Optimal. Leaf size=65 \[ -\frac{\text{CosIntegral}\left (\sin ^{-1}(a+b x)\right )}{2 b}+\frac{a+b x}{2 b \sin ^{-1}(a+b x)}-\frac{\sqrt{1-(a+b x)^2}}{2 b \sin ^{-1}(a+b x)^2} \]
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Rubi [A] time = 0.0855354, antiderivative size = 65, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.625, Rules used = {4803, 4621, 4719, 4623, 3302} \[ -\frac{\text{CosIntegral}\left (\sin ^{-1}(a+b x)\right )}{2 b}+\frac{a+b x}{2 b \sin ^{-1}(a+b x)}-\frac{\sqrt{1-(a+b x)^2}}{2 b \sin ^{-1}(a+b x)^2} \]
Antiderivative was successfully verified.
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Rule 4803
Rule 4621
Rule 4719
Rule 4623
Rule 3302
Rubi steps
\begin{align*} \int \frac{1}{\sin ^{-1}(a+b x)^3} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1}{\sin ^{-1}(x)^3} \, dx,x,a+b x\right )}{b}\\ &=-\frac{\sqrt{1-(a+b x)^2}}{2 b \sin ^{-1}(a+b x)^2}-\frac{\operatorname{Subst}\left (\int \frac{x}{\sqrt{1-x^2} \sin ^{-1}(x)^2} \, dx,x,a+b x\right )}{2 b}\\ &=-\frac{\sqrt{1-(a+b x)^2}}{2 b \sin ^{-1}(a+b x)^2}+\frac{a+b x}{2 b \sin ^{-1}(a+b x)}-\frac{\operatorname{Subst}\left (\int \frac{1}{\sin ^{-1}(x)} \, dx,x,a+b x\right )}{2 b}\\ &=-\frac{\sqrt{1-(a+b x)^2}}{2 b \sin ^{-1}(a+b x)^2}+\frac{a+b x}{2 b \sin ^{-1}(a+b x)}-\frac{\operatorname{Subst}\left (\int \frac{\cos (x)}{x} \, dx,x,\sin ^{-1}(a+b x)\right )}{2 b}\\ &=-\frac{\sqrt{1-(a+b x)^2}}{2 b \sin ^{-1}(a+b x)^2}+\frac{a+b x}{2 b \sin ^{-1}(a+b x)}-\frac{\text{Ci}\left (\sin ^{-1}(a+b x)\right )}{2 b}\\ \end{align*}
Mathematica [A] time = 0.0675659, size = 65, normalized size = 1. \[ -\frac{\text{CosIntegral}\left (\sin ^{-1}(a+b x)\right )}{2 b}+\frac{a+b x}{2 b \sin ^{-1}(a+b x)}-\frac{\sqrt{1-(a+b x)^2}}{2 b \sin ^{-1}(a+b x)^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.027, size = 53, normalized size = 0.8 \begin{align*}{\frac{1}{b} \left ( -{\frac{1}{2\, \left ( \arcsin \left ( bx+a \right ) \right ) ^{2}}\sqrt{1- \left ( bx+a \right ) ^{2}}}+{\frac{bx+a}{2\,\arcsin \left ( bx+a \right ) }}-{\frac{{\it Ci} \left ( \arcsin \left ( bx+a \right ) \right ) }{2}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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 )^{3}}, 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}^{3}{\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 = 1.15734, size = 77, normalized size = 1.18 \begin{align*} -\frac{\operatorname{Ci}\left (\arcsin \left (b x + a\right )\right )}{2 \, b} + \frac{b x + a}{2 \, b \arcsin \left (b x + a\right )} - \frac{\sqrt{-{\left (b x + a\right )}^{2} + 1}}{2 \, b \arcsin \left (b x + a\right )^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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