Optimal. Leaf size=31 \[ \frac {\text {Ci}\left (2 \sin ^{-1}(a+b x)\right )}{2 b}+\frac {\log \left (\sin ^{-1}(a+b x)\right )}{2 b} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.12, antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.121, Rules used = {4807, 4661, 3312, 3302} \[ \frac {\text {CosIntegral}\left (2 \sin ^{-1}(a+b x)\right )}{2 b}+\frac {\log \left (\sin ^{-1}(a+b x)\right )}{2 b} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3302
Rule 3312
Rule 4661
Rule 4807
Rubi steps
\begin {align*} \int \frac {\sqrt {1-a^2-2 a b x-b^2 x^2}}{\sin ^{-1}(a+b x)} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {\sqrt {1-x^2}}{\sin ^{-1}(x)} \, dx,x,a+b x\right )}{b}\\ &=\frac {\operatorname {Subst}\left (\int \frac {\cos ^2(x)}{x} \, dx,x,\sin ^{-1}(a+b x)\right )}{b}\\ &=\frac {\operatorname {Subst}\left (\int \left (\frac {1}{2 x}+\frac {\cos (2 x)}{2 x}\right ) \, dx,x,\sin ^{-1}(a+b x)\right )}{b}\\ &=\frac {\log \left (\sin ^{-1}(a+b x)\right )}{2 b}+\frac {\operatorname {Subst}\left (\int \frac {\cos (2 x)}{x} \, dx,x,\sin ^{-1}(a+b x)\right )}{2 b}\\ &=\frac {\text {Ci}\left (2 \sin ^{-1}(a+b x)\right )}{2 b}+\frac {\log \left (\sin ^{-1}(a+b x)\right )}{2 b}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.08, size = 24, normalized size = 0.77 \[ \frac {\text {Ci}\left (2 \sin ^{-1}(a+b x)\right )+\log \left (\sin ^{-1}(a+b x)\right )}{2 b} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.42, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1}}{\arcsin \left (b x + a\right )}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 1.25, size = 27, normalized size = 0.87 \[ \frac {\operatorname {Ci}\left (2 \, \arcsin \left (b x + a\right )\right )}{2 \, b} + \frac {\log \left (\arcsin \left (b x + a\right )\right )}{2 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.18, size = 28, normalized size = 0.90 \[ \frac {\Ci \left (2 \arcsin \left (b x +a \right )\right )}{2 b}+\frac {\ln \left (\arcsin \left (b x +a \right )\right )}{2 b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1}}{\arcsin \left (b x + a\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.03 \[ \int \frac {\sqrt {-a^2-2\,a\,b\,x-b^2\,x^2+1}}{\mathrm {asin}\left (a+b\,x\right )} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {- \left (a + b x - 1\right ) \left (a + b x + 1\right )}}{\operatorname {asin}{\left (a + b x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________