Optimal. Leaf size=67 \[ -\frac {x^2}{b c \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )}+\frac {2 \text {Int}\left (\frac {x}{\left (1+c^2 x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )},x\right )}{b c} \]
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Rubi [A]
time = 0.13, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps
used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {}
\begin {gather*} \int \frac {x^2}{\left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {x^2}{\left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2} \, dx &=-\frac {x^2}{b c \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )}+\frac {2 \int \frac {x}{\left (1+c^2 x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )} \, dx}{b c}\\ \end {align*}
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Mathematica [A]
time = 2.45, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^2}{\left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2} \, dx \end {gather*}
Verification is not applicable to the result.
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Maple [A]
time = 180.00, size = 0, normalized size = 0.00 \[\int \frac {x^{2}}{\left (c^{2} x^{2}+1\right )^{\frac {3}{2}} \left (a +b \arcsinh \left (c x \right )\right )^{2}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
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 [A]
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 [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{2}}{\left (a + b \operatorname {asinh}{\left (c x \right )}\right )^{2} \left (c^{2} x^{2} + 1\right )^{\frac {3}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
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 [A]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {x^2}{{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2\,{\left (c^2\,x^2+1\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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