Optimal. Leaf size=163 \[ -\frac {3 \text {Ci}(2 b x)}{2 b^4}+\frac {3 \text {Ci}(b x) \cos (b x)}{b^4}-\frac {3 \log (x)}{2 b^4}-\frac {13 \sin ^2(b x)}{8 b^4}+\frac {3 \cos ^2(b x)}{8 b^4}+\frac {3 x \text {Ci}(b x) \sin (b x)}{b^3}+\frac {x \sin (b x) \cos (b x)}{b^3}-\frac {3 x^2 \text {Ci}(b x) \cos (b x)}{2 b^2}+\frac {x^2}{4 b^2}+\frac {x^2 \sin ^2(b x)}{4 b^2}+\frac {1}{4} x^4 \text {Ci}(b x)^2-\frac {x^3 \text {Ci}(b x) \sin (b x)}{2 b} \]
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Rubi [A] time = 0.23, antiderivative size = 163, normalized size of antiderivative = 1.00, number of steps used = 19, number of rules used = 11, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.100, Rules used = {6508, 6514, 12, 3443, 3310, 30, 6520, 2564, 6518, 3312, 3302} \[ -\frac {3 x^2 \text {CosIntegral}(b x) \cos (b x)}{2 b^2}-\frac {3 \text {CosIntegral}(2 b x)}{2 b^4}+\frac {3 x \text {CosIntegral}(b x) \sin (b x)}{b^3}+\frac {3 \text {CosIntegral}(b x) \cos (b x)}{b^4}+\frac {x^2}{4 b^2}+\frac {x^2 \sin ^2(b x)}{4 b^2}-\frac {3 \log (x)}{2 b^4}-\frac {13 \sin ^2(b x)}{8 b^4}+\frac {3 \cos ^2(b x)}{8 b^4}+\frac {x \sin (b x) \cos (b x)}{b^3}+\frac {1}{4} x^4 \text {CosIntegral}(b x)^2-\frac {x^3 \text {CosIntegral}(b x) \sin (b x)}{2 b} \]
Antiderivative was successfully verified.
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Rule 12
Rule 30
Rule 2564
Rule 3302
Rule 3310
Rule 3312
Rule 3443
Rule 6508
Rule 6514
Rule 6518
Rule 6520
Rubi steps
\begin {align*} \int x^3 \text {Ci}(b x)^2 \, dx &=\frac {1}{4} x^4 \text {Ci}(b x)^2-\frac {1}{2} \int x^3 \cos (b x) \text {Ci}(b x) \, dx\\ &=\frac {1}{4} x^4 \text {Ci}(b x)^2-\frac {x^3 \text {Ci}(b x) \sin (b x)}{2 b}+\frac {1}{2} \int \frac {x^2 \cos (b x) \sin (b x)}{b} \, dx+\frac {3 \int x^2 \text {Ci}(b x) \sin (b x) \, dx}{2 b}\\ &=-\frac {3 x^2 \cos (b x) \text {Ci}(b x)}{2 b^2}+\frac {1}{4} x^4 \text {Ci}(b x)^2-\frac {x^3 \text {Ci}(b x) \sin (b x)}{2 b}+\frac {3 \int x \cos (b x) \text {Ci}(b x) \, dx}{b^2}+\frac {\int x^2 \cos (b x) \sin (b x) \, dx}{2 b}+\frac {3 \int \frac {x \cos ^2(b x)}{b} \, dx}{2 b}\\ &=-\frac {3 x^2 \cos (b x) \text {Ci}(b x)}{2 b^2}+\frac {1}{4} x^4 \text {Ci}(b x)^2+\frac {3 x \text {Ci}(b x) \sin (b x)}{b^3}-\frac {x^3 \text {Ci}(b x) \sin (b x)}{2 b}+\frac {x^2 \sin ^2(b x)}{4 b^2}-\frac {3 \int \text {Ci}(b x) \sin (b x) \, dx}{b^3}-\frac {\int x \sin ^2(b