Optimal. Leaf size=118 \[ -\frac {2 \cosh (a+b x)}{3 b^3}-\frac {a^2 \cosh (a+b x)}{3 b^3}+\frac {a x \cosh (a+b x)}{3 b^2}-\frac {x^2 \cosh (a+b x)}{3 b}-\frac {a \sinh (a+b x)}{3 b^3}+\frac {2 x \sinh (a+b x)}{3 b^2}+\frac {a^3 \text {Shi}(a+b x)}{3 b^3}+\frac {1}{3} x^3 \text {Shi}(a+b x) \]
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Rubi [A]
time = 0.22, antiderivative size = 118, normalized size of antiderivative = 1.00, number of steps
used = 10, number of rules used = 6, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {6667, 6874,
2718, 3377, 2717, 3379} \begin {gather*} \frac {a^3 \text {Shi}(a+b x)}{3 b^3}-\frac {a^2 \cosh (a+b x)}{3 b^3}-\frac {a \sinh (a+b x)}{3 b^3}-\frac {2 \cosh (a+b x)}{3 b^3}+\frac {2 x \sinh (a+b x)}{3 b^2}+\frac {a x \cosh (a+b x)}{3 b^2}+\frac {1}{3} x^3 \text {Shi}(a+b x)-\frac {x^2 \cosh (a+b x)}{3 b} \end {gather*}
Antiderivative was successfully verified.
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Rule 2717
Rule 2718
Rule 3377
Rule 3379
Rule 6667
Rule 6874
Rubi steps
\begin {align*} \int x^2 \text {Shi}(a+b x) \, dx &=\frac {1}{3} x^3 \text {Shi}(a+b x)-\frac {1}{3} b \int \frac {x^3 \sinh (a+b x)}{a+b x} \, dx\\ &=\frac {1}{3} x^3 \text {Shi}(a+b x)-\frac {1}{3} b \int \left (\frac {a^2 \sinh (a+b x)}{b^3}-\frac {a x \sinh (a+b x)}{b^2}+\frac {x^2 \sinh (a+b x)}{b}-\frac {a^3 \sinh (a+b x)}{b^3 (a+b x)}\right ) \, dx\\ &=\frac {1}{3} x^3 \text {Shi}(a+b x)-\frac {1}{3} \int x^2 \sinh (a+b x) \, dx-\frac {a^2 \int \sinh (a+b x) \, dx}{3 b^2}+\frac {a^3 \int \frac {\sinh (a+b x)}{a+b x} \, dx}{3 b^2}+\frac {a \int x \sinh (a+b x) \, dx}{3 b}\\ &=-\frac {a^2 \cosh (a+b x)}{3 b^3}+\frac {a x \cosh (a+b x)}{3 b^2}-\frac {x^2 \cosh (a+b x)}{3 b}+\frac {a^3 \text {Shi}(a+b x)}{3 b^3}+\frac {1}{3} x^3 \text {Shi}(a+b x)-\frac {a \int \cosh (a+b x) \, dx}{3 b^2}+\frac {2 \int x \cosh (a+b x) \, dx}{3 b}\\ &=-\frac {a^2 \cosh (a+b x)}{3 b^3}+\frac {a x \cosh (a+b x)}{3 b^2}-\frac {x^2 \cosh (a+b x)}{3 b}-\frac {a \sinh (a+b x)}{3 b^3}+\frac {2 x \sinh (a+b x)}{3 b^2}+\frac {a^3 \text {Shi}(a+b x)}{3 b^3}+\frac {1}{3} x^3 \text {Shi}(a+b x)-\frac {2 \int \sinh (a+b x) \, dx}{3 b^2}\\ &=-\frac {2 \cosh (a+b x)}{3 b^3}-\frac {a^2 \cosh (a+b x)}{3 b^3}+\frac {a x \cosh (a+b x)}{3 b^2}-\frac {x^2 \cosh (a+b x)}{3 b}-\frac {a \sinh (a+b x)}{3 b^3}+\frac {2 x \sinh (a+b x)}{3 b^2}+\frac {a^3 \text {Shi}(a+b x)}{3 b^3}+\frac {1}{3} x^3 \text {Shi}(a+b x)\\ \end {align*}
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Mathematica [A]
time = 0.13, size = 64, normalized size = 0.54 \begin {gather*} -\frac {\left (2+a^2-a b x+b^2 x^2\right ) \cosh (a+b x)+(a-2 b x) \sinh (a+b x)-\left (a^3+b^3 x^3\right ) \text {Shi}(a+b x)}{3 b^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.32, size = 101, normalized size = 0.86
method | result | size |
derivativedivides | \(\frac {\frac {\hyperbolicSineIntegral \left (b x +a \right ) b^{3} x^{3}}{3}+\frac {a^{3} \hyperbolicSineIntegral \left (b x +a \right )}{3}-a^{2} \cosh \left (b x +a \right )+a \left (\left (b x +a \right ) \cosh \left (b x +a \right )-\sinh \left (b x +a \right )\right )-\frac {\left (b x +a \right )^{2} \cosh \left (b x +a \right )}{3}+\frac {2 \left (b x +a \right ) \sinh \left (b x +a \right )}{3}-\frac {2 \cosh \left (b x +a \right )}{3}}{b^{3}}\) | \(101\) |
default | \(\frac {\frac {\hyperbolicSineIntegral \left (b x +a \right ) b^{3} x^{3}}{3}+\frac {a^{3} \hyperbolicSineIntegral \left (b x +a \right )}{3}-a^{2} \cosh \left (b x +a \right )+a \left (\left (b x +a \right ) \cosh \left (b x +a \right )-\sinh \left (b x +a \right )\right )-\frac {\left (b x +a \right )^{2} \cosh \left (b x +a \right )}{3}+\frac {2 \left (b x +a \right ) \sinh \left (b x +a \right )}{3}-\frac {2 \cosh \left (b x +a \right )}{3}}{b^{3}}\) | \(101\) |
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} \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 ) \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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