Optimal. Leaf size=11 \[ \frac{\text{CosIntegral}\left (\sin ^{-1}(a+b x)\right )}{b} \]
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Rubi [A] time = 0.0222556, antiderivative size = 11, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.375, Rules used = {4803, 4623, 3302} \[ \frac{\text{CosIntegral}\left (\sin ^{-1}(a+b x)\right )}{b} \]
Antiderivative was successfully verified.
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Rule 4803
Rule 4623
Rule 3302
Rubi steps
\begin{align*} \int \frac{1}{\sin ^{-1}(a+b x)} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1}{\sin ^{-1}(x)} \, dx,x,a+b x\right )}{b}\\ &=\frac{\operatorname{Subst}\left (\int \frac{\cos (x)}{x} \, dx,x,\sin ^{-1}(a+b x)\right )}{b}\\ &=\frac{\text{Ci}\left (\sin ^{-1}(a+b x)\right )}{b}\\ \end{align*}
Mathematica [A] time = 0.0159863, size = 11, normalized size = 1. \[ \frac{\text{CosIntegral}\left (\sin ^{-1}(a+b x)\right )}{b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.03, size = 12, normalized size = 1.1 \begin{align*}{\frac{{\it Ci} \left ( \arcsin \left ( bx+a \right ) \right ) }{b}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\arcsin \left (b x + a\right )}\,{d x} \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{1}{\arcsin \left (b x + a\right )}, 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{1}{\operatorname{asin}{\left (a + b x \right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.19932, size = 15, normalized size = 1.36 \begin{align*} \frac{\operatorname{Ci}\left (\arcsin \left (b x + a\right )\right )}{b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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