Optimal. Leaf size=97 \[ -\frac {i \text {Li}_2\left (-e^{2 i \sin ^{-1}(a+b x)}\right )}{b}+\frac {(a+b x) \sin ^{-1}(a+b x)^2}{b \sqrt {1-(a+b x)^2}}-\frac {i \sin ^{-1}(a+b x)^2}{b}+\frac {2 \sin ^{-1}(a+b x) \log \left (1+e^{2 i \sin ^{-1}(a+b x)}\right )}{b} \]
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Rubi [A] time = 0.16, antiderivative size = 97, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.212, Rules used = {4807, 4651, 4675, 3719, 2190, 2279, 2391} \[ -\frac {i \text {PolyLog}\left (2,-e^{2 i \sin ^{-1}(a+b x)}\right )}{b}+\frac {(a+b x) \sin ^{-1}(a+b x)^2}{b \sqrt {1-(a+b x)^2}}-\frac {i \sin ^{-1}(a+b x)^2}{b}+\frac {2 \sin ^{-1}(a+b x) \log \left (1+e^{2 i \sin ^{-1}(a+b x)}\right )}{b} \]
Antiderivative was successfully verified.
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Rule 2190
Rule 2279
Rule 2391
Rule 3719
Rule 4651
Rule 4675
Rule 4807
Rubi steps
\begin {align*} \int \frac {\sin ^{-1}(a+b x)^2}{\left (1-a^2-2 a b x-b^2 x^2\right )^{3/2}} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {\sin ^{-1}(x)^2}{\left (1-x^2\right )^{3/2}} \, dx,x,a+b x\right )}{b}\\ &=\frac {(a+b x) \sin ^{-1}(a+b x)^2}{b \sqrt {1-(a+b x)^2}}-\frac {2 \operatorname {Subst}\left (\int \frac {x \sin ^{-1}(x)}{1-x^2} \, dx,x,a+b x\right )}{b}\\ &=\frac {(a+b x) \sin ^{-1}(a+b x)^2}{b \sqrt {1-(a+b x)^2}}-\frac {2 \operatorname {Subst}\left (\int x \tan (x) \, dx,x,\sin ^{-1}(a+b x)\right )}{b}\\ &=-\frac {i \sin ^{-1}(a+b x)^2}{b}+\frac {(a+b x) \sin ^{-1}(a+b x)^2}{b \sqrt {1-(a+b x)^2}}+\frac {(4 i) \operatorname {Subst}\left (\int \frac {e^{2 i x} x}{1+e^{2 i x}} \, dx,x,\sin ^{-1}(a+b x)\right )}{b}\\ &=-\frac {i \sin ^{-1}(a+b x)^2}{b}+\frac {(a+b x) \sin ^{-1}(a+b x)^2}{b \sqrt {1-(a+b x)^2}}+\frac {2 \sin ^{-1}(a+b x) \log \left (1+e^{2 i \sin ^{-1}(a+b x)}\right )}{b}-\frac {2 \operatorname {Subst}\left (\int \log \left (1+e^{2 i x}\right ) \, dx,x,\sin ^{-1}(a+b x)\right )}{b}\\ &=-\frac {i \sin ^{-1}(a+b x)^2}{b}+\frac {(a+b x) \sin ^{-1}(a+b x)^2}{b \sqrt {1-(a+b x)^2}}+\frac {2 \sin ^{-1}(a+b x) \log \left (1+e^{2 i \sin ^{-1}(a+b x)}\right )}{b}+\frac {i \operatorname {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{2 i \sin ^{-1}(a+b x)}\right )}{b}\\ &=-\frac {i \sin ^{-1}(a+b x)^2}{b}+\frac {(a+b x) \sin ^{-1}(a+b x)^2}{b \sqrt {1-(a+b x)^2}}+\frac {2 \sin ^{-1}(a+b x) \log \left (1+e^{2 i \sin ^{-1}(a+b x)}\right )}{b}-\frac {i \text {Li}_2\left (-e^{2 i \sin ^{-1}(a+b x)}\right )}{b}\\ \end {align*}
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Mathematica [A] time = 0.39, size = 114, normalized size = 1.18 \[ \frac {\sin ^{-1}(a+b x) \left (\frac {\left (-i \sqrt {-a^2-2 a b x-b^2 x^2+1}+a+b x\right ) \sin ^{-1}(a+b x)}{\sqrt {-a^2-2 a b x-b^2 x^2+1}}+2 \log \left (1+e^{2 i \sin ^{-1}(a+b x)}\right )\right )-i \text {Li}_2\left (-e^{2 i \sin ^{-1}(a+b x)}\right )}{b} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.51, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} \arcsin \left (b x + a\right )^{2}}{b^{4} x^{4} + 4 \, a b^{3} x^{3} + 2 \, {\left (3 \, a^{2} - 1\right )} b^{2} x^{2} + a^{4} + 4 \, {\left (a^{3} - a\right )} b x - 2 \, a^{2} + 1}, x\right ) \]
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 \[ \int \frac {\arcsin \left (b x + a\right )^{2}}{{\left (-b^{2} x^{2} - 2 \, a b x - a^{2} + 1\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.31, size = 194, normalized size = 2.00 \[ \frac {\left (-\sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}\, x b +i x^{2} b^{2}-\sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}\, a +2 i x a b +i a^{2}-i\right ) \arcsin \left (b x +a \right )^{2}}{b \left (b^{2} x^{2}+2 a b x +a^{2}-1\right )}+\frac {2 \arcsin \left (b x +a \right ) \ln \left (1+\left (i \left (b x +a \right )+\sqrt {1-\left (b x +a \right )^{2}}\right )^{2}\right )}{b}-\frac {2 i \arcsin \left (b x +a \right )^{2}}{b}-\frac {i \polylog \left (2, -\left (i \left (b x +a \right )+\sqrt {1-\left (b x +a \right )^{2}}\right )^{2}\right )}{b} \]
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 \[ \int \frac {\arcsin \left (b x + a\right )^{2}}{{\left (-b^{2} x^{2} - 2 \, a b x - a^{2} + 1\right )}^{\frac {3}{2}}}\,{d x} \]
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 \[ \int \frac {{\mathrm {asin}\left (a+b\,x\right )}^2}{{\left (-a^2-2\,a\,b\,x-b^2\,x^2+1\right )}^{3/2}} \,d x \]
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 \[ \int \frac {\operatorname {asin}^{2}{\left (a + b x \right )}}{\left (- \left (a + b x - 1\right ) \left (a + b x + 1\right )\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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