Integrand size = 10, antiderivative size = 184 \[ \int x^3 \operatorname {CosIntegral}(a+b x) \, dx=\frac {3 \cos (a+b x)}{2 b^4}-\frac {a^2 \cos (a+b x)}{4 b^4}+\frac {a x \cos (a+b x)}{2 b^3}-\frac {3 x^2 \cos (a+b x)}{4 b^2}-\frac {a^4 \operatorname {CosIntegral}(a+b x)}{4 b^4}+\frac {1}{4} x^4 \operatorname {CosIntegral}(a+b x)-\frac {a \sin (a+b x)}{2 b^4}+\frac {a^3 \sin (a+b x)}{4 b^4}+\frac {3 x \sin (a+b x)}{2 b^3}-\frac {a^2 x \sin (a+b x)}{4 b^3}+\frac {a x^2 \sin (a+b x)}{4 b^2}-\frac {x^3 \sin (a+b x)}{4 b} \]
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Time = 0.32 (sec) , antiderivative size = 184, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {6639, 6874, 2717, 3377, 2718, 3383} \[ \int x^3 \operatorname {CosIntegral}(a+b x) \, dx=-\frac {a^4 \operatorname {CosIntegral}(a+b x)}{4 b^4}+\frac {a^3 \sin (a+b x)}{4 b^4}-\frac {a^2 \cos (a+b x)}{4 b^4}-\frac {a^2 x \sin (a+b x)}{4 b^3}-\frac {a \sin (a+b x)}{2 b^4}+\frac {3 \cos (a+b x)}{2 b^4}+\frac {3 x \sin (a+b x)}{2 b^3}+\frac {a x \cos (a+b x)}{2 b^3}+\frac {a x^2 \sin (a+b x)}{4 b^2}-\frac {3 x^2 \cos (a+b x)}{4 b^2}+\frac {1}{4} x^4 \operatorname {CosIntegral}(a+b x)-\frac {x^3 \sin (a+b x)}{4 b} \]
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Rule 2717
Rule 2718
Rule 3377
Rule 3383
Rule 6639
Rule 6874
Rubi steps \begin{align*} \text {integral}& = \frac {1}{4} x^4 \operatorname {CosIntegral}(a+b x)-\frac {1}{4} b \int \frac {x^4 \cos (a+b x)}{a+b x} \, dx \\ & = \frac {1}{4} x^4 \operatorname {CosIntegral}(a+b x)-\frac {1}{4} b \int \left (-\frac {a^3 \cos (a+b x)}{b^4}+\frac {a^2 x \cos (a+b x)}{b^3}-\frac {a x^2 \cos (a+b x)}{b^2}+\frac {x^3 \cos (a+b x)}{b}+\frac {a^4 \cos (a+b x)}{b^4 (a+b x)}\right ) \, dx \\ & = \frac {1}{4} x^4 \operatorname {CosIntegral}(a+b x)-\frac {1}{4} \int x^3 \cos (a+b x) \, dx+\frac {a^3 \int \cos (a+b x) \, dx}{4 b^3}-\frac {a^4 \int \frac {\cos (a+b x)}{a+b x} \, dx}{4 b^3}-\frac {a^2 \int x \cos (a+b x) \, dx}{4 b^2}+\frac {a \int x^2 \cos (a+b x) \, dx}{4 b} \\ & = -\frac {a^4 \operatorname {CosIntegral}(a+b x)}{4 b^4}+\frac {1}{4} x^4 \operatorname {CosIntegral}(a+b x)+\frac {a^3 \sin (a+b x)}{4 b^4}-\frac {a^2 x \sin (a+b x)}{4 b^3}+\frac {a x^2 \sin (a+b x)}{4 b^2}-\frac {x^3 \sin (a+b x)}{4 b}+\frac {a^2 \int \sin (a+b x) \, dx}{4 b^3}-\frac {a \int x \sin (a+b x) \, dx}{2 b^2}+\frac {3 \int x^2 \sin (a+b x) \, dx}{4 b} \\ & = -\frac {a^2 \cos (a+b x)}{4 b^4}+\frac {a x \cos (a+b x)}{2 