Optimal. Leaf size=186 \[ -\frac {x}{b^2}+\frac {a \cosh (2 a+2 b x)}{4 b^3}-\frac {x \cosh (2 a+2 b x)}{4 b^2}-\frac {a \text {Chi}(2 a+2 b x)}{b^3}+\frac {a \log (a+b x)}{b^3}+\frac {\cosh (a+b x) \sinh (a+b x)}{b^3}+\frac {\sinh (2 a+2 b x)}{8 b^3}+\frac {2 \cosh (a+b x) \text {Shi}(a+b x)}{b^3}+\frac {x^2 \cosh (a+b x) \text {Shi}(a+b x)}{b}-\frac {2 x \sinh (a+b x) \text {Shi}(a+b x)}{b^2}-\frac {\text {Shi}(2 a+2 b x)}{b^3}-\frac {a^2 \text {Shi}(2 a+2 b x)}{2 b^3} \]
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Rubi [A]
time = 0.49, antiderivative size = 186, normalized size of antiderivative = 1.00, number of steps
used = 21, number of rules used = 16, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {6677,
5736, 6873, 6874, 2718, 3377, 2717, 3379, 6683, 2715, 8, 3393, 3382, 6675, 5556, 12}
\begin {gather*} -\frac {a^2 \text {Shi}(2 a+2 b x)}{2 b^3}-\frac {a \text {Chi}(2 a+2 b x)}{b^3}-\frac {\text {Shi}(2 a+2 b x)}{b^3}+\frac {2 \text {Shi}(a+b x) \cosh (a+b x)}{b^3}+\frac {a \log (a+b x)}{b^3}+\frac {\sinh (2 a+2 b x)}{8 b^3}+\frac {a \cosh (2 a+2 b x)}{4 b^3}+\frac {\sinh (a+b x) \cosh (a+b x)}{b^3}-\frac {2 x \text {Shi}(a+b x) \sinh (a+b x)}{b^2}-\frac {x \cosh (2 a+2 b x)}{4 b^2}+\frac {x^2 \text {Shi}(a+b x) \cosh (a+b x)}{b}-\frac {x}{b^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 8
Rule 12
Rule 2715
Rule 2717
Rule 2718
Rule 3377
Rule 3379
Rule 3382
Rule 3393
Rule 5556
Rule 5736
Rule 6675
Rule 6677
Rule 6683
Rule 6873
Rule 6874
Rubi steps
\begin {align*} \int x^2 \sinh (a+b x) \text {Shi}(a+b x) \, dx &=\frac {x^2 \cosh (a+b x) \text {Shi}(a+b x)}{b}-\frac {2 \int x \cosh (a+b x) \text {Shi}(a+b x) \, dx}{b}-\int \frac {x^2 \cosh (a+b x) \sinh (a+b x)}{a+b x} \, dx\\ &=\frac {x^2 \cosh (a+b x) \text {Shi}(a+b x)}{b}-\frac {2 x \sinh (a+b x) \text {Shi}(a+b x)}{b^2}-\frac {1}{2} \int \frac {x^2 \sinh (2 (a+b x))}{a+b x} \, dx+\frac {2 \int \sinh (a+b x) \text {Shi}(a+b x) \, dx}{b^2}+\frac {2 \int \frac {x \sinh ^2(a+b x)}{a+b x} \, dx}{b}\\ &=\frac {2 \cosh (a+b x) \text {Shi}(a+b x)}{b^3}+\frac {x^2 \cosh (a+b x) \text {Shi}(a+b x)}{b}-\frac {2 x \sinh (a+b x) \text {Shi}(a+b x)}{b^2}-\frac {1}{2} \int \frac {x^2 \sinh (2 a+2 b x)}{a+b x} \, dx-\frac {2 \int \frac {\cosh (a+b x) \sinh (a+b x)}{a+b x} \, dx}{b^2}+\frac {2 \int \left (\frac {\sinh ^2(a+b x)}{b}-\frac {a \sinh ^2(a+b x)}{b (a+b x)}\right ) \, dx}{b}\\ &=\frac {2 \cosh (a+b x) \text {Shi}(a+b x)}{b^3}+\frac {x^2 \cosh (a+b x) \text {Shi}(a+b x)}{b}-\frac {2 x \sinh (a+b x) \text {Shi}(a+b x)}{b^2}-\frac {1}{2} \int \left (-\frac {a \sinh (2 a+2 b x)}{b^2}+\frac {x \sinh (2 a+2 b x)}{b}+\frac {a^2 \sinh (2 a+2 b x)}{b^2 (a+b x)}\right ) \, dx+\frac {2 \int \sinh ^2(a+b x) \, dx}{b^2}-\frac {2 \int \frac {\sinh (2 a+2 b x)}{2 (a+b x)} \, dx}{b^2}-\frac {(2 a) \int \frac {\sinh ^2(a+b x)}{a+b x} \, dx}{b^2}\\ &=\frac {\cosh (a+b x) \sinh (a+b x)}{b^3}+\frac {2 \cosh (a+b x) \text {Shi}(a+b x)}{b^3}+\frac {x^2 \cosh (a+b x) \text {Shi}(a+b x)}{b}-\frac {2 x \sinh (a+b x) \text {Shi}(a+b x)}{b^2}-\frac {\int 1 \, dx}{b^2}-\frac {\int \frac {\sinh (2 a+2 b x)}{a+b x} \, dx}{b^2}+\frac {a \int \sinh (2 a+2 b x) \, dx}{2 b^2}+\frac {(2 a) \int \left (\frac {1}{2 (a+b x)}-\frac {\cosh (2 a+2 b x)}{2 (a+b x)}\right ) \, dx}{b^2}-\frac {a^2 \int \frac {\sinh (2 a+2 b x)}{a+b x} \, dx}{2 b^2}-\frac {\int x \sinh (2 a+2 b x) \, dx}{2 b}\\ &=-\frac {x}{b^2}+\frac {a \cosh (2 a+2 b x)}{4 