Optimal. Leaf size=220 \[ \frac {a x}{2 b^2}-\frac {x^2}{4 b}+\frac {\cosh ^2(a+b x)}{4 b^3}+\frac {\cosh (2 a+2 b x)}{2 b^3}+\frac {2 \cosh (a+b x) \text {Chi}(a+b x)}{b^3}+\frac {x^2 \cosh (a+b x) \text {Chi}(a+b x)}{b}-\frac {\text {Chi}(2 a+2 b x)}{b^3}-\frac {a^2 \text {Chi}(2 a+2 b x)}{2 b^3}-\frac {\log (a+b x)}{b^3}-\frac {a^2 \log (a+b x)}{2 b^3}+\frac {a \cosh (a+b x) \sinh (a+b x)}{2 b^3}-\frac {x \cosh (a+b x) \sinh (a+b x)}{2 b^2}-\frac {2 x \text {Chi}(a+b x) \sinh (a+b x)}{b^2}-\frac {a \text {Shi}(2 a+2 b x)}{b^3} \]
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Rubi [A]
time = 0.57, antiderivative size = 220, normalized size of antiderivative = 1.00, number of
steps used = 21, number of rules used = 14, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.875, Rules used
= {6684, 6874, 2715, 8, 3391, 30, 3393, 3382, 6678, 5736, 6873, 2718, 3379, 6682}
\begin {gather*} -\frac {a^2 \text {Chi}(2 a+2 b x)}{2 b^3}-\frac {a^2 \log (a+b x)}{2 b^3}-\frac {\text {Chi}(2 a+2 b x)}{b^3}+\frac {2 \text {Chi}(a+b x) \cosh (a+b x)}{b^3}-\frac {a \text {Shi}(2 a+2 b x)}{b^3}-\frac {\log (a+b x)}{b^3}+\frac {\cosh ^2(a+b x)}{4 b^3}+\frac {\cosh (2 a+2 b x)}{2 b^3}+\frac {a \sinh (a+b x) \cosh (a+b x)}{2 b^3}-\frac {2 x \text {Chi}(a+b x) \sinh (a+b x)}{b^2}+\frac {a x}{2 b^2}-\frac {x \sinh (a+b x) \cosh (a+b x)}{2 b^2}+\frac {x^2 \text {Chi}(a+b x) \cosh (a+b x)}{b}-\frac {x^2}{4 b} \end {gather*}
Antiderivative was successfully verified.
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Rule 8
Rule 30
Rule 2715
Rule 2718
Rule 3379
Rule 3382
Rule 3391
Rule 3393
Rule 5736
Rule 6678
Rule 6682
Rule 6684
Rule 6873
Rule 6874
Rubi steps
\begin {align*} \int x^2 \text {Chi}(a+b x) \sinh (a+b x) \, dx &=\frac {x^2 \cosh (a+b x) \text {Chi}(a+b x)}{b}-\frac {2 \int x \cosh (a+b x) \text {Chi}(a+b x) \, dx}{b}-\int \frac {x^2 \cosh ^2(a+b x)}{a+b x} \, dx\\ &=\frac {x^2 \cosh (a+b x) \text {Chi}(a+b x)}{b}-\frac {2 x \text {Chi}(a+b x) \sinh (a+b x)}{b^2}+\frac {2 \int \text {Chi}(a+b x) \sinh (a+b x) \, dx}{b^2}+\frac {2 \int \frac {x \cosh (a+b x) \sinh (a+b x)}{a+b x} \, dx}{b}-\int \left (-\frac {a \cosh ^2(a+b x)}{b^2}+\frac {x \cosh ^2(a+b x)}{b}+\frac {a^2 \cosh ^2(a+b x)}{b^2 (a+b x)}\right ) \, dx\\ &=\frac {2 \cosh (a+b x) \text {Chi}(a+b x)}{b^3}+\frac {x^2 \cosh (a+b x) \text {Chi}(a+b x)}{b}-\frac {2 x \text {Chi}(a+b x) \sinh (a+b x)}{b^2}-\frac {2 \int \frac {\cosh ^2(a+b x)}{a+b x} \, dx}{b^2}+\frac {a \int \cosh ^2(a+b x) \, dx}{b^2}-\frac {a^2 \int \frac {\cosh ^2(a+b x)}{a+b x} \, dx}{b^2}-\frac {\int x \cosh ^2(a+b x) \, dx}{b}+\frac {\int \frac {x \sinh (2 (a+b x))}{a+b x} \, dx}{b}\\ &=\frac {\cosh ^2(a+b x)}{4 b^3}+\frac {2 \cosh (a+b x) \text {Chi}(a+b x)}{b^3}+\frac {x^2 \cosh (a+b x) \text {Chi}(a+b x)}{b}+\frac {a \cosh (a+b x) \sinh (a+b x)}{2 b^3}-\frac {x \cosh (a+b x) \sinh (a+b x)}{2 b^2}-\frac {2 x \text {Chi}(a+b x) \sinh (a+b x)}{b^2}-\frac {2 \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 \int 1 \, dx}{2 b^2}-\frac {a^2 \int \left (\frac {1}{2 (a+b x)}+\frac {\cosh (2 a+2 b x)}{2 (a+b x)}\right ) \, dx}{b^2}-\frac {\int x \, dx}{2 b}+\frac {\int \frac {x \sinh (2 a+2 b x)}{a+b x} \, dx}{b}\\ &=\frac {a x}{2 b^2}-\frac {x^2}{4 b}+\frac {\cosh ^2(a+b x)}{4 b^3}+\frac {2 \cosh (a+b x) \text {Chi}(a+b x)}{b^3}+\frac {x^2 \cosh (a+b x) \text {Chi}(a+b x)}{b}-\frac {\log (a+b x)}{b^3}-\frac {a^2 \log (a+b x)}{2 b^3}+\frac {a \cosh (a+b x) \sinh (a+b x)}{2 b^3}-\frac {x \cosh (a+b x) \sinh (a+b x)}{2 b^2}-\frac {2 x \text {Chi}(a+b x) \sinh (a+b x)}{b^2}-\frac {\int \frac {\cosh (2 a+2 b x)}{a+b x} \, dx}{b^2}-\frac {a^2 \int \frac {\cosh (2 a+2 b x)}{a+b x} \, dx}{2 b^2}+\frac {\int \left (\frac {\sinh (2 a+2 b x)}{b}+\frac {a \sinh (2 a+2 b x)}{b (-a-b x)}\right ) \, dx}{b}\\ &=\frac {a x}{2 b^2}-\frac {x^2}{4 b}+\frac {\cosh ^2(a+b x)}{4 b^3}+\frac {2 \cosh (a+b x) \text {Chi}(a+b x)}{b^3}+\frac {x^2 \cosh (a+b x) \text {Chi}(a+b x)}{b}-\frac {\text {Chi}(2 a+2 b x)}{b^3}-\frac {a^2 \text {Chi}(2 a+2 b x)}{2 b^3}-\frac {\log (a+b x)}{b^3}-\frac {a^2 \log (a+b x)}{2 b^3}+\frac {a \cosh (a+b x) \sinh (a+b x)}{2 b^3}-\frac {x \cosh (a+b x) \sinh (a+b x)}{2 b^2}-\frac {2 x \text {Chi}(a+b x) \sinh (a+b x)}{b^2}+\frac {\int \sinh (2 a+2 b x) \, dx}{b^2}+\frac {a \int \frac {\sinh (2 a+2 b x)}{-a-b x} \, dx}{b^2}\\ &=\frac {a x}{2 b^2}-\frac {x^2}{4 b}+\frac {\cosh ^2(a+b x)}{4 b^3}+\frac {\cosh (2 a+2 b x)}{2 b^3}+\frac {2 \cosh (a+b x) \text {Chi}(a+b x)}{b^3}+\frac {x^2 \cosh (a+b x) \text {Chi}(a+b x)}{b}-\frac {\text {Chi}(2 a+2 b x)}{b^3}-\frac {a^2 \text {Chi}(2 a+2 b x)}{2 b^3}-\frac {\log (a+b x)}{b^3}-\frac {a^2 \log (a+b x)}{2 b^3}+\frac {a \cosh (a+b x) \sinh (a+b x)}{2 b^3}-\frac {x \cosh (a+b x) \sinh (a+b x)}{2 b^2}-\frac {2 x \text {Chi}(a+b x) \sinh (a+b x)}{b^2}-\frac {a \text {Shi}(2 a+2 b x)}{b^3}\\ \end {align*}
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Mathematica [A]