b^3}-\frac {3 x^2 \cos (a+b x)}{4 b^2}-\frac {a^4 \operatorname {CosIntegral}(a+b x)}{4 b^4}+\frac {1}{4} x^4 \operatorname {CosIntegral}(a+b x)+\frac {a^3 \sin (a+b x)}{4 b^4}-\frac {a^2 x \sin (a+b x)}{4 b^3}+\frac {a x^2 \sin (a+b x)}{4 b^2}-\frac {x^3 \sin (a+b x)}{4 b}-\frac {a \int \cos (a+b x) \, dx}{2 b^3}+\frac {3 \int x \cos (a+b x) \, dx}{2 b^2} \\ & = -\frac {a^2 \cos (a+b x)}{4 b^4}+\frac {a x \cos (a+b x)}{2 b^3}-\frac {3 x^2 \cos (a+b x)}{4 b^2}-\frac {a^4 \operatorname {CosIntegral}(a+b x)}{4 b^4}+\frac {1}{4} x^4 \operatorname {CosIntegral}(a+b x)-\frac {a \sin (a+b x)}{2 b^4}+\frac {a^3 \sin (a+b x)}{4 b^4}+\frac {3 x \sin (a+b x)}{2 b^3}-\frac {a^2 x \sin (a+b x)}{4 b^3}+\frac {a x^2 \sin (a+b x)}{4 b^2}-\frac {x^3 \sin (a+b x)}{4 b}-\frac {3 \int \sin (a+b x) \, dx}{2 b^3} \\ & = \frac {3 \cos (a+b x)}{2 b^4}-\frac {a^2 \cos (a+b x)}{4 b^4}+\frac {a x \cos (a+b x)}{2 b^3}-\frac {3 x^2 \cos (a+b x)}{4 b^2}-\frac {a^4 \operatorname {CosIntegral}(a+b x)}{4 b^4}+\frac {1}{4} x^4 \operatorname {CosIntegral}(a+b x)-\frac {a \sin (a+b x)}{2 b^4}+\frac {a^3 \sin (a+b x)}{4 b^4}+\frac {3 x \sin (a+b x)}{2 b^3}-\frac {a^2 x \sin (a+b x)}{4 b^3}+\frac {a x^2 \sin (a+b x)}{4 b^2}-\frac {x^3 \sin (a+b x)}{4 b} \\ \end{align*}
Time = 0.16 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.52 \[ \int x^3 \operatorname {CosIntegral}(a+b x) \, dx=\frac {-\left (\left (-6+a^2-2 a b x+3 b^2 x^2\right ) \cos (a+b x)\right )+\left (-a^4+b^4 x^4\right ) \operatorname {CosIntegral}(a+b x)+\left (-2 a+a^3+6 b x-a^2 b x+a b^2 x^2-b^3 x^3\right ) \sin (a+b x)}{4 b^4} \]
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Time = 0.44 (sec) , antiderivative size = 153, normalized size of antiderivative = 0.83
method | result | size |
parts | \(\frac {x^{4} \operatorname {Ci}\left (b x +a \right )}{4}-\frac {a^{4} \operatorname {Ci}\left (b x +a \right )-4 a^{3} \sin \left (b x +a \right )+6 a^{2} \left (\cos \left (b x +a \right )+\left (b x +a \right ) \sin \left (b x +a \right )\right )-4 a \left (\left (b x +a \right )^{2} \sin \left (b x +a \right )-2 \sin \left (b x +a \right )+2 \left (b x +a \right ) \cos \left (b x +a \right )\right )+\left (b x +a \right )^{3} \sin \left (b x +a \right )+3 \left (b x +a \right )^{2} \cos \left (b x +a \right )-6 \cos \left (b x +a \right )-6 \left (b x +a \right ) \sin \left (b x +a \right )}{4 b^{4}}\) | \(153\) |
