Optimal. Leaf size=85 \[ \frac {\text {Shi}\left (\cosh ^{-1}(a x)\right )}{8 a^3}+\frac {9 \text {Shi}\left (3 \cosh ^{-1}(a x)\right )}{8 a^3}+\frac {x}{a^2 \cosh ^{-1}(a x)}-\frac {3 x^3}{2 \cosh ^{-1}(a x)}-\frac {x^2 \sqrt {a x-1} \sqrt {a x+1}}{2 a \cosh ^{-1}(a x)^2} \]
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Rubi [A] time = 0.50, antiderivative size = 85, 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 = {5668, 5775, 5670, 5448, 3298, 5658} \[ \frac {\text {Shi}\left (\cosh ^{-1}(a x)\right )}{8 a^3}+\frac {9 \text {Shi}\left (3 \cosh ^{-1}(a x)\right )}{8 a^3}+\frac {x}{a^2 \cosh ^{-1}(a x)}-\frac {3 x^3}{2 \cosh ^{-1}(a x)}-\frac {x^2 \sqrt {a x-1} \sqrt {a x+1}}{2 a \cosh ^{-1}(a x)^2} \]
Antiderivative was successfully verified.
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Rule 3298
Rule 5448
Rule 5658
Rule 5668
Rule 5670
Rule 5775
Rubi steps
\begin {align*} \int \frac {x^2}{\cosh ^{-1}(a x)^3} \, dx &=-\frac {x^2 \sqrt {-1+a x} \sqrt {1+a x}}{2 a \cosh ^{-1}(a x)^2}-\frac {\int \frac {x}{\sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^2} \, dx}{a}+\frac {1}{2} (3 a) \int \frac {x^3}{\sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^2} \, dx\\ &=-\frac {x^2 \sqrt {-1+a x} \sqrt {1+a x}}{2 a \cosh ^{-1}(a x)^2}+\frac {x}{a^2 \cosh ^{-1}(a x)}-\frac {3 x^3}{2 \cosh ^{-1}(a x)}+\frac {9}{2} \int \frac {x^2}{\cosh ^{-1}(a x)} \, dx-\frac {\int \frac {1}{\cosh ^{-1}(a x)} \, dx}{a^2}\\ &=-\frac {x^2 \sqrt {-1+a x} \sqrt {1+a x}}{2 a \cosh ^{-1}(a x)^2}+\frac {x}{a^2 \cosh ^{-1}(a x)}-\frac {3 x^3}{2 \cosh ^{-1}(a x)}-\frac {\operatorname {Subst}\left (\int \frac {\sinh (x)}{x} \, dx,x,\cosh ^{-1}(a x)\right )}{a^3}+\frac {9 \operatorname {Subst}\left (\int \frac {\cosh ^2(x) \sinh (x)}{x} \, dx,x,\cosh ^{-1}(a x)\right )}{2 a^3}\\ &=-\frac {x^2 \sqrt {-1+a x} \sqrt {1+a x}}{2 a \cosh ^{-1}(a x)^2}+\frac {x}{a^2 \cosh ^{-1}(a x)}-\frac {3 x^3}{2 \cosh ^{-1}(a x)}-\frac {\text {Shi}\left (\cosh ^{-1}(a x)\right )}{a^3}+\frac {9 \operatorname {Subst}\left (\int \left (\frac {\sinh (x)}{4 x}+\frac {\sinh (3 x)}{4 x}\right ) \, dx,x,\cosh ^{-1}(a x)\right )}{2 a^3}\\ &=-\frac {x^2 \sqrt {-1+a x} \sqrt {1+a x}}{2 a \cosh ^{-1}(a x)^2}+\frac {x}{a^2 \cosh ^{-1}(a x)}-\frac {3 x^3}{2 \cosh ^{-1}(a x)}-\frac {\text {Shi}\left (\cosh ^{-1}(a x)\right )}{a^3}+\frac {9 \operatorname {Subst}\left (\int \frac {\sinh (x)}{x} \, dx,x,\cosh ^{-1}(a x)\right )}{8 a^3}+\frac {9 \operatorname {Subst}\left (\int \frac {\sinh (3 x)}{x} \, dx,x,\cosh ^{-1}(a x)\right )}{8 a^3}\\ &=-\frac {x^2 \sqrt {-1+a x} \sqrt {1+a x}}{2 a \cosh ^{-1}(a x)^2}+\frac {x}{a^2 \cosh ^{-1}(a x)}-\frac {3 x^3}{2 \cosh ^{-1}(a x)}+\frac {\text {Shi}\left (\cosh ^{-1}(a x)\right )}{8 a^3}+\frac {9 \text {Shi}\left (3 \cosh ^{-1}(a x)\right )}{8 a^3}\\ \end {align*}
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Mathematica [A] time = 0.16, size = 69, normalized size = 0.81 \[ \frac {-\frac {4 a x \left (\left (3 a^2 x^2-2\right ) \cosh ^{-1}(a x)+a x \sqrt {a x-1} \sqrt {a x+1}\right )}{\cosh ^{-1}(a x)^2}+\text {Shi}\left (\cosh ^{-1}(a x)\right )+9 \text {Shi}\left (3 \cosh ^{-1}(a x)\right )}{8 a^3} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.56, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {x^{2}}{\operatorname {arcosh}\left (a x\right )^{3}}, x\right ) \]
