Optimal. Leaf size=25 \[ b \text {Ci}(b x)-\frac {\text {Si}(b x)}{x}-\frac {\sin (b x)}{x} \]
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Rubi [A] time = 0.05, antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6503, 12, 3297, 3302} \[ b \text {CosIntegral}(b x)-\frac {\text {Si}(b x)}{x}-\frac {\sin (b x)}{x} \]
Antiderivative was successfully verified.
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Rule 12
Rule 3297
Rule 3302
Rule 6503
Rubi steps
\begin {align*} \int \frac {\text {Si}(b x)}{x^2} \, dx &=-\frac {\text {Si}(b x)}{x}+b \int \frac {\sin (b x)}{b x^2} \, dx\\ &=-\frac {\text {Si}(b x)}{x}+\int \frac {\sin (b x)}{x^2} \, dx\\ &=-\frac {\sin (b x)}{x}-\frac {\text {Si}(b x)}{x}+b \int \frac {\cos (b x)}{x} \, dx\\ &=b \text {Ci}(b x)-\frac {\sin (b x)}{x}-\frac {\text {Si}(b x)}{x}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 25, normalized size = 1.00 \[ b \text {Ci}(b x)-\frac {\text {Si}(b x)}{x}-\frac {\sin (b x)}{x} \]
Antiderivative was successfully verified.
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fricas [F] time = 2.96, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\operatorname {Si}\left (b x\right )}{x^{2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.84, size = 37, normalized size = 1.48 \[ \frac {b x \operatorname {Ci}\left (b x\right ) + b x \operatorname {Ci}\left (-b x\right ) - 2 \, \sin \left (b x\right )}{2 \, x} - \frac {\operatorname {Si}\left (b x\right )}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 32, normalized size = 1.28 \[ b \left (-\frac {\Si \left (b x \right )}{b x}-\frac {\sin \left (b x \right )}{b x}+\Ci \left (b x \right )\right ) \]
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 \[ \int \frac {{\rm Si}\left (b x\right )}{x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.04 \[ b\,\mathrm {cosint}\left (b\,x\right )-\frac {\mathrm {sinint}\left (b\,x\right )}{x}-\frac {\sin \left (b\,x\right )}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.85, size = 36, normalized size = 1.44 \[ - \frac {b^{3} x^{2} {{}_{3}F_{4}\left (\begin {matrix} 1, 1, \frac {3}{2} \\ 2, 2, \frac {5}{2}, \frac {5}{2} \end {matrix}\middle | {- \frac {b^{2} x^{2}}{4}} \right )}}{36} + \frac {b \log {\left (b^{2} x^{2} \right )}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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