Optimal. Leaf size=137 \[ \frac {x^{1+m} \sinh ^{-1}(a x)^2}{1+m}-\frac {2 a x^{2+m} \sinh ^{-1}(a x) \, _2F_1\left (\frac {1}{2},\frac {2+m}{2};\frac {4+m}{2};-a^2 x^2\right )}{2+3 m+m^2}+\frac {2 a^2 x^{3+m} \, _3F_2\left (1,\frac {3}{2}+\frac {m}{2},\frac {3}{2}+\frac {m}{2};2+\frac {m}{2},\frac {5}{2}+\frac {m}{2};-a^2 x^2\right )}{6+11 m+6 m^2+m^3} \]
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Rubi [A]
time = 0.07, antiderivative size = 137, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {5776, 5817}
\begin {gather*} \frac {2 a^2 x^{m+3} \, _3F_2\left (1,\frac {m}{2}+\frac {3}{2},\frac {m}{2}+\frac {3}{2};\frac {m}{2}+2,\frac {m}{2}+\frac {5}{2};-a^2 x^2\right )}{m^3+6 m^2+11 m+6}-\frac {2 a x^{m+2} \sinh ^{-1}(a x) \, _2F_1\left (\frac {1}{2},\frac {m+2}{2};\frac {m+4}{2};-a^2 x^2\right )}{m^2+3 m+2}+\frac {x^{m+1} \sinh ^{-1}(a x)^2}{m+1} \end {gather*}
Antiderivative was successfully verified.
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Rule 5776
Rule 5817
Rubi steps
\begin {align*} \int x^m \sinh ^{-1}(a x)^2 \, dx &=\frac {x^{1+m} \sinh ^{-1}(a x)^2}{1+m}-\frac {(2 a) \int \frac {x^{1+m} \sinh ^{-1}(a x)}{\sqrt {1+a^2 x^2}} \, dx}{1+m}\\ &=\frac {x^{1+m} \sinh ^{-1}(a x)^2}{1+m}-\frac {2 a x^{2+m} \sinh ^{-1}(a x) \, _2F_1\left (\frac {1}{2},\frac {2+m}{2};\frac {4+m}{2};-a^2 x^2\right )}{2+3 m+m^2}+\frac {2 a^2 x^{3+m} \, _3F_2\left (1,\frac {3}{2}+\frac {m}{2},\frac {3}{2}+\frac {m}{2};2+\frac {m}{2},\frac {5}{2}+\frac {m}{2};-a^2 x^2\right )}{6+11 m+6 m^2+m^3}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 123, normalized size = 0.90 \begin {gather*} \frac {x^{1+m} \left ((3+m) \sinh ^{-1}(a x) \left ((2+m) \sinh ^{-1}(a x)-2 a x \, _2F_1\left (\frac {1}{2},\frac {2+m}{2};\frac {4+m}{2};-a^2 x^2\right )\right )+2 a^2 x^2 \, _3F_2\left (1,\frac {3}{2}+\frac {m}{2},\frac {3}{2}+\frac {m}{2};2+\frac {m}{2},\frac {5}{2}+\frac {m}{2};-a^2 x^2\right )\right )}{(1+m) (2+m) (3+m)} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 3.54, size = 0, normalized size = 0.00 \[\int x^{m} \arcsinh \left (a x \right )^{2}\, dx\]
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 \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^{m} \operatorname {asinh}^{2}{\left (a 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^m\,{\mathrm {asinh}\left (a\,x\right )}^2 \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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