Integrand size = 8, antiderivative size = 63 \[ \int x^3 \operatorname {CosIntegral}(b x) \, dx=\frac {3 \cos (b x)}{2 b^4}-\frac {3 x^2 \cos (b x)}{4 b^2}+\frac {1}{4} x^4 \operatorname {CosIntegral}(b x)+\frac {3 x \sin (b x)}{2 b^3}-\frac {x^3 \sin (b x)}{4 b} \]
[Out]
Time = 0.05 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6639, 12, 3377, 2718} \[ \int x^3 \operatorname {CosIntegral}(b x) \, dx=\frac {3 \cos (b x)}{2 b^4}+\frac {3 x \sin (b x)}{2 b^3}-\frac {3 x^2 \cos (b x)}{4 b^2}+\frac {1}{4} x^4 \operatorname {CosIntegral}(b x)-\frac {x^3 \sin (b x)}{4 b} \]
[In]
[Out]
Rule 12
Rule 2718
Rule 3377
Rule 6639
Rubi steps \begin{align*} \text {integral}& = \frac {1}{4} x^4 \operatorname {CosIntegral}(b x)-\frac {1}{4} b \int \frac {x^3 \cos (b x)}{b} \, dx \\ & = \frac {1}{4} x^4 \operatorname {CosIntegral}(b x)-\frac {1}{4} \int x^3 \cos (b x) \, dx \\ & = \frac {1}{4} x^4 \operatorname {CosIntegral}(b x)-\frac {x^3 \sin (b x)}{4 b}+\frac {3 \int x^2 \sin (b x) \, dx}{4 b} \\ & = -\frac {3 x^2 \cos (b x)}{4 b^2}+\frac {1}{4} x^4 \operatorname {CosIntegral}(b x)-\frac {x^3 \sin (b x)}{4 b}+\frac {3 \int x \cos (b x) \, dx}{2 b^2} \\ & = -\frac {3 x^2 \cos (b x)}{4 b^2}+\frac {1}{4} x^4 \operatorname {CosIntegral}(b x)+\frac {3 x \sin (b x)}{2 b^3}-\frac {x^3 \sin (b x)}{4 b}-\frac {3 \int \sin (b x) \, dx}{2 b^3} \\ & = \frac {3 \cos (b x)}{2 b^4}-\frac {3 x^2 \cos (b x)}{4 b^2}+\frac {1}{4} x^4 \operatorname {CosIntegral}(b x)+\frac {3 x \sin (b x)}{2 b^3}-\frac {x^3 \sin (b x)}{4 b} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.84 \[ \int x^3 \operatorname {CosIntegral}(b x) \, dx=-\frac {3 \left (-2+b^2 x^2\right ) \cos (b x)}{4 b^4}+\frac {1}{4} x^4 \operatorname {CosIntegral}(b x)-\frac {x \left (-6+b^2 x^2\right ) \sin (b x)}{4 b^3} \]
[In]
[Out]
Time = 0.36 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.86
method | result | size |
parts | \(\frac {x^{4} \operatorname {Ci}\left (b x \right )}{4}-\frac {b^{3} x^{3} \sin \left (b x \right )+3 b^{2} x^{2} \cos \left (b x \right )-6 \cos \left (b x \right )-6 b x \sin \left (b x \right )}{4 b^{4}}\) | \(54\) |
derivativedivides | \(\frac {\frac {b^{4} x^{4} \operatorname {Ci}\left (b x \right )}{4}-\frac {b^{3} x^{3} \sin \left (b x \right )}{4}-\frac {3 b^{2} x^{2} \cos \left (b x \right )}{4}+\frac {3 \cos \left (b x \right )}{2}+\frac {3 b x \sin \left (b x \right )}{2}}{b^{4}}\) | \(56\) |
default | \(\frac {\frac {b^{4} x^{4} \operatorname {Ci}\left (b x \right )}{4}-\frac {b^{3} x^{3} \sin \left (b x \right )}{4}-\frac {3 b^{2} x^{2} \cos \left (b x \right )}{4}+\frac {3 \cos \left (b x \right )}{2}+\frac {3 b x \sin \left (b x \right )}{2}}{b^{4}}\) | \(56\) |
meijerg | \(\frac {4 \sqrt {\pi }\, \left (-\frac {b^{6} x^{6} \operatorname {hypergeom}\left (\left [1, 1, 3\right ], \left [\frac {3}{2}, 2, 2, 4\right ], -\frac {b^{2} x^{2}}{4}\right )}{96 \sqrt {\pi }}+\frac {\left (-\frac {1}{2}+2 \gamma +2 \ln \left (x \right )+2 \ln \left (b \right )\right ) x^{4} b^{4}}{32 \sqrt {\pi }}\right )}{b^{4}}\) | \(63\) |
[In]
[Out]
none
Time = 0.26 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.94 \[ \int x^3 \operatorname {CosIntegral}(b x) \, dx=-\frac {\pi b^{3} x^{3} \sin \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) + 3 \, b x \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) - {\left (\pi ^{2} b^{4} x^{4} + 3\right )} \operatorname {C}\left (b x\right )}{4 \, \pi ^{2} b^{4}} \]
[In]
[Out]
Time = 1.52 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.35 \[ \int x^3 \operatorname {CosIntegral}(b x) \, dx=- \frac {x^{4} \log {\left (b x \right )}}{4} + \frac {x^{4} \log {\left (b^{2} x^{2} \right )}}{8} + \frac {x^{4} \operatorname {Ci}{\left (b x \right )}}{4} - \frac {x^{3} \sin {\left (b x \right )}}{4 b} - \frac {3 x^{2} \cos {\left (b x \right )}}{4 b^{2}} + \frac {3 x \sin {\left (b x \right )}}{2 b^{3}} + \frac {3 \cos {\left (b x \right )}}{2 b^{4}} \]
[In]
[Out]
Result contains complex when optimal does not.
Time = 0.29 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.49 \[ \int x^3 \operatorname {CosIntegral}(b x) \, dx=\frac {1}{4} \, x^{4} \operatorname {C}\left (b x\right ) - \frac {\sqrt {\frac {1}{2}} {\left (4 \, \sqrt {\frac {1}{2}} \pi ^{2} b^{3} x^{3} \sin \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) + 12 \, \sqrt {\frac {1}{2}} \pi b x \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) + \left (3 i - 3\right ) \, \left (\frac {1}{4}\right )^{\frac {1}{4}} \pi \operatorname {erf}\left (\sqrt {\frac {1}{2} i \, \pi } b x\right ) - \left (3 i + 3\right ) \, \left (\frac {1}{4}\right )^{\frac {1}{4}} \pi \operatorname {erf}\left (\sqrt {-\frac {1}{2} i \, \pi } b x\right )\right )}}{8 \, \pi ^{3} b^{4}} \]
[In]
[Out]
\[ \int x^3 \operatorname {CosIntegral}(b x) \, dx=\int { x^{3} \operatorname {C}\left (b x\right ) \,d x } \]
[In]
[Out]
Timed out. \[ \int x^3 \operatorname {CosIntegral}(b x) \, dx=\frac {6\,\cos \left (b\,x\right )-3\,b^2\,x^2\,\cos \left (b\,x\right )-b^3\,x^3\,\sin \left (b\,x\right )+6\,b\,x\,\sin \left (b\,x\right )}{4\,b^4}+\frac {x^4\,\mathrm {cosint}\left (b\,x\right )}{4} \]
[In]
[Out]