Optimal. Leaf size=90 \[ \frac{i x^m (-i b x)^{-m} \text{Gamma}(m+1,-i b x)}{2 b (m+1)}-\frac{i x^m (i b x)^{-m} \text{Gamma}(m+1,i b x)}{2 b (m+1)}+\frac{x^{m+1} \text{CosIntegral}(b x)}{m+1} \]
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Rubi [A] time = 0.0823221, antiderivative size = 90, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {6504, 12, 3307, 2181} \[ \frac{i x^m (-i b x)^{-m} \text{Gamma}(m+1,-i b x)}{2 b (m+1)}-\frac{i x^m (i b x)^{-m} \text{Gamma}(m+1,i b x)}{2 b (m+1)}+\frac{x^{m+1} \text{CosIntegral}(b x)}{m+1} \]
Antiderivative was successfully verified.
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Rule 6504
Rule 12
Rule 3307
Rule 2181
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*}
Mathematica [A] time = 0.0762758, size = 78, normalized size = 0.87 \[ \frac{x^m \left (2 x \text{CosIntegral}(b x)+\frac{i \left (b^2 x^2\right )^{-m} \left ((i b x)^m \text{Gamma}(m+1,-i b x)-(-i b x)^m \text{Gamma}(m+1,i b x)\right )}{b}\right )}{2 (m+1)} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.099, size = 124, normalized size = 1.4 \begin{align*}{2}^{-1+m}{b}^{-1-m}\sqrt{\pi } \left ( 2\,{\frac{ \left ( \Psi \left ( 1/2+m/2 \right ) +2\,\gamma -\Psi \left ( 3/2+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 ) }}-{\frac{{2}^{-1-m}{x}^{3+m}{b}^{3+m}}{\sqrt{\pi } \left ( 3+m \right ) }{\mbox{$_3$F$_4$}(1,1,{\frac{3}{2}}+{\frac{m}{2}};\,{\frac{3}{2}},2,2,{\frac{5}{2}}+{\frac{m}{2}};\,-{\frac{{b}^{2}{x}^{2}}{4}})}} \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{m}{\rm Ci}\left (b x\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (x^{m} \operatorname{Ci}\left (b x\right ), x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 2.17449, size = 654, normalized size = 7.27 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{m}{\rm Ci}\left (b x\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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