Optimal. Leaf size=103 \[ -\frac {1}{2} b^2 \text {Int}\left (\frac {\text {Ci}(b x) \sin (b x)}{x},x\right )+b^2 (-\text {Si}(2 b x))-\frac {\text {Ci}(b x) \sin (b x)}{2 x^2}-\frac {b \text {Ci}(b x) \cos (b x)}{2 x}-\frac {\sin (2 b x)}{8 x^2}-\frac {b \cos ^2(b x)}{2 x}-\frac {b \cos (2 b x)}{4 x} \]
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Rubi [A] time = 0.25, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {\text {CosIntegral}(b x) \sin (b x)}{x^3} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {\text {Ci}(b x) \sin (b x)}{x^3} \, dx &=-\frac {\text {Ci}(b x) \sin (b x)}{2 x^2}+\frac {1}{2} b \int \frac {\cos (b x) \text {Ci}(b x)}{x^2} \, dx+\frac {1}{2} b \int \frac {\cos (b x) \sin (b x)}{b x^3} \, dx\\ &=-\frac {b \cos (b x) \text {Ci}(b x)}{2 x}-\frac {\text {Ci}(b x) \sin (b x)}{2 x^2}+\frac {1}{2} \int \frac {\cos (b x) \sin (b x)}{x^3} \, dx+\frac {1}{2} b^2 \int \frac {\cos ^2(b x)}{b x^2} \, dx-\frac {1}{2} b^2 \int \frac {\text {Ci}(b x) \sin (b x)}{x} \, dx\\ &=-\frac {b \cos (b x) \text {Ci}(b x)}{2 x}-\frac {\text {Ci}(b x) \sin (b x)}{2 x^2}+\frac {1}{2} \int \frac {\sin (2 b x)}{2 x^3} \, dx+\frac {1}{2} b \int \frac {\cos ^2(b x)}{x^2} \, dx-\frac {1}{2} b^2 \int \frac {\text {Ci}(b x) \sin (b x)}{x} \, dx\\ &=-\frac {b \cos ^2(b x)}{2 x}-\frac {b \cos (b x) \text {Ci}(b x)}{2 x}-\frac {\text {Ci}(b x) \sin (b x)}{2 x^2}+\frac {1}{4} \int \frac {\sin (2 b x)}{x^3} \, dx-\frac {1}{2} b^2 \int \frac {\text {Ci}(b x) \sin (b x)}{x} \, dx+b^2 \int -\frac {\sin (2 b x)}{2 x} \, dx\\ &=-\frac {b \cos ^2(b x)}{2 x}-\frac {b \cos (b x) \text {Ci}(b x)}{2 x}-\frac {\text {Ci}(b x) \sin (b x)}{2 x^2}-\frac {\sin (2 b x)}{8 x^2}+\frac {1}{4} b \int \frac {\cos (2 b x)}{x^2} \, dx-\frac {1}{2} b^2 \int \frac {\text {Ci}(b x) \sin (b x)}{x} \, dx-\frac {1}{2} b^2 \int \frac {\sin (2 b x)}{x} \, dx\\ &=-\frac {b \cos ^2(b x)}{2 x}-\frac {b \cos (2 b x)}{4 x}-\frac {b \cos (b x) \text {Ci}(b x)}{2 x}-\frac {\text {Ci}(b x) \sin (b x)}{2 x^2}-\frac {\sin (2 b x)}{8 x^2}-\frac {1}{2} b^2 \text {Si}(2 b x)-\frac {1}{2} b^2 \int \frac {\text {Ci}(b x) \sin (b x)}{x} \, dx-\frac {1}{2} b^2 \int \frac {\sin (2 b x)}{x} \, dx\\ &=-\frac {b \cos ^2(b x)}{2 x}-\frac {b \cos (2 b x)}{4 x}-\frac {b \cos (b x) \text {Ci}(b x)}{2 x}-\frac {\text {Ci}(b x) \sin (b x)}{2 x^2}-\frac {\sin (2 b x)}{8 x^2}-b^2 \text {Si}(2 b x)-\frac {1}{2} b^2 \int \frac {\text {Ci}(b x) \sin (b x)}{x} \, dx\\ \end {align*}
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Mathematica [A] time = 1.63, size = 0, normalized size = 0.00 \[ \int \frac {\text {Ci}(b x) \sin (b x)}{x^3} \, dx \]
Verification is Not applicable to the result.
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fricas [A] time = 0.75, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\operatorname {Ci}\left (b x\right ) \sin \left (b x\right )}{x^{3}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\rm Ci}\left (b x\right ) \sin \left (b x\right )}{x^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 0, normalized size = 0.00 \[ \int \frac {\Ci \left (b x \right ) \sin \left (b x \right )}{x^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\rm Ci}\left (b x\right ) \sin \left (b x\right )}{x^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [A] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\mathrm {cosint}\left (b\,x\right )\,\sin \left (b\,x\right )}{x^3} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sin {\left (b x \right )} \operatorname {Ci}{\left (b x \right )}}{x^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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