Optimal. Leaf size=103 \[ -\frac {b \sqrt {1-(a+b x)^2}}{2 \left (1-a^2\right ) x}-\frac {\text {ArcSin}(a+b x)}{2 x^2}-\frac {a b^2 \tanh ^{-1}\left (\frac {1-a (a+b x)}{\sqrt {1-a^2} \sqrt {1-(a+b x)^2}}\right )}{2 \left (1-a^2\right )^{3/2}} \]
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Rubi [A]
time = 0.08, antiderivative size = 103, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {4889, 4827,
745, 739, 212} \begin {gather*} -\frac {a b^2 \tanh ^{-1}\left (\frac {1-a (a+b x)}{\sqrt {1-a^2} \sqrt {1-(a+b x)^2}}\right )}{2 \left (1-a^2\right )^{3/2}}-\frac {b \sqrt {1-(a+b x)^2}}{2 \left (1-a^2\right ) x}-\frac {\text {ArcSin}(a+b x)}{2 x^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 739
Rule 745
Rule 4827
Rule 4889
Rubi steps
\begin {align*} \int \frac {\sin ^{-1}(a+b x)}{x^3} \, dx &=\frac {\text {Subst}\left (\int \frac {\sin ^{-1}(x)}{\left (-\frac {a}{b}+\frac {x}{b}\right )^3} \, dx,x,a+b x\right )}{b}\\ &=-\frac {\sin ^{-1}(a+b x)}{2 x^2}+\frac {1}{2} \text {Subst}\left (\int \frac {1}{\left (-\frac {a}{b}+\frac {x}{b}\right )^2 \sqrt {1-x^2}} \, dx,x,a+b x\right )\\ &=-\frac {b \sqrt {1-(a+b x)^2}}{2 \left (1-a^2\right ) x}-\frac {\sin ^{-1}(a+b x)}{2 x^2}+\frac {(a b) \text {Subst}\left (\int \frac {1}{\left (-\frac {a}{b}+\frac {x}{b}\right ) \sqrt {1-x^2}} \, dx,x,a+b x\right )}{2 \left (1-a^2\right )}\\ &=-\frac {b \sqrt {1-(a+b x)^2}}{2 \left (1-a^2\right ) x}-\frac {\sin ^{-1}(a+b x)}{2 x^2}-\frac {(a b) \text {Subst}\left (\int \frac {1}{\frac {1}{b^2}-\frac {a^2}{b^2}-x^2} \, dx,x,\frac {\frac {1}{b}-\frac {a (a+b x)}{b}}{\sqrt {1-(a+b x)^2}}\right )}{2 \left (1-a^2\right )}\\ &=-\frac {b \sqrt {1-(a+b x)^2}}{2 \left (1-a^2\right ) x}-\frac {\sin ^{-1}(a+b x)}{2 x^2}-\frac {a b^2 \tanh ^{-1}\left (\frac {1-a (a+b x)}{\sqrt {1-a^2} \sqrt {1-(a+b x)^2}}\right )}{2 \left (1-a^2\right )^{3/2}}\\ \end {align*}
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Mathematica [A]
time = 0.14, size = 125, normalized size = 1.21 \begin {gather*} -\frac {\text {ArcSin}(a+b x)+\frac {b x \left (\sqrt {1-a^2} \sqrt {1-a^2-2 a b x-b^2 x^2}-a b x \log (x)+a b x \log \left (1-a^2-a b x+\sqrt {1-a^2} \sqrt {1-a^2-2 a b x-b^2 x^2}\right )\right )}{\left (1-a^2\right )^{3/2}}}{2 x^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.01, size = 124, normalized size = 1.20
method | result | size |
derivativedivides | \(b^{2} \left (-\frac {\arcsin \left (b x +a \right )}{2 b^{2} x^{2}}-\frac {\sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}{2 \left (-a^{2}+1\right ) b x}-\frac {a \ln \left (\frac {-2 a^{2}+2-2 a b x +2 \sqrt {-a^{2}+1}\, \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}{b x}\right )}{2 \left (-a^{2}+1\right )^{\frac {3}{2}}}\right )\) | \(124\) |
default | \(b^{2} \left (-\frac {\arcsin \left (b x +a \right )}{2 b^{2} x^{2}}-\frac {\sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}{2 \left (-a^{2}+1\right ) b x}-\frac {a \ln \left (\frac {-2 a^{2}+2-2 a b x +2 \sqrt {-a^{2}+1}\, \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}{b x}\right )}{2 \left (-a^{2}+1\right )^{\frac {3}{2}}}\right )\) | \(124\) |
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 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 3.45, size = 325, normalized size = 3.16 \begin {gather*} \left [-\frac {\sqrt {-a^{2} + 1} a b^{2} x^{2} \log \left (\frac {{\left (2 \, a^{2} - 1\right )} b^{2} x^{2} + 2 \, a^{4} + 4 \, {\left (a^{3} - a\right )} b x - 2 \, \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left (a b x + a^{2} - 1\right )} \sqrt {-a^{2} + 1} - 4 \, a^{2} + 2}{x^{2}}\right ) - 2 \, \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left (a^{2} - 1\right )} b x + 2 \, {\left (a^{4} - 2 \, a^{2} + 1\right )} \arcsin \left (b x + a\right )}{4 \, {\left (a^{4} - 2 \, a^{2} + 1\right )} x^{2}}, -\frac {\sqrt {a^{2} - 1} a b^{2} x^{2} \arctan \left (\frac {\sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left (a b x + a^{2} - 1\right )} \sqrt {a^{2} - 1}}{{\left (a^{2} - 1\right )} b^{2} x^{2} + a^{4} + 2 \, {\left (a^{3} - a\right )} b x - 2 \, a^{2} + 1}\right ) - \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left (a^{2} - 1\right )} b x + {\left (a^{4} - 2 \, a^{2} + 1\right )} \arcsin \left (b x + a\right )}{2 \, {\left (a^{4} - 2 \, a^{2} + 1\right )} x^{2}}\right ] \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 {\operatorname {asin}{\left (a + b x \right )}}{x^{3}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 243 vs.
\(2 (87) = 174\).
time = 0.42, size = 243, normalized size = 2.36 \begin {gather*} -{\left (\frac {a b^{2} \arctan \left (\frac {\frac {{\left (\sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left | b \right |} + b\right )} a}{b^{2} x + a b} - 1}{\sqrt {a^{2} - 1}}\right )}{{\left (a^{2} {\left | b \right |} - {\left | b \right |}\right )} \sqrt {a^{2} - 1}} - \frac {a b^{2} - \frac {{\left (\sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left | b \right |} + b\right )} b^{2}}{b^{2} x + a b}}{{\left (a^{3} {\left | b \right |} - a {\left | b \right |}\right )} {\left (\frac {{\left (\sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left | b \right |} + b\right )}^{2} a}{{\left (b^{2} x + a b\right )}^{2}} + a - \frac {2 \, {\left (\sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left | b \right |} + b\right )}}{b^{2} x + a b}\right )}}\right )} b - \frac {\arcsin \left (b x + a\right )}{2 \, x^{2}} \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 = -0.01 \begin {gather*} \int \frac {\mathrm {asin}\left (a+b\,x\right )}{x^3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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