Optimal. Leaf size=149 \[ \frac {x^2}{2 b^2}+\frac {3 \text {CosIntegral}(2 b x)}{2 b^4}-\frac {3 \log (x)}{2 b^4}-\frac {x \cos (b x) \sin (b x)}{b^3}+\frac {2 \sin ^2(b x)}{b^4}-\frac {x^2 \sin ^2(b x)}{4 b^2}-\frac {3 x \cos (b x) \text {Si}(b x)}{b^3}+\frac {x^3 \cos (b x) \text {Si}(b x)}{2 b}+\frac {3 \sin (b x) \text {Si}(b x)}{b^4}-\frac {3 x^2 \sin (b x) \text {Si}(b x)}{2 b^2}+\frac {1}{4} x^4 \text {Si}(b x)^2 \]
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Rubi [A]
time = 0.15, antiderivative size = 149, 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 = {6642, 6648,
12, 3524, 3391, 30, 6654, 2644, 6652, 3393, 3383} \begin {gather*} \frac {3 \text {CosIntegral}(2 b x)}{2 b^4}+\frac {3 \text {Si}(b x) \sin (b x)}{b^4}-\frac {3 \log (x)}{2 b^4}+\frac {2 \sin ^2(b x)}{b^4}-\frac {3 x \text {Si}(b x) \cos (b x)}{b^3}-\frac {x \sin (b x) \cos (b x)}{b^3}-\frac {3 x^2 \text {Si}(b x) \sin (b x)}{2 b^2}+\frac {x^2}{2 b^2}-\frac {x^2 \sin ^2(b x)}{4 b^2}+\frac {1}{4} x^4 \text {Si}(b x)^2+\frac {x^3 \text {Si}(b x) \cos (b x)}{2 b} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 30
Rule 2644
Rule 3383
Rule 3391
Rule 3393
Rule 3524
Rule 6642
Rule 6648
Rule 6652
Rule 6654
Rubi steps
\begin {align*} \int x^3 \text {Si}(b x)^2 \, dx &=\frac {1}{4} x^4 \text {Si}(b x)^2-\frac {1}{2} \int x^3 \sin (b x) \text {Si}(b x) \, dx\\ &=\frac {x^3 \cos (b x) \text {Si}(b x)}{2 b}+\frac {1}{4} x^4 \text {Si}(b x)^2-\frac {1}{2} \int \frac {x^2 \cos (b x) \sin (b x)}{b} \, dx-\frac {3 \int x^2 \cos (b x) \text {Si}(b x) \, dx}{2 b}\\ &=\frac {x^3 \cos (b x) \text {Si}(b x)}{2 b}-\frac {3 x^2 \sin (b x) \text {Si}(b x)}{2 b^2}+\frac {1}{4} x^4 \text {Si}(b x)^2+\frac {3 \int x \sin (b x) \text {Si}(b x) \, dx}{b^2}-\frac {\int x^2 \cos (b x) \sin (b x) \, dx}{2 b}+\frac {3 \int \frac {x \sin ^2(b x)}{b} \, dx}{2 b}\\ &=-\frac {x^2 \sin ^2(b x)}{4 b^2}-\frac {3 x \cos (b x) \text {Si}(b x)}{b^3}+\frac {x^3 \cos (b x) \text {Si}(b x)}{2 b}-\frac {3 x^2 \sin (b x) \text {Si}(b x)}{2 b^2}+\frac {1}{4} x^4 \text {Si}(b x)^2+\frac {3 \int \cos (b x) \text {Si}(b x) \, dx}{b^3}+\frac {\int x \sin ^2(b x) \, dx}{2 b^2}+\frac {3 \int x \sin ^2(b x) \, dx}{2 b^2}+\frac {3 \int \frac {\cos (b x) \sin (b x)}{b} \, dx}{b^2}\\ &=-\frac {x \cos (b x) \sin (b x)}{b^3}+\frac {\sin ^2(b x)}{2 b^4}-\frac {x^2 \sin ^2(b x)}{4 b^2}-\frac {3 x \cos (b x) \text {Si}(b x)}{b^3}+\frac {x^3 \cos (b x) \text {Si}(b x)}{2 b}+\frac {3 \sin (b x) \text {Si}(b x)}{b^4}-\frac {3 x^2 \sin (b x) \text {Si}(b x)}{2 b^2}+\frac {1}{4} x^4 \text {Si}(b x)^2+\frac {3 \int \cos (b x) \sin (b x) \, dx}{b^3}-\frac {3 \int \frac {\sin ^2(b x)}{b