Optimal. Leaf size=146 \[ -\frac {5 x}{2 b^3}-\frac {x^3}{6 b}+\frac {x \cosh ^2(b x)}{2 b^3}+\frac {6 x \cosh (b x) \text {Chi}(b x)}{b^3}+\frac {x^3 \cosh (b x) \text {Chi}(b x)}{b}-\frac {4 \cosh (b x) \sinh (b x)}{b^4}-\frac {x^2 \cosh (b x) \sinh (b x)}{2 b^2}-\frac {6 \text {Chi}(b x) \sinh (b x)}{b^4}-\frac {3 x^2 \text {Chi}(b x) \sinh (b x)}{b^2}+\frac {3 x \sinh ^2(b x)}{2 b^3}+\frac {3 \text {Shi}(2 b x)}{b^4} \]
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Rubi [A]
time = 0.14, antiderivative size = 146, normalized size of antiderivative = 1.00, number of steps
used = 20, number of rules used = 11, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.917, Rules used = {6684, 12,
3392, 30, 2715, 8, 6678, 5480, 6676, 5556, 3379} \begin {gather*} -\frac {6 \text {Chi}(b x) \sinh (b x)}{b^4}+\frac {3 \text {Shi}(2 b x)}{b^4}-\frac {4 \sinh (b x) \cosh (b x)}{b^4}+\frac {6 x \text {Chi}(b x) \cosh (b x)}{b^3}-\frac {5 x}{2 b^3}+\frac {3 x \sinh ^2(b x)}{2 b^3}+\frac {x \cosh ^2(b x)}{2 b^3}-\frac {3 x^2 \text {Chi}(b x) \sinh (b x)}{b^2}-\frac {x^2 \sinh (b x) \cosh (b x)}{2 b^2}+\frac {x^3 \text {Chi}(b x) \cosh (b x)}{b}-\frac {x^3}{6 b} \end {gather*}
Antiderivative was successfully verified.
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Rule 8
Rule 12
Rule 30
Rule 2715
Rule 3379
Rule 3392
Rule 5480
Rule 5556
Rule 6676
Rule 6678
Rule 6684
Rubi steps
\begin {align*} \int x^3 \text {Chi}(b x) \sinh (b x) \, dx &=\frac {x^3 \cosh (b x) \text {Chi}(b x)}{b}-\frac {3 \int x^2 \cosh (b x) \text {Chi}(b x) \, dx}{b}-\int \frac {x^2 \cosh ^2(b x)}{b} \, dx\\ &=\frac {x^3 \cosh (b x) \text {Chi}(b x)}{b}-\frac {3 x^2 \text {Chi}(b x) \sinh (b x)}{b^2}+\frac {6 \int x \text {Chi}(b x) \sinh (b x) \, dx}{b^2}-\frac {\int x^2 \cosh ^2(b x) \, dx}{b}+\frac {3 \int \frac {x \cosh (b x) \sinh (b x)}{b} \, dx}{b}\\ &=\frac {x \cosh ^2(b x)}{2 b^3}+\frac {6 x \cosh (b x) \text {Chi}(b x)}{b^3}+\frac {x^3 \cosh (b x) \text {Chi}(b x)}{b}-\frac {x^2 \cosh (b x) \sinh (b x)}{2 b^2}-\frac {3 x^2 \text {Chi}(b x) \sinh (b x)}{b^2}-\frac {\int \cosh ^2(b x) \, dx}{2 b^3}-\frac {6 \int \cosh (b x) \text {Chi}(b x) \, dx}{b^3}+\frac {3 \int x \cosh (b x) \sinh (b x) \, dx}{b^2}-\frac {6 \int \frac {\cosh ^2(b x)}{b} \, dx}{b^2}-\frac {\int x^2 \, dx}{2 b}\\ &=-\frac {x^3}{6 b}+\frac {x \cosh ^2(b x)}{2 b^3}+\frac {6 x \cosh (b x) \text {Chi}(b x)}{b^3}+\frac {x^3 \cosh (b x) \text {Chi}(b x)}{b}-\frac {\cosh (b x) \sinh (b x)}{4 b^4}-\frac {x^2 \cosh (b x) \sinh (b x)}{2 b^2}-\frac {6 \text {Chi}(b x) \sinh (b x)}{b^4}-\frac {3 x^2 \text {Chi}(b x) \sinh (b x)}{b^2}+\frac {3 x \sinh ^2(b x)}{2 b^3}-\frac {\int 1 \, dx}{4 b^3}-\frac {3 \int \sinh ^2(b x) \, dx}{2 b^3}-\frac {6 \int \cosh ^2(b x) \, dx}{b^3}+\frac {6 \int \frac {\cosh (b x) \sinh (b x)}{b x} \, dx}{b^3}\\ &=-\frac {x}{4 b^3}-\frac {x^3}{6 b}+\frac {x \cosh ^2(b x)}{2 b^3}+\frac {6 x \cosh (b x) \text {Chi}(b x)}{b^3}+\frac {x^3 \cosh (b x) \text {Chi}(b x)}{b}-\frac {4 \cosh (b x) \sinh (b x)}{b^4}-\frac {x^2 \cosh (b x) \sinh (b x)}{2 b^2}-\frac {6 \text {Chi}(b x) \sinh (b x)}{b^4}-\frac {3 x^2 \text {Chi}(b x) \sinh (b x)}{b^2}+\frac {3 x \sinh ^2(b x)}{2 b^3}+\frac {6 \int \frac {\cosh (b x) \sinh (b x)}{x} \, dx}{b^4}+\frac {3 \int 1 \, dx}{4 b^3}-\frac {3 \int 1 \, dx}{b^3}\\ &=-\frac {5 x}{2 b^3}-\frac {x^3}{6 b}+\frac {x \cosh ^2(b x)}{2 b^3}+\frac {6 x \cosh (b x) \text {Chi}(b x)}{b^3}+\frac {x^3 \cosh (b x) \text {Chi}(b x)}{b}-\frac {4 \cosh (b x) \sinh (b x)}{b^4}-\frac {x^2 \cosh (b x) \sinh (b x)}{2 b^2}-\frac {6 \text {Chi}(b x) \sinh (b x)}{b^4}-\frac {3 x^2 \text {Chi}(b x) \sinh (b x)}{b^2}+\frac {3 x \sinh ^2(b x)}{2 b^3}+\frac {6 \int \frac {\sinh (2 b x)}{2 x} \, dx}{b^4}\\ &=-\frac {5 x}{2 b^3}-\frac {x^3}{6 b}+\frac {x \cosh ^2(b x)}{2 b^3}+\frac {6 x \cosh (b x) \text {Chi}(b x)}{b^3}+\frac {x^3 \cosh (b x) \text {Chi}(b x)}{b}-\frac {4 \cosh (b x) \sinh (b x)}{b^4}-\frac {x^2 \cosh (b x) \sinh (b x)}{2 b^2}-\frac {6 \text {Chi}(b x) \sinh (b x)}{b^4}-\frac {3 x^2 \text {Chi}(b x) \sinh (b x)}{b^2}+\frac {3 x \sinh ^2(b x)}{2 b^3}+\frac {3 \int \frac {\sinh (2 b x)}{x} \, dx}{b^4}\\ &=-\frac {5 x}{2 b^3}-\frac {x^3}{6 b}+\frac {x \cosh ^2(b x)}{2 b^3}+\frac {6 x \cosh (b x) \text {Chi}(b x)}{b^3}+\frac {x^3 \cosh (b x) \text {Chi}(b x)}{b}-\frac {4 \cosh (b x) \sinh (b x)}{b^4}-\frac {x^2 \cosh (b x) \sinh (b x)}{2 b^2}-\frac {6 \text {Chi}(b x) \sinh (b x)}{b^4}-\frac {3 x^2 \text {Chi}(b x) \sinh (b x)}{b^2}+\frac {3 x \sinh ^2(b x)}{2 b^3}+\frac {3 \text {Shi}(2 b x)}{b^4}\\ \end {align*}
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Mathematica [A]
time = 0.08, size = 94, normalized size = 0.64 \begin {gather*} \frac {-36 b x-2 b^3 x^3+12 b x \cosh (2 b x)+12 \text {Chi}(b x) \left (b x \left (6+b^2 x^2\right ) \cosh (b x)-3 \left (2+b^2 x^2\right ) \sinh (b x)\right )-24 \sinh (2 b x)-3 b^2 x^2 \sinh (2 b x)+36 \text {Shi}(2 b x)}{12 b^4} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.36, size = 104, normalized size = 0.71
method | result | size |
derivativedivides | \(\frac {\hyperbolicCosineIntegral \left (b x \right ) \left (b^{3} x^{3} \cosh \left (b x \right )-3 b^{2} x^{2} \sinh \left (b x \right )+6 b x \cosh \left (b x \right )-6 \sinh \left (b x \right )\right )-\frac {b^{2} x^{2} \cosh \left (b x \right ) \sinh \left (b x \right )}{2}-\frac {b^{3} x^{3}}{6}+2 \left (\cosh ^{2}\left (b x \right )\right ) b x -4 \cosh \left (b x \right ) \sinh \left (b x \right )-4 b x +3 \hyperbolicSineIntegral \left (2 b x \right )}{b^{4}}\) | \(104\) |
default | \(\frac {\hyperbolicCosineIntegral \left (b x \right ) \left (b^{3} x^{3} \cosh \left (b x \right )-3 b^{2} x^{2} \sinh \left (b x \right )+6 b x \cosh \left (b x \right )-6 \sinh \left (b x \right )\right )-\frac {b^{2} x^{2} \cosh \left (b x \right ) \sinh \left (b x \right )}{2}-\frac {b^{3} x^{3}}{6}+2 \left (\cosh ^{2}\left (b x \right )\right ) b x -4 \cosh \left (b x \right ) \sinh \left (b x \right )-4 b x +3 \hyperbolicSineIntegral \left (2 b x \right )}{b^{4}}\) | \(104\) |
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 [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int x^{3} \sinh {\left (b x \right )} \operatorname {Chi}\left (b x\right )\, dx \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^3\,\mathrm {coshint}\left (b\,x\right )\,\mathrm {sinh}\left (b\,x\right ) \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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