Optimal. Leaf size=90 \[ \frac {x^{1+m} \text {CosIntegral}(b x)}{1+m}+\frac {i x^m (-i b x)^{-m} \Gamma (1+m,-i b x)}{2 b (1+m)}-\frac {i x^m (i b x)^{-m} \Gamma (1+m,i b x)}{2 b (1+m)} \]
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Rubi [A]
time = 0.06, antiderivative size = 90, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 4, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6639, 12, 3388,
2212} \begin {gather*} \frac {i x^m (-i b x)^{-m} \text {Gamma}(m+1,-i b x)}{2 b (m+1)}-\frac {i x^m (i b x)^{-m} \text {Gamma}(m+1,i b x)}{2 b (m+1)}+\frac {x^{m+1} \text {CosIntegral}(b x)}{m+1} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 2212
Rule 3388
Rule 6639
Rubi steps
\begin {align*} \int x^m \text {Ci}(b x) \, dx &=\frac {x^{1+m} \text {Ci}(b x)}{1+m}-\frac {b \int \frac {x^m \cos (b x)}{b} \, dx}{1+m}\\ &=\frac {x^{1+m} \text {Ci}(b x)}{1+m}-\frac {\int x^m \cos (b x) \, dx}{1+m}\\ &=\frac {x^{1+m} \text {Ci}(b x)}{1+m}-\frac {\int e^{-i b x} x^m \, dx}{2 (1+m)}-\frac {\int e^{i b x} x^m \, dx}{2 (1+m)}\\ &=\frac {x^{1+m} \text {Ci}(b x)}{1+m}+\frac {i x^m (-i b x)^{-m} \Gamma (1+m,-i b x)}{2 b (1+m)}-\frac {i x^m (i b x)^{-m} \Gamma (1+m,i b x)}{2 b (1+m)}\\ \end {align*}
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Mathematica [A]
time = 0.06, size = 78, normalized size = 0.87 \begin {gather*} \frac {x^m \left (2 x \text {CosIntegral}(b x)+\frac {i \left (b^2 x^2\right )^{-m} \left ((i b x)^m \Gamma (1+m,-i b x)-(-i b x)^m \Gamma (1+m,i b x)\right )}{b}\right )}{2 (1+m)} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 5 vs. order
4.
time = 0.55, size = 124, normalized size = 1.38
method | result | size |
meijerg | \(2^{-1+m} b^{-1-m} \sqrt {\pi }\, \left (-\frac {2^{-1-m} x^{3+m} b^{3+m} \hypergeom \left (\left [1, 1, \frac {3}{2}+\frac {m}{2}\right ], \left [\frac {3}{2}, 2, 2, \frac {5}{2}+\frac {m}{2}\right ], -\frac {b^{2} x^{2}}{4}\right )}{\sqrt {\pi }\, \left (3+m \right )}+\frac {2 \left (\Psi \left (\frac {1}{2}+\frac {m}{2}\right )+2 \gamma -\Psi \left (\frac {3}{2}+\frac {m}{2}\right )+2 \ln \left (x \right )+2 \ln \left (b \right )\right ) x^{1+m} 2^{-1-m} b^{1+m}}{\sqrt {\pi }\, \left (1+m \right )}\right )\) | \(124\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.11, size = 109, normalized size = 1.21 \begin {gather*} \frac {2 \, \pi b x x^{m} \operatorname {C}\left (b x\right ) - i \, {\left (\cosh \left (\frac {1}{2} \, m \log \left (\frac {1}{2} i \, \pi b^{2}\right )\right ) - \sinh \left (\frac {1}{2} \, m \log \left (\frac {1}{2} i \, \pi b^{2}\right )\right )\right )} \Gamma \left (\frac {1}{2} \, m + 1, \frac {1}{2} i \, \pi b^{2} x^{2}\right ) + i \, {\left (\cosh \left (\frac {1}{2} \, m \log \left (-\frac {1}{2} i \, \pi b^{2}\right )\right ) - \sinh \left (\frac {1}{2} \, m \log \left (-\frac {1}{2} i \, \pi b^{2}\right )\right )\right )} \Gamma \left (\frac {1}{2} \, m + 1, -\frac {1}{2} i \, \pi b^{2} x^{2}\right )}{2 \, \pi {\left (b m + b\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 654 vs.
