Optimal. Leaf size=86 \[ \frac {x^{m+1} \text {Si}(b x)}{m+1}+\frac {x^m (-i b x)^{-m} \Gamma (m+1,-i b x)}{2 b (m+1)}+\frac {x^m (i b x)^{-m} \Gamma (m+1,i b x)}{2 b (m+1)} \]
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Rubi [A] time = 0.07, antiderivative size = 86, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6503, 12, 3308, 2181} \[ \frac {x^m (-i b x)^{-m} \text {Gamma}(m+1,-i b x)}{2 b (m+1)}+\frac {x^m (i b x)^{-m} \text {Gamma}(m+1,i b x)}{2 b (m+1)}+\frac {x^{m+1} \text {Si}(b x)}{m+1} \]
Antiderivative was successfully verified.
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Rule 12
Rule 2181
Rule 3308
Rule 6503
Rubi steps
\begin {align*} \int x^m \text {Si}(b x) \, dx &=\frac {x^{1+m} \text {Si}(b x)}{1+m}-\frac {b \int \frac {x^m \sin (b x)}{b} \, dx}{1+m}\\ &=\frac {x^{1+m} \text {Si}(b x)}{1+m}-\frac {\int x^m \sin (b x) \, dx}{1+m}\\ &=\frac {x^{1+m} \text {Si}(b x)}{1+m}-\frac {i \int e^{-i b x} x^m \, dx}{2 (1+m)}+\frac {i \int e^{i b x} x^m \, dx}{2 (1+m)}\\ &=\frac {x^m (-i b x)^{-m} \Gamma (1+m,-i b x)}{2 b (1+m)}+\frac {x^m (i b x)^{-m} \Gamma (1+m,i b x)}{2 b (1+m)}+\frac {x^{1+m} \text {Si}(b x)}{1+m}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 82, normalized size = 0.95 \[ \frac {x^m \left (b^2 x^2\right )^{-m} \left (2 b x \left (b^2 x^2\right )^m \text {Si}(b x)+(-i b x)^m \Gamma (m+1,i b x)+(i b x)^m \Gamma (m+1,-i b x)\right )}{2 b (m+1)} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.57, size = 0, normalized size = 0.00 \[ {\rm integral}\left (x^{m} \operatorname {Si}\left (b x\right ), x\right ) \]
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 \[ \int x^{m} {\rm Si}\left (b x\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.08, size = 37, normalized size = 0.43 \[ \frac {b \,x^{m +2} \hypergeom \left (\left [\frac {1}{2}, 1+\frac {m}{2}\right ], \left [\frac {3}{2}, \frac {3}{2}, 2+\frac {m}{2}\right ], -\frac {b^{2} x^{2}}{4}\right )}{m +2} \]
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 \[ \int x^{m} {\rm Si}\left (b x\right )\,{d x} \]
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 \[ \int x^m\,\mathrm {sinint}\left (b\,x\right ) \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.87, size = 46, normalized size = 0.53 \[ \frac {b x^{2} x^{m} \Gamma \left (\frac {m}{2} + 1\right ) {{}_{2}F_{3}\left (\begin {matrix} \frac {1}{2}, \frac {m}{2} + 1 \\ \frac {3}{2}, \frac {3}{2}, \frac {m}{2} + 2 \end {matrix}\middle | {- \frac {b^{2} x^{2}}{4}} \right )}}{2 \Gamma \left (\frac {m}{2} + 2\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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