Optimal. Leaf size=272 \[ -\frac{a b^2 \text{PolyLog}\left (2,-\frac{i e^{i \sin ^{-1}(a+b x)}}{a-\sqrt{a^2-1}}\right )}{\left (a^2-1\right )^{3/2}}+\frac{a b^2 \text{PolyLog}\left (2,-\frac{i e^{i \sin ^{-1}(a+b x)}}{\sqrt{a^2-1}+a}\right )}{\left (a^2-1\right )^{3/2}}+\frac{b^2 \log (x)}{1-a^2}-\frac{i a b^2 \sin ^{-1}(a+b x) \log \left (1+\frac{i e^{i \sin ^{-1}(a+b x)}}{a-\sqrt{a^2-1}}\right )}{\left (a^2-1\right )^{3/2}}+\frac{i a b^2 \sin ^{-1}(a+b x) \log \left (1+\frac{i e^{i \sin ^{-1}(a+b x)}}{\sqrt{a^2-1}+a}\right )}{\left (a^2-1\right )^{3/2}}-\frac{b \sqrt{1-(a+b x)^2} \sin ^{-1}(a+b x)}{\left (1-a^2\right ) x}-\frac{\sin ^{-1}(a+b x)^2}{2 x^2} \]
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Rubi [A] time = 0.585408, antiderivative size = 272, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 11, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.917, Rules used = {4805, 4743, 4773, 3324, 3323, 2264, 2190, 2279, 2391, 2668, 31} \[ -\frac{a b^2 \text{PolyLog}\left (2,-\frac{i e^{i \sin ^{-1}(a+b x)}}{a-\sqrt{a^2-1}}\right )}{\left (a^2-1\right )^{3/2}}+\frac{a b^2 \text{PolyLog}\left (2,-\frac{i e^{i \sin ^{-1}(a+b x)}}{\sqrt{a^2-1}+a}\right )}{\left (a^2-1\right )^{3/2}}+\frac{b^2 \log (x)}{1-a^2}-\frac{i a b^2 \sin ^{-1}(a+b x) \log \left (1+\frac{i e^{i \sin ^{-1}(a+b x)}}{a-\sqrt{a^2-1}}\right )}{\left (a^2-1\right )^{3/2}}+\frac{i a b^2 \sin ^{-1}(a+b x) \log \left (1+\frac{i e^{i \sin ^{-1}(a+b x)}}{\sqrt{a^2-1}+a}\right )}{\left (a^2-1\right )^{3/2}}-\frac{b \sqrt{1-(a+b x)^2} \sin ^{-1}(a+b x)}{\left (1-a^2\right ) x}-\frac{\sin ^{-1}(a+b x)^2}{2 x^2} \]
Antiderivative was successfully verified.
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Rule 4805
Rule 4743
Rule 4773
Rule 3324
Rule 3323
Rule 2264
Rule 2190
Rule 2279
Rule 2391
Rule 2668
Rule 31
Rubi steps
\begin{align*} \int \frac{\sin ^{-1}(a+b x)^2}{x^3} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\sin ^{-1}(x)^2}{\left (-\frac{a}{b}+\frac{x}{b}\right )^3} \, dx,x,a+b x\right )}{b}\\ &=-\frac{\sin ^{-1}(a+b x)^2}{2 x^2}+\operatorname{Subst}\left (\int \frac{\sin ^{-1}(x)}{\left (-\frac{a}{b}+\frac{x}{b}\right )^2 \sqrt{1-x^2}} \, dx,x,a+b x\right )\\ &=-\frac{\sin ^{-1}(a+b x)^2}{2 x^2}+\operatorname{Subst}\left (\int \frac{x}{\left (-\frac{a}{b}+\frac{\sin (x)}{b}\right )^2} \, dx,x,\sin ^{-1}(a+b x)\right )\\ &=-\frac{b \sqrt{1-(a+b x)^2} \sin ^{-1}(a+b x)}{\left (1-a^2\right ) x}-\frac{\sin ^{-1}(a+b x)^2}{2 x^2}+\frac{b \operatorname{Subst}\left (\int \frac{\cos (x)}{-\frac{a}{b}+\frac{\sin (x)}{b}} \, dx,x,\sin ^{-1}(a+b x)\right )}{1-a^2}+\frac{(a b) \operatorname{Subst}\left (\int \frac{x}{-\frac{a}{b}+\frac{\sin (x)}{b}} \, dx,x,\sin ^{-1}(a+b x)\right )}{1-a^2}\\ &=-\frac{b \sqrt{1-(a+b x)^2} \sin ^{-1}(a+b x)}{\left (1-a^2\right ) x}-\frac{\sin ^{-1}(a+b x)^2}{2 x^2}+\frac{(2 a b) \operatorname{Subst}\left (\int \frac{e^{i x} x}{\frac{i}{b}-\frac{2 a e^{i x}}{b}-\frac{i e^{2 i x}}{b}} \, dx,x,\sin ^{-1}(a+b x)\right )}{1-a^2}+\frac{b^2 \operatorname{Subst}\left (\int \frac{1}{-\frac{a}{b}+x} \, dx,x,\frac{a}{b}+x\right )}{1-a^2}\\ &=-\frac{b \sqrt{1-(a+b x)^2} \sin ^{-1}(a+b x)}{\left (1-a^2\right ) x}-\frac{\sin ^{-1}(a+b x)^2}{2 x^2}+\frac{b^2 \log (x)}{1-a^2}+\frac{(2 i a b) \operatorname{Subst}\left (\int \frac{e^{i x} x}{-\frac{2 a}{b}-\frac{2 \sqrt{-1+a^2}}{b}-\frac{2 i e^{i x}}{b}} \, dx,x,\sin ^{-1}(a+b x)\right )}{\left (-1+a^2\right )^{3/2}}-\frac{(2 i a b) \operatorname{Subst}\left (\int \frac{e^{i x} x}{-\frac{2 a}{b}+\frac{2 \sqrt{-1+a^2}}{b}-\frac{2 i e^{i x}}{b}} \, dx,x,\sin ^{-1}(a+b x)\right )}{\left (-1+a^2\right )^{3/2}}\\ &=-\frac{b \sqrt{1-(a+b x)^2} \sin ^{-1}(a+b x)}{\left (1-a^2\right ) x}-\frac{\sin ^{-1}(a+b x)^2}{2 x^2}-\frac{i a b^2 \sin ^{-1}(a+b x) \log \left (1+\frac{i e^{i \sin ^{-1}(a+b x)}}{a-\sqrt{-1+a^2}}\right )}{\left (-1+a^2\right )^{3/2}}+\frac{i a b^2 \sin ^{-1}(a+b x) \log \left (1+\frac{i e^{i \sin ^{-1}(a+b x)}}{a+\sqrt{-1+a^2}}\right )}{\left (-1+a^2\right )^{3/2}}+\frac{b^2 \log (x)}{1-a^2}-\frac{\left (i a b^2\right ) \operatorname{Subst}\left (\int \log \left (1-\frac{2 i e^{i x}}{\left (-\frac{2 a}{b}-\frac{2 \sqrt{-1+a^2}}{b}\right ) b}\right ) \, dx,x,\sin ^{-1}(a+b x)\right )}{\left (-1+a^2\right )^{3/2}}+\frac{\left (i a b^2\right ) \operatorname{Subst}\left (\int \log \left (1-\frac{2 i e^{i x}}{\left (-\frac{2 a}{b}+\frac{2 \sqrt{-1+a^2}}{b}\right ) b}\right ) \, dx,x,\sin ^{-1}(a+b x)\right )}{\left (-1+a^2\right )^{3/2}}\\ &=-\frac{b \sqrt{1-(a+b x)^2} \sin ^{-1}(a+b x)}{\left (1-a^2\right ) x}-\frac{\sin ^{-1}(a+b x)^2}{2 x^2}-\frac{i a b^2 \sin ^{-1}(a+b x) \log \left (1+\frac{i e^{i \sin ^{-1}(a+b x)}}{a-\sqrt{-1+a^2}}\right )}{\left (-1+a^2\right )^{3/2}}+\frac{i a b^2 \sin ^{-1}(a+b x) \log \left (1+\frac{i e^{i \sin ^{-1}(a+b x)}}{a+\sqrt{-1+a^2}}\right )}{\left (-1+a^2\right )^{3/2}}+\frac{b^2 \log (x)}{1-a^2}-\frac{\left (a b^2\right ) \operatorname{Subst}\left (\int \frac{\log \left (1-\frac{2 i x}{\left (-\frac{2 a}{b}-\frac{2 \sqrt{-1+a^2}}{b}\right ) b}\right )}{x} \, dx,x,e^{i \sin ^{-1}(a+b x)}\right )}{\left (-1+a^2\right )^{3/2}}+\frac{\left (a b^2\right ) \operatorname{Subst}\left (\int \frac{\log \left (1-\frac{2 i x}{\left (-\frac{2 a}{b}+\frac{2 \sqrt{-1+a^2}}{b}\right ) b}\right )}{x} \, dx,x,e^{i \sin ^{-1}(a+b x)}\right )}{\left (-1+a^2\right )^{3/2}}\\ &=-\frac{b \sqrt{1-(a+b x)^2} \sin ^{-1}(a+b x)}{\left (1-a^2\right ) x}-\frac{\sin ^{-1}(a+b x)^2}{2 x^2}-\frac{i a b^2 \sin ^{-1}(a+b x) \log \left (1+\frac{i e^{i \sin ^{-1}(a+b x)}}{a-\sqrt{-1+a^2}}\right )}{\left (-1+a^2\right )^{3/2}}+\frac{i a b^2 \sin ^{-1}(a+b x) \log \left (1+\frac{i e^{i \sin ^{-1}(a+b x)}}{a+\sqrt{-1+a^2}}\right )}{\left (-1+a^2\right )^{3/2}}+\frac{b^2 \log (x)}{1-a^2}-\frac{a b^2 \text{Li}_2\left (-\frac{i e^{i \sin ^{-1}(a+b x)}}{a-\sqrt{-1+a^2}}\right )}{\left (-1+a^2\right )^{3/2}}+\frac{a b^2 \text{Li}_2\left (-\frac{i e^{i \sin ^{-1}(a+b