Optimal. Leaf size=49 \[ \frac {2 x \cosh (b x)}{3 b^2}+\frac {1}{3} x^3 \text {Chi}(b x)-\frac {2 \sinh (b x)}{3 b^3}-\frac {x^2 \sinh (b x)}{3 b} \]
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Rubi [A]
time = 0.04, antiderivative size = 49, 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 = {6668, 12, 3377,
2717} \begin {gather*} -\frac {2 \sinh (b x)}{3 b^3}+\frac {2 x \cosh (b x)}{3 b^2}+\frac {1}{3} x^3 \text {Chi}(b x)-\frac {x^2 \sinh (b x)}{3 b} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 2717
Rule 3377
Rule 6668
Rubi steps
\begin {align*} \int x^2 \text {Chi}(b x) \, dx &=\frac {1}{3} x^3 \text {Chi}(b x)-\frac {1}{3} b \int \frac {x^2 \cosh (b x)}{b} \, dx\\ &=\frac {1}{3} x^3 \text {Chi}(b x)-\frac {1}{3} \int x^2 \cosh (b x) \, dx\\ &=\frac {1}{3} x^3 \text {Chi}(b x)-\frac {x^2 \sinh (b x)}{3 b}+\frac {2 \int x \sinh (b x) \, dx}{3 b}\\ &=\frac {2 x \cosh (b x)}{3 b^2}+\frac {1}{3} x^3 \text {Chi}(b x)-\frac {x^2 \sinh (b x)}{3 b}-\frac {2 \int \cosh (b x) \, dx}{3 b^2}\\ &=\frac {2 x \cosh (b x)}{3 b^2}+\frac {1}{3} x^3 \text {Chi}(b x)-\frac {2 \sinh (b x)}{3 b^3}-\frac {x^2 \sinh (b x)}{3 b}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 44, normalized size = 0.90 \begin {gather*} \frac {2 x \cosh (b x)}{3 b^2}+\frac {1}{3} x^3 \text {Chi}(b x)-\frac {\left (2+b^2 x^2\right ) \sinh (b x)}{3 b^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.26, size = 44, normalized size = 0.90
method | result | size |
derivativedivides | \(\frac {\frac {b^{3} x^{3} \hyperbolicCosineIntegral \left (b x \right )}{3}-\frac {b^{2} x^{2} \sinh \left (b x \right )}{3}+\frac {2 b x \cosh \left (b x \right )}{3}-\frac {2 \sinh \left (b x \right )}{3}}{b^{3}}\) | \(44\) |
default | \(\frac {\frac {b^{3} x^{3} \hyperbolicCosineIntegral \left (b x \right )}{3}-\frac {b^{2} x^{2} \sinh \left (b x \right )}{3}+\frac {2 b x \cosh \left (b x \right )}{3}-\frac {2 \sinh \left (b x \right )}{3}}{b^{3}}\) | \(44\) |
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 [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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Sympy [A]
time = 1.22, size = 70, normalized size = 1.43 \begin {gather*} - \frac {x^{3} \log {\left (b x \right )}}{3} + \frac {x^{3} \log {\left (b^{2} x^{2} \right )}}{6} + \frac {x^{3} \operatorname {Chi}\left (b x\right )}{3} - \frac {x^{2} \sinh {\left (b x \right )}}{3 b} + \frac {2 x \cosh {\left (b x \right )}}{3 b^{2}} - \frac {2 \sinh {\left (b x \right )}}{3 b^{3}} \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 = -0.02 \begin {gather*} \frac {x^3\,\mathrm {coshint}\left (b\,x\right )}{3}-\frac {\frac {2\,\mathrm {sinh}\left (b\,x\right )}{3}+\frac {b^2\,x^2\,\mathrm {sinh}\left (b\,x\right )}{3}-\frac {2\,b\,x\,\mathrm {cosh}\left (b\,x\right )}{3}}{b^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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