Integrand size = 14, antiderivative size = 97 \[ \int x \sin (a+b x) \text {Si}(a+b x) \, dx=-\frac {\cos (2 a+2 b x)}{4 b^2}+\frac {\operatorname {CosIntegral}(2 a+2 b x)}{2 b^2}-\frac {\log (a+b x)}{2 b^2}-\frac {x \cos (a+b x) \text {Si}(a+b x)}{b}+\frac {\sin (a+b x) \text {Si}(a+b x)}{b^2}-\frac {a \text {Si}(2 a+2 b x)}{2 b^2} \]
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Time = 0.19 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.643, Rules used = {6648, 4669, 6873, 6874, 2718, 3380, 6652, 3393, 3383} \[ \int x \sin (a+b x) \text {Si}(a+b x) \, dx=\frac {\operatorname {CosIntegral}(2 a+2 b x)}{2 b^2}-\frac {a \text {Si}(2 a+2 b x)}{2 b^2}+\frac {\text {Si}(a+b x) \sin (a+b x)}{b^2}-\frac {\log (a+b x)}{2 b^2}-\frac {\cos (2 a+2 b x)}{4 b^2}-\frac {x \text {Si}(a+b x) \cos (a+b x)}{b} \]
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Rule 2718
Rule 3380
Rule 3383
Rule 3393
Rule 4669
Rule 6648
Rule 6652
Rule 6873
Rule 6874
Rubi steps \begin{align*} \text {integral}& = -\frac {x \cos (a+b x) \text {Si}(a+b x)}{b}+\frac {\int \cos (a+b x) \text {Si}(a+b x) \, dx}{b}+\int \frac {x \cos (a+b x) \sin (a+b x)}{a+b x} \, dx \\ & = -\frac {x \cos (a+b x) \text {Si}(a+b x)}{b}+\frac {\sin (a+b x) \text {Si}(a+b x)}{b^2}+\frac {1}{2} \int \frac {x \sin (2 (a+b x))}{a+b x} \, dx-\frac {\int \frac {\sin ^2(a+b x)}{a+b x} \, dx}{b} \\ & = -\frac {x \cos (a+b x) \text {Si}(a+b x)}{b}+\frac {\sin (a+b x) \text {Si}(a+b x)}{b^2}+\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 {\log (a+b x)}{2 b^2}-\frac {x \cos (a+b x) \text {Si}(a+b x)}{b}+\frac {\sin (a+b x) \text {Si}(a+b x)}{b^2}+\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 {\operatorname {CosIntegral}(2 a+2 b x)}{2 b^2}-\frac {\log (a+b x)}{2 b^2}-\frac {x \cos (a+b x) \text {Si}(a+b x)}{b}+\frac {\sin (a+b x) \text {Si}(a+b x)}{b^2}+\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 {\operatorname {CosIntegral}(2 a+2 b x)}{2 b^2}-\frac {\log (a+b x)}{2 b^2}-\frac {x \cos (a+b x) \text {Si}(a+b x)}{b}+\frac {\sin (a+b x) \text {Si}(a+b x)}{b^2}-\frac {a \text {Si}(2 a+2 b x)}{2 b^2} \\ \end{align*}
Time = 0.18 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.73 \[ \int x \sin (a+b x) \text {Si}(a+b x) \, dx=-\frac {\cos (2 (a+b x))-2 \operatorname {CosIntegral}(2 (a+b x))+2 \log (a+b x)+4 (b x \cos (a+b x)-\sin (a+b x)) \text {Si}(a+b x)+2 a \text {Si}(2 (a+b x))}{4 b^2} \]
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Time = 1.58 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.85
method | result | size |
derivativedivides | \(\frac {\operatorname {Si}\left (b x +a \right ) \left (a \cos \left (b x +a \right )+\sin \left (b x +a \right )-\left (b x +a \right ) \cos \left (b x +a \right )\right )-\frac {a \,\operatorname {Si}\left (2 b x +2 a \right )}{2}-\frac {\ln \left (b x +a \right )}{2}+\frac {\operatorname {Ci}\left (2 b x +2 a \right )}{2}-\frac {\cos \left (b x +a \right )^{2}}{2}}{b^{2}}\) | \(82\) |
default | \(\frac {\operatorname {Si}\left (b x +a \right ) \left (a \cos \left (b x +a \right )+\sin \left (b x +a \right )-\left (b x +a \right ) \cos \left (b x +a \right )\right )-\frac {a \,\operatorname {Si}\left (2 b x +2 a \right )}{2}-\frac {\ln \left (b x +a \right )}{2}+\frac {\operatorname {Ci}\left (2 b x +2 a \right )}{2}-\frac {\cos \left (b x +a \right )^{2}}{2}}{b^{2}}\) | \(82\) |
