Optimal. Leaf size=26 \[ -\frac {\text {Ci}(b x)}{x}-b \text {Si}(b x)-\frac {\cos (b x)}{x} \]
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Rubi [A] time = 0.05, antiderivative size = 26, 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 = {6504, 12, 3297, 3299} \[ -\frac {\text {CosIntegral}(b x)}{x}-b \text {Si}(b x)-\frac {\cos (b x)}{x} \]
Antiderivative was successfully verified.
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Rule 12
Rule 3297
Rule 3299
Rule 6504
Rubi steps
\begin {align*} \int \frac {\text {Ci}(b x)}{x^2} \, dx &=-\frac {\text {Ci}(b x)}{x}+b \int \frac {\cos (b x)}{b x^2} \, dx\\ &=-\frac {\text {Ci}(b x)}{x}+\int \frac {\cos (b x)}{x^2} \, dx\\ &=-\frac {\cos (b x)}{x}-\frac {\text {Ci}(b x)}{x}-b \int \frac {\sin (b x)}{x} \, dx\\ &=-\frac {\cos (b x)}{x}-\frac {\text {Ci}(b x)}{x}-b \text {Si}(b x)\\ \end {align*}
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Mathematica [A] time = 0.01, size = 26, normalized size = 1.00 \[ -\frac {\text {Ci}(b x)}{x}-b \text {Si}(b x)-\frac {\cos (b x)}{x} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.69, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\operatorname {Ci}\left (b x\right )}{x^{2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [C] time = 0.51, size = 109, normalized size = 4.19 \[ -\frac {b x \Im \left (\operatorname {Ci}\left (b x\right ) \right ) \tan \left (\frac {1}{2} \, b x\right )^{2} - b x \Im \left (\operatorname {Ci}\left (-b x\right ) \right ) \tan \left (\frac {1}{2} \, b x\right )^{2} + 2 \, b x \operatorname {Si}\left (b x\right ) \tan \left (\frac {1}{2} \, b x\right )^{2} + b x \Im \left (\operatorname {Ci}\left (b x\right ) \right ) - b x \Im \left (\operatorname {Ci}\left (-b x\right ) \right ) + 2 \, b x \operatorname {Si}\left (b x\right ) - 2 \, \tan \left (\frac {1}{2} \, b x\right )^{2} + 2}{2 \, {\left (x \tan \left (\frac {1}{2} \, b x\right )^{2} + x\right )}} - \frac {\operatorname {Ci}\left (b x\right )}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 34, normalized size = 1.31 \[ b \left (-\frac {\Ci \left (b x \right )}{b x}-\frac {\cos \left (b x \right )}{b x}-\Si \left (b x \right )\right ) \]
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 \[ \int \frac {{\rm Ci}\left (b x\right )}{x^{2}}\,{d x} \]
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 \[ -b\,\mathrm {sinint}\left (b\,x\right )-\frac {\mathrm {cosint}\left (b\,x\right )}{x}-\frac {\cos \left (b\,x\right )}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.90, size = 42, normalized size = 1.62 \[ - \frac {b^{2} x {{}_{3}F_{4}\left (\begin {matrix} \frac {1}{2}, 1, 1 \\ \frac {3}{2}, \frac {3}{2}, 2, 2 \end {matrix}\middle | {- \frac {b^{2} x^{2}}{4}} \right )}}{4} - \frac {\log {\left (b^{2} x^{2} \right )}}{2 x} - \frac {1}{x} - \frac {\gamma }{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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