Optimal. Leaf size=125 \[ -\frac {x^2}{b^2}+\frac {3 \text {Chi}(2 b x)}{b^4}-\frac {3 \log (x)}{b^4}+\frac {2 x \cosh (b x) \sinh (b x)}{b^3}-\frac {4 \sinh ^2(b x)}{b^4}-\frac {x^2 \sinh ^2(b x)}{2 b^2}+\frac {6 x \cosh (b x) \text {Shi}(b x)}{b^3}+\frac {x^3 \cosh (b x) \text {Shi}(b x)}{b}-\frac {6 \sinh (b x) \text {Shi}(b x)}{b^4}-\frac {3 x^2 \sinh (b x) \text {Shi}(b x)}{b^2} \]
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Rubi [A]
time = 0.14, antiderivative size = 125, 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 = {6677, 12,
5480, 3391, 30, 6683, 2644, 6681, 3393, 3382} \begin {gather*} \frac {3 \text {Chi}(2 b x)}{b^4}-\frac {6 \text {Shi}(b x) \sinh (b x)}{b^4}-\frac {3 \log (x)}{b^4}-\frac {4 \sinh ^2(b x)}{b^4}+\frac {6 x \text {Shi}(b x) \cosh (b x)}{b^3}+\frac {2 x \sinh (b x) \cosh (b x)}{b^3}-\frac {3 x^2 \text {Shi}(b x) \sinh (b x)}{b^2}-\frac {x^2}{b^2}-\frac {x^2 \sinh ^2(b x)}{2 b^2}+\frac {x^3 \text {Shi}(b x) \cosh (b x)}{b} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 30
Rule 2644
Rule 3382
Rule 3391
Rule 3393
Rule 5480
Rule 6677
Rule 6681
Rule 6683
Rubi steps
\begin {align*} \int x^3 \sinh (b x) \text {Shi}(b x) \, dx &=\frac {x^3 \cosh (b x) \text {Shi}(b x)}{b}-\frac {3 \int x^2 \cosh (b x) \text {Shi}(b x) \, dx}{b}-\int \frac {x^2 \cosh (b x) \sinh (b x)}{b} \, dx\\ &=\frac {x^3 \cosh (b x) \text {Shi}(b x)}{b}-\frac {3 x^2 \sinh (b x) \text {Shi}(b x)}{b^2}+\frac {6 \int x \sinh (b x) \text {Shi}(b x) \, dx}{b^2}-\frac {\int x^2 \cosh (b x) \sinh (b x) \, dx}{b}+\frac {3 \int \frac {x \sinh ^2(b x)}{b} \, dx}{b}\\ &=-\frac {x^2 \sinh ^2(b x)}{2 b^2}+\frac {6 x \cosh (b x) \text {Shi}(b x)}{b^3}+\frac {x^3 \cosh (b x) \text {Shi}(b x)}{b}-\frac {3 x^2 \sinh (b x) \text {Shi}(b x)}{b^2}-\frac {6 \int \cosh (b x) \text {Shi}(b x) \, dx}{b^3}+\frac {\int x \sinh ^2(b x) \, dx}{b^2}+\frac {3 \int x \sinh ^2(b x) \, dx}{b^2}-\frac {6 \int \frac {\cosh (b x) \sinh (b x)}{b} \, dx}{b^2}\\ &=\frac {2 x \cosh (b x) \sinh (b x)}{b^3}-\frac {\sinh ^2(b x)}{b^4}-\frac {x^2 \sinh ^2(b x)}{2 b^2}+\frac {6 x \cosh (b x) \text {Shi}(b x)}{b^3}+\frac {x^3 \cosh (b x) \text {Shi}(b x)}{b}-\frac {6 \sinh (b x) \text {Shi}(b x)}{b^4}-\frac {3 x^2 \sinh (b x) \text {Shi}(b x)}{b^2}-\frac {6 \int \cosh (b x) \sinh (b x) \, dx}{b^3}+\frac {6 \int \frac {\sinh ^2(b x)}{b x} \, dx}{b^3}-\frac {\int x \, dx}{2 b^2}-\frac {3 \int x \, dx}{2 b^2}\\ &=-\frac {x^2}{b^2}+\frac {2 x \cosh (b x) \sinh (b x)}{b^3}-\frac {\sinh ^2(b x)}{b^4}-\frac {x^2 \sinh ^2(b x)}{2 b^2}+\frac {6 x \cosh (b x) \text {Shi}(b x)}{b^3}+\frac {x^3 \cosh (b x) \text {Shi}(b x)}{b}-\frac {6 \sinh (b x) \text {Shi}(b x)}{b^4}-\frac {3 x^2 \sinh (b x) \text {Shi}(b x)}{b^2}+\frac {6 \int \frac {\sinh ^2(b x)}{x} \, dx}{b^4}+\frac {6 \text {Subst}(\int x \, dx,x,i \sinh (b