Optimal. Leaf size=32 \[ \frac {2 \cos (b x) \text {Si}(b x)}{b}+x \text {Si}(b x)^2-\frac {\text {Si}(2 b x)}{b} \]
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Rubi [A]
time = 0.03, antiderivative size = 32, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 5, integrand size = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.833, Rules used = {6640, 6646, 12,
4491, 3380} \begin {gather*} x \text {Si}(b x)^2-\frac {\text {Si}(2 b x)}{b}+\frac {2 \text {Si}(b x) \cos (b x)}{b} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 3380
Rule 4491
Rule 6640
Rule 6646
Rubi steps
\begin {align*} \int \text {Si}(b x)^2 \, dx &=x \text {Si}(b x)^2-2 \int \sin (b x) \text {Si}(b x) \, dx\\ &=\frac {2 \cos (b x) \text {Si}(b x)}{b}+x \text {Si}(b x)^2-2 \int \frac {\cos (b x) \sin (b x)}{b x} \, dx\\ &=\frac {2 \cos (b x) \text {Si}(b x)}{b}+x \text {Si}(b x)^2-\frac {2 \int \frac {\cos (b x) \sin (b x)}{x} \, dx}{b}\\ &=\frac {2 \cos (b x) \text {Si}(b x)}{b}+x \text {Si}(b x)^2-\frac {2 \int \frac {\sin (2 b x)}{2 x} \, dx}{b}\\ &=\frac {2 \cos (b x) \text {Si}(b x)}{b}+x \text {Si}(b x)^2-\frac {\int \frac {\sin (2 b x)}{x} \, dx}{b}\\ &=\frac {2 \cos (b x) \text {Si}(b x)}{b}+x \text {Si}(b x)^2-\frac {\text {Si}(2 b x)}{b}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 32, normalized size = 1.00 \begin {gather*} \frac {2 \cos (b x) \text {Si}(b x)}{b}+x \text {Si}(b x)^2-\frac {\text {Si}(2 b x)}{b} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.36, size = 32, normalized size = 1.00
method | result | size |
derivativedivides | \(\frac {\sinIntegral \left (b x \right )^{2} b x +2 \cos \left (b x \right ) \sinIntegral \left (b x \right )-\sinIntegral \left (2 b x \right )}{b}\) | \(32\) |
default | \(\frac {\sinIntegral \left (b x \right )^{2} b x +2 \cos \left (b x \right ) \sinIntegral \left (b x \right )-\sinIntegral \left (2 b x \right )}{b}\) | \(32\) |
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 = 0.37, size = 31, normalized size = 0.97 \begin {gather*} \frac {b x \operatorname {Si}\left (b x\right )^{2} + 2 \, \cos \left (b x\right ) \operatorname {Si}\left (b x\right ) - \operatorname {Si}\left (2 \, b x\right )}{b} \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 \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 [C] Result contains higher order function than in optimal. Order 9 vs. order
4.
time = 0.40, size = 49, normalized size = 1.53 \begin {gather*} x \operatorname {Si}\left (b x\right )^{2} + \frac {2 \, \cos \left (b x\right ) \operatorname {Si}\left (b x\right )}{b} - \frac {\Im \left ( \operatorname {Ci}\left (2 \, b x\right ) \right ) - \Im \left ( \operatorname {Ci}\left (-2 \, b x\right ) \right ) + 2 \, \operatorname {Si}\left (2 \, b x\right )}{2 \, b} \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 = -0.03 \begin {gather*} \int {\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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