derivativedivides | \(\frac {\frac {\operatorname {Ci}\left (b x +a \right ) b^{4} x^{4}}{4}-\frac {a^{4} \operatorname {Ci}\left (b x +a \right )}{4}+a^{3} \sin \left (b x +a \right )-\frac {3 a^{2} \left (\cos \left (b x +a \right )+\left (b x +a \right ) \sin \left (b x +a \right )\right )}{2}+a \left (\left (b x +a \right )^{2} \sin \left (b x +a \right )-2 \sin \left (b x +a \right )+2 \left (b x +a \right ) \cos \left (b x +a \right )\right )-\frac {\left (b x +a \right )^{3} \sin \left (b x +a \right )}{4}-\frac {3 \left (b x +a \right )^{2} \cos \left (b x +a \right )}{4}+\frac {3 \cos \left (b x +a \right )}{2}+\frac {3 \left (b x +a \right ) \sin \left (b x +a \right )}{2}}{b^{4}}\) | \(154\) |
default | \(\frac {\frac {\operatorname {Ci}\left (b x +a \right ) b^{4} x^{4}}{4}-\frac {a^{4} \operatorname {Ci}\left (b x +a \right )}{4}+a^{3} \sin \left (b x +a \right )-\frac {3 a^{2} \left (\cos \left (b x +a \right )+\left (b x +a \right ) \sin \left (b x +a \right )\right )}{2}+a \left (\left (b x +a \right )^{2} \sin \left (b x +a \right )-2 \sin \left (b x +a \right )+2 \left (b x +a \right ) \cos \left (b x +a \right )\right )-\frac {\left (b x +a \right )^{3} \sin \left (b x +a \right )}{4}-\frac {3 \left (b x +a \right )^{2} \cos \left (b x +a \right )}{4}+\frac {3 \cos \left (b x +a \right )}{2}+\frac {3 \left (b x +a \right ) \sin \left (b x +a \right )}{2}}{b^{4}}\) | \(154\) |
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Time = 0.27 (sec) , antiderivative size = 176, normalized size of antiderivative = 0.96 \[ \int x^3 \operatorname {CosIntegral}(a+b x) \, dx=\frac {\pi ^{2} b^{5} x^{4} \operatorname {C}\left (b x + a\right ) + 6 \, \pi a^{2} \sqrt {b^{2}} \operatorname {S}\left (\frac {\sqrt {b^{2}} {\left (b x + a\right )}}{b}\right ) - {\left (\pi ^{2} a^{4} - 3\right )} \sqrt {b^{2}} \operatorname {C}\left (\frac {\sqrt {b^{2}} {\left (b x + a\right )}}{b}\right ) - {\left (3 \, b^{2} x - 5 \, a b\right )} \cos \left (\frac {1}{2} \, \pi b^{2} x^{2} + \pi a b x + \frac {1}{2} \, \pi a^{2}\right ) - {\left (\pi b^{4} x^{3} - \pi a b^{3} x^{2} + \pi a^{2} b^{2} x - \pi a^{3} b\right )} \sin \left (\frac {1}{2} \, \pi b^{2} x^{2} + \pi a b x + \frac {1}{2} \, \pi a^{2}\right )}{4 \, \pi ^{2} b^{5}} \]
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\[ \int x^3 \operatorname {CosIntegral}(a+b x) \, dx=\int x^{3} \operatorname {Ci}{\left (a + b x \right )}\, dx \]
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Result contains complex when optimal does not.
