Optimal. Leaf size=102 \[ \frac {\text {Shi}\left (\cosh ^{-1}(a x)\right )}{16 a^5}+\frac {27 \text {Shi}\left (3 \cosh ^{-1}(a x)\right )}{32 a^5}+\frac {25 \text {Shi}\left (5 \cosh ^{-1}(a x)\right )}{32 a^5}+\frac {2 x^3}{a^2 \cosh ^{-1}(a x)}-\frac {5 x^5}{2 \cosh ^{-1}(a x)}-\frac {x^4 \sqrt {a x-1} \sqrt {a x+1}}{2 a \cosh ^{-1}(a x)^2} \]
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Rubi [A] time = 0.64, antiderivative size = 102, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 5, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5668, 5775, 5670, 5448, 3298} \[ \frac {\text {Shi}\left (\cosh ^{-1}(a x)\right )}{16 a^5}+\frac {27 \text {Shi}\left (3 \cosh ^{-1}(a x)\right )}{32 a^5}+\frac {25 \text {Shi}\left (5 \cosh ^{-1}(a x)\right )}{32 a^5}+\frac {2 x^3}{a^2 \cosh ^{-1}(a x)}-\frac {5 x^5}{2 \cosh ^{-1}(a x)}-\frac {x^4 \sqrt {a x-1} \sqrt {a x+1}}{2 a \cosh ^{-1}(a x)^2} \]
Antiderivative was successfully verified.
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Rule 3298
Rule 5448
Rule 5668
Rule 5670
Rule 5775
Rubi steps
\begin {align*} \int \frac {x^4}{\cosh ^{-1}(a x)^3} \, dx &=-\frac {x^4 \sqrt {-1+a x} \sqrt {1+a x}}{2 a \cosh ^{-1}(a x)^2}-\frac {2 \int \frac {x^3}{\sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^2} \, dx}{a}+\frac {1}{2} (5 a) \int \frac {x^5}{\sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^2} \, dx\\ &=-\frac {x^4 \sqrt {-1+a x} \sqrt {1+a x}}{2 a \cosh ^{-1}(a x)^2}+\frac {2 x^3}{a^2 \cosh ^{-1}(a x)}-\frac {5 x^5}{2 \cosh ^{-1}(a x)}+\frac {25}{2} \int \frac {x^4}{\cosh ^{-1}(a x)} \, dx-\frac {6 \int \frac {x^2}{\cosh ^{-1}(a x)} \, dx}{a^2}\\ &=-\frac {x^4 \sqrt {-1+a x} \sqrt {1+a x}}{2 a \cosh ^{-1}(a x)^2}+\frac {2 x^3}{a^2 \cosh ^{-1}(a x)}-\frac {5 x^5}{2 \cosh ^{-1}(a x)}-\frac {6 \operatorname {Subst}\left (\int \frac {\cosh ^2(x) \sinh (x)}{x} \, dx,x,\cosh ^{-1}(a x)\right )}{a^5}+\frac {25 \operatorname {Subst}\left (\int \frac {\cosh ^4(x) \sinh (x)}{x} \, dx,x,\cosh ^{-1}(a x)\right )}{2 a^5}\\ &=-\frac {x^4 \sqrt {-1+a x} \sqrt {1+a x}}{2 a \cosh ^{-1}(a x)^2}+\frac {2 x^3}{a^2 \cosh ^{-1}(a x)}-\frac {5 x^5}{2 \cosh ^{-1}(a x)}-\frac {6 \operatorname {Subst}\left (\int \left (\frac {\sinh (x)}{4 x}+\frac {\sinh (3 x)}{4 x}\right ) \, dx,x,\cosh ^{-1}(a x)\right )}{a^5}+\frac {25 \operatorname {Subst}\left (\int \left (\frac {\sinh (x)}{8 x}+\frac {3 \sinh (3 x)}{16 x}+\frac {\sinh (5 x)}{16 x}\right ) \, dx,x,\cosh ^{-1}(a x)\right )}{2 a^5}\\ &=-\frac {x^4 \sqrt {-1+a x} \sqrt {1+a x}}{2 a \cosh ^{-1}(a x)^2}+\frac {2 x^3}{a^2 \cosh ^{-1}(a x)}-\frac {5 x^5}{2 \cosh ^{-1}(a x)}+\frac {25 \operatorname {Subst}\left (\int \frac {\sinh (5 x)}{x} \, dx,x,\cosh ^{-1}(a x)\right )}{32 a^5}-\frac {3 \operatorname {Subst}\left (\int \frac {\sinh (x)}{x} \, dx,x,\cosh ^{-1}(a x)\right )}{2 a^5}-\frac {3 \operatorname {Subst}\left (\int \frac {\sinh (3 x)}{x} \, dx,x,\cosh ^{-1}(a x)\right )}{2 a^5}+\frac {25 \operatorname {Subst}\left (\int \frac {\sinh (x)}{x} \, dx,x,\cosh ^{-1}(a x)\right )}{16 a^5}+\frac {75 \operatorname {Subst}\left (\int \frac {\sinh (3 x)}{x} \, dx,x,\cosh ^{-1}(a x)\right )}{32 a^5}\\ &=-\frac {x^4 \sqrt {-1+a x} \sqrt {1+a x}}{2 a \cosh ^{-1}(a x)^2}+\frac {2 x^3}{a^2 \cosh ^{-1}(a x)}-\frac {5 x^5}{2 \cosh ^{-1}(a x)}+\frac {\text {Shi}\left (\cosh ^{-1}(a x)\right )}{16 a^5}+\frac {27 \text {Shi}\left (3 \cosh ^{-1}(a x)\right )}{32 a^5}+\frac {25 \text {Shi}\left (5 \cosh ^{-1}(a x)\right )}{32 a^5}\\ \end {align*}
