Integrand size = 12, antiderivative size = 147 \[ \int x^3 \operatorname {CosIntegral}(b x) \sin (b x) \, dx=-\frac {5 x}{2 b^3}+\frac {x^3}{6 b}+\frac {x \cos ^2(b x)}{2 b^3}+\frac {6 x \cos (b x) \operatorname {CosIntegral}(b x)}{b^3}-\frac {x^3 \cos (b x) \operatorname {CosIntegral}(b x)}{b}-\frac {4 \cos (b x) \sin (b x)}{b^4}+\frac {x^2 \cos (b x) \sin (b x)}{2 b^2}-\frac {6 \operatorname {CosIntegral}(b x) \sin (b x)}{b^4}+\frac {3 x^2 \operatorname {CosIntegral}(b x) \sin (b x)}{b^2}-\frac {3 x \sin ^2(b x)}{2 b^3}+\frac {3 \text {Si}(2 b x)}{b^4} \]
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Time = 0.13 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.00, number of steps used = 20, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.917, Rules used = {6655, 12, 3392, 30, 2715, 8, 6649, 3524, 6647, 4491, 3380} \[ \int x^3 \operatorname {CosIntegral}(b x) \sin (b x) \, dx=-\frac {6 \operatorname {CosIntegral}(b x) \sin (b x)}{b^4}+\frac {3 \text {Si}(2 b x)}{b^4}-\frac {4 \sin (b x) \cos (b x)}{b^4}+\frac {6 x \operatorname {CosIntegral}(b x) \cos (b x)}{b^3}-\frac {5 x}{2 b^3}-\frac {3 x \sin ^2(b x)}{2 b^3}+\frac {x \cos ^2(b x)}{2 b^3}+\frac {3 x^2 \operatorname {CosIntegral}(b x) \sin (b x)}{b^2}+\frac {x^2 \sin (b x) \cos (b x)}{2 b^2}-\frac {x^3 \operatorname {CosIntegral}(b x) \cos (b x)}{b}+\frac {x^3}{6 b} \]
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Rule 8
Rule 12
Rule 30
Rule 2715
Rule 3380
Rule 3392
Rule 3524
Rule 4491
Rule 6647
Rule 6649
Rule 6655
Rubi steps \begin{align*} \text {integral}& = -\frac {x^3 \cos (b x) \operatorname {CosIntegral}(b x)}{b}+\frac {3 \int x^2 \cos (b x) \operatorname {CosIntegral}(b x) \, dx}{b}+\int \frac {x^2 \cos ^2(b x)}{b} \, dx \\ & = -\frac {x^3 \cos (b x) \operatorname {CosIntegral}(b x)}{b}+\frac {3 x^2 \operatorname {CosIntegral}(b x) \sin (b x)}{b^2}-\frac {6 \int x \operatorname {CosIntegral}(b x) \sin (b x) \, dx}{b^2}+\frac {\int x^2 \cos ^2(b x) \, dx}{b}-\frac {3 \int \frac {x \cos (b x) \sin (b x)}{b} \, dx}{b} \\ & = \frac {x \cos ^2(b x)}{2 b^3}+\frac {6 x \cos (b x) \operatorname {CosIntegral}(b x)}{b^3}-\frac {x^3 \cos (b x) \operatorname {CosIntegral}(b x)}{b}+\frac {x^2 \cos (b x) \sin (b x)}{2 b^2}+\frac {3 x^2 \operatorname {CosIntegral}(b x) \sin (b x)}{b^2}-\frac {\int \cos ^2(b x) \, dx}{2 b^3}-\frac {6 \int \cos (b x) \operatorname {CosIntegral}(b x) \, dx}{b^3}-\frac {3 \int x \cos (b x) \sin (b x) \, dx}{b^2}-\frac {6 \int \frac {\cos ^2(b x)}{b} \, dx}{b^2}+\frac {\int x^2 \, dx}{2 b} \\ & = \frac {x^3}{6 b}+\frac {x \cos ^2(b x)}{2 b^3}+\frac {6 x \cos (b x) \operatorname {CosIntegral}(b x)}{b^3}-\frac {x^3 \cos (b x) \operatorname {CosIntegral}(b x)}{b}-\frac {\cos (b x) \sin (b x)}{4 b^4}+\frac {x^2 \cos (b x) \sin (b x)}{2 b^2}-\frac {6 \operatorname {CosIntegral}(b x) \sin (b x)}{b^4}+\frac {3 x^2 \operatorname {CosIntegral}(b x) \sin (b x)}{b^2}-\frac {3 x \sin ^2(b x)}{2 b^3}-\frac {\int 1 \, dx}{4 b^3}+\frac {3 \int \sin ^2(b x) \, dx}{2 b^3}-\frac {6 \int \cos ^2(b x) \, dx}{b^3}+\frac {6 \int \frac {\cos (b x) \sin (b x)}{b x} \, dx}{b^3} \\ & = -\frac {x}{4 b^3}+\frac {x^3}{6 b}+\frac {x \cos ^2(b x)}{2 b^3}+\frac {6 x \cos (b x) \operatorname {CosIntegral}(b x)}{b^3}-\frac {x^3 \cos (b x) \operatorname {CosIntegral}(b x)}{b}-\frac {4 \cos (b x) \sin (b x)}{b^4}+\frac {x^2 \cos (b x) \sin (b x)}{2 