x} \, dx}{b^3}+\frac {\int x \, dx}{4 b^2}+\frac {3 \int x \, dx}{4 b^2}\\ &=\frac {x^2}{2 b^2}-\frac {x \cos (b x) \sin (b x)}{b^3}+\frac {\sin ^2(b x)}{2 b^4}-\frac {x^2 \sin ^2(b x)}{4 b^2}-\frac {3 x \cos (b x) \text {Si}(b x)}{b^3}+\frac {x^3 \cos (b x) \text {Si}(b x)}{2 b}+\frac {3 \sin (b x) \text {Si}(b x)}{b^4}-\frac {3 x^2 \sin (b x) \text {Si}(b x)}{2 b^2}+\frac {1}{4} x^4 \text {Si}(b x)^2-\frac {3 \int \frac {\sin ^2(b x)}{x} \, dx}{b^4}+\frac {3 \text {Subst}(\int x \, dx,x,\sin (b x))}{b^4}\\ &=\frac {x^2}{2 b^2}-\frac {x \cos (b x) \sin (b x)}{b^3}+\frac {2 \sin ^2(b x)}{b^4}-\frac {x^2 \sin ^2(b x)}{4 b^2}-\frac {3 x \cos (b x) \text {Si}(b x)}{b^3}+\frac {x^3 \cos (b x) \text {Si}(b x)}{2 b}+\frac {3 \sin (b x) \text {Si}(b x)}{b^4}-\frac {3 x^2 \sin (b x) \text {Si}(b x)}{2 b^2}+\frac {1}{4} x^4 \text {Si}(b x)^2-\frac {3 \int \left (\frac {1}{2 x}-\frac {\cos (2 b x)}{2 x}\right ) \, dx}{b^4}\\ &=\frac {x^2}{2 b^2}-\frac {3 \log (x)}{2 b^4}-\frac {x \cos (b x) \sin (b x)}{b^3}+\frac {2 \sin ^2(b x)}{b^4}-\frac {x^2 \sin ^2(b x)}{4 b^2}-\frac {3 x \cos (b x) \text {Si}(b x)}{b^3}+\frac {x^3 \cos (b x) \text {Si}(b x)}{2 b}+\frac {3 \sin (b x) \text {Si}(b x)}{b^4}-\frac {3 x^2 \sin (b x) \text {Si}(b x)}{2 b^2}+\frac {1}{4} x^4 \text {Si}(b x)^2+\frac {3 \int \frac {\cos (2 b x)}{x} \, dx}{2 b^4}\\ &=\frac {x^2}{2 b^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 {2 \sin ^2(b x)}{b^4}-\frac {x^2 \sin ^2(b x)}{4 b^2}-\frac {3 x \cos (b x) \text {Si}(b x)}{b^3}+\frac {x^3 \cos (b x) \text {Si}(b x)}{2 b}+\frac {3 \sin (b x) \text {Si}(b x)}{b^4}-\frac {3 x^2 \sin (b x) \text {Si}(b x)}{2 b^2}+\frac {1}{4} x^4 \text {Si}(b x)^2\\ \end {align*}
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Mathematica [A]
time = 0.09, size = 107, normalized size = 0.72 \begin {gather*} \frac {3 b^2 x^2-8 \cos (2 b x)+b^2 x^2 \cos (2 b x)+12 \text {CosIntegral}(2 b x)-12 \log (x)-4 b x \sin (2 b x)+4 \left (b x \left (-6+b^2 x^2\right ) \cos (b x)-3 \left (-2+b^2 x^2\right ) \sin (b x)\right ) \text {Si}(b x)+2 b^4 x^4 \text {Si}(b x)^2}{8 b^4} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.45, size = 154, normalized size = 1.03
method | result | size |
derivativedivides | \(\frac {\frac {b^{4} x^{4} \sinIntegral \left (b x \right )^{2}}{4}-2 \sinIntegral \left (b x \right ) \left (-\frac {b^{3} x^{3} \cos \left (b x \right )}{4}+\frac {3 b^{2} x^{2} \sin \left (b x \right )}{4}-\frac {3 \sin \left (b x \right )}{2}+\frac {3 b x \cos \left (b x \right )}{2}\right )+\frac {b^{2} x^{2} \left (\cos ^{2}\left (b x \right )\right )}{4}-\frac {b x \left (\frac {\sin \left (b x \right ) \cos \left (b x \right )}{2}+\frac {b x}{2}\right )}{2}-\frac {b^{2} x^{2}}{4}+\frac {\left (\sin ^{2}\left (b x \right )\right )}{2}+\frac {3 b x \left (-\frac {\sin \left (b x \right ) \cos \left (b x \right )}{2}+\frac {b x}{2}\right )}{2}-\frac {3 \left (\cos ^{2}\left (b x \right )\right )}{2}-\frac {3 \ln \left (b x \right )}{2}+\frac {3 \cosineIntegral \left (2 b x \right )}{2}}{b^{4}}\) | \(154\) |