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Mathematica [A] time = 0.15, size = 107, normalized size = 1.05 \[ \frac {-80 a^5 x^5 \cosh ^{-1}(a x)-16 a^4 x^4 \sqrt {a x-1} \sqrt {a x+1}+64 a^3 x^3 \cosh ^{-1}(a x)+2 \cosh ^{-1}(a x)^2 \text {Shi}\left (\cosh ^{-1}(a x)\right )+27 \cosh ^{-1}(a x)^2 \text {Shi}\left (3 \cosh ^{-1}(a x)\right )+25 \cosh ^{-1}(a x)^2 \text {Shi}\left (5 \cosh ^{-1}(a x)\right )}{32 a^5 \cosh ^{-1}(a x)^2} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.72, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {x^{4}}{\operatorname {arcosh}\left (a x\right )^{3}}, x\right ) \]
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 \[ \int \frac {x^{4}}{\operatorname {arcosh}\left (a x\right )^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.11, size = 123, normalized size = 1.21 \[ \frac {-\frac {\sqrt {a x -1}\, \sqrt {a x +1}}{16 \mathrm {arccosh}\left (a x \right )^{2}}-\frac {a x}{16 \,\mathrm {arccosh}\left (a x \right )}+\frac {\Shi \left (\mathrm {arccosh}\left (a x \right )\right )}{16}-\frac {3 \sinh \left (3 \,\mathrm {arccosh}\left (a x \right )\right )}{32 \mathrm {arccosh}\left (a x \right )^{2}}-\frac {9 \cosh \left (3 \,\mathrm {arccosh}\left (a x \right )\right )}{32 \,\mathrm {arccosh}\left (a x \right )}+\frac {27 \Shi \left (3 \,\mathrm {arccosh}\left (a x \right )\right )}{32}-\frac {\sinh \left (5 \,\mathrm {arccosh}\left (a x \right )\right )}{32 \mathrm {arccosh}\left (a x \right )^{2}}-\frac {5 \cosh \left (5 \,\mathrm {arccosh}\left (a x \right )\right )}{32 \,\mathrm {arccosh}\left (a x \right )}+\frac {25 \Shi \left (5 \,\mathrm {arccosh}\left (a x \right )\right )}{32}}{a^{5}} \]
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^{11} - 3 \, a^{6} x^{9} + 3 \, a^{4} x^{7} - a^{2} x^{5} + {\left (a^{5} x^{8} - a^{3} x^{6}\right )} {\left (a x + 1\right )}^{\frac {3}{2}} {\left (a x - 1\right )}^{\frac {3}{2}} + {\left (3 \, a^{6} x^{9} - 5 \, a^{4} x^{7} + 2 \, a^{2} x^{5}\right )} {\left (a x + 1\right )} {\left (a x - 1\right )} + {\left (3 \, a^{7} x^{10} - 7 \, a^{5} x^{8} + 5 \, a^{3} x^{6} - a x^{4}\right )} \sqrt {a x + 1} \sqrt {a x - 1} + {\left (5 \, a^{8} x^{11} - 15 \, a^{6} x^{9} + 15 \, a^{4} x^{7} - 5 \, a^{2} x^{5} + {\left (5 \, a^{5} x^{8} - 8 \, a^{3} x^{6} + 3 \, a x^{4}\right )} {\left (a x + 1\right )}^{\frac {3}{2}} {\left (a x - 1\right )}^{\frac {3}{2}} + {\left (15 \, a^{6} x^{9} - 31 \, a^{4} x^{7} + 20 \, a^{2} x^{5} - 4 \, x^{3}\right )} {\left (a x + 1\right )} {\left (a x - 1\right )} + {\left (15 \, a^{7} x^{10} - 38 \, a^{5} x^{8} + 32 \, a^{3} x^{6} - 9 \, a x^{4}\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 {25 \, a^{10} x^{12} - 100 \, a^{8} x^{10} + 150 \, a^{6} x^{8} - 100 \, a^{4} x^{6} + 25 \, a^{2} x^{4} + {\left (25 \, a^{6} x^{8} - 24 \, a^{4} x^{6} + 3 \, a^{2} x^{4}\right )} {\left (a x + 1\right )}^{2} {\left (a x - 1\right )}^{2} + {\left (100 \, a^{7} x^{9} - 172 \, a^{5} x^{7} + 87 \, a^{3} x^{5} - 12 \, a x^{3}\right )} {\left (a x + 1\right )}^{\frac {3}{2}} {\left (a x - 1\right )}^{\frac {3}{2}} + 3 \, {\left (50 \, a^{8} x^{10} - 124 \, a^{6} x^{8} + 105 \, a^{4} x^{6} - 35 \, a^{2} x^{4} + 4 \, x^{2}\right )} {\left (a x + 1\right )} {\left (a x - 1\right )} + {\left (100 \, a^{9} x^{11} - 324 \, a^{7} x^{9} + 381 \, a^{5} x^{7} - 193 \, a^{3} x^{5} + 36 \, a x^{3}\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 \[ \int \frac {x^4}{{\mathrm {acosh}\left (a\,x\right )}^3} \,d x \]
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 \[ \int \frac {x^{4}}{\operatorname {acosh}^{3}{\left (a x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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