\(2 (70) = 140\).
time = 0.98, size = 654, normalized size = 7.27 \begin {gather*} \frac {4 \cdot 2^{m} b^{- m} m x \sqrt {e^{- 2 m \log {\left (2 \right )}} e^{m \log {\left (b^{2} x^{2} \right )}}} \log {\left (b^{2} x^{2} \right )} \Gamma \left (\frac {m}{2} + \frac {5}{2}\right )}{8 m^{2} \Gamma \left (\frac {m}{2} + \frac {5}{2}\right ) + 16 m \Gamma \left (\frac {m}{2} + \frac {5}{2}\right ) + 8 \Gamma \left (\frac {m}{2} + \frac {5}{2}\right )} + \frac {8 \cdot 2^{m} \gamma b^{- m} m x \sqrt {e^{- 2 m \log {\left (2 \right )}} e^{m \log {\left (b^{2} x^{2} \right )}}} \Gamma \left (\frac {m}{2} + \frac {5}{2}\right )}{8 m^{2} \Gamma \left (\frac {m}{2} + \frac {5}{2}\right ) + 16 m \Gamma \left (\frac {m}{2} + \frac {5}{2}\right ) + 8 \Gamma \left (\frac {m}{2} + \frac {5}{2}\right )} + \frac {4 \cdot 2^{m} b^{- m} x \sqrt {e^{- 2 m \log {\left (2 \right )}} e^{m \log {\left (b^{2} x^{2} \right )}}} \log {\left (b^{2} x^{2} \right )} \Gamma \left (\frac {m}{2} + \frac {5}{2}\right )}{8 m^{2} \Gamma \left (\frac {m}{2} + \frac {5}{2}\right ) + 16 m \Gamma \left (\frac {m}{2} + \frac {5}{2}\right ) + 8 \Gamma \left (\frac {m}{2} + \frac {5}{2}\right )} - \frac {8 \cdot 2^{m} b^{- m} x \sqrt {e^{- 2 m \log {\left (2 \right )}} e^{m \log {\left (b^{2} x^{2} \right )}}} \Gamma \left (\frac {m}{2} + \frac {5}{2}\right )}{8 m^{2} \Gamma \left (\frac {m}{2} + \frac {5}{2}\right ) + 16 m \Gamma \left (\frac {m}{2} + \frac {5}{2}\right ) + 8 \Gamma \left (\frac {m}{2} + \frac {5}{2}\right )} + \frac {8 \cdot 2^{m} \gamma b^{- m} x \sqrt {e^{- 2 m \log {\left (2 \right )}} e^{m \log {\left (b^{2} x^{2} \right )}}} \Gamma \left (\frac {m}{2} + \frac {5}{2}\right )}{8 m^{2} \Gamma \left (\frac {m}{2} + \frac {5}{2}\right ) + 16 m \Gamma \left (\frac {m}{2} + \frac {5}{2}\right ) + 8 \Gamma \left (\frac {m}{2} + \frac {5}{2}\right )} - \frac {b^{2} m^{2} x^{3} x^{m} \Gamma \left (\frac {m}{2} + \frac {3}{2}\right ) {{}_{3}F_{4}\left (\begin {matrix} 1, 1, \frac {m}{2} + \frac {3}{2} \\ \frac {3}{2}, 2, 2, \frac {m}{2} + \frac {5}{2} \end {matrix}\middle | {- \frac {b^{2} x^{2}}{4}} \right )}}{8 m^{2} \Gamma \left (\frac {m}{2} + \frac {5}{2}\right ) + 16 m \Gamma \left (\frac {m}{2} + \frac {5}{2}\right ) + 8 \Gamma \left (\frac {m}{2} + \frac {5}{2}\right )} - \frac {2 b^{2} m x^{3} x^{m} \Gamma \left (\frac {m}{2} + \frac {3}{2}\right ) {{}_{3}F_{4}\left (\begin {matrix} 1, 1, \frac {m}{2} + \frac {3}{2} \\ \frac {3}{2}, 2, 2, \frac {m}{2} + \frac {5}{2} \end {matrix}\middle | {- \frac {b^{2} x^{2}}{4}} \right )}}{8 m^{2} \Gamma \left (\frac {m}{2} + \frac {5}{2}\right ) + 16 m \Gamma \left (\frac {m}{2} + \frac {5}{2}\right ) + 8 \Gamma \left (\frac {m}{2} + \frac {5}{2}\right )} - \frac {b^{2} x^{3} x^{m} \Gamma \left (\frac {m}{2} + \frac {3}{2}\right ) {{}_{3}F_{4}\left (\begin {matrix} 1, 1, \frac {m}{2} + \frac {3}{2} \\ \frac {3}{2}, 2, 2, \frac {m}{2} + \frac {5}{2} \end {matrix}\middle | {- \frac {b^{2} x^{2}}{4}} \right )}}{8 m^{2} \Gamma \left (\frac {m}{2} + \frac {5}{2}\right ) + 16 m \Gamma \left (\frac {m}{2} + \frac {5}{2}\right ) + 8 \Gamma \left (\frac {m}{2} + \frac {5}{2}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int x^m\,\mathrm {cosint}\left (b\,x\right ) \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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