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 \[ \int \frac {x^{2}}{\operatorname {arcosh}\left (a x\right )^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 84, normalized size = 0.99 \[ \frac {-\frac {\sqrt {a x -1}\, \sqrt {a x +1}}{8 \mathrm {arccosh}\left (a x \right )^{2}}-\frac {a x}{8 \,\mathrm {arccosh}\left (a x \right )}+\frac {\Shi \left (\mathrm {arccosh}\left (a x \right )\right )}{8}-\frac {\sinh \left (3 \,\mathrm {arccosh}\left (a x \right )\right )}{8 \mathrm {arccosh}\left (a x \right )^{2}}-\frac {3 \cosh \left (3 \,\mathrm {arccosh}\left (a x \right )\right )}{8 \,\mathrm {arccosh}\left (a x \right )}+\frac {9 \Shi \left (3 \,\mathrm {arccosh}\left (a x \right )\right )}{8}}{a^{3}} \]
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 \[ -\frac {a^{8} x^{9} - 3 \, a^{6} x^{7} + 3 \, a^{4} x^{5} - a^{2} x^{3} + {\left (a^{5} x^{6} - a^{3} x^{4}\right )} {\left (a x + 1\right )}^{\frac {3}{2}} {\left (a x - 1\right )}^{\frac {3}{2}} + {\left (3 \, a^{6} x^{7} - 5 \, a^{4} x^{5} + 2 \, a^{2} x^{3}\right )} {\left (a x + 1\right )} {\left (a x - 1\right )} + {\left (3 \, a^{7} x^{8} - 7 \, a^{5} x^{6} + 5 \, a^{3} x^{4} - a x^{2}\right )} \sqrt {a x + 1} \sqrt {a x - 1} + {\left (3 \, a^{8} x^{9} - 9 \, a^{6} x^{7} + 9 \, a^{4} x^{5} - 3 \, a^{2} x^{3} + {\left (3 \, a^{5} x^{6} - 4 \, a^{3} x^{4} + a x^{2}\right )} {\left (a x + 1\right )}^{\frac {3}{2}} {\left (a x - 1\right )}^{\frac {3}{2}} + {\left (9 \, a^{6} x^{7} - 17 \, a^{4} x^{5} + 10 \, a^{2} x^{3} - 2 \, x\right )} {\left (a x + 1\right )} {\left (a x - 1\right )} + {\left (9 \, a^{7} x^{8} - 22 \, a^{5} x^{6} + 18 \, a^{3} x^{4} - 5 \, a x^{2}\right )} \sqrt {a x + 1} \sqrt {a x - 1}\right )} \log \left (a x + \sqrt {a x + 1} \sqrt {a x - 1}\right )}{2 \, {\left (a^{8} x^{6} + {\left (a x + 1\right )}^{\frac {3}{2}} {\left (a x - 1\right )}^{\frac {3}{2}} a^{5} x^{3} - 3 \, a^{6} x^{4} + 3 \, a^{4} x^{2} + 3 \, {\left (a^{6} x^{4} - a^{4} x^{2}\right )} {\left (a x + 1\right )} {\left (a x - 1\right )} + 3 \, {\left (a^{7} x^{5} - 2 \, a^{5} x^{3} + a^{3} x\right )} \sqrt {a x + 1} \sqrt {a x - 1} - a^{2}\right )} \log \left (a x + \sqrt {a x + 1} \sqrt {a x - 1}\right )^{2}} + \int \frac {9 \, a^{10} x^{10} - 36 \, a^{8} x^{8} + 54 \, a^{6} x^{6} - 36 \, a^{4} x^{4} + {\left (9 \, a^{6} x^{6} - 4 \, a^{4} x^{4} - a^{2} x^{2}\right )} {\left (a x + 1\right )}^{2} {\left (a x - 1\right )}^{2} + {\left (36 \, a^{7} x^{7} - 48 \, a^{5} x^{5} + 13 \, a^{3} x^{3} + 2 \, a x\right )} {\left (a x + 1\right )}^{\frac {3}{2}} {\left (a x - 1\right )}^{\frac {3}{2}} + 9 \, a^{2} x^{2} + {\left (54 \, a^{8} x^{8} - 120 \, a^{6} x^{6} + 83 \, a^{4} x^{4} - 19 \, a^{2} x^{2} + 2\right )} {\left (a x + 1\right )} {\left (a x - 1\right )} + {\left (36 \, a^{9} x^{9} - 112 \, a^{7} x^{7} + 123 \, a^{5} x^{5} - 57 \, a^{3} x^{3} + 10 \, a x\right )} \sqrt {a x + 1} \sqrt {a x - 1}}{2 \, {\left (a^{10} x^{8} + {\left (a x + 1\right )}^{2} {\left (a x - 1\right )}^{2} a^{6} x^{4} - 4 \, a^{8} x^{6} + 6 \, a^{6} x^{4} - 4 \, a^{4} x^{2} + 4 \, {\left (a^{7} x^{5} - a^{5} x^{3}\right )} {\left (a x + 1\right )}^{\frac {3}{2}} {\left (a x - 1\right )}^{\frac {3}{2}} + 6 \, {\left (a^{8} x^{6} - 2 \, a^{6} x^{4} + a^{4} x^{2}\right )} {\left (a x + 1\right )} {\left (a x - 1\right )} + 4 \, {\left (a^{9} x^{7} - 3 \, a^{7} x^{5} + 3 \, a^{5} x^{3} - a^{3} x\right )} \sqrt {a x + 1} \sqrt {a x - 1} + a^{2}\right )} \log \left (a x + \sqrt {a x + 1} \sqrt {a x - 1}\right )}\,{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 = -0.01 \[ \int \frac {x^2}{{\mathrm {acosh}\left (a\,x\right )}^3} \,d x \]
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 \[ \int \frac {x^{2}}{\operatorname {acosh}^{3}{\left (a x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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