Integrand size = 16, antiderivative size = 187 \[ \int x^2 \sin (a+b x) \text {Si}(a+b x) \, dx=-\frac {x}{b^2}+\frac {a \cos (2 a+2 b x)}{4 b^3}-\frac {x \cos (2 a+2 b x)}{4 b^2}-\frac {a \operatorname {CosIntegral}(2 a+2 b x)}{b^3}+\frac {a \log (a+b x)}{b^3}+\frac {\cos (a+b x) \sin (a+b x)}{b^3}+\frac {\sin (2 a+2 b x)}{8 b^3}+\frac {2 \cos (a+b x) \text {Si}(a+b x)}{b^3}-\frac {x^2 \cos (a+b x) \text {Si}(a+b x)}{b}+\frac {2 x \sin (a+b x) \text {Si}(a+b x)}{b^2}-\frac {\text {Si}(2 a+2 b x)}{b^3}+\frac {a^2 \text {Si}(2 a+2 b x)}{2 b^3} \]
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Time = 0.47 (sec) , antiderivative size = 187, normalized size of antiderivative = 1.00, number of steps used = 21, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {6648, 4669, 6873, 6874, 2718, 3377, 2717, 3380, 6654, 2715, 8, 3393, 3383, 6646, 4491, 12} \[ \int x^2 \sin (a+b x) \text {Si}(a+b x) \, dx=\frac {a^2 \text {Si}(2 a+2 b x)}{2 b^3}-\frac {a \operatorname {CosIntegral}(2 a+2 b x)}{b^3}-\frac {\text {Si}(2 a+2 b x)}{b^3}+\frac {2 \text {Si}(a+b x) \cos (a+b x)}{b^3}+\frac {a \log (a+b x)}{b^3}+\frac {\sin (2 a+2 b x)}{8 b^3}+\frac {a \cos (2 a+2 b x)}{4 b^3}+\frac {\sin (a+b x) \cos (a+b x)}{b^3}+\frac {2 x \text {Si}(a+b x) \sin (a+b x)}{b^2}-\frac {x \cos (2 a+2 b x)}{4 b^2}-\frac {x^2 \text {Si}(a+b x) \cos (a+b x)}{b}-\frac {x}{b^2} \]
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Rule 8
Rule 12
Rule 2715
Rule 2717
Rule 2718
Rule 3377
Rule 3380
Rule 3383
Rule 3393
Rule 4491
Rule 4669
Rule 6646
Rule 6648
Rule 6654
Rule 6873
Rule 6874
Rubi steps \begin{align*} \text {integral}& = -\frac {x^2 \cos (a+b x) \text {Si}(a+b x)}{b}+\frac {2 \int x \cos (a+b x) \text {Si}(a+b x) \, dx}{b}+\int \frac {x^2 \cos (a+b x) \sin (a+b x)}{a+b x} \, dx \\ & = -\frac {x^2 \cos (a+b x) \text {Si}(a+b x)}{b}+\frac {2 x \sin (a+b x) \text {Si}(a+b x)}{b^2}+\frac {1}{2} \int \frac {x^2 \sin (2 (a+b x))}{a+b x} \, dx-\frac {2 \int \sin (a+b x) \text {Si}(a+b x) \, dx}{b^2}-\frac {2 \int \frac {x \sin ^2(a+b x)}{a+b x} \, dx}{b} \\ & = \frac {2 \cos (a+b x) \text {Si}(a+b x)}{b^3}-\frac {x^2 \cos (a+b x) \text {Si}(a+b x)}{b}+\frac {2 x \sin (a+b x) \text {Si}(a+b x)}{b^2}+\frac {1}{2} \int \frac {x^2 \sin (2 a+2 b x)}{a+b x} \, dx-\frac {2 \int \frac {\cos (a+b x) \sin (a+b x)}{a+b x} \, dx}{b^2}-\frac {2 \int \left (\frac {\sin ^2(a+b x)}{b}-\frac {a \sin ^2(a+b x)}{b (a+b x)}\right ) \, dx}{b} \\ & = \frac {2 \cos (a+b x) \text {Si}(a+b x)}{b^3}-\frac {x^2 \cos (a+b x) \text {Si}(a+b x)}{b}+\frac {2 x \sin (a+b x) \text {Si}(a+b x)}{b^2}+\frac {1}{2} \int \left (-\frac {a \sin (2 a+2 b x)}{b^2}+\frac {x \sin (2 a+2 b x)}{b}+\frac {a^2 \sin (2 a+2 b x)}{b^2 (a+b x)}\right ) \, dx-\frac {2 \int \sin ^2(a+b x) \, dx}{b^2}-\frac {2 \int \frac {\sin (2 a+2 b x)}{2 (a+b x)} \, dx}{b^2}+\frac {(2 a) \int \frac {\sin ^2(a+b x)}{a+b x} \, dx}{b^2} \\ & = \frac {\cos (a+b x) \sin (a+b x)}{b^3}+\frac {2 \cos (a+b x) \text {Si}(a+b x)}{b^3}-\frac {x^2 \cos (a+b x) \text {Si}(a+b x)}{b}+\frac {2 x \sin (a+b x) \text {Si}(a+b x)}{b^2}-\frac {\int 1 \, dx}{b^2}-\frac {\int \frac {\sin (2 a+2 b x)}{a+b x} \, dx}{b^2}-\frac {a \int \sin (2 a+2 b x) \, dx}{2 b^2}+\frac {(2 a) \int \left (\frac {1}{2 (a+b x)}-\frac {\cos (2 a+2 b x)}{2 (a+b x)}\right ) \, dx}{b^2}+\frac {a^2 \int \frac {\sin (2 a+2 