x) \, dx}{2 b^2}+\frac {3 \int x \cos ^2(b x) \, dx}{2 b^2}-\frac {3 \int \frac {\cos (b x) \sin (b x)}{b} \, dx}{b^2}\\ &=\frac {3 \cos ^2(b x)}{8 b^4}+\frac {3 \cos (b x) \text {Ci}(b x)}{b^4}-\frac {3 x^2 \cos (b x) \text {Ci}(b x)}{2 b^2}+\frac {1}{4} x^4 \text {Ci}(b x)^2+\frac {x \cos (b x) \sin (b x)}{b^3}+\frac {3 x \text {Ci}(b x) \sin (b x)}{b^3}-\frac {x^3 \text {Ci}(b x) \sin (b x)}{2 b}-\frac {\sin ^2(b x)}{8 b^4}+\frac {x^2 \sin ^2(b x)}{4 b^2}-\frac {3 \int \frac {\cos ^2(b x)}{b x} \, dx}{b^3}-\frac {3 \int \cos (b x) \sin (b x) \, dx}{b^3}-\frac {\int x \, dx}{4 b^2}+\frac {3 \int x \, dx}{4 b^2}\\ &=\frac {x^2}{4 b^2}+\frac {3 \cos ^2(b x)}{8 b^4}+\frac {3 \cos (b x) \text {Ci}(b x)}{b^4}-\frac {3 x^2 \cos (b x) \text {Ci}(b x)}{2 b^2}+\frac {1}{4} x^4 \text {Ci}(b x)^2+\frac {x \cos (b x) \sin (b x)}{b^3}+\frac {3 x \text {Ci}(b x) \sin (b x)}{b^3}-\frac {x^3 \text {Ci}(b x) \sin (b x)}{2 b}-\frac {\sin ^2(b x)}{8 b^4}+\frac {x^2 \sin ^2(b x)}{4 b^2}-\frac {3 \int \frac {\cos ^2(b x)}{x} \, dx}{b^4}-\frac {3 \operatorname {Subst}(\int x \, dx,x,\sin (b x))}{b^4}\\ &=\frac {x^2}{4 b^2}+\frac {3 \cos ^2(b x)}{8 b^4}+\frac {3 \cos (b x) \text {Ci}(b x)}{b^4}-\frac {3 x^2 \cos (b x) \text {Ci}(b x)}{2 b^2}+\frac {1}{4} x^4 \text {Ci}(b x)^2+\frac {x \cos (b x) \sin (b x)}{b^3}+\frac {3 x \text {Ci}(b x) \sin (b x)}{b^3}-\frac {x^3 \text {Ci}(b x) \sin (b x)}{2 b}-\frac {13 \sin ^2(b x)}{8 b^4}+\frac {x^2 \sin ^2(b x)}{4 b^2}-\frac {3 \int \left (\frac {1}{2 x}+\frac {\cos (2 b x)}{2 x}\right ) \, dx}{b^4}\\ &=\frac {x^2}{4 b^2}+\frac {3 \cos ^2(b x)}{8 b^4}+\frac {3 \cos (b x) \text {Ci}(b x)}{b^4}-\frac {3 x^2 \cos (b x) \text {Ci}(b x)}{2 b^2}+\frac {1}{4} x^4 \text {Ci}(b x)^2-\frac {3 \log (x)}{2 b^4}+\frac {x \cos (b x) \sin (b x)}{b^3}+\frac {3 x \text {Ci}(b x) \sin (b x)}{b^3}-\frac {x^3 \text {Ci}(b x) \sin (b x)}{2 b}-\frac {13 \sin ^2(b x)}{8 b^4}+\frac {x^2 \sin ^2(b x)}{4 b^2}-\frac {3 \int \frac {\cos (2 b x)}{x} \, dx}{2 b^4}\\ &=\frac {x^2}{4 b^2}+\frac {3 \cos ^2(b x)}{8 b^4}+\frac {3 \cos (b x) \text {Ci}(b x)}{b^4}-\frac {3 x^2 \cos (b x) \text {Ci}(b x)}{2 b^2}+\frac {1}{4} x^4 \text {Ci}(b x)^2-\frac {3 \text {Ci}(2 b x)}{2 b^4}-\frac {3 \log (x)}{2 b^4}+\frac {x \cos (b x) \sin (b x)}{b^3}+\frac {3 x \text {Ci}(b x) \sin (b x)}{b^3}-\frac {x^3 \text {Ci}(b x) \sin (b x)}{2 b}-\frac {13 \sin ^2(b x)}{8 b^4}+\frac {x^2 \sin ^2(b x)}{4 b^2}\\ \end {align*}
