^2}+\frac {b \sin (b x)}{4 x} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.90, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\operatorname {Ci}\left (b x\right )}{x^{3}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.37, size = 53, normalized size = 1.15 \[ -\frac {b^{2} x^{2} \operatorname {Ci}\left (b x\right ) + b^{2} x^{2} \operatorname {Ci}\left (-b x\right ) - 2 \, b x \sin \left (b x\right ) + 2 \, \cos \left (b x\right )}{8 \, x^{2}} - \frac {\operatorname {Ci}\left (b x\right )}{2 \, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 48, normalized size = 1.04 \[ b^{2} \left (-\frac {\Ci \left (b x \right )}{2 b^{2} x^{2}}-\frac {\cos \left (b x \right )}{4 b^{2} x^{2}}+\frac {\sin \left (b x \right )}{4 b x}-\frac {\Ci \left (b x \right )}{4}\right ) \]
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 \[ \int \frac {{\rm Ci}\left (b x\right )}{x^{3}}\,{d x} \]
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 \[ -\frac {\frac {\cos \left (b\,x\right )}{2}-\frac {b\,x\,\sin \left (b\,x\right )}{2}}{2\,x^2}-\frac {b^2\,\mathrm {cosint}\left (b\,x\right )}{4}-\frac {\mathrm {cosint}\left (b\,x\right )}{2\,x^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 2.24, size = 87, normalized size = 1.89 \[ \frac {b^{2} \log {\left (b x \right )}}{4} - \frac {b^{2} \log {\left (b^{2} x^{2} \right )}}{8} - \frac {b^{2} \operatorname {Ci}{\left (b x \right )}}{4} + \frac {b \sin {\left (b x \right )}}{4 x} + \frac {\log {\left (b x \right )}}{2 x^{2}} - \frac {\log {\left (b^{2} x^{2} \right )}}{4 x^{2}} - \frac {\cos {\left (b x \right )}}{4 x^{2}} - \frac {\operatorname {Ci}{\left (b x \right )}}{2 x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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