Time = 1.10 (sec) , antiderivative size = 502, normalized size of antiderivative = 2.73 \[ \int x^3 \operatorname {CosIntegral}(a+b x) \, dx=\frac {1}{4} \, x^{4} \operatorname {C}\left (b x + a\right ) + \frac {{\left (16 \, {\left (-i \, \pi ^{2} e^{\left (\frac {1}{2} i \, \pi b^{2} x^{2} + i \, \pi a b x + \frac {1}{2} i \, \pi a^{2}\right )} + i \, \pi ^{2} e^{\left (-\frac {1}{2} i \, \pi b^{2} x^{2} - i \, \pi a b x - \frac {1}{2} i \, \pi a^{2}\right )}\right )} a^{4} + 32 \, {\left (\pi \Gamma \left (2, \frac {1}{2} i \, \pi b^{2} x^{2} + i \, \pi a b x + \frac {1}{2} i \, \pi a^{2}\right ) + \pi \Gamma \left (2, -\frac {1}{2} i \, \pi b^{2} x^{2} - i \, \pi a b x - \frac {1}{2} i \, \pi a^{2}\right )\right )} a^{2} + 16 \, {\left ({\left (-i \, \pi ^{2} e^{\left (\frac {1}{2} i \, \pi b^{2} x^{2} + i \, \pi a b x + \frac {1}{2} i \, \pi a^{2}\right )} + i \, \pi ^{2} e^{\left (-\frac {1}{2} i \, \pi b^{2} x^{2} - i \, \pi a b x - \frac {1}{2} i \, \pi a^{2}\right )}\right )} a^{3} + 2 \, {\left (\pi \Gamma \left (2, \frac {1}{2} i \, \pi b^{2} x^{2} + i \, \pi a b x + \frac {1}{2} i \, \pi a^{2}\right ) + \pi \Gamma \left (2, -\frac {1}{2} i \, \pi b^{2} x^{2} - i \, \pi a b x - \frac {1}{2} i \, \pi a^{2}\right )\right )} a\right )} b x + {\left ({\left (\left (i - 1\right ) \, \sqrt {2} \pi ^{\frac {5}{2}} {\left (\operatorname {erf}\left (\sqrt {\frac {1}{2} i \, \pi b^{2} x^{2} + i \, \pi a b x + \frac {1}{2} i \, \pi a^{2}}\right ) - 1\right )} - \left (i + 1\right ) \, \sqrt {2} \pi ^{\frac {5}{2}} {\left (\operatorname {erf}\left (\sqrt {-\frac {1}{2} i \, \pi b^{2} x^{2} - i \, \pi a b x - \frac {1}{2} i \, \pi a^{2}}\right ) - 1\right )}\right )} a^{4} + 12 \, {\left (-\left (i + 1\right ) \, \sqrt {2} \pi \Gamma \left (\frac {3}{2}, \frac {1}{2} i \, \pi b^{2} x^{2} + i \, \pi a b x + \frac {1}{2} i \, \pi a^{2}\right ) + \left (i - 1\right ) \, \sqrt {2} \pi \Gamma \left (\frac {3}{2}, -\frac {1}{2} i \, \pi b^{2} x^{2} - i \, \pi a b x - \frac {1}{2} i \, \pi a^{2}\right )\right )} a^{2} + \left (4 i - 4\right ) \, \sqrt {2} \Gamma \left (\frac {5}{2}, \frac {1}{2} i \, \pi b^{2} x^{2} + i \, \pi a b x + \frac {1}{2} i \, \pi a^{2}\right ) - \left (4 i + 4\right ) \, \sqrt {2} \Gamma \left (\frac {5}{2}, -\frac {1}{2} i \, \pi b^{2} x^{2} - i \, \pi a b x - \frac {1}{2} i \, \pi a^{2}\right )\right )} \sqrt {2 \, \pi b^{2} x^{2} + 4 \, \pi a b x + 2 \, \pi a^{2}}\right )} b}{32 \, {\left (\pi ^{3} b^{6} x + \pi ^{3} a b^{5}\right )}} \]
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\[ \int x^3 \operatorname {CosIntegral}(a+b x) \, dx=\int { x^{3} \operatorname {C}\left (b x + a\right ) \,d x } \]
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Timed out. \[ \int x^3 \operatorname {CosIntegral}(a+b x) \, dx=\int x^3\,\mathrm {cosint}\left (a+b\,x\right ) \,d x \]
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