time = 0.25, size = 134, normalized size = 0.61 \begin {gather*} -\frac {-4 a b x+2 b^2 x^2-5 \cosh (2 (a+b x))+4 \left (2+a^2\right ) \text {Chi}(2 (a+b x))+8 \log (a+b x)+4 a^2 \log (a+b x)-8 \text {Chi}(a+b x) \left (\left (2+b^2 x^2\right ) \cosh (a+b x)-2 b x \sinh (a+b x)\right )-2 a \sinh (2 (a+b x))+2 b x \sinh (2 (a+b x))+8 a \text {Shi}(2 (a+b x))}{8 b^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.55, size = 198, normalized size = 0.90
method | result | size |
derivativedivides | \(\frac {\hyperbolicCosineIntegral \left (b x +a \right ) \left (a^{2} \cosh \left (b x +a \right )-2 a \left (\left (b x +a \right ) \cosh \left (b x +a \right )-\sinh \left (b x +a \right )\right )+\left (b x +a \right )^{2} \cosh \left (b x +a \right )-2 \left (b x +a \right ) \sinh \left (b x +a \right )+2 \cosh \left (b x +a \right )\right )-\frac {a^{2} \ln \left (b x +a \right )}{2}-\frac {a^{2} \hyperbolicCosineIntegral \left (2 b x +2 a \right )}{2}+a \cosh \left (b x +a \right ) \sinh \left (b x +a \right )+a \left (b x +a \right )-a \hyperbolicSineIntegral \left (2 b x +2 a \right )-\frac {\left (b x +a \right ) \cosh \left (b x +a \right ) \sinh \left (b x +a \right )}{2}-\frac {\left (b x +a \right )^{2}}{4}+\frac {5 \left (\cosh ^{2}\left (b x +a \right )\right )}{4}-\ln \left (b x +a \right )-\hyperbolicCosineIntegral \left (2 b x +2 a \right )}{b^{3}}\) | \(198\) |
default | \(\frac {\hyperbolicCosineIntegral \left (b x +a \right ) \left (a^{2} \cosh \left (b x +a \right )-2 a \left (\left (b x +a \right ) \cosh \left (b x +a \right )-\sinh \left (b x +a \right )\right )+\left (b x +a \right )^{2} \cosh \left (b x +a \right )-2 \left (b x +a \right ) \sinh \left (b x +a \right )+2 \cosh \left (b x +a \right )\right )-\frac {a^{2} \ln \left (b x +a \right )}{2}-\frac {a^{2} \hyperbolicCosineIntegral \left (2 b x +2 a \right )}{2}+a \cosh \left (b x +a \right ) \sinh \left (b x +a \right )+a \left (b x +a \right )-a \hyperbolicSineIntegral \left (2 b x +2 a \right )-\frac {\left (b x +a \right ) \cosh \left (b x +a \right ) \sinh \left (b x +a \right )}{2}-\frac {\left (b x +a \right )^{2}}{4}+\frac {5 \left (\cosh ^{2}\left (b x +a \right )\right )}{4}-\ln \left (b x +a \right )-\hyperbolicCosineIntegral \left (2 b x +2 a \right )}{b^{3}}\) | \(198\) |
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 {Chi}\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 = -0.00 \begin {gather*} \int x^2\,\mathrm {coshint}\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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