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Time = 0.26 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.74 \[ \int x \sin (a+b x) \text {Si}(a+b x) \, dx=-\frac {2 \, b x \cos \left (b x + a\right ) \operatorname {Si}\left (b x + a\right ) + \cos \left (b x + a\right )^{2} + a \operatorname {Si}\left (2 \, b x + 2 \, a\right ) - 2 \, \sin \left (b x + a\right ) \operatorname {Si}\left (b x + a\right ) - \operatorname {Ci}\left (2 \, b x + 2 \, a\right ) + \log \left (b x + a\right )}{2 \, b^{2}} \]
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\[ \int x \sin (a+b x) \text {Si}(a+b x) \, dx=\int x \sin {\left (a + b x \right )} \operatorname {Si}{\left (a + b x \right )}\, dx \]
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\[ \int x \sin (a+b x) \text {Si}(a+b x) \, dx=\int { x \sin \left (b x + a\right ) \operatorname {Si}\left (b x + a\right ) \,d x } \]
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.31 (sec) , antiderivative size = 507, normalized size of antiderivative = 5.23 \[ \int x \sin (a+b x) \text {Si}(a+b x) \, dx=-{\left (\frac {x \cos \left (b x + a\right )}{b} - \frac {\sin \left (b x + a\right )}{b^{2}}\right )} \operatorname {Si}\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 \left (a\right )^{2} - a \Im \left ( \operatorname {Ci}\left (-2 \, b x - 2 \, a\right ) \right ) \tan \left (b x\right )^{2} \tan \left (a\right )^{2} + 2 \, a \operatorname {Si}\left (2 \, b x + 2 \, a\right ) \tan \left (b x\right )^{2} \tan \left (a\right )^{2} + 2 \, \log \left ({\left | b x + a \right |}\right ) \tan \left (b x\right )^{2} \tan \left (a\right )^{2} - \Re \left ( \operatorname {Ci}\left (2 \, b x + 2 \, a\right ) \right ) \tan \left (b x\right )^{2} \tan \left (a\right )^{2} - \Re \left ( \operatorname {Ci}\left (-2 \, b x - 2 \, a\right ) \right ) \tan \left (b x\right )^{2} \tan \left (a\right )^{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 \left (a\right )^{2} - a \Im \left ( \operatorname {Ci}\left (-2 \, b x - 2 \, a\right ) \right ) \tan \left (a\right )^{2} + 2 \, a \operatorname {Si}\left (2 \, b x + 2 \, a\right ) \tan \left (a\right )^{2} + \tan \left (b x\right )^{2} \tan \left (a\right )^{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 \left (a\right )^{2} - \Re \left ( \operatorname {Ci}\left (2 \, b x + 2 \, a\right ) \right ) \tan \left (a\right )^{2} - \Re \left ( \operatorname {Ci}\left (-2 \, b x - 2 \, a\right ) \right ) \tan \left (a\right )^{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 \left (a\right ) - \tan \left (a\right )^{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 \left (a\right )^{2} + b^{2} \tan \left (b x\right )^{2} + b^{2} \tan \left (a\right )^{2} + b^{2}\right )}} \]
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Timed out. \[ \int x \sin (a+b x) \text {Si}(a+b x) \, dx=\int x\,\mathrm {sinint}\left (a+b\,x\right )\,\sin \left (a+b\,x\right ) \,d x \]
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