Optimal. Leaf size=142 \[ \frac {3 \text {Chi}(2 b x)}{b^4}-\frac {6 \text {Chi}(b x) \cosh (b x)}{b^4}+\frac {3 \log (x)}{b^4}-\frac {13 \sinh ^2(b x)}{4 b^4}-\frac {3 \cosh ^2(b x)}{4 b^4}+\frac {6 x \text {Chi}(b x) \sinh (b x)}{b^3}+\frac {2 x \sinh (b x) \cosh (b x)}{b^3}-\frac {3 x^2 \text {Chi}(b x) \cosh (b x)}{b^2}+\frac {x^2}{2 b^2}-\frac {x^2 \sinh ^2(b x)}{2 b^2}+\frac {x^3 \text {Chi}(b x) \sinh (b x)}{b} \]
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Rubi [A] time = 0.21, antiderivative size = 142, normalized size of antiderivative = 1.00, number of steps used = 18, number of rules used = 10, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.833, Rules used = {6543, 12, 5372, 3310, 30, 6549, 2564, 6547, 3312, 3301} \[ -\frac {3 x^2 \text {Chi}(b x) \cosh (b x)}{b^2}+\frac {3 \text {Chi}(2 b x)}{b^4}+\frac {6 x \text {Chi}(b x) \sinh (b x)}{b^3}-\frac {6 \text {Chi}(b x) \cosh (b x)}{b^4}+\frac {x^2}{2 b^2}-\frac {x^2 \sinh ^2(b x)}{2 b^2}+\frac {3 \log (x)}{b^4}-\frac {13 \sinh ^2(b x)}{4 b^4}-\frac {3 \cosh ^2(b x)}{4 b^4}+\frac {2 x \sinh (b x) \cosh (b x)}{b^3}+\frac {x^3 \text {Chi}(b x) \sinh (b x)}{b} \]
Antiderivative was successfully verified.
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Rule 12
Rule 30
Rule 2564
Rule 3301
Rule 3310
Rule 3312
Rule 5372
Rule 6543
Rule 6547
Rule 6549
Rubi steps
\begin {align*} \int x^3 \cosh (b x) \text {Chi}(b x) \, dx &=\frac {x^3 \text {Chi}(b x) \sinh (b x)}{b}-\frac {3 \int x^2 \text {Chi}(b x) \sinh (b x) \, dx}{b}-\int \frac {x^2 \cosh (b x) \sinh (b x)}{b} \, dx\\ &=-\frac {3 x^2 \cosh (b x) \text {Chi}(b x)}{b^2}+\frac {x^3 \text {Chi}(b x) \sinh (b x)}{b}+\frac {6 \int x \cosh (b x) \text {Chi}(b x) \, dx}{b^2}-\frac {\int x^2 \cosh (b x) \sinh (b x) \, dx}{b}+\frac {3 \int \frac {x \cosh ^2(b x)}{b} \, dx}{b}\\ &=-\frac {3 x^2 \cosh (b x) \text {Chi}(b x)}{b^2}+\frac {6 x \text {Chi}(b x) \sinh (b x)}{b^3}+\frac {x^3 \text {Chi}(b x) \sinh (b x)}{b}-\frac {x^2 \sinh ^2(b x)}{2 b^2}-\frac {6 \int \text {Chi}(b x) \sinh (b x) \, dx}{b^3}+\frac {\int x \sinh ^2(b x) \, dx}{b^2}+\frac {3 \int x \cosh ^2(b x) \, dx}{b^2}-\frac {6 \int \frac {\cosh (b x) \sinh (b x)}{b} \, dx}{b^2}\\ &=-\frac {3 \cosh ^2(b x)}{4 b^4}-\frac {6 \cosh (b x) \text {Chi}(b x)}{b^4}-\frac {3 x^2 \cosh (b x) \text {Chi}(b x)}{b^2}+\frac {2 x \cosh (b x) \sinh (b x)}{b^3}+\frac {6 x \text {Chi}(b x) \sinh (b x)}{b^3}+\frac {x^3 \text {Chi}(b x) \sinh (b x)}{b}-\frac {\sinh ^2(b x)}{4 b^4}-\frac {x^2 \sinh ^2(b x)}{2 b^2}+\frac {6 \int \frac {\cosh ^2(b x)}{b x} \, dx}{b^3}-\frac {6 \int \cosh (b x) \sinh (b x) \, dx}{b^3}-\frac {\int x \, dx}{2 b^2}+\frac {3 \int x \, dx}{2 b^2}\\ &=\frac {x^2}{2 b^2}-\frac {3 \cosh ^2(b x)}{4 b^4}-\frac {6 \cosh (b x) \text {Chi}(b x)}{b^4}-\frac {3 x^2 \cosh (b x) \text {Chi}(b x)}{b^2}+\frac {2 x \cosh (b x) \sinh (b x)}{b^3}+\frac {6 x \text {Chi}(b x) \sinh (b x)}{b^3}+\frac {x^3 \text {Chi}(b x) \sinh (b x)}{b}-\frac {\sinh ^2(b x)}{4 b^4}-\frac {x^2 \sinh ^2(b x)}{2 b^2}+\frac {6 \int \frac {\cosh ^2(b x)}{x} \, dx}{b^4}+\frac {6 \operatorname {Subst}(\int x \, dx,x,i \sinh (b x))}{b^4}\\ &=\frac {x^2}{2 b^2}-\frac {3 \cosh ^2(b x)}{4 b^4}-\frac {6 \cosh (b x) \text {Chi}(b x)}{b^4}-\frac {3 x^2 \cosh (b x) \text {Chi}(b x)}{b^2}+\frac {2 x \cosh (b x) \sinh (b x)}{b^3}+\frac {6 x \text {Chi}(b x) \sinh (b x)}{b^3}+\frac {x^3 \text {Chi}(b x) \sinh (b x)}{b}-\frac {13 \sinh ^2(b x)}{4 b^4}-\frac {x^2 \sinh ^2(b x)}{2 b^2}+\frac {6 \int \left (\frac {1}{2 x}+\frac {\cosh (2 b x)}{2 x}\right ) \, dx}{b^4}\\ &=\frac {x^2}{2 b^2}-\frac {3 \cosh ^2(b x)}{4 b^4}-\frac {6 \cosh (b x) \text {Chi}(b x)}{b^4}-\frac {3 x^2 \cosh (b x) \text {Chi}(b x)}{b^2}+\frac {3 \log (x)}{b^4}+\frac {2 x \cosh (b x) \sinh (b x)}{b^3}+\frac {6 x \text {Chi}(b x) \sinh (b x)}{b^3}+\frac {x^3 \text {Chi}(b x) \sinh (b x)}{b}-\frac {13 \sinh ^2(b x)}{4 b^4}-\frac {x^2 \sinh ^2(b x)}{2 b^2}+\frac {3 \int \frac {\cosh (2 b x)}{x} \, dx}{b^4}\\ &=\frac {x^2}{2 b^2}-\frac {3 \cosh ^2(b x)}{4 b^4}-\frac {6 \cosh (b x) \text {Chi}(b x)}{b^4}-\frac {3 x^2 \cosh (b x) \text {Chi}(b x)}{b^2}+\frac {3 \text {Chi}(2 b x)}{b^4}+\frac {3 \log (x)}{b^4}+\frac {2 x \cosh (b x) \sinh (b x)}{b^3}+\frac {6 x \text {Chi}(b x) \sinh (b x)}{b^3}+\frac {x^3 \text {Chi}(b x) \sinh (b x)}{b}-\frac {13 \sinh ^2(b x)}{4 b^4}-\frac {x^2 \sinh ^2(b x)}{2 b^2}\\ \end {align*}
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Mathematica [A] time = 0.10, size = 94, normalized size = 0.66 \[ \frac {4 \text {Chi}(b x) \left (b x \left (b^2 x^2+6\right ) \sinh (b x)-3 \left (b^2 x^2+2\right ) \cosh (b x)\right )+3 b^2 x^2-b^2 x^2 \cosh (2 b x)+12 \text {Chi}(2 b x)+4 b x \sinh (2 b x)-8 \cosh (2 b x)+12 \log (x)}{4 b^4} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.51, size = 0, normalized size = 0.00 \[ {\rm integral}\left (x^{3} \cosh \left (b x\right ) \operatorname {Chi}\left (b x\right ), x\right ) \]
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 \[ \int x^{3} {\rm Chi}\left (b x\right ) \cosh \left (b x\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 125, normalized size = 0.88 \[ \frac {x^{3} \Chi \left (b x \right ) \sinh \left (b x \right )}{b}-\frac {3 x^{2} \Chi \left (b x \right ) \cosh \left (b x \right )}{b^{2}}+\frac {6 x \Chi \left (b x \right ) \sinh \left (b x \right )}{b^{3}}-\frac {6 \Chi \left (b x \right ) \cosh \left (b x \right )}{b^{4}}-\frac {x^{2} \left (\cosh ^{2}\left (b x \right )\right )}{2 b^{2}}+\frac {2 x \cosh \left (b x \right ) \sinh \left (b x \right )}{b^{3}}+\frac {x^{2}}{b^{2}}-\frac {4 \left (\cosh ^{2}\left (b x \right )\right )}{b^{4}}+\frac {3 \ln \left (b x \right )}{b^{4}}+\frac {3 \Chi \left (2 b x \right )}{b^{4}} \]
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 \[ \int x^{3} {\rm Chi}\left (b x\right ) \cosh \left (b x\right )\,{d x} \]
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 \[ \int x^3\,\mathrm {coshint}\left (b\,x\right )\,\mathrm {cosh}\left (b\,x\right ) \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{3} \cosh {\left (b x \right )} \operatorname {Chi}\left (b x\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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