x)}}{a+\sqrt{-1+a^2}}\right )}{\left (-1+a^2\right )^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.114261, size = 314, normalized size = 1.15 \[ \frac{-2 a b^2 x^2 \text{PolyLog}\left (2,\frac{i e^{i \sin ^{-1}(a+b x)}}{\sqrt{a^2-1}-a}\right )+2 a b^2 x^2 \text{PolyLog}\left (2,-\frac{i e^{i \sin ^{-1}(a+b x)}}{\sqrt{a^2-1}+a}\right )-2 \sqrt{a^2-1} b^2 x^2 \log (x)-2 i a b^2 x^2 \sin ^{-1}(a+b x) \log \left (\frac{-\sqrt{a^2-1}+i e^{i \sin ^{-1}(a+b x)}+a}{a-\sqrt{a^2-1}}\right )+2 i a b^2 x^2 \sin ^{-1}(a+b x) \log \left (\frac{\sqrt{a^2-1}+i e^{i \sin ^{-1}(a+b x)}+a}{\sqrt{a^2-1}+a}\right )+2 \sqrt{a^2-1} b x \sqrt{1-(a+b x)^2} \sin ^{-1}(a+b x)-a^2 \sqrt{a^2-1} \sin ^{-1}(a+b x)^2+\sqrt{a^2-1} \sin ^{-1}(a+b x)^2}{2 \left (a^2-1\right )^{3/2} x^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.57, size = 526, normalized size = 1.9 \begin{align*}{\frac{-i{b}^{2}\arcsin \left ( bx+a \right ) }{{a}^{2}-1}}-{\frac{ \left ( \arcsin \left ( bx+a \right ) \right ) ^{2}{a}^{2}}{ \left ( 2\,{a}^{2}-2 \right ){x}^{2}}}+{\frac{b\arcsin \left ( bx+a \right ) }{ \left ({a}^{2}-1 \right ) x}\sqrt{1- \left ( bx+a \right ) ^{2}}}+{\frac{ \left ( \arcsin \left ( bx+a \right ) \right ) ^{2}}{ \left ( 2\,{a}^{2}-2 \right ){x}^{2}}}-{\frac{{b}^{2}}{{a}^{2}-1}\ln \left ( 2\,ia \left ( i \left ( bx+a \right ) +\sqrt{1- \left ( bx+a \right ) ^{2}} \right ) - \left ( i \left ( bx+a \right ) +\sqrt{1- \left ( bx+a \right ) ^{2}} \right ) ^{2}+1 \right ) }+2\,{\frac{{b}^{2}\ln \left ( i \left ( bx+a \right ) +\sqrt{1- \left ( bx+a \right ) ^{2}} \right ) }{{a}^{2}-1}}+{\frac{{b}^{2}a\arcsin \left ( bx+a \right ) }{ \left ({a}^{2}-1 \right ) ^{2}}\sqrt{-{a}^{2}+1}\ln \left ({ \left ( ia+\sqrt{-{a}^{2}+1}-i \left ( bx+a \right ) -\sqrt{1- \left ( bx+a \right ) ^{2}} \right ) \left ( ia+\sqrt{-{a}^{2}+1} \right ) ^{-1}} \right ) }-{\frac{{b}^{2}a\arcsin \left ( bx+a \right ) }{ \left ({a}^{2}-1 \right ) ^{2}}\sqrt{-{a}^{2}+1}\ln \left ({ \left ( ia-\sqrt{-{a}^{2}+1}-i \left ( bx+a \right ) -\sqrt{1- \left ( bx+a \right ) ^{2}} \right ) \left ( ia-\sqrt{-{a}^{2}+1} \right ) ^{-1}} \right ) }-{\frac{i{b}^{2}a}{ \left ({a}^{2}-1 \right ) ^{2}}\sqrt{-{a}^{2}+1}{\it dilog} \left ({ \left ( ia+\sqrt{-{a}^{2}+1}-i \left ( bx+a \right ) -\sqrt{1- \left ( bx+a \right ) ^{2}} \right ) \left ( ia+\sqrt{-{a}^{2}+1} \right ) ^{-1}} \right ) }+{\frac{i{b}^{2}a}{ \left ({a}^{2}-1 \right ) ^{2}}\sqrt{-{a}^{2}+1}{\it dilog} \left ({ \left ( ia-\sqrt{-{a}^{2}+1}-i \left ( bx+a \right ) -\sqrt{1- \left ( bx+a \right ) ^{2}} \right ) \left ( ia-\sqrt{-{a}^{2}+1} \right ) ^{-1}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\arcsin \left (b x + a\right )^{2}}{x^{3}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{asin}^{2}{\left (a + b x \right )}}{x^{3}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\arcsin \left (b x + a\right )^{2}}{x^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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