Optimal. Leaf size=272 \[ -\frac {a b^2 \text {Li}_2\left (-\frac {i e^{i \sin ^{-1}(a+b x)}}{a-\sqrt {a^2-1}}\right )}{\left (a^2-1\right )^{3/2}}+\frac {a b^2 \text {Li}_2\left (-\frac {i e^{i \sin ^{-1}(a+b x)}}{a+\sqrt {a^2-1}}\right )}{\left (a^2-1\right )^{3/2}}+\frac {b^2 \log (x)}{1-a^2}-\frac {i a b^2 \sin ^{-1}(a+b x) \log \left (1+\frac {i e^{i \sin ^{-1}(a+b x)}}{a-\sqrt {a^2-1}}\right )}{\left (a^2-1\right )^{3/2}}+\frac {i a b^2 \sin ^{-1}(a+b x) \log \left (1+\frac {i e^{i \sin ^{-1}(a+b x)}}{\sqrt {a^2-1}+a}\right )}{\left (a^2-1\right )^{3/2}}-\frac {b \sqrt {1-(a+b x)^2} \sin ^{-1}(a+b x)}{\left (1-a^2\right ) x}-\frac {\sin ^{-1}(a+b x)^2}{2 x^2} \]
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Rubi [A] time = 0.59, antiderivative size = 272, normalized size of antiderivative = 1.00, 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 31
Rule 2190
Rule 2264
Rule 2279
Rule 2391
Rule 2668
Rule 3323
Rule 3324
Rule 4743
Rule 4773
Rule 4805
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*}
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Mathematica [A] time = 0.14, size = 314, normalized size = 1.15 \[ \frac {-2 a b^2 x^2 \text {Li}_2\left (\frac {i e^{i \sin ^{-1}(a+b x)}}{\sqrt {a^2-1}-a}\right )+2 a b^2 x^2 \text {Li}_2\left (-\frac {i e^{i \sin ^{-1}(a+b x)}}{a+\sqrt {a^2-1}}\right )-2 \sqrt {a^2-1} b^2 x^2 \log (x)-2 i a b^2 x^2 \sin ^{-1}(a+b x) \log \left (\frac {-\sqrt {a^2-1}+i e^{i \sin ^{-1}(a+b x)}+a}{a-\sqrt {a^2-1}}\right )+2 i a b^2 x^2 \sin ^{-1}(a+b x) \log \left (\frac {\sqrt {a^2-1}+i e^{i \sin ^{-1}(a+b x)}+a}{\sqrt {a^2-1}+a}\right )+2 \sqrt {a^2-1} b x \sqrt {1-(a+b x)^2} \sin ^{-1}(a+b x)-a^2 \sqrt {a^2-1} \sin ^{-1}(a+b x)^2+\sqrt {a^2-1} \sin ^{-1}(a+b x)^2}{2 \left (a^2-1\right )^{3/2} x^2} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.42, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\arcsin \left (b x + a\right )^{2}}{x^{3}}, x\right ) \]
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 \[ \int \frac {\arcsin \left (b x + a\right )^{2}}{x^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.98, size = 526, normalized size = 1.93 \[ -\frac {i b^{2} \arcsin \left (b x +a \right )}{a^{2}-1}+\frac {b \arcsin \left (b x +a \right ) \sqrt {1-\left (b x +a \right )^{2}}}{\left (a^{2}-1\right ) x}-\frac {\arcsin \left (b x +a \right )^{2} a^{2}}{2 \left (a^{2}-1\right ) x^{2}}+\frac {\arcsin \left (b x +a \right )^{2}}{2 \left (a^{2}-1\right ) x^{2}}+\frac {2 b^{2} \ln \left (i \left (b x +a \right )+\sqrt {1-\left (b x +a \right )^{2}}\right )}{a^{2}-1}-\frac {b^{2} \ln \left (2 i a \left (i \left (b x +a \right )+\sqrt {1-\left (b x +a \right )^{2}}\right )-\left (i \left (b x +a \right )+\sqrt {1-\left (b x +a \right )^{2}}\right )^{2}+1\right )}{a^{2}-1}+\frac {b^{2} \sqrt {-a^{2}+1}\, a \arcsin \left (b x +a \right ) \ln \left (\frac {i a +\sqrt {-a^{2}+1}-i \left (b x +a \right )-\sqrt {1-\left (b x +a \right )^{2}}}{i a +\sqrt {-a^{2}+1}}\right )}{\left (a^{2}-1\right )^{2}}-\frac {b^{2} \sqrt {-a^{2}+1}\, a \arcsin \left (b x +a \right ) \ln \left (\frac {i a -\sqrt {-a^{2}+1}-i \left (b x +a \right )-\sqrt {1-\left (b x +a \right )^{2}}}{i a -\sqrt {-a^{2}+1}}\right )}{\left (a^{2}-1\right )^{2}}-\frac {i b^{2} \sqrt {-a^{2}+1}\, \dilog \left (\frac {i a +\sqrt {-a^{2}+1}-i \left (b x +a \right )-\sqrt {1-\left (b x +a \right )^{2}}}{i a +\sqrt {-a^{2}+1}}\right ) a}{\left (a^{2}-1\right )^{2}}+\frac {i b^{2} \sqrt {-a^{2}+1}\, \dilog \left (\frac {i a -\sqrt {-a^{2}+1}-i \left (b x +a \right )-\sqrt {1-\left (b x +a \right )^{2}}}{i a -\sqrt {-a^{2}+1}}\right ) a}{\left (a^{2}-1\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\mathrm {asin}\left (a+b\,x\right )}^2}{x^3} \,d x \]
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 \[ \int \frac {\operatorname {asin}^{2}{\left (a + b x \right )}}{x^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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