default | \(\frac {\frac {b^{4} x^{4} \sinIntegral \left (b x \right )^{2}}{4}-2 \sinIntegral \left (b x \right ) \left (-\frac {b^{3} x^{3} \cos \left (b x \right )}{4}+\frac {3 b^{2} x^{2} \sin \left (b x \right )}{4}-\frac {3 \sin \left (b x \right )}{2}+\frac {3 b x \cos \left (b x \right )}{2}\right )+\frac {b^{2} x^{2} \left (\cos ^{2}\left (b x \right )\right )}{4}-\frac {b x \left (\frac {\sin \left (b x \right ) \cos \left (b x \right )}{2}+\frac {b x}{2}\right )}{2}-\frac {b^{2} x^{2}}{4}+\frac {\left (\sin ^{2}\left (b x \right )\right )}{2}+\frac {3 b x \left (-\frac {\sin \left (b x \right ) \cos \left (b x \right )}{2}+\frac {b x}{2}\right )}{2}-\frac {3 \left (\cos ^{2}\left (b x \right )\right )}{2}-\frac {3 \ln \left (b x \right )}{2}+\frac {3 \cosineIntegral \left (2 b x \right )}{2}}{b^{4}}\) | \(154\) |
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 [A]
time = 0.39, size = 112, normalized size = 0.75 \begin {gather*} \frac {b^{4} x^{4} \operatorname {Si}\left (b x\right )^{2} + b^{2} x^{2} + {\left (b^{2} x^{2} - 8\right )} \cos \left (b x\right )^{2} + 2 \, {\left (b^{3} x^{3} - 6 \, b x\right )} \cos \left (b x\right ) \operatorname {Si}\left (b x\right ) - 2 \, {\left (2 \, b x \cos \left (b x\right ) + 3 \, {\left (b^{2} x^{2} - 2\right )} \operatorname {Si}\left (b x\right )\right )} \sin \left (b x\right ) + 3 \, \operatorname {Ci}\left (2 \, b x\right ) + 3 \, \operatorname {Ci}\left (-2 \, b x\right ) - 6 \, \log \left (x\right )}{4 \, b^{4}} \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 x^{3} \operatorname {Si}^{2}{\left (b x \right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.41, size = 117, normalized size = 0.79 \begin {gather*} \frac {1}{4} \, x^{4} \operatorname {Si}\left (b x\right )^{2} + \frac {1}{2} \, {\left (\frac {{\left (b^{3} x^{3} - 6 \, b x\right )} \cos \left (b x\right )}{b^{4}} - \frac {3 \, {\left (b^{2} x^{2} - 2\right )} \sin \left (b x\right )}{b^{4}}\right )} \operatorname {Si}\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 \left (x\right )}{8 \, b^{4}} \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 x^3\,{\mathrm {sinint}\left (b\,x\right )}^2 \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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