Optimal. Leaf size=96 \[ -\frac {\text {Ci}(2 a+2 b x)}{2 b^2}+\frac {\text {Ci}(a+b x) \cos (a+b x)}{b^2}+\frac {a \text {Si}(2 a+2 b x)}{2 b^2}-\frac {\log (a+b x)}{2 b^2}+\frac {\cos (2 a+2 b x)}{4 b^2}+\frac {x \text {Ci}(a+b x) \sin (a+b x)}{b} \]
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Rubi [A] time = 0.25, antiderivative size = 96, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 9, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.643, Rules used = {6514, 4573, 6741, 6742, 2638, 3299, 6518, 3312, 3302} \[ -\frac {\text {CosIntegral}(2 a+2 b x)}{2 b^2}+\frac {\text {CosIntegral}(a+b x) \cos (a+b x)}{b^2}+\frac {a \text {Si}(2 a+2 b x)}{2 b^2}-\frac {\log (a+b x)}{2 b^2}+\frac {\cos (2 a+2 b x)}{4 b^2}+\frac {x \text {CosIntegral}(a+b x) \sin (a+b x)}{b} \]
Antiderivative was successfully verified.
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Rule 2638
Rule 3299
Rule 3302
Rule 3312
Rule 4573
Rule 6514
Rule 6518
Rule 6741
Rule 6742
Rubi steps
\begin {align*} \int x \cos (a+b x) \text {Ci}(a+b x) \, dx &=\frac {x \text {Ci}(a+b x) \sin (a+b x)}{b}-\frac {\int \text {Ci}(a+b x) \sin (a+b x) \, dx}{b}-\int \frac {x \cos (a+b x) \sin (a+b x)}{a+b x} \, dx\\ &=\frac {\cos (a+b x) \text {Ci}(a+b x)}{b^2}+\frac {x \text {Ci}(a+b x) \sin (a+b x)}{b}-\frac {1}{2} \int \frac {x \sin (2 (a+b x))}{a+b x} \, dx-\frac {\int \frac {\cos ^2(a+b x)}{a+b x} \, dx}{b}\\ &=\frac {\cos (a+b x) \text {Ci}(a+b x)}{b^2}+\frac {x \text {Ci}(a+b x) \sin (a+b x)}{b}-\frac {1}{2} \int \frac {x \sin (2 a+2 b x)}{a+b x} \, dx-\frac {\int \left (\frac {1}{2 (a+b x)}+\frac {\cos (2 a+2 b x)}{2 (a+b x)}\right ) \, dx}{b}\\ &=\frac {\cos (a+b x) \text {Ci}(a+b x)}{b^2}-\frac {\log (a+b x)}{2 b^2}+\frac {x \text {Ci}(a+b x) \sin (a+b x)}{b}-\frac {1}{2} \int \left (\frac {\sin (2 a+2 b x)}{b}+\frac {a \sin (2 a+2 b x)}{b (-a-b x)}\right ) \, dx-\frac {\int \frac {\cos (2 a+2 b x)}{a+b x} \, dx}{2 b}\\ &=\frac {\cos (a+b x) \text {Ci}(a+b x)}{b^2}-\frac {\text {Ci}(2 a+2 b x)}{2 b^2}-\frac {\log (a+b x)}{2 b^2}+\frac {x \text {Ci}(a+b x) \sin (a+b x)}{b}-\frac {\int \sin (2 a+2 b x) \, dx}{2 b}-\frac {a \int \frac {\sin (2 a+2 b x)}{-a-b x} \, dx}{2 b}\\ &=\frac {\cos (2 a+2 b x)}{4 b^2}+\frac {\cos (a+b x) \text {Ci}(a+b x)}{b^2}-\frac {\text {Ci}(2 a+2 b x)}{2 b^2}-\frac {\log (a+b x)}{2 b^2}+\frac {x \text {Ci}(a+b x) \sin (a+b x)}{b}+\frac {a \text {Si}(2 a+2 b x)}{2 b^2}\\ \end {align*}
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Mathematica [A] time = 0.17, size = 69, normalized size = 0.72 \[ \frac {-2 \text {Ci}(2 (a+b x))+4 \text {Ci}(a+b x) (b x \sin (a+b x)+\cos (a+b x))+2 a \text {Si}(2 (a+b x))-2 \log (a+b x)+\cos (2 (a+b x))}{4 b^2} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.13, size = 0, normalized size = 0.00 \[ {\rm integral}\left (x \cos \left (b x + a\right ) \operatorname {Ci}\left (b x + a\right ), x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [C] time = 0.17, size = 505, normalized size = 5.26 \[ {\left (\frac {x \sin \left (b x + a\right )}{b} + \frac {\cos \left (b x + a\right )}{b^{2}}\right )} \operatorname {Ci}\left (b x + a\right ) + \frac {a \Im \left (\operatorname {Ci}\left (2 \, b x + 2 \, a\right ) \right ) \tan \left (b x\right )^{2} \tan \relax (a)^{2} - a \Im \left (\operatorname {Ci}\left (-2 \, b x - 2 \, a\right ) \right ) \tan \left (b x\right )^{2} \tan \relax (a)^{2} + 2 \, a \operatorname {Si}\left (2 \, b x + 2 \, a\right ) \tan \left (b x\right )^{2} \tan \relax (a)^{2} - 2 \, \log \left ({\left | b x + a \right |}\right ) \tan \left (b x\right )^{2} \tan \relax (a)^{2} - \Re \left (\operatorname {Ci}\left (2 \, b x + 2 \, a\right ) \right ) \tan \left (b x\right )^{2} \tan \relax (a)^{2} - \Re \left (\operatorname {Ci}\left (-2 \, b x - 2 \, a\right ) \right ) \tan \left (b x\right )^{2} \tan \relax (a)^{2} + a \Im \left (\operatorname {Ci}\left (2 \, b x + 2 \, a\right ) \right ) \tan \left (b x\right )^{2} - a \Im \left (\operatorname {Ci}\left (-2 \, b x - 2 \, a\right ) \right ) \tan \left (b x\right )^{2} + 2 \, a \operatorname {Si}\left (2 \, b x + 2 \, a\right ) \tan \left (b x\right )^{2} + a \Im \left (\operatorname {Ci}\left (2 \, b x + 2 \, a\right ) \right ) \tan \relax (a)^{2} - a \Im \left (\operatorname {Ci}\left (-2 \, b x - 2 \, a\right ) \right ) \tan \relax (a)^{2} + 2 \, a \operatorname {Si}\left (2 \, b x + 2 \, a\right ) \tan \relax (a)^{2} + \tan \left (b x\right )^{2} \tan \relax (a)^{2} - 2 \, \log \left ({\left | b x + a \right |}\right ) \tan \left (b x\right )^{2} - \Re \left (\operatorname {Ci}\left (2 \, b x + 2 \, a\right ) \right ) \tan \left (b x\right )^{2} - \Re \left (\operatorname {Ci}\left (-2 \, b x - 2 \, a\right ) \right ) \tan \left (b x\right )^{2} - 2 \, \log \left ({\left | b x + a \right |}\right ) \tan \relax (a)^{2} - \Re \left (\operatorname {Ci}\left (2 \, b x + 2 \, a\right ) \right ) \tan \relax (a)^{2} - \Re \left (\operatorname {Ci}\left (-2 \, b x - 2 \, a\right ) \right ) \tan \relax (a)^{2} + a \Im \left (\operatorname {Ci}\left (2 \, b x + 2 \, a\right ) \right ) - a \Im \left (\operatorname {Ci}\left (-2 \, b x - 2 \, a\right ) \right ) + 2 \, a \operatorname {Si}\left (2 \, b x + 2 \, a\right ) - \tan \left (b x\right )^{2} - 4 \, \tan \left (b x\right ) \tan \relax (a) - \tan \relax (a)^{2} - 2 \, \log \left ({\left | b x + a \right |}\right ) - \Re \left (\operatorname {Ci}\left (2 \, b x + 2 \, a\right ) \right ) - \Re \left (\operatorname {Ci}\left (-2 \, b x - 2 \, a\right ) \right ) + 1}{4 \, {\left (b^{2} \tan \left (b x\right )^{2} \tan \relax (a)^{2} + b^{2} \tan \left (b x\right )^{2} + b^{2} \tan \relax (a)^{2} + b^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 88, normalized size = 0.92 \[ \frac {x \Ci \left (b x +a \right ) \sin \left (b x +a \right )}{b}+\frac {\Ci \left (b x +a \right ) \cos \left (b x +a \right )}{b^{2}}-\frac {\ln \left (b x +a \right )}{2 b^{2}}-\frac {\Ci \left (2 b x +2 a \right )}{2 b^{2}}+\frac {\cos ^{2}\left (b x +a \right )}{2 b^{2}}+\frac {a \Si \left (2 b x +2 a \right )}{2 b^{2}} \]
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 \[ \int x {\rm Ci}\left (b x + a\right ) \cos \left (b x + a\right )\,{d x} \]
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 \[ \int x\,\mathrm {cosint}\left (a+b\,x\right )\,\cos \left (a+b\,x\right ) \,d x \]
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 \[ \int x \cos {\left (a + b x \right )} \operatorname {Ci}{\left (a + b x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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