b^3}-\frac {x \cosh (2 a+2 b x)}{4 b^2}+\frac {a \log (a+b x)}{b^3}+\frac {\cosh (a+b x) \sinh (a+b x)}{b^3}+\frac {2 \cosh (a+b x) \text {Shi}(a+b x)}{b^3}+\frac {x^2 \cosh (a+b x) \text {Shi}(a+b x)}{b}-\frac {2 x \sinh (a+b x) \text {Shi}(a+b x)}{b^2}-\frac {\text {Shi}(2 a+2 b x)}{b^3}-\frac {a^2 \text {Shi}(2 a+2 b x)}{2 b^3}+\frac {\int \cosh (2 a+2 b x) \, dx}{4 b^2}-\frac {a \int \frac {\cosh (2 a+2 b x)}{a+b x} \, dx}{b^2}\\ &=-\frac {x}{b^2}+\frac {a \cosh (2 a+2 b x)}{4 b^3}-\frac {x \cosh (2 a+2 b x)}{4 b^2}-\frac {a \text {Chi}(2 a+2 b x)}{b^3}+\frac {a \log (a+b x)}{b^3}+\frac {\cosh (a+b x) \sinh (a+b x)}{b^3}+\frac {\sinh (2 a+2 b x)}{8 b^3}+\frac {2 \cosh (a+b x) \text {Shi}(a+b x)}{b^3}+\frac {x^2 \cosh (a+b x) \text {Shi}(a+b x)}{b}-\frac {2 x \sinh (a+b x) \text {Shi}(a+b x)}{b^2}-\frac {\text {Shi}(2 a+2 b x)}{b^3}-\frac {a^2 \text {Shi}(2 a+2 b x)}{2 b^3}\\ \end {align*}
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Mathematica [A]
time = 0.21, size = 123, normalized size = 0.66 \begin {gather*} \frac {-8 b x+2 a \cosh (2 (a+b x))-2 b x \cosh (2 (a+b x))-8 a \text {Chi}(2 (a+b x))+8 a \log (a+b x)+5 \sinh (2 (a+b x))+8 \left (\left (2+b^2 x^2\right ) \cosh (a+b x)-2 b x \sinh (a+b x)\right ) \text {Shi}(a+b x)-8 \text {Shi}(2 (a+b x))-4 a^2 \text {Shi}(2 (a+b x))}{8 b^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.60, size = 174, normalized size = 0.94
method | result | size |
derivativedivides | \(\frac {\hyperbolicSineIntegral \left (b x +a \right ) \left (a^{2} \cosh \left (b x +a \right )-2 a \left (\left (b x +a \right ) \cosh \left (b x +a \right )-\sinh \left (b x +a \right )\right )+\left (b x +a \right )^{2} \cosh \left (b x +a \right )-2 \left (b x +a \right ) \sinh \left (b x +a \right )+2 \cosh \left (b x +a \right )\right )-\frac {a^{2} \hyperbolicSineIntegral \left (2 b x +2 a \right )}{2}+a \left (\cosh ^{2}\left (b x +a \right )\right )+a \ln \left (b x +a \right )-a \hyperbolicCosineIntegral \left (2 b x +2 a \right )-\frac {\left (b x +a \right ) \left (\cosh ^{2}\left (b x +a \right )\right )}{2}+\frac {5 \cosh \left (b x +a \right ) \sinh \left (b x +a \right )}{4}-\frac {3 b x}{4}-\frac {3 a}{4}-\hyperbolicSineIntegral \left (2 b x +2 a \right )}{b^{3}}\) | \(174\) |
default | \(\frac {\hyperbolicSineIntegral \left (b x +a \right ) \left (a^{2} \cosh \left (b x +a \right )-2 a \left (\left (b x +a \right ) \cosh \left (b x +a \right )-\sinh \left (b x +a \right )\right )+\left (b x +a \right )^{2} \cosh \left (b x +a \right )-2 \left (b x +a \right ) \sinh \left (b x +a \right )+2 \cosh \left (b x +a \right )\right )-\frac {a^{2} \hyperbolicSineIntegral \left (2 b x +2 a \right )}{2}+a \left (\cosh ^{2}\left (b x +a \right )\right )+a \ln \left (b x +a \right )-a \hyperbolicCosineIntegral \left (2 b x +2 a \right )-\frac {\left (b x +a \right ) \left (\cosh ^{2}\left (b x +a \right )\right )}{2}+\frac {5 \cosh \left (b x +a \right ) \sinh \left (b x +a \right )}{4}-\frac {3 b x}{4}-\frac {3 a}{4}-\hyperbolicSineIntegral \left (2 b x +2 a \right )}{b^{3}}\) | \(174\) |
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^{2} \sinh {\left (a + b x \right )} \operatorname {Shi}{\left (a + 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^2\,\mathrm {sinhint}\left (a+b\,x\right )\,\mathrm {sinh}\left (a+b\,x\right ) \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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