Optimal. Leaf size=79 \[ \text {Int}\left (\frac {1}{x \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )},x\right )+\frac {\sin \left (\frac {a}{b}\right ) \text {Ci}\left (\frac {a+b \sin ^{-1}(c x)}{b}\right )}{b}-\frac {\cos \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a+b \sin ^{-1}(c x)}{b}\right )}{b} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.40, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {\sqrt {1-c^2 x^2}}{x \left (a+b \sin ^{-1}(c x)\right )} \, dx \]
Verification is Not applicable to the result.
[In]
[Out]
Rubi steps
\begin {align*} \int \frac {\sqrt {1-c^2 x^2}}{x \left (a+b \sin ^{-1}(c x)\right )} \, dx &=\int \left (\frac {1}{x \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}-\frac {c^2 x}{\sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}\right ) \, dx\\ &=-\left (c^2 \int \frac {x}{\sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )} \, dx\right )+\int \frac {1}{x \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )} \, dx\\ &=\int \frac {1}{x \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )} \, dx-\operatorname {Subst}\left (\int \frac {\sin (x)}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )\\ &=-\left (\cos \left (\frac {a}{b}\right ) \operatorname {Subst}\left (\int \frac {\sin \left (\frac {a}{b}+x\right )}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )\right )+\sin \left (\frac {a}{b}\right ) \operatorname {Subst}\left (\int \frac {\cos \left (\frac {a}{b}+x\right )}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )+\int \frac {1}{x \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )} \, dx\\ &=\frac {\text {Ci}\left (\frac {a}{b}+\sin ^{-1}(c x)\right ) \sin \left (\frac {a}{b}\right )}{b}-\frac {\cos \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a}{b}+\sin ^{-1}(c x)\right )}{b}+\int \frac {1}{x \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )} \, dx\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 3.18, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {1-c^2 x^2}}{x \left (a+b \sin ^{-1}(c x)\right )} \, dx \]
Verification is Not applicable to the result.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 1.24, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {-c^{2} x^{2} + 1}}{b x \arcsin \left (c x\right ) + a x}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.50, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {-c^{2} x^{2}+1}}{x \left (a +b \arcsin \left (c x \right )\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {-c^{2} x^{2} + 1}}{{\left (b \arcsin \left (c x\right ) + a\right )} x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [A] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\sqrt {1-c^2\,x^2}}{x\,\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {- \left (c x - 1\right ) \left (c x + 1\right )}}{x \left (a + b \operatorname {asin}{\left (c x \right )}\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________