x))}{b^4}\\ &=-\frac {x^2}{b^2}+\frac {2 x \cosh (b x) \sinh (b x)}{b^3}-\frac {4 \sinh ^2(b x)}{b^4}-\frac {x^2 \sinh ^2(b x)}{2 b^2}+\frac {6 x \cosh (b x) \text {Shi}(b x)}{b^3}+\frac {x^3 \cosh (b x) \text {Shi}(b x)}{b}-\frac {6 \sinh (b x) \text {Shi}(b x)}{b^4}-\frac {3 x^2 \sinh (b x) \text {Shi}(b x)}{b^2}-\frac {6 \int \left (\frac {1}{2 x}-\frac {\cosh (2 b x)}{2 x}\right ) \, dx}{b^4}\\ &=-\frac {x^2}{b^2}-\frac {3 \log (x)}{b^4}+\frac {2 x \cosh (b x) \sinh (b x)}{b^3}-\frac {4 \sinh ^2(b x)}{b^4}-\frac {x^2 \sinh ^2(b x)}{2 b^2}+\frac {6 x \cosh (b x) \text {Shi}(b x)}{b^3}+\frac {x^3 \cosh (b x) \text {Shi}(b x)}{b}-\frac {6 \sinh (b x) \text {Shi}(b x)}{b^4}-\frac {3 x^2 \sinh (b x) \text {Shi}(b x)}{b^2}+\frac {3 \int \frac {\cosh (2 b x)}{x} \, dx}{b^4}\\ &=-\frac {x^2}{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 {4 \sinh ^2(b x)}{b^4}-\frac {x^2 \sinh ^2(b x)}{2 b^2}+\frac {6 x \cosh (b x) \text {Shi}(b x)}{b^3}+\frac {x^3 \cosh (b x) \text {Shi}(b x)}{b}-\frac {6 \sinh (b x) \text {Shi}(b x)}{b^4}-\frac {3 x^2 \sinh (b x) \text {Shi}(b x)}{b^2}\\ \end {align*}
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Mathematica [A]
time = 0.12, size = 93, normalized size = 0.74 \begin {gather*} -\frac {3 b^2 x^2+8 \cosh (2 b x)+b^2 x^2 \cosh (2 b x)-12 \text {Chi}(2 b x)+12 \log (x)-4 b x \sinh (2 b x)-4 \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 ) \text {Shi}(b x)}{4 b^4} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.32, size = 104, normalized size = 0.83
method | result | size |
derivativedivides | \(\frac {\hyperbolicSineIntegral \left (b x \right ) \left (b^{3} x^{3} \cosh \left (b x \right )-3 b^{2} x^{2} \sinh \left (b x \right )+6 b x \cosh \left (b x \right )-6 \sinh \left (b x \right )\right )-\frac {b^{2} x^{2} \left (\cosh ^{2}\left (b x \right )\right )}{2}+2 b x \cosh \left (b x \right ) \sinh \left (b x \right )-\frac {b^{2} x^{2}}{2}-4 \left (\cosh ^{2}\left (b x \right )\right )-3 \ln \left (b x \right )+3 \hyperbolicCosineIntegral \left (2 b x \right )}{b^{4}}\) | \(104\) |
default | \(\frac {\hyperbolicSineIntegral \left (b x \right ) \left (b^{3} x^{3} \cosh \left (b x \right )-3 b^{2} x^{2} \sinh \left (b x \right )+6 b x \cosh \left (b x \right )-6 \sinh \left (b x \right )\right )-\frac {b^{2} x^{2} \left (\cosh ^{2}\left (b x \right )\right )}{2}+2 b x \cosh \left (b x \right ) \sinh \left (b x \right )-\frac {b^{2} x^{2}}{2}-4 \left (\cosh ^{2}\left (b x \right )\right )-3 \ln \left (b x \right )+3 \hyperbolicCosineIntegral \left (2 b x \right )}{b^{4}}\) | \(104\) |
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 {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 {sinh}\left (b\,x\right ) \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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