Optimal. Leaf size=328 \[ \frac {a^2 (a+b x) \text {Shi}(a+b x)^2}{3 b^3}+\frac {a^2 \text {Shi}(2 a+2 b x)}{b^3}-\frac {2 a^2 \text {Shi}(a+b x) \cosh (a+b x)}{3 b^3}+\frac {a \text {Chi}(2 a+2 b x)}{b^3}+\frac {2 \text {Shi}(2 a+2 b x)}{3 b^3}-\frac {2 a \text {Shi}(a+b x) \sinh (a+b x)}{3 b^3}-\frac {4 \text {Shi}(a+b x) \cosh (a+b x)}{3 b^3}-\frac {a \log (a+b x)}{b^3}-\frac {\sinh (2 a+2 b x)}{12 b^3}-\frac {a \cosh (2 a+2 b x)}{3 b^3}-\frac {2 \sinh (a+b x) \cosh (a+b x)}{3 b^3}-\frac {a x (a+b x) \text {Shi}(a+b x)^2}{3 b^2}+\frac {4 x \text {Shi}(a+b x) \sinh (a+b x)}{3 b^2}+\frac {2 a x \text {Shi}(a+b x) \cosh (a+b x)}{3 b^2}+\frac {x \cosh (2 a+2 b x)}{6 b^2}+\frac {x^2 (a+b x) \text {Shi}(a+b x)^2}{3 b}-\frac {2 x^2 \text {Shi}(a+b x) \cosh (a+b x)}{3 b}+\frac {2 x}{3 b^2} \]
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Rubi [A] time = 1.49, antiderivative size = 328, normalized size of antiderivative = 1.00, number of steps used = 39, number of rules used = 19, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.583, Rules used = {6538, 6542, 5617, 6741, 6742, 2638, 3296, 2637, 3298, 6548, 2635, 8, 3312, 3301, 6540, 5448, 12, 6546, 6534} \[ \frac {a^2 (a+b x) \text {Shi}(a+b x)^2}{3 b^3}+\frac {a^2 \text {Shi}(2 a+2 b x)}{b^3}-\frac {2 a^2 \text {Shi}(a+b x) \cosh (a+b x)}{3 b^3}+\frac {a \text {Chi}(2 a+2 b x)}{b^3}-\frac {a x (a+b x) \text {Shi}(a+b x)^2}{3 b^2}+\frac {2 \text {Shi}(2 a+2 b x)}{3 b^3}-\frac {2 a \text {Shi}(a+b x) \sinh (a+b x)}{3 b^3}+\frac {4 x \text {Shi}(a+b x) \sinh (a+b x)}{3 b^2}+\frac {2 a x \text {Shi}(a+b x) \cosh (a+b x)}{3 b^2}-\frac {4 \text {Shi}(a+b x) \cosh (a+b x)}{3 b^3}-\frac {a \log (a+b x)}{b^3}-\frac {\sinh (2 a+2 b x)}{12 b^3}-\frac {a \cosh (2 a+2 b x)}{3 b^3}+\frac {x \cosh (2 a+2 b x)}{6 b^2}-\frac {2 \sinh (a+b x) \cosh (a+b x)}{3 b^3}+\frac {x^2 (a+b x) \text {Shi}(a+b x)^2}{3 b}-\frac {2 x^2 \text {Shi}(a+b x) \cosh (a+b x)}{3 b}+\frac {2 x}{3 b^2} \]
Antiderivative was successfully verified.
