Optimal. Leaf size=97 \[ -\frac {\cos ^2(b x)}{4 x^2}-\frac {\cos (b x) \text {CosIntegral}(b x)}{2 x^2}-\frac {1}{4} b^2 \text {CosIntegral}(b x)^2-b^2 \text {CosIntegral}(2 b x)+\frac {b \cos (b x) \sin (b x)}{2 x}+\frac {b \text {CosIntegral}(b x) \sin (b x)}{2 x}+\frac {b \sin (2 b x)}{4 x} \]
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Rubi [A]
time = 0.14, antiderivative size = 97, normalized size of antiderivative = 1.00, 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 = {6651, 6657,
6818, 12, 4491, 3378, 3383, 3395, 29, 3393} \begin {gather*} -\frac {1}{4} b^2 \text {CosIntegral}(b x)^2-b^2 \text {CosIntegral}(2 b x)-\frac {\text {CosIntegral}(b x) \cos (b x)}{2 x^2}+\frac {b \text {CosIntegral}(b x) \sin (b x)}{2 x}-\frac {\cos ^2(b x)}{4 x^2}+\frac {b \sin (2 b x)}{4 x}+\frac {b \sin (b x) \cos (b x)}{2 x} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 29
Rule 3378
Rule 3383
Rule 3393
Rule 3395
Rule 4491
Rule 6651
Rule 6657
Rule 6818
Rubi steps
\begin {align*} \int \frac {\cos (b x) \text {Ci}(b x)}{x^3} \, dx &=-\frac {\cos (b x) \text {Ci}(b x)}{2 x^2}+\frac {1}{2} b \int \frac {\cos ^2(b x)}{b x^3} \, dx-\frac {1}{2} b \int \frac {\text {Ci}(b x) \sin (b x)}{x^2} \, dx\\ &=-\frac {\cos (b x) \text {Ci}(b x)}{2 x^2}+\frac {b \text {Ci}(b x) \sin (b x)}{2 x}+\frac {1}{2} \int \frac {\cos ^2(b x)}{x^3} \, dx-\frac {1}{2} b^2 \int \frac {\cos (b x) \text {Ci}(b x)}{x} \, dx-\frac {1}{2} b^2 \int \frac {\cos (b x) \sin (b x)}{b x^2} \, dx\\ &=-\frac {\cos ^2(b x)}{4 x^2}-\frac {\cos (b x) \text {Ci}(b x)}{2 x^2}-\frac {1}{4} b^2 \text {Ci}(b x)^2+\frac {b \cos (b x) \sin (b x)}{2 x}+\frac {b \text {Ci}(b x) \sin (b x)}{2 x}-\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 {\cos ^2(b x)}{x} \, dx\\ &=-\frac {\cos ^2(b x)}{4 x^2}-\frac {\cos (b x) \text {Ci}(b x)}{2 x^2}-\frac {1}{4} b^2 \text {Ci}(b x)^2+\frac {1}{2} b^2 \log (x)+\frac {b \cos (b x) \sin (b x)}{2 x}+\frac {b \text {Ci}(b x) \sin (b x)}{2 x}-\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 {\cos ^2(b x)}{4 x^2}-\frac {\cos (b x) \text {Ci}(b x)}{2 x^2}-\frac {1}{4} b^2 \text {Ci}(b x)^2+\frac {b \cos (b x) \sin (b x)}{2 x}+\frac {b \text {Ci}(b x) \sin (b x)}{2 x}-\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 {\cos ^2(b x)}{4 x^2}-\frac {\cos (b x) \text {Ci}(b x)}{2 x^2}-\frac {1}{4} b^2 \text {Ci}(b x)^2-\frac {1}{2} b^2 \text {Ci}(2 b x)+\frac {b \cos (b x) \sin (b x)}{2 x}+\frac {b \text {Ci}(b x) \sin (b x)}{2 x}+\frac {b \sin (2 b x)}{4 x}-\frac {1}{2} b^2 \int \frac {\cos (2 b x)}{x} \, dx\\ &=-\frac {\cos ^2(b x)}{4 x^2}-\frac {\cos (b x) \text {Ci}(b x)}{2 x^2}-\frac {1}{4} b^2 \text {Ci}(b x)^2-b^2 \text {Ci}(2 b x)+\frac {b \cos (b x) \sin (b x)}{2 x}+\frac {b \text {Ci}(b x) \sin (b x)}{2 x}+\frac {b \sin (2 b x)}{4 x}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 97, normalized size = 1.00 \begin {gather*} -\frac {\cos ^2(b x)}{4 x^2}-\frac {\cos (b x) \text {CosIntegral}(b x)}{2 x^2}-\frac {1}{4} b^2 \text {CosIntegral}(b x)^2-b^2 \text {CosIntegral}(2 b x)+\frac {b \cos (b x) \sin (b x)}{2 x}+\frac {b \text {CosIntegral}(b x) \sin (b x)}{2 x}+\frac {b \sin (2 b x)}{4 x} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.19, size = 0, normalized size = 0.00 \[\int \frac {\cosineIntegral \left (b x \right ) \cos \left (b x \right )}{x^{3}}\, dx\]
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\cos {\left (b x \right )} \operatorname {Ci}{\left (b x \right )}}{x^{3}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\mathrm {cosint}\left (b\,x\right )\,\cos \left (b\,x\right )}{x^3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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