b^2}-\frac {6 \operatorname {CosIntegral}(b x) \sin (b x)}{b^4}+\frac {3 x^2 \operatorname {CosIntegral}(b x) \sin (b x)}{b^2}-\frac {3 x \sin ^2(b x)}{2 b^3}+\frac {6 \int \frac {\cos (b x) \sin (b x)}{x} \, dx}{b^4}+\frac {3 \int 1 \, dx}{4 b^3}-\frac {3 \int 1 \, dx}{b^3} \\ & = -\frac {5 x}{2 b^3}+\frac {x^3}{6 b}+\frac {x \cos ^2(b x)}{2 b^3}+\frac {6 x \cos (b x) \operatorname {CosIntegral}(b x)}{b^3}-\frac {x^3 \cos (b x) \operatorname {CosIntegral}(b x)}{b}-\frac {4 \cos (b x) \sin (b x)}{b^4}+\frac {x^2 \cos (b x) \sin (b x)}{2 b^2}-\frac {6 \operatorname {CosIntegral}(b x) \sin (b x)}{b^4}+\frac {3 x^2 \operatorname {CosIntegral}(b x) \sin (b x)}{b^2}-\frac {3 x \sin ^2(b x)}{2 b^3}+\frac {6 \int \frac {\sin (2 b x)}{2 x} \, dx}{b^4} \\ & = -\frac {5 x}{2 b^3}+\frac {x^3}{6 b}+\frac {x \cos ^2(b x)}{2 b^3}+\frac {6 x \cos (b x) \operatorname {CosIntegral}(b x)}{b^3}-\frac {x^3 \cos (b x) \operatorname {CosIntegral}(b x)}{b}-\frac {4 \cos (b x) \sin (b x)}{b^4}+\frac {x^2 \cos (b x) \sin (b x)}{2 b^2}-\frac {6 \operatorname {CosIntegral}(b x) \sin (b x)}{b^4}+\frac {3 x^2 \operatorname {CosIntegral}(b x) \sin (b x)}{b^2}-\frac {3 x \sin ^2(b x)}{2 b^3}+\frac {3 \int \frac {\sin (2 b x)}{x} \, dx}{b^4} \\ & = -\frac {5 x}{2 b^3}+\frac {x^3}{6 b}+\frac {x \cos ^2(b x)}{2 b^3}+\frac {6 x \cos (b x) \operatorname {CosIntegral}(b x)}{b^3}-\frac {x^3 \cos (b x) \operatorname {CosIntegral}(b x)}{b}-\frac {4 \cos (b x) \sin (b x)}{b^4}+\frac {x^2 \cos (b x) \sin (b x)}{2 b^2}-\frac {6 \operatorname {CosIntegral}(b x) \sin (b x)}{b^4}+\frac {3 x^2 \operatorname {CosIntegral}(b x) \sin (b x)}{b^2}-\frac {3 x \sin ^2(b x)}{2 b^3}+\frac {3 \text {Si}(2 b x)}{b^4} \\ \end{align*}
Time = 0.07 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.64 \[ \int x^3 \operatorname {CosIntegral}(b x) \sin (b x) \, dx=\frac {-36 b x+2 b^3 x^3+12 b x \cos (2 b x)-12 \operatorname {CosIntegral}(b x) \left (b x \left (-6+b^2 x^2\right ) \cos (b x)-3 \left (-2+b^2 x^2\right ) \sin (b x)\right )-24 \sin (2 b x)+3 b^2 x^2 \sin (2 b x)+36 \text {Si}(2 b x)}{12 b^4} \]
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Time = 1.46 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.76
method | result | size |
derivativedivides | \(\frac {\operatorname {Ci}\left (b x \right ) \left (-b^{3} x^{3} \cos \left (b x \right )+3 b^{2} x^{2} \sin \left (b x \right )-6 \sin \left (b x \right )+6 b x \cos \left (b x \right )\right )+b^{2} x^{2} \left (\frac {\sin \left (b x \right ) \cos \left (b x \right )}{2}+\frac {b x}{2}\right )+2 b x \cos \left (b x \right )^{2}-4 \sin \left (b x \right ) \cos \left (b x \right )-4 b x -\frac {b^{3} x^{3}}{3}+3 \,\operatorname {Si}\left (2 b x \right )}{b^{4}}\) | \(111\) |
default | \(\frac {\operatorname {Ci}\left (b x \right ) \left (-b^{3} x^{3} \cos \left (b x \right )+3 b^{2} x^{2} \sin \left (b x \right )-6 \sin \left (b x \right )+6 b x \cos \left (b x \right )\right )+b^{2} x^{2} \left (\frac {\sin \left (b x \right ) \cos \left (b x \right )}{2}+\frac {b x}{2}\right )+2 b x \cos \left (b x \right )^{2}-4 \sin \left (b x \right ) \cos \left (b x \right )-4 b x -\frac {b^{3} x^{3}}{3}+3 \,\operatorname {Si}\left (2 b x \right )}{b^{4}}\) | \(111\) |
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Leaf count of result is larger than twice the leaf count of optimal. 361 vs. \(2 (137) = 274\).
