Optimal. Leaf size=32 \[ x \text {Si}(b x)^2-\frac {\text {Si}(2 b x)}{b}+\frac {2 \text {Si}(b x) \cos (b x)}{b} \]
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Rubi [A] time = 0.05, 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 = {6505, 6511, 12, 4406, 3299} \[ x \text {Si}(b x)^2-\frac {\text {Si}(2 b x)}{b}+\frac {2 \text {Si}(b x) \cos (b x)}{b} \]
Antiderivative was successfully verified.
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Rule 12
Rule 3299
Rule 4406
Rule 6505
Rule 6511
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 \[ x \text {Si}(b x)^2-\frac {\text {Si}(2 b x)}{b}+\frac {2 \text {Si}(b x) \cos (b x)}{b} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.62, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\operatorname {Si}\left (b x\right )^{2}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [C] time = 0.64, size = 49, normalized size = 1.53 \[ x \operatorname {Si}\left (b x\right )^{2} + \frac {2 \, \cos \left (b x\right ) \operatorname {Si}\left (b x\right )}{b} - \frac {\Im \left (\operatorname {Ci}\left (2 \, b x\right ) \right ) - \Im \left (\operatorname {Ci}\left (-2 \, b x\right ) \right ) + 2 \, \operatorname {Si}\left (2 \, b x\right )}{2 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 32, normalized size = 1.00 \[ \frac {b x \Si \left (b x \right )^{2}+2 \cos \left (b x \right ) \Si \left (b x \right )-\Si \left (2 b x \right )}{b} \]
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 \[ \int {\rm Si}\left (b x\right )^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.03 \[ \int {\mathrm {sinint}\left (b\,x\right )}^2 \,d x \]
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 \[ \int \operatorname {Si}^{2}{\left (b x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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