Optimal. Leaf size=47 \[ \frac {\text {CosIntegral}(2 \text {ArcSin}(a+b x))}{2 b}+\frac {\text {CosIntegral}(4 \text {ArcSin}(a+b x))}{8 b}+\frac {3 \log (\text {ArcSin}(a+b x))}{8 b} \]
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Rubi [A]
time = 0.10, antiderivative size = 47, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 4, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.121, Rules used = {4891, 4753,
3393, 3383} \begin {gather*} \frac {\text {CosIntegral}(2 \text {ArcSin}(a+b x))}{2 b}+\frac {\text {CosIntegral}(4 \text {ArcSin}(a+b x))}{8 b}+\frac {3 \log (\text {ArcSin}(a+b x))}{8 b} \end {gather*}
Antiderivative was successfully verified.
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Rule 3383
Rule 3393
Rule 4753
Rule 4891
Rubi steps
\begin {align*} \int \frac {\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 {\left (1-x^2\right )^{3/2}}{\sin ^{-1}(x)} \, dx,x,a+b x\right )}{b}\\ &=\frac {\text {Subst}\left (\int \frac {\cos ^4(x)}{x} \, dx,x,\sin ^{-1}(a+b x)\right )}{b}\\ &=\frac {\text {Subst}\left (\int \left (\frac {3}{8 x}+\frac {\cos (2 x)}{2 x}+\frac {\cos (4 x)}{8 x}\right ) \, dx,x,\sin ^{-1}(a+b x)\right )}{b}\\ &=\frac {3 \log \left (\sin ^{-1}(a+b x)\right )}{8 b}+\frac {\text {Subst}\left (\int \frac {\cos (4 x)}{x} \, dx,x,\sin ^{-1}(a+b x)\right )}{8 b}+\frac {\text {Subst}\left (\int \frac {\cos (2 x)}{x} \, dx,x,\sin ^{-1}(a+b x)\right )}{2 b}\\ &=\frac {\text {Ci}\left (2 \sin ^{-1}(a+b x)\right )}{2 b}+\frac {\text {Ci}\left (4 \sin ^{-1}(a+b x)\right )}{8 b}+\frac {3 \log \left (\sin ^{-1}(a+b x)\right )}{8 b}\\ \end {align*}
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Mathematica [A]
time = 0.19, size = 37, normalized size = 0.79 \begin {gather*} \frac {4 \text {CosIntegral}(2 \text {ArcSin}(a+b x))+\text {CosIntegral}(4 \text {ArcSin}(a+b x))+3 \log (\text {ArcSin}(a+b x))}{8 b} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.42, size = 36, normalized size = 0.77
method | result | size |
default | \(\frac {3 \ln \left (\arcsin \left (b x +a \right )\right )+4 \cosineIntegral \left (2 \arcsin \left (b x +a \right )\right )+\cosineIntegral \left (4 \arcsin \left (b x +a \right )\right )}{8 b}\) | \(36\) |
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 \frac {\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 = 0.46, size = 41, normalized size = 0.87 \begin {gather*} \frac {\operatorname {Ci}\left (4 \, \arcsin \left (b x + a\right )\right )}{8 \, b} + \frac {\operatorname {Ci}\left (2 \, \arcsin \left (b x + a\right )\right )}{2 \, b} + \frac {3 \, \log \left (\arcsin \left (b x + a\right )\right )}{8 \, b} \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 = -0.02 \begin {gather*} \int \frac {{\left (-a^2-2\,a\,b\,x-b^2\,x^2+1\right )}^{3/2}}{\mathrm {asin}\left (a+b\,x\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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