b x)}{a+b x} \, dx}{2 b^2}+\frac {\int x \sin (2 a+2 b x) \, dx}{2 b} \\ & = -\frac {x}{b^2}+\frac {a \cos (2 a+2 b x)}{4 b^3}-\frac {x \cos (2 a+2 b x)}{4 b^2}+\frac {a \log (a+b x)}{b^3}+\frac {\cos (a+b x) \sin (a+b x)}{b^3}+\frac {2 \cos (a+b x) \text {Si}(a+b x)}{b^3}-\frac {x^2 \cos (a+b x) \text {Si}(a+b x)}{b}+\frac {2 x \sin (a+b x) \text {Si}(a+b x)}{b^2}-\frac {\text {Si}(2 a+2 b x)}{b^3}+\frac {a^2 \text {Si}(2 a+2 b x)}{2 b^3}+\frac {\int \cos (2 a+2 b x) \, dx}{4 b^2}-\frac {a \int \frac {\cos (2 a+2 b x)}{a+b x} \, dx}{b^2} \\ & = -\frac {x}{b^2}+\frac {a \cos (2 a+2 b x)}{4 b^3}-\frac {x \cos (2 a+2 b x)}{4 b^2}-\frac {a \operatorname {CosIntegral}(2 a+2 b x)}{b^3}+\frac {a \log (a+b x)}{b^3}+\frac {\cos (a+b x) \sin (a+b x)}{b^3}+\frac {\sin (2 a+2 b x)}{8 b^3}+\frac {2 \cos (a+b x) \text {Si}(a+b x)}{b^3}-\frac {x^2 \cos (a+b x) \text {Si}(a+b x)}{b}+\frac {2 x \sin (a+b x) \text {Si}(a+b x)}{b^2}-\frac {\text {Si}(2 a+2 b x)}{b^3}+\frac {a^2 \text {Si}(2 a+2 b x)}{2 b^3} \\ \end{align*}
Time = 0.25 (sec) , antiderivative size = 123, normalized size of antiderivative = 0.66 \[ \int x^2 \sin (a+b x) \text {Si}(a+b x) \, dx=\frac {-8 b x+2 a \cos (2 (a+b x))-2 b x \cos (2 (a+b x))-8 a \operatorname {CosIntegral}(2 (a+b x))+8 a \log (a+b x)+5 \sin (2 (a+b x))-8 \left (\left (-2+b^2 x^2\right ) \cos (a+b x)-2 b x \sin (a+b x)\right ) \text {Si}(a+b x)-8 \text {Si}(2 (a+b x))+4 a^2 \text {Si}(2 (a+b x))}{8 b^3} \]
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Time = 2.51 (sec) , antiderivative size = 175, normalized size of antiderivative = 0.94
method | result | size |
derivativedivides | \(\frac {\operatorname {Si}\left (b x +a \right ) \left (-a^{2} \cos \left (b x +a \right )-2 a \left (\sin \left (b x +a \right )-\left (b x +a \right ) \cos \left (b x +a \right )\right )-\left (b x +a \right )^{2} \cos \left (b x +a \right )+2 \cos \left (b x +a \right )+2 \left (b x +a \right ) \sin \left (b x +a \right )\right )+\frac {a^{2} \operatorname {Si}\left (2 b x +2 a \right )}{2}+a \cos \left (b x +a \right )^{2}-\frac {\left (b x +a \right ) \cos \left (b x +a \right )^{2}}{2}+\frac {5 \sin \left (b x +a \right ) \cos \left (b x +a \right )}{4}-\frac {3 b x}{4}-\frac {3 a}{4}+a \ln \left (b x +a \right )-a \,\operatorname {Ci}\left (2 b x +2 a \right )-\operatorname {Si}\left (2 b x +2 a \right )}{b^{3}}\) | \(175\) |
default | \(\frac {\operatorname {Si}\left (b x +a \right ) \left (-a^{2} \cos \left (b x +a \right )-2 a \left (\sin \left (b x +a \right )-\left (b x +a \right ) \cos \left (b x +a \right )\right )-\left (b x +a \right )^{2} \cos \left (b x +a \right )+2 \cos \left (b x +a \right )+2 \left (b x +a \right ) \sin \left (b x +a \right )\right )+\frac {a^{2} \operatorname {Si}\left (2 b x +2 a \right )}{2}+a \cos \left (b x +a \right )^{2}-\frac {\left (b x +a \right ) \cos \left (b x +a \right )^{2}}{2}+\frac {5 \sin \left (b x +a \right ) \cos \left (b x +a \right )}{4}-\frac {3 b x}{4}-\frac {3 a}{4}+a \ln \left (b x +a \right )-a \,\operatorname {Ci}\left (2 b x +2 a \right )-\operatorname {Si}\left (2 b x +2 a \right )}{b^{3}}\) | \(175\) |
