Optimal. Leaf size=49 \[ \frac {(a+b x) \text {Si}(a+b x)^2}{b}-\frac {\text {Si}(2 a+2 b x)}{b}+\frac {2 \text {Si}(a+b x) \cos (a+b x)}{b} \]
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Rubi [A] time = 0.06, antiderivative size = 49, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.625, Rules used = {6505, 6511, 4406, 12, 3299} \[ \frac {(a+b x) \text {Si}(a+b x)^2}{b}-\frac {\text {Si}(2 a+2 b x)}{b}+\frac {2 \text {Si}(a+b x) \cos (a+b x)}{b} \]
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}(a+b x)^2 \, dx &=\frac {(a+b x) \text {Si}(a+b x)^2}{b}-2 \int \sin (a+b x) \text {Si}(a+b x) \, dx\\ &=\frac {2 \cos (a+b x) \text {Si}(a+b x)}{b}+\frac {(a+b x) \text {Si}(a+b x)^2}{b}-2 \int \frac {\cos (a+b x) \sin (a+b x)}{a+b x} \, dx\\ &=\frac {2 \cos (a+b x) \text {Si}(a+b x)}{b}+\frac {(a+b x) \text {Si}(a+b x)^2}{b}-2 \int \frac {\sin (2 a+2 b x)}{2 (a+b x)} \, dx\\ &=\frac {2 \cos (a+b x) \text {Si}(a+b x)}{b}+\frac {(a+b x) \text {Si}(a+b x)^2}{b}-\int \frac {\sin (2 a+2 b x)}{a+b x} \, dx\\ &=\frac {2 \cos (a+b x) \text {Si}(a+b x)}{b}+\frac {(a+b x) \text {Si}(a+b x)^2}{b}-\frac {\text {Si}(2 a+2 b x)}{b}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 43, normalized size = 0.88 \[ \frac {(a+b x) \text {Si}(a+b x)^2-\text {Si}(2 (a+b x))+2 \text {Si}(a+b x) \cos (a+b x)}{b} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.40, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\operatorname {Si}\left (b x + a\right )^{2}, x\right ) \]
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 \[ \int {\rm Si}\left (b x + a\right )^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 45, normalized size = 0.92 \[ \frac {\left (b x +a \right ) \Si \left (b x +a \right )^{2}+2 \cos \left (b x +a \right ) \Si \left (b x +a \right )-\Si \left (2 b x +2 a \right )}{b} \]
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 \[ \int {\rm Si}\left (b x + a\right )^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int {\mathrm {sinint}\left (a+b\,x\right )}^2 \,d x \]
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 \[ \int \operatorname {Si}^{2}{\left (a + b x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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