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Rule 8
Rule 12
Rule 2635
Rule 2637
Rule 2638
Rule 3296
Rule 3298
Rule 3301
Rule 3312
Rule 5448
Rule 5617
Rule 6534
Rule 6538
Rule 6540
Rule 6542
Rule 6546
Rule 6548
Rule 6741
Rule 6742
Rubi steps
\begin {align*} \int x^2 \text {Shi}(a+b x)^2 \, dx &=\frac {x^2 (a+b x) \text {Shi}(a+b x)^2}{3 b}-\frac {2}{3} \int x^2 \sinh (a+b x) \text {Shi}(a+b x) \, dx-\frac {(2 a) \int x \text {Shi}(a+b x)^2 \, dx}{3 b}\\ &=-\frac {2 x^2 \cosh (a+b x) \text {Shi}(a+b x)}{3 b}-\frac {a x (a+b x) \text {Shi}(a+b x)^2}{3 b^2}+\frac {x^2 (a+b x) \text {Shi}(a+b x)^2}{3 b}+\frac {2}{3} \int \frac {x^2 \cosh (a+b x) \sinh (a+b x)}{a+b x} \, dx+\frac {a^2 \int \text {Shi}(a+b x)^2 \, dx}{3 b^2}+\frac {4 \int x \cosh (a+b x) \text {Shi}(a+b x) \, dx}{3 b}+\frac {(2 a) \int x \sinh (a+b x) \text {Shi}(a+b x) \, dx}{3 b}\\ &=\frac {2 a x \cosh (a+b x) \text {Shi}(a+b x)}{3 b^2}-\frac {2 x^2 \cosh (a+b x) \text {Shi}(a+b x)}{3 b}+\frac {4 x \sinh (a+b x) \text {Shi}(a+b x)}{3 b^2}+\frac {a^2 (a+b x) \text {Shi}(a+b x)^2}{3 b^3}-\frac {a x (a+b x) \text {Shi}(a+b x)^2}{3 b^2}+\frac {x^2 (a+b x) \text {Shi}(a+b x)^2}{3 b}+\frac {1}{3} \int \frac {x^2 \sinh (2 (a+b x))}{a+b x} \, dx-\frac {4 \int \sinh (a+b x) \text {Shi}(a+b x) \, dx}{3 b^2}-\frac {(2 a) \int \cosh (a+b x) \text {Shi}(a+b x) \, dx}{3 b^2}-\frac {\left (2 a^2\right ) \int \sinh (a+b x) \text {Shi}(a+b x) \, dx}{3 b^2}-\frac {4 \int \frac {x \sinh ^2(a+b x)}{a+b x} \, dx}{3 b}-\frac {(2 a) \int \frac {x \cosh (a+b x) \sinh (a+b x)}{a+b x} \, dx}{3 b}\\ &=-\frac {4 \cosh (a+b x) \text {Shi}(a+b x)}{3 b^3}-\frac {2 a^2 \cosh (a+b x) \text {Shi}(a+b x)}{3 b^3}+\frac {2 a x \cosh (a+b x) \text {Shi}(a+b x)}{3 b^2}-\frac {2 x^2 \cosh (a+b x) \text {Shi}(a+b x)}{3 b}-\frac {2 a \sinh (a+b x) \text {Shi}(a+b x)}{3 b^3}+\frac {4 x \sinh (a+b x) \text {Shi}(a+b x)}{3 b^2}+\frac {a^2 (a+b x) \text {Shi}(a+b x)^2}{3 b^3}-\frac {a x (a+b x) \text {Shi}(a+b x)^2}{3 b^2}+\frac {x^2 (a+b x) \text {Shi}(a+b x)^2}{3 b}+\frac {1}{3} \int \frac {x^2 \sinh (2 a+2 b x)}{a+b x} \, dx+\frac {4 \int \frac {\cosh (a+b x) \sinh (a+b x)}{a+b x} \, dx}{3 b^2}+\frac {(2 a) \int \frac {\sinh ^2(a+b x)}{a+b x} \, dx}{3 b^2}+\frac {\left (2 a^2\right ) \int \frac {\cosh (a+b x) \sinh (a+b x)}{a+b x} \, dx}{3 b^2}-\frac {4 \int \left (\frac {\sinh ^2(a+b x)}{b}-\frac {a \sinh ^2(a+b x)}{b (a+b x)}\right ) \, dx}{3 b}-\frac {a \int \frac {x \sinh (2 (a+b x))}{a+b x} \, dx}{3 b}\\ &=-\frac {4 \cosh (a+b x) \text {Shi}(a+b x)}{3 b^3}-\frac {2 a^2 \cosh (a+b x) \text {Shi}(a+b x)}{3 b^3}+\frac {2 a x \cosh (a+b x) \text {Shi}(a+b x)}{3 b^2}-\frac {2 x^2 \cosh (a+b x) \text {Shi}(a+b x)}{3 b}-\frac {2 a \sinh (a+b x) \text {Shi}(a+b x)}{3 b^3}+\frac {4 x \sinh (a+b x) \text {Shi}(a+b x)}{3 b^2}+\frac {a^2 (a+b x) \text {Shi}(a+b x)^2}{3 b^3}-\frac {a x (a+b x) \text {Shi}(a+b x)^2}{3 b^2}+\frac {x^2 (a+b x) \text {Shi}(a+b x)^2}{3 b}+\frac {1}{3} \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 {4 \int \sinh ^2(a+b x) \, dx}{3 b^2}+\frac {4 \int \frac {\sinh (2 a+2 b x)}{2 (a+b x)} \, dx}{3 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}{3 b^2}+\frac {(4 a) \int \frac {\sinh ^2(a+b x)}{a+b x} \, dx}{3 b^2}+\frac {\left (2 a^2\right ) \int \frac {\sinh (2 a+2 b x)}{2 (a+b x)} \, dx}{3 b^2}-\frac {a \int \frac {x \sinh (2 a+2 b x)}{a+b x} \, dx}{3 b}\\ &=-\frac {a \log (a+b x)}{3 b^3}-\frac {2 \cosh (a+b x) \sinh (a+b x)}{3 b^3}-\frac {4 \cosh (a+b x) \text {Shi}(a+b x)}{3 b^3}-\frac {2 a^2 \cosh (a+b x) \text {Shi}(a+b x)}{3 b^3}+\frac {2 a x \cosh (a+b x) \text {Shi}(a+b x)}{3 b^2}-\frac {2 x^2 \cosh (a+b x) \text {Shi}(a+b x)}{3 b}-\frac {2 a \sinh (a+b x) \text {Shi}(a+b x)}{3 b^3}+\frac {4 x \sinh (a+b x) \text {Shi}(a+b x)}{3 b^2}+\frac {a^2 (a+b x) \text {Shi}(a+b x)^2}{3 b^3}-\frac {a x (a+b x) \text {Shi}(a+b x)^2}{3 b^2}+\frac {x^2 (a+b x) \text {Shi}(a+b x)^2}{3 b}+\frac {2 \int 1 \, dx}{3 b^2}+\frac {2 \int \frac {\sinh (2 a+2 b x)}{a+b x} \, dx}{3 b^2}+\frac {a \int \frac {\cosh (2 a+2 b x)}{a+b x} \, dx}{3 b^2}-\frac {a \int \sinh (2 a+2 b x) \, dx}{3 b^2}-\frac {(4 a) \int \left (\frac {1}{2 (a+b x)}-\frac {\cosh (2 a+2 b x)}{2 (a+b x)}\right ) \, dx}{3 b^2}+2 \frac {a^2 \int \frac {\sinh (2 a+2 b x)}{a+b x} \, dx}{3 b^2}+\frac {\int x \sinh (2 a+2 b x) \, dx}{3 b}-\frac {a \int \left (\frac {\sinh (2 a+2 b x)}{b}+\frac {a \sinh (2 a+2 b x)}{b (-a-b x)}\right ) \, dx}{3 b}\\ &=\frac {2 x}{3 b^2}-\frac {a \cosh (2 a+2 b x)}{6 b^3}+\frac {x \cosh (2 a+2 b x)}{6 b^2}+\frac {a \text {Chi}(2 a+2 b x)}{3 b^3}-\frac {a \log (a+b x)}{b^3}-\frac {2 \cosh (a+b x) \sinh (a+b x)}{3 b^3}-\frac {4 \cosh (a+b x) \text {Shi}(a+b x)}{3 b^3}-\frac {2 a^2 \cosh (a+b x) \text {Shi}(a+b x)}{3 b^3}+\frac {2 a x \cosh (a+b x) \text {Shi}(a+b x)}{3 b^2}-\frac {2 x^2 \cosh (a+b x) \text {Shi}(a+b x)}{3 b}-\frac {2 a \sinh (a+b x) \text {Shi}(a+b x)}{3 b^3}+\frac {4 x \sinh (a+b x) \text {Shi}(a+b x)}{3 b^2}+\frac {a^2 (a+b x) \text {Shi}(a+b x)^2}{3 b^3}-\frac {a x (a+b