Optimal. Leaf size=27 \[ \text {Int}\left (\frac {1}{\left (1-(a+b x)^2\right )^{3/2} \text {ArcSin}(a+b x)},x\right ) \]
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Rubi [A]
time = 0.05, 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 {1}{\left (1-a^2-2 a b x-b^2 x^2\right )^{3/2} \text {ArcSin}(a+b x)} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {1}{\left (1-a^2-2 a b x-b^2 x^2\right )^{3/2} \sin ^{-1}(a+b x)} \, dx &=\frac {\text {Subst}\left (\int \frac {1}{\left (1-x^2\right )^{3/2} \sin ^{-1}(x)} \, dx,x,a+b x\right )}{b}\\ \end {align*}
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Mathematica [A]
time = 0.55, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\left (1-a^2-2 a b x-b^2 x^2\right )^{3/2} \text {ArcSin}(a+b x)} \, dx \end {gather*}
Verification is not applicable to the result.
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Maple [A]
time = 1.35, size = 0, normalized size = 0.00 \[\int \frac {1}{\left (-b^{2} x^{2}-2 a b x -a^{2}+1\right )^{\frac {3}{2}} \arcsin \left (b x +a \right )}\, dx\]
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 {1}{\left (- \left (a + b x - 1\right ) \left (a + b x + 1\right )\right )^{\frac {3}{2}} \operatorname {asin}{\left (a + b x \right )}}\, 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 = -0.04 \begin {gather*} \int \frac {1}{\mathrm {asin}\left (a+b\,x\right )\,{\left (-a^2-2\,a\,b\,x-b^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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