Time = 0.30 (sec) , antiderivative size = 361, normalized size of antiderivative = 2.46 \[ \int x^3 \operatorname {CosIntegral}(b x) \sin (b x) \, dx=-\frac {2 \, \pi b \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) \cos \left (b x\right ) + 2 \, {\left (\pi ^{3} b^{4} x^{3} - 6 \, \pi ^{3} b^{2} x\right )} \cos \left (b x\right ) \operatorname {C}\left (b x\right ) + {\left (6 \, \pi ^{3} \sin \left (\frac {1}{2 \, \pi }\right ) - {\left (3 \, \pi ^{2} - 1\right )} \cos \left (\frac {1}{2 \, \pi }\right )\right )} \sqrt {b^{2}} \operatorname {C}\left (\frac {{\left (\pi b x + 1\right )} \sqrt {b^{2}}}{\pi b}\right ) - {\left (6 \, \pi ^{3} \sin \left (\frac {1}{2 \, \pi }\right ) - {\left (3 \, \pi ^{2} - 1\right )} \cos \left (\frac {1}{2 \, \pi }\right )\right )} \sqrt {b^{2}} \operatorname {C}\left (\frac {{\left (\pi b x - 1\right )} \sqrt {b^{2}}}{\pi b}\right ) - {\left (6 \, \pi ^{3} \cos \left (\frac {1}{2 \, \pi }\right ) + {\left (3 \, \pi ^{2} - 1\right )} \sin \left (\frac {1}{2 \, \pi }\right )\right )} \sqrt {b^{2}} \operatorname {S}\left (\frac {{\left (\pi b x + 1\right )} \sqrt {b^{2}}}{\pi b}\right ) + {\left (6 \, \pi ^{3} \cos \left (\frac {1}{2 \, \pi }\right ) + {\left (3 \, \pi ^{2} - 1\right )} \sin \left (\frac {1}{2 \, \pi }\right )\right )} \sqrt {b^{2}} \operatorname {S}\left (\frac {{\left (\pi b x - 1\right )} \sqrt {b^{2}}}{\pi b}\right ) + 2 \, {\left (3 \, \pi ^{2} b^{2} x \sin \left (b x\right ) - {\left (\pi ^{2} b^{3} x^{2} - 6 \, \pi ^{2} b + b\right )} \cos \left (b x\right )\right )} \sin \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) + 2 \, {\left (\pi b^{2} x \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) - 3 \, {\left (\pi ^{3} b^{3} x^{2} - 2 \, \pi ^{3} b\right )} \operatorname {C}\left (b x\right )\right )} \sin \left (b x\right )}{2 \, \pi ^{3} b^{5}} \]
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\[ \int x^3 \operatorname {CosIntegral}(b x) \sin (b x) \, dx=\int x^{3} \sin {\left (b x \right )} \operatorname {Ci}{\left (b x \right )}\, dx \]
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\[ \int x^3 \operatorname {CosIntegral}(b x) \sin (b x) \, dx=\int { x^{3} \operatorname {C}\left (b x\right ) \sin \left (b x\right ) \,d x } \]
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\[ \int x^3 \operatorname {CosIntegral}(b x) \sin (b x) \, dx=\int { x^{3} \operatorname {C}\left (b x\right ) \sin \left (b x\right ) \,d x } \]
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Timed out. \[ \int x^3 \operatorname {CosIntegral}(b x) \sin (b x) \, dx=\int x^3\,\mathrm {cosint}\left (b\,x\right )\,\sin \left (b\,x\right ) \,d x \]
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