Optimal. Leaf size=64 \[ -\frac {\text {ArcSin}(a+b x)}{x}-\frac {b \tanh ^{-1}\left (\frac {1-a (a+b x)}{\sqrt {1-a^2} \sqrt {1-(a+b x)^2}}\right )}{\sqrt {1-a^2}} \]
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Rubi [A]
time = 0.06, antiderivative size = 64, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {4889, 4827,
739, 212} \begin {gather*} -\frac {b \tanh ^{-1}\left (\frac {1-a (a+b x)}{\sqrt {1-a^2} \sqrt {1-(a+b x)^2}}\right )}{\sqrt {1-a^2}}-\frac {\text {ArcSin}(a+b x)}{x} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 739
Rule 4827
Rule 4889
Rubi steps
\begin {align*} \int \frac {\sin ^{-1}(a+b x)}{x^2} \, dx &=\frac {\text {Subst}\left (\int \frac {\sin ^{-1}(x)}{\left (-\frac {a}{b}+\frac {x}{b}\right )^2} \, dx,x,a+b x\right )}{b}\\ &=-\frac {\sin ^{-1}(a+b x)}{x}+\text {Subst}\left (\int \frac {1}{\left (-\frac {a}{b}+\frac {x}{b}\right ) \sqrt {1-x^2}} \, dx,x,a+b x\right )\\ &=-\frac {\sin ^{-1}(a+b x)}{x}-\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 )\\ &=-\frac {\sin ^{-1}(a+b x)}{x}-\frac {b \tanh ^{-1}\left (\frac {b \left (\frac {1}{b}-\frac {a (a+b x)}{b}\right )}{\sqrt {1-a^2} \sqrt {1-(a+b x)^2}}\right )}{\sqrt {1-a^2}}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 66, normalized size = 1.03 \begin {gather*} -\frac {\text {ArcSin}(a+b x)}{x}-\frac {b \tanh ^{-1}\left (\frac {1-a^2-a b x}{\sqrt {1-a^2} \sqrt {1-(a+b x)^2}}\right )}{\sqrt {1-a^2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.15, size = 82, normalized size = 1.28
method | result | size |
derivativedivides | \(b \left (-\frac {\arcsin \left (b x +a \right )}{b x}-\frac {\ln \left (\frac {-2 a^{2}+2-2 a b x +2 \sqrt {-a^{2}+1}\, \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}{b x}\right )}{\sqrt {-a^{2}+1}}\right )\) | \(82\) |
default | \(b \left (-\frac {\arcsin \left (b x +a \right )}{b x}-\frac {\ln \left (\frac {-2 a^{2}+2-2 a b x +2 \sqrt {-a^{2}+1}\, \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}{b x}\right )}{\sqrt {-a^{2}+1}}\right )\) | \(82\) |
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 = 2.90, size = 233, normalized size = 3.64 \begin {gather*} \left [-\frac {\sqrt {-a^{2} + 1} b x \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 \, {\left (a^{2} - 1\right )} \arcsin \left (b x + a\right )}{2 \, {\left (a^{2} - 1\right )} x}, \frac {\sqrt {a^{2} - 1} b x \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 ) - {\left (a^{2} - 1\right )} \arcsin \left (b x + a\right )}{{\left (a^{2} - 1\right )} x}\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^{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.40, size = 79, normalized size = 1.23 \begin {gather*} \frac {2 \, 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 )}{\sqrt {a^{2} - 1} {\left | b \right |}} - \frac {\arcsin \left (b x + a\right )}{x} \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 {\mathrm {asin}\left (a+b\,x\right )}{x^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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