Optimal. Leaf size=76 \[ -\frac{x^m (-b x)^{-m} \text{Gamma}(m+1,-b x)}{2 b (m+1)}+\frac{x^m (b x)^{-m} \text{Gamma}(m+1,b x)}{2 b (m+1)}+\frac{x^{m+1} \text{Chi}(b x)}{m+1} \]
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Rubi [A] time = 0.0771015, antiderivative size = 76, 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 = {6533, 12, 3307, 2181} \[ -\frac{x^m (-b x)^{-m} \text{Gamma}(m+1,-b x)}{2 b (m+1)}+\frac{x^m (b x)^{-m} \text{Gamma}(m+1,b x)}{2 b (m+1)}+\frac{x^{m+1} \text{Chi}(b x)}{m+1} \]
Antiderivative was successfully verified.
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Rule 6533
Rule 12
Rule 3307
Rule 2181
Rubi steps
\begin{align*} \int x^m \text{Chi}(b x) \, dx &=\frac{x^{1+m} \text{Chi}(b x)}{1+m}-\frac{b \int \frac{x^m \cosh (b x)}{b} \, dx}{1+m}\\ &=\frac{x^{1+m} \text{Chi}(b x)}{1+m}-\frac{\int x^m \cosh (b x) \, dx}{1+m}\\ &=\frac{x^{1+m} \text{Chi}(b x)}{1+m}-\frac{\int e^{-b x} x^m \, dx}{2 (1+m)}-\frac{\int e^{b x} x^m \, dx}{2 (1+m)}\\ &=\frac{x^{1+m} \text{Chi}(b x)}{1+m}-\frac{x^m (-b x)^{-m} \Gamma (1+m,-b x)}{2 b (1+m)}+\frac{x^m (b x)^{-m} \Gamma (1+m,b x)}{2 b (1+m)}\\ \end{align*}
Mathematica [A] time = 0.0565109, size = 74, normalized size = 0.97 \[ \frac{x^{m+1} \text{Chi}(b x)}{m+1}-\frac{-x^{m+1} (-b x)^{-m-1} \text{Gamma}(m+1,-b x)-x^{m+1} (b x)^{-m-1} \text{Gamma}(m+1,b x)}{2 (m+1)} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.069, size = 0, normalized size = 0. \begin{align*} \int{x}^{m}{\it Chi} \left ( bx \right ) \, dx \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 Chi}\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{Chi}\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 = 2.34736, size = 649, normalized size = 8.54 \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 Chi}\left (b x\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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