Optimal. Leaf size=41 \[ -\frac {\text {Si}\left (\sin ^{-1}(a+b x)\right )}{b}-\frac {\sqrt {1-(a+b x)^2}}{b \sin ^{-1}(a+b x)} \]
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Rubi [A] time = 0.08, antiderivative size = 41, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {4803, 4621, 4723, 3299} \[ -\frac {\text {Si}\left (\sin ^{-1}(a+b x)\right )}{b}-\frac {\sqrt {1-(a+b x)^2}}{b \sin ^{-1}(a+b x)} \]
Antiderivative was successfully verified.
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Rule 3299
Rule 4621
Rule 4723
Rule 4803
Rubi steps
\begin {align*} \int \frac {1}{\sin ^{-1}(a+b x)^2} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {1}{\sin ^{-1}(x)^2} \, dx,x,a+b x\right )}{b}\\ &=-\frac {\sqrt {1-(a+b x)^2}}{b \sin ^{-1}(a+b x)}-\frac {\operatorname {Subst}\left (\int \frac {x}{\sqrt {1-x^2} \sin ^{-1}(x)} \, dx,x,a+b x\right )}{b}\\ &=-\frac {\sqrt {1-(a+b x)^2}}{b \sin ^{-1}(a+b x)}-\frac {\operatorname {Subst}\left (\int \frac {\sin (x)}{x} \, dx,x,\sin ^{-1}(a+b x)\right )}{b}\\ &=-\frac {\sqrt {1-(a+b x)^2}}{b \sin ^{-1}(a+b x)}-\frac {\text {Si}\left (\sin ^{-1}(a+b x)\right )}{b}\\ \end {align*}
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Mathematica [A] time = 0.08, size = 37, normalized size = 0.90 \[ -\frac {\text {Si}\left (\sin ^{-1}(a+b x)\right )+\frac {\sqrt {1-(a+b x)^2}}{\sin ^{-1}(a+b x)}}{b} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.40, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {1}{\arcsin \left (b x + a\right )^{2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.35, size = 39, normalized size = 0.95 \[ -\frac {\operatorname {Si}\left (\arcsin \left (b x + a\right )\right )}{b} - \frac {\sqrt {-{\left (b x + a\right )}^{2} + 1}}{b \arcsin \left (b x + a\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.09, size = 38, normalized size = 0.93 \[ \frac {-\frac {\sqrt {1-\left (b x +a \right )^{2}}}{\arcsin \left (b x +a \right )}-\Si \left (\arcsin \left (b x +a \right )\right )}{b} \]
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 \[ \frac {b \arctan \left (b x + a, \sqrt {b x + a + 1} \sqrt {-b x - a + 1}\right ) \int \frac {{\left (b x + a\right )} \sqrt {-b x - a + 1}}{\sqrt {b x + a + 1} {\left (b x + a - 1\right )} \arctan \left (b x + a, \sqrt {b x + a + 1} \sqrt {-b x - a + 1}\right )}\,{d x} - \sqrt {b x + a + 1} \sqrt {-b x - a + 1}}{b \arctan \left (b x + a, \sqrt {b x + a + 1} \sqrt {-b x - a + 1}\right )} \]
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 \[ \int \frac {1}{{\mathrm {asin}\left (a+b\,x\right )}^2} \,d x \]
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 \[ \int \frac {1}{\operatorname {asin}^{2}{\left (a + b x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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