Integrand size = 25, antiderivative size = 82 \[ \int \frac {\sqrt {1-c^2 x^2}}{a+b \arcsin (c x)} \, dx=\frac {\cos \left (\frac {2 a}{b}\right ) \operatorname {CosIntegral}\left (\frac {2 (a+b \arcsin (c x))}{b}\right )}{2 b c}+\frac {\log (a+b \arcsin (c x))}{2 b c}+\frac {\sin \left (\frac {2 a}{b}\right ) \text {Si}\left (\frac {2 (a+b \arcsin (c x))}{b}\right )}{2 b c} \]
[Out]
Time = 0.09 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {4753, 3393, 3384, 3380, 3383} \[ \int \frac {\sqrt {1-c^2 x^2}}{a+b \arcsin (c x)} \, dx=\frac {\cos \left (\frac {2 a}{b}\right ) \operatorname {CosIntegral}\left (\frac {2 (a+b \arcsin (c x))}{b}\right )}{2 b c}+\frac {\sin \left (\frac {2 a}{b}\right ) \text {Si}\left (\frac {2 (a+b \arcsin (c x))}{b}\right )}{2 b c}+\frac {\log (a+b \arcsin (c x))}{2 b c} \]
[In]
[Out]
Rule 3380
Rule 3383
Rule 3384
Rule 3393
Rule 4753
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {\cos ^2\left (\frac {a}{b}-\frac {x}{b}\right )}{x} \, dx,x,a+b \arcsin (c x)\right )}{b c} \\ & = \frac {\text {Subst}\left (\int \left (\frac {1}{2 x}+\frac {\cos \left (\frac {2 a}{b}-\frac {2 x}{b}\right )}{2 x}\right ) \, dx,x,a+b \arcsin (c x)\right )}{b c} \\ & = \frac {\log (a+b \arcsin (c x))}{2 b c}+\frac {\text {Subst}\left (\int \frac {\cos \left (\frac {2 a}{b}-\frac {2 x}{b}\right )}{x} \, dx,x,a+b \arcsin (c x)\right )}{2 b c} \\ & = \frac {\log (a+b \arcsin (c x))}{2 b c}+\frac {\cos \left (\frac {2 a}{b}\right ) \text {Subst}\left (\int \frac {\cos \left (\frac {2 x}{b}\right )}{x} \, dx,x,a+b \arcsin (c x)\right )}{2 b c}+\frac {\sin \left (\frac {2 a}{b}\right ) \text {Subst}\left (\int \frac {\sin \left (\frac {2 x}{b}\right )}{x} \, dx,x,a+b \arcsin (c x)\right )}{2 b c} \\ & = \frac {\cos \left (\frac {2 a}{b}\right ) \operatorname {CosIntegral}\left (\frac {2 (a+b \arcsin (c x))}{b}\right )}{2 b c}+\frac {\log (a+b \arcsin (c x))}{2 b c}+\frac {\sin \left (\frac {2 a}{b}\right ) \text {Si}\left (\frac {2 (a+b \arcsin (c x))}{b}\right )}{2 b c} \\ \end{align*}
Time = 0.28 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.76 \[ \int \frac {\sqrt {1-c^2 x^2}}{a+b \arcsin (c x)} \, dx=\frac {\cos \left (\frac {2 a}{b}\right ) \operatorname {CosIntegral}\left (2 \left (\frac {a}{b}+\arcsin (c x)\right )\right )+\log (a+b \arcsin (c x))+\sin \left (\frac {2 a}{b}\right ) \text {Si}\left (2 \left (\frac {a}{b}+\arcsin (c x)\right )\right )}{2 b c} \]
[In]
[Out]
Time = 0.12 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.77
method | result | size |
default | \(\frac {\operatorname {Si}\left (2 \arcsin \left (c x \right )+\frac {2 a}{b}\right ) \sin \left (\frac {2 a}{b}\right )+\operatorname {Ci}\left (2 \arcsin \left (c x \right )+\frac {2 a}{b}\right ) \cos \left (\frac {2 a}{b}\right )+\ln \left (a +b \arcsin \left (c x \right )\right )}{2 c b}\) | \(63\) |
[In]
[Out]
\[ \int \frac {\sqrt {1-c^2 x^2}}{a+b \arcsin (c x)} \, dx=\int { \frac {\sqrt {-c^{2} x^{2} + 1}}{b \arcsin \left (c x\right ) + a} \,d x } \]
[In]
[Out]
\[ \int \frac {\sqrt {1-c^2 x^2}}{a+b \arcsin (c x)} \, dx=\int \frac {\sqrt {- \left (c x - 1\right ) \left (c x + 1\right )}}{a + b \operatorname {asin}{\left (c x \right )}}\, dx \]
[In]
[Out]
\[ \int \frac {\sqrt {1-c^2 x^2}}{a+b \arcsin (c x)} \, dx=\int { \frac {\sqrt {-c^{2} x^{2} + 1}}{b \arcsin \left (c x\right ) + a} \,d x } \]
[In]
[Out]
none
Time = 0.32 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.24 \[ \int \frac {\sqrt {1-c^2 x^2}}{a+b \arcsin (c x)} \, dx=\frac {\cos \left (\frac {a}{b}\right )^{2} \operatorname {Ci}\left (\frac {2 \, a}{b} + 2 \, \arcsin \left (c x\right )\right )}{b c} + \frac {\cos \left (\frac {a}{b}\right ) \sin \left (\frac {a}{b}\right ) \operatorname {Si}\left (\frac {2 \, a}{b} + 2 \, \arcsin \left (c x\right )\right )}{b c} - \frac {\operatorname {Ci}\left (\frac {2 \, a}{b} + 2 \, \arcsin \left (c x\right )\right )}{2 \, b c} + \frac {\log \left (b \arcsin \left (c x\right ) + a\right )}{2 \, b c} \]
[In]
[Out]
Timed out. \[ \int \frac {\sqrt {1-c^2 x^2}}{a+b \arcsin (c x)} \, dx=\int \frac {\sqrt {1-c^2\,x^2}}{a+b\,\mathrm {asin}\left (c\,x\right )} \,d x \]
[In]
[Out]