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Time = 0.26 (sec) , antiderivative size = 114, normalized size of antiderivative = 0.61 \[ \int x^2 \sin (a+b x) \text {Si}(a+b x) \, dx=-\frac {2 \, {\left (b x - a\right )} \cos \left (b x + a\right )^{2} + 4 \, {\left (b^{2} x^{2} - 2\right )} \cos \left (b x + a\right ) \operatorname {Si}\left (b x + a\right ) + 3 \, b x + 4 \, a \operatorname {Ci}\left (2 \, b x + 2 \, a\right ) - 4 \, a \log \left (b x + a\right ) - {\left (8 \, b x \operatorname {Si}\left (b x + a\right ) + 5 \, \cos \left (b x + a\right )\right )} \sin \left (b x + a\right ) - 2 \, {\left (a^{2} - 2\right )} \operatorname {Si}\left (2 \, b x + 2 \, a\right )}{4 \, b^{3}} \]
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\[ \int x^2 \sin (a+b x) \text {Si}(a+b x) \, dx=\int x^{2} \sin {\left (a + b x \right )} \operatorname {Si}{\left (a + b x \right )}\, dx \]
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\[ \int x^2 \sin (a+b x) \text {Si}(a+b x) \, dx=\int { x^{2} \sin \left (b x + a\right ) \operatorname {Si}\left (b x + a\right ) \,d x } \]
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.33 (sec) , antiderivative size = 398, normalized size of antiderivative = 2.13 \[ \int x^2 \sin (a+b x) \text {Si}(a+b x) \, dx={\left (\frac {2 \, x \sin \left (b x + a\right )}{b^{2}} - \frac {{\left (b^{2} x^{2} - 2\right )} \cos \left (b x + a\right )}{b^{3}}\right )} \operatorname {Si}\left (b x + a\right ) + \frac {a^{2} \Im \left ( \operatorname {Ci}\left (2 \, b x + 2 \, a\right ) \right ) \tan \left (b x + a\right )^{2} - a^{2} \Im \left ( \operatorname {Ci}\left (-2 \, b x - 2 \, a\right ) \right ) \tan \left (b x + a\right )^{2} + 2 \, a^{2} \operatorname {Si}\left (2 \, b x + 2 \, a\right ) \tan \left (b x + a\right )^{2} - 3 \, b x \tan \left (b x + a\right )^{2} + 4 \, a \log \left ({\left | b x + a \right |}\right ) \tan \left (b x + a\right )^{2} - 2 \, a \Re \left ( \operatorname {Ci}\left (2 \, b x + 2 \, a\right ) \right ) \tan \left (b x + a\right )^{2} - 2 \, a \Re \left ( \operatorname {Ci}\left (-2 \, b x - 2 \, a\right ) \right ) \tan \left (b x + a\right )^{2} + a^{2} \Im \left ( \operatorname {Ci}\left (2 \, b x + 2 \, a\right ) \right ) - a^{2} \Im \left ( \operatorname {Ci}\left (-2 \, b x - 2 \, a\right ) \right ) + 2 \, a^{2} \operatorname {Si}\left (2 \, b x + 2 \, a\right ) - a \tan \left (b x + a\right )^{2} - 2 \, \Im \left ( \operatorname {Ci}\left (2 \, b x + 2 \, a\right ) \right ) \tan \left (b x + a\right )^{2} + 2 \, \Im \left ( \operatorname {Ci}\left (-2 \, b x - 2 \, a\right ) \right ) \tan \left (b x + a\right )^{2} - 4 \, \operatorname {Si}\left (2 \, b x + 2 \, a\right ) \tan \left (b x + a\right )^{2} - 5 \, b x + 4 \, a \log \left ({\left | b x + a \right |}\right ) - 2 \, a \Re \left ( \operatorname {Ci}\left (2 \, b x + 2 \, a\right ) \right ) - 2 \, a \Re \left ( \operatorname {Ci}\left (-2 \, b x - 2 \, a\right ) \right ) + a - 2 \, \Im \left ( \operatorname {Ci}\left (2 \, b x + 2 \, a\right ) \right ) + 2 \, \Im \left ( \operatorname {Ci}\left (-2 \, b x - 2 \, a\right ) \right ) - 4 \, \operatorname {Si}\left (2 \, b x + 2 \, a\right ) + 5 \, \tan \left (b x + a\right )}{4 \, {\left (b^{3} \tan \left (b x + a\right )^{2} + b^{3}\right )}} \]
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Timed out. \[ \int x^2 \sin (a+b x) \text {Si}(a+b x) \, dx=\int x^2\,\mathrm {sinint}\left (a+b\,x\right )\,\sin \left (a+b\,x\right ) \,d x \]
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