Optimal. Leaf size=128 \[ \frac {4 x}{b^3}+\frac {x^3}{6 b}-\frac {4 \cosh (b x) \sinh (b x)}{b^4}-\frac {x^2 \cosh (b x) \sinh (b x)}{2 b^2}+\frac {2 x \sinh ^2(b x)}{b^3}-\frac {6 \cosh (b x) \text {Shi}(b x)}{b^4}-\frac {3 x^2 \cosh (b x) \text {Shi}(b x)}{b^2}+\frac {6 x \sinh (b x) \text {Shi}(b x)}{b^3}+\frac {x^3 \sinh (b x) \text {Shi}(b x)}{b}+\frac {3 \text {Shi}(2 b x)}{b^4} \]
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Rubi [A]
time = 0.13, antiderivative size = 128, 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 = {6683, 12,
3392, 30, 2715, 8, 6677, 5480, 6675, 5556, 3379} \begin {gather*} \frac {3 \text {Shi}(2 b x)}{b^4}-\frac {6 \text {Shi}(b x) \cosh (b x)}{b^4}-\frac {4 \sinh (b x) \cosh (b x)}{b^4}+\frac {6 x \text {Shi}(b x) \sinh (b x)}{b^3}+\frac {4 x}{b^3}+\frac {2 x \sinh ^2(b x)}{b^3}-\frac {3 x^2 \text {Shi}(b x) \cosh (b x)}{b^2}-\frac {x^2 \sinh (b x) \cosh (b x)}{2 b^2}+\frac {x^3 \text {Shi}(b x) \sinh (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 6675
Rule 6677
Rule 6683
Rubi steps
\begin {align*} \int x^3 \cosh (b x) \text {Shi}(b x) \, dx &=\frac {x^3 \sinh (b x) \text {Shi}(b x)}{b}-\frac {3 \int x^2 \sinh (b x) \text {Shi}(b x) \, dx}{b}-\int \frac {x^2 \sinh ^2(b x)}{b} \, dx\\ &=-\frac {3 x^2 \cosh (b x) \text {Shi}(b x)}{b^2}+\frac {x^3 \sinh (b x) \text {Shi}(b x)}{b}+\frac {6 \int x \cosh (b x) \text {Shi}(b x) \, dx}{b^2}-\frac {\int x^2 \sinh ^2(b x) \, dx}{b}+\frac {3 \int \frac {x \cosh (b x) \sinh (b x)}{b} \, dx}{b}\\ &=-\frac {x^2 \cosh (b x) \sinh (b x)}{2 b^2}+\frac {x \sinh ^2(b x)}{2 b^3}-\frac {3 x^2 \cosh (b x) \text {Shi}(b x)}{b^2}+\frac {6 x \sinh (b x) \text {Shi}(b x)}{b^3}+\frac {x^3 \sinh (b x) \text {Shi}(b x)}{b}-\frac {\int \sinh ^2(b x) \, dx}{2 b^3}-\frac {6 \int \sinh (b x) \text {Shi}(b x) \, dx}{b^3}+\frac {3 \int x \cosh (b x) \sinh (b x) \, dx}{b^2}-\frac {6 \int \frac {\sinh ^2(b x)}{b} \, dx}{b^2}+\frac {\int x^2 \, dx}{2 b}\\ &=\frac {x^3}{6 b}-\frac {\cosh (b x) \sinh (b x)}{4 b^4}-\frac {x^2 \cosh (b x) \sinh (b x)}{2 b^2}+\frac {2 x \sinh ^2(b x)}{b^3}-\frac {6 \cosh (b x) \text {Shi}(b x)}{b^4}-\frac {3 x^2 \cosh (b x) \text {Shi}(b x)}{b^2}+\frac {6 x \sinh (b x) \text {Shi}(b x)}{b^3}+\frac {x^3 \sinh (b x) \text {Shi}(b x)}{b}+\frac {\int 1 \, dx}{4 b^3}-\frac {3 \int \sinh ^2(b x) \, dx}{2 b^3}+\frac {6 \int \frac {\cosh (b x) \sinh (b x)}{b x} \, dx}{b^3}-\frac {6 \int \sinh ^2(b x) \, dx}{b^3}\\ &=\frac {x}{4 b^3}+\frac {x^3}{6 b}-\frac {4 \cosh (b x) \sinh (b x)}{b^4}-\frac {x^2 \cosh (b x) \sinh (b x)}{2 b^2}+\frac {2 x \sinh ^2(b x)}{b^3}-\frac {6 \cosh (b x) \text {Shi}(b x)}{b^4}-\frac {3 x^2 \cosh (b x) \text {Shi}(b x)}{b^2}+\frac {6 x \sinh (b x) \text {Shi}(b x)}{b^3}+\frac {x^3 \sinh (b x) \text {Shi}(b x)}{b}+\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 {4 x}{b^3}+\frac {x^3}{6 b}-\frac {4 \cosh (b x) \sinh (b