Integrand size = 14, antiderivative size = 99 \[ \int \frac {x}{\sqrt {a+b \arcsin (c x)}} \, dx=\frac {\sqrt {\pi } \cos \left (\frac {2 a}{b}\right ) \operatorname {FresnelS}\left (\frac {2 \sqrt {a+b \arcsin (c x)}}{\sqrt {b} \sqrt {\pi }}\right )}{2 \sqrt {b} c^2}-\frac {\sqrt {\pi } \operatorname {FresnelC}\left (\frac {2 \sqrt {a+b \arcsin (c x)}}{\sqrt {b} \sqrt {\pi }}\right ) \sin \left (\frac {2 a}{b}\right )}{2 \sqrt {b} c^2} \]
[Out]
Time = 0.08 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.571, Rules used = {4731, 4491, 12, 3387, 3386, 3432, 3385, 3433} \[ \int \frac {x}{\sqrt {a+b \arcsin (c x)}} \, dx=\frac {\sqrt {\pi } \cos \left (\frac {2 a}{b}\right ) \operatorname {FresnelS}\left (\frac {2 \sqrt {a+b \arcsin (c x)}}{\sqrt {b} \sqrt {\pi }}\right )}{2 \sqrt {b} c^2}-\frac {\sqrt {\pi } \sin \left (\frac {2 a}{b}\right ) \operatorname {FresnelC}\left (\frac {2 \sqrt {a+b \arcsin (c x)}}{\sqrt {b} \sqrt {\pi }}\right )}{2 \sqrt {b} c^2} \]
[In]
[Out]
Rule 12
Rule 3385
Rule 3386
Rule 3387
Rule 3432
Rule 3433
Rule 4491
Rule 4731
Rubi steps \begin{align*} \text {integral}& = -\frac {\text {Subst}\left (\int \frac {\cos \left (\frac {a}{b}-\frac {x}{b}\right ) \sin \left (\frac {a}{b}-\frac {x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \arcsin (c x)\right )}{b c^2} \\ & = -\frac {\text {Subst}\left (\int \frac {\sin \left (\frac {2 a}{b}-\frac {2 x}{b}\right )}{2 \sqrt {x}} \, dx,x,a+b \arcsin (c x)\right )}{b c^2} \\ & = -\frac {\text {Subst}\left (\int \frac {\sin \left (\frac {2 a}{b}-\frac {2 x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \arcsin (c x)\right )}{2 b c^2} \\ & = \frac {\cos \left (\frac {2 a}{b}\right ) \text {Subst}\left (\int \frac {\sin \left (\frac {2 x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \arcsin (c x)\right )}{2 b c^2}-\frac {\sin \left (\frac {2 a}{b}\right ) \text {Subst}\left (\int \frac {\cos \left (\frac {2 x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \arcsin (c x)\right )}{2 b c^2} \\ & = \frac {\cos \left (\frac {2 a}{b}\right ) \text {Subst}\left (\int \sin \left (\frac {2 x^2}{b}\right ) \, dx,x,\sqrt {a+b \arcsin (c x)}\right )}{b c^2}-\frac {\sin \left (\frac {2 a}{b}\right ) \text {Subst}\left (\int \cos \left (\frac {2 x^2}{b}\right ) \, dx,x,\sqrt {a+b \arcsin (c x)}\right )}{b c^2} \\ & = \frac {\sqrt {\pi } \cos \left (\frac {2 a}{b}\right ) \operatorname {FresnelS}\left (\frac {2 \sqrt {a+b \arcsin (c x)}}{\sqrt {b} \sqrt {\pi }}\right )}{2 \sqrt {b} c^2}-\frac {\sqrt {\pi } \operatorname {FresnelC}\left (\frac {2 \sqrt {a+b \arcsin (c x)}}{\sqrt {b} \sqrt {\pi }}\right ) \sin \left (\frac {2 a}{b}\right )}{2 \sqrt {b} c^2} \\ \end{align*}
Result contains complex when optimal does not.
Time = 0.05 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.24 \[ \int \frac {x}{\sqrt {a+b \arcsin (c x)}} \, dx=-\frac {e^{-\frac {2 i a}{b}} \left (\sqrt {-\frac {i (a+b \arcsin (c x))}{b}} \Gamma \left (\frac {1}{2},-\frac {2 i (a+b \arcsin (c x))}{b}\right )+e^{\frac {4 i a}{b}} \sqrt {\frac {i (a+b \arcsin (c x))}{b}} \Gamma \left (\frac {1}{2},\frac {2 i (a+b \arcsin (c x))}{b}\right )\right )}{4 \sqrt {2} c^2 \sqrt {a+b \arcsin (c x)}} \]
[In]
[Out]
Time = 0.05 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.92
method | result | size |
default | \(-\frac {\sqrt {\pi }\, \sqrt {-\frac {1}{b}}\, \left (\cos \left (\frac {2 a}{b}\right ) \operatorname {FresnelS}\left (\frac {2 \sqrt {2}\, \sqrt {a +b \arcsin \left (c x \right )}}{\sqrt {\pi }\, \sqrt {-\frac {2}{b}}\, b}\right )+\sin \left (\frac {2 a}{b}\right ) \operatorname {FresnelC}\left (\frac {2 \sqrt {2}\, \sqrt {a +b \arcsin \left (c x \right )}}{\sqrt {\pi }\, \sqrt {-\frac {2}{b}}\, b}\right )\right )}{2 c^{2}}\) | \(91\) |
[In]
[Out]
Exception generated. \[ \int \frac {x}{\sqrt {a+b \arcsin (c x)}} \, dx=\text {Exception raised: TypeError} \]
[In]
[Out]
\[ \int \frac {x}{\sqrt {a+b \arcsin (c x)}} \, dx=\int \frac {x}{\sqrt {a + b \operatorname {asin}{\left (c x \right )}}}\, dx \]
[In]
[Out]
\[ \int \frac {x}{\sqrt {a+b \arcsin (c x)}} \, dx=\int { \frac {x}{\sqrt {b \arcsin \left (c x\right ) + a}} \,d x } \]
[In]
[Out]
Result contains complex when optimal does not.
Time = 0.40 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.33 \[ \int \frac {x}{\sqrt {a+b \arcsin (c x)}} \, dx=\frac {i \, \sqrt {\pi } \operatorname {erf}\left (-\frac {\sqrt {b \arcsin \left (c x\right ) + a}}{\sqrt {b}} + \frac {i \, \sqrt {b \arcsin \left (c x\right ) + a} \sqrt {b}}{{\left | b \right |}}\right ) e^{\left (-\frac {2 i \, a}{b}\right )}}{4 \, c^{2} {\left (\sqrt {b} - \frac {i \, b^{\frac {3}{2}}}{{\left | b \right |}}\right )}} - \frac {i \, \sqrt {\pi } \operatorname {erf}\left (-\frac {\sqrt {b \arcsin \left (c x\right ) + a}}{\sqrt {b}} - \frac {i \, \sqrt {b \arcsin \left (c x\right ) + a} \sqrt {b}}{{\left | b \right |}}\right ) e^{\left (\frac {2 i \, a}{b}\right )}}{4 \, \sqrt {b} c^{2} {\left (\frac {i \, b}{{\left | b \right |}} + 1\right )}} \]
[In]
[Out]
Timed out. \[ \int \frac {x}{\sqrt {a+b \arcsin (c x)}} \, dx=\int \frac {x}{\sqrt {a+b\,\mathrm {asin}\left (c\,x\right )}} \,d x \]
[In]
[Out]