Optimal. Leaf size=57 \[ -\frac {\left (1-(a+b x)^2\right )^2}{b \text {ArcSin}(a+b x)}-\frac {\text {Si}(2 \text {ArcSin}(a+b x))}{b}-\frac {\text {Si}(4 \text {ArcSin}(a+b x))}{2 b} \]
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Rubi [A]
time = 0.11, antiderivative size = 57, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 5, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.152, Rules used = {4891, 4751,
4809, 4491, 3380} \begin {gather*} -\frac {\text {Si}(2 \text {ArcSin}(a+b x))}{b}-\frac {\text {Si}(4 \text {ArcSin}(a+b x))}{2 b}-\frac {\left (1-(a+b x)^2\right )^2}{b \text {ArcSin}(a+b x)} \end {gather*}
Antiderivative was successfully verified.
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Rule 3380
Rule 4491
Rule 4751
Rule 4809
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)^2} \, dx &=\frac {\text {Subst}\left (\int \frac {\left (1-x^2\right )^{3/2}}{\sin ^{-1}(x)^2} \, dx,x,a+b x\right )}{b}\\ &=-\frac {\left (1-(a+b x)^2\right )^2}{b \sin ^{-1}(a+b x)}-\frac {4 \text {Subst}\left (\int \frac {x \left (1-x^2\right )}{\sin ^{-1}(x)} \, dx,x,a+b x\right )}{b}\\ &=-\frac {\left (1-(a+b x)^2\right )^2}{b \sin ^{-1}(a+b x)}-\frac {4 \text {Subst}\left (\int \frac {\cos ^3(x) \sin (x)}{x} \, dx,x,\sin ^{-1}(a+b x)\right )}{b}\\ &=-\frac {\left (1-(a+b x)^2\right )^2}{b \sin ^{-1}(a+b x)}-\frac {4 \text {Subst}\left (\int \left (\frac {\sin (2 x)}{4 x}+\frac {\sin (4 x)}{8 x}\right ) \, dx,x,\sin ^{-1}(a+b x)\right )}{b}\\ &=-\frac {\left (1-(a+b x)^2\right )^2}{b \sin ^{-1}(a+b x)}-\frac {\text {Subst}\left (\int \frac {\sin (4 x)}{x} \, dx,x,\sin ^{-1}(a+b x)\right )}{2 b}-\frac {\text {Subst}\left (\int \frac {\sin (2 x)}{x} \, dx,x,\sin ^{-1}(a+b x)\right )}{b}\\ &=-\frac {\left (1-(a+b x)^2\right )^2}{b \sin ^{-1}(a+b x)}-\frac {\text {Si}\left (2 \sin ^{-1}(a+b x)\right )}{b}-\frac {\text {Si}\left (4 \sin ^{-1}(a+b x)\right )}{2 b}\\ \end {align*}
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Mathematica [A]
time = 0.19, size = 70, normalized size = 1.23 \begin {gather*} -\frac {2 \left (-1+a^2+2 a b x+b^2 x^2\right )^2+2 \text {ArcSin}(a+b x) \text {Si}(2 \text {ArcSin}(a+b x))+\text {ArcSin}(a+b x) \text {Si}(4 \text {ArcSin}(a+b x))}{2 b \text {ArcSin}(a+b x)} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.44, size = 70, normalized size = 1.23
method | result | size |
default | \(-\frac {8 \sinIntegral \left (2 \arcsin \left (b x +a \right )\right ) \arcsin \left (b x +a \right )+4 \sinIntegral \left (4 \arcsin \left (b x +a \right )\right ) \arcsin \left (b x +a \right )+4 \cos \left (2 \arcsin \left (b x +a \right )\right )+\cos \left (4 \arcsin \left (b x +a \right )\right )+3}{8 b \arcsin \left (b x +a \right )}\) | \(70\) |
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}^{2}{\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 = 0.51, size = 61, normalized size = 1.07 \begin {gather*} -\frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2} - 1\right )}^{2}}{b \arcsin \left (b x + a\right )} - \frac {\operatorname {Si}\left (4 \, \arcsin \left (b x + a\right )\right )}{2 \, b} - \frac {\operatorname {Si}\left (2 \, \arcsin \left (b x + a\right )\right )}{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 )}^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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