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Mathematica [A] time = 0.14, size = 108, normalized size = 0.66 \[ \frac {2 b^4 x^4 \text {Ci}(b x)^2-4 \text {Ci}(b x) \left (b x \left (b^2 x^2-6\right ) \sin (b x)+3 \left (b^2 x^2-2\right ) \cos (b x)\right )+3 b^2 x^2-b^2 x^2 \cos (2 b x)-12 \text {Ci}(2 b x)+4 b x \sin (2 b x)+8 \cos (2 b x)-12 \log (x)}{8 b^4} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.24, size = 0, normalized size = 0.00 \[ {\rm integral}\left (x^{3} \operatorname {Ci}\left (b x\right )^{2}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.24, size = 117, normalized size = 0.72 \[ \frac {1}{4} \, x^{4} \operatorname {Ci}\left (b x\right )^{2} - \frac {1}{2} \, {\left (\frac {3 \, {\left (b^{2} x^{2} - 2\right )} \cos \left (b x\right )}{b^{4}} + \frac {{\left (b^{3} x^{3} - 6 \, b x\right )} \sin \left (b x\right )}{b^{4}}\right )} \operatorname {Ci}\left (b x\right ) - \frac {b^{2} x^{2} \cos \left (2 \, b x\right ) - 3 \, b^{2} x^{2} - 4 \, b x \sin \left (2 \, b x\right ) - 8 \, \cos \left (2 \, b x\right ) + 6 \, \operatorname {Ci}\left (2 \, b x\right ) + 6 \, \operatorname {Ci}\left (-2 \, b x\right ) + 12 \, \log \relax (x)}{8 \, b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 148, normalized size = 0.91 \[ \frac {x^{4} \Ci \left (b x \right )^{2}}{4}-\frac {x^{3} \Ci \left (b x \right ) \sin \left (b x \right )}{2 b}-\frac {3 x^{2} \Ci \left (b x \right ) \cos \left (b x \right )}{2 b^{2}}+\frac {3 \Ci \left (b x \right ) \cos \left (b x \right )}{b^{4}}+\frac {3 x \Ci \left (b x \right ) \sin \left (b x \right )}{b^{3}}-\frac {x^{2} \left (\cos ^{2}\left (b x \right )\right )}{4 b^{2}}+\frac {x \cos \left (b x \right ) \sin \left (b x \right )}{b^{3}}+\frac {x^{2}}{2 b^{2}}-\frac {\sin ^{2}\left (b x \right )}{2 b^{4}}-\frac {3 \ln \left (b x \right )}{2 b^{4}}-\frac {3 \Ci \left (2 b x \right )}{2 b^{4}}+\frac {3 \left (\cos ^{2}\left (b x \right )\right )}{2 b^{4}} \]
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 \[ \int x^{3} {\rm Ci}\left (b x\right )^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int x^3\,{\mathrm {cosint}\left (b\,x\right )}^2 \,d x \]
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 \[ \int x^{3} \operatorname {Ci}^{2}{\left (b x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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