x)}{b^4}-\frac {x^2 \cosh (b x) \sinh (b x)}{2 b^2}+\frac {2 x \sinh ^2(b x)}{b^3}-\frac {6 \cosh (b x) \text {Shi}(b x)}{b^4}-\frac {3 x^2 \cosh (b x) \text {Shi}(b x)}{b^2}+\frac {6 x \sinh (b x) \text {Shi}(b x)}{b^3}+\frac {x^3 \sinh (b x) \text {Shi}(b x)}{b}+\frac {6 \int \frac {\sinh (2 b x)}{2 x} \, dx}{b^4}\\ &=\frac {4 x}{b^3}+\frac {x^3}{6 b}-\frac {4 \cosh (b x) \sinh (b x)}{b^4}-\frac {x^2 \cosh (b x) \sinh (b x)}{2 b^2}+\frac {2 x \sinh ^2(b x)}{b^3}-\frac {6 \cosh (b x) \text {Shi}(b x)}{b^4}-\frac {3 x^2 \cosh (b x) \text {Shi}(b x)}{b^2}+\frac {6 x \sinh (b x) \text {Shi}(b x)}{b^3}+\frac {x^3 \sinh (b x) \text {Shi}(b x)}{b}+\frac {3 \int \frac {\sinh (2 b x)}{x} \, dx}{b^4}\\ &=\frac {4 x}{b^3}+\frac {x^3}{6 b}-\frac {4 \cosh (b x) \sinh (b x)}{b^4}-\frac {x^2 \cosh (b x) \sinh (b x)}{2 b^2}+\frac {2 x \sinh ^2(b x)}{b^3}-\frac {6 \cosh (b x) \text {Shi}(b x)}{b^4}-\frac {3 x^2 \cosh (b x) \text {Shi}(b x)}{b^2}+\frac {6 x \sinh (b x) \text {Shi}(b x)}{b^3}+\frac {x^3 \sinh (b x) \text {Shi}(b x)}{b}+\frac {3 \text {Shi}(2 b x)}{b^4}\\ \end {align*}
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Mathematica [A]
time = 0.07, size = 94, normalized size = 0.73 \begin {gather*} \frac {36 b x+2 b^3 x^3+12 b x \cosh (2 b x)-24 \sinh (2 b x)-3 b^2 x^2 \sinh (2 b x)+12 \left (-3 \left (2+b^2 x^2\right ) \cosh (b x)+b x \left (6+b^2 x^2\right ) \sinh (b x)\right ) \text {Shi}(b x)+36 \text {Shi}(2 b x)}{12 b^4} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.30, size = 104, normalized size = 0.81
method | result | size |
derivativedivides | \(\frac {\hyperbolicSineIntegral \left (b x \right ) \left (b^{3} x^{3} \sinh \left (b x \right )-3 b^{2} x^{2} \cosh \left (b x \right )+6 b x \sinh \left (b x \right )-6 \cosh \left (b x \right )\right )-\frac {b^{2} x^{2} \cosh \left (b x \right ) \sinh \left (b x \right )}{2}+\frac {b^{3} x^{3}}{6}+2 b x \left (\cosh ^{2}\left (b x \right )\right )-4 \cosh \left (b x \right ) \sinh \left (b x \right )+2 b x +3 \hyperbolicSineIntegral \left (2 b x \right )}{b^{4}}\) | \(104\) |
default | \(\frac {\hyperbolicSineIntegral \left (b x \right ) \left (b^{3} x^{3} \sinh \left (b x \right )-3 b^{2} x^{2} \cosh \left (b x \right )+6 b x \sinh \left (b x \right )-6 \cosh \left (b x \right )\right )-\frac {b^{2} x^{2} \cosh \left (b x \right ) \sinh \left (b x \right )}{2}+\frac {b^{3} x^{3}}{6}+2 b x \left (\cosh ^{2}\left (b x \right )\right )-4 \cosh \left (b x \right ) \sinh \left (b x \right )+2 b x +3 \hyperbolicSineIntegral \left (2 b x \right )}{b^{4}}\) | \(104\) |
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} \cosh {\left (b x \right )} \operatorname {Shi}{\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 {sinhint}\left (b\,x\right )\,\mathrm {cosh}\left (b\,x\right ) \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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