x) \text {Shi}(a+b x)^2}{3 b^2}+\frac {x^2 (a+b x) \text {Shi}(a+b x)^2}{3 b}+\frac {2 \text {Shi}(2 a+2 b x)}{3 b^3}+\frac {2 a^2 \text {Shi}(2 a+2 b x)}{3 b^3}-\frac {\int \cosh (2 a+2 b x) \, dx}{6 b^2}-\frac {a \int \sinh (2 a+2 b x) \, dx}{3 b^2}+\frac {(2 a) \int \frac {\cosh (2 a+2 b x)}{a+b x} \, dx}{3 b^2}-\frac {a^2 \int \frac {\sinh (2 a+2 b x)}{-a-b x} \, dx}{3 b^2}\\ &=\frac {2 x}{3 b^2}-\frac {a \cosh (2 a+2 b x)}{3 b^3}+\frac {x \cosh (2 a+2 b x)}{6 b^2}+\frac {a \text {Chi}(2 a+2 b x)}{b^3}-\frac {a \log (a+b x)}{b^3}-\frac {2 \cosh (a+b x) \sinh (a+b x)}{3 b^3}-\frac {\sinh (2 a+2 b x)}{12 b^3}-\frac {4 \cosh (a+b x) \text {Shi}(a+b x)}{3 b^3}-\frac {2 a^2 \cosh (a+b x) \text {Shi}(a+b x)}{3 b^3}+\frac {2 a x \cosh (a+b x) \text {Shi}(a+b x)}{3 b^2}-\frac {2 x^2 \cosh (a+b x) \text {Shi}(a+b x)}{3 b}-\frac {2 a \sinh (a+b x) \text {Shi}(a+b x)}{3 b^3}+\frac {4 x \sinh (a+b x) \text {Shi}(a+b x)}{3 b^2}+\frac {a^2 (a+b x) \text {Shi}(a+b x)^2}{3 b^3}-\frac {a x (a+b x) \text {Shi}(a+b x)^2}{3 b^2}+\frac {x^2 (a+b x) \text {Shi}(a+b x)^2}{3 b}+\frac {2 \text {Shi}(2 a+2 b x)}{3 b^3}+\frac {a^2 \text {Shi}(2 a+2 b x)}{b^3}\\ \end {align*}
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Mathematica [A] time = 1.38, size = 158, normalized size = 0.48 \[ \frac {4 \left (a^3+b^3 x^3\right ) \text {Shi}(a+b x)^2-8 \text {Shi}(a+b x) \left (\left (a^2-a b x+b^2 x^2+2\right ) \cosh (a+b x)+(a-2 b x) \sinh (a+b x)\right )+12 a^2 \text {Shi}(2 (a+b x))+12 a \text {Chi}(2 (a+b x))+8 \text {Shi}(2 (a+b x))-12 a \log (a+b x)-5 \sinh (2 (a+b x))-4 a \cosh (2 (a+b x))+2 b x \cosh (2 (a+b x))+8 a+8 b x}{12 b^3} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.53, size = 0, normalized size = 0.00 \[ {\rm integral}\left (x^{2} \operatorname {Shi}\left (b x + a\right )^{2}, x\right ) \]
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 \[ \int x^{2} {\rm Shi}\left (b x + a\right )^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.02, size = 0, normalized size = 0.00 \[ \int x^{2} \Shi \left (b x +a \right )^{2}\, dx \]
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 \[ \int x^{2} {\rm Shi}\left (b x + a\right )^{2}\,{d x} \]
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 \[ \int x^2\,{\mathrm {sinhint}\left (a+b\,x\right )}^2 \,d x \]
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 \[ \int x^{2} \operatorname {Shi}^{2}{\left (a + b x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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