Integrand size = 14, antiderivative size = 130 \[ \int \frac {x}{(a+b \arcsin (c x))^{3/2}} \, dx=-\frac {2 x \sqrt {1-c^2 x^2}}{b c \sqrt {a+b \arcsin (c x)}}+\frac {2 \sqrt {\pi } \cos \left (\frac {2 a}{b}\right ) \operatorname {FresnelC}\left (\frac {2 \sqrt {a+b \arcsin (c x)}}{\sqrt {b} \sqrt {\pi }}\right )}{b^{3/2} c^2}+\frac {2 \sqrt {\pi } \operatorname {FresnelS}\left (\frac {2 \sqrt {a+b \arcsin (c x)}}{\sqrt {b} \sqrt {\pi }}\right ) \sin \left (\frac {2 a}{b}\right )}{b^{3/2} c^2} \]
[Out]
Time = 0.07 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {4727, 3387, 3386, 3432, 3385, 3433} \[ \int \frac {x}{(a+b \arcsin (c x))^{3/2}} \, dx=\frac {2 \sqrt {\pi } \cos \left (\frac {2 a}{b}\right ) \operatorname {FresnelC}\left (\frac {2 \sqrt {a+b \arcsin (c x)}}{\sqrt {b} \sqrt {\pi }}\right )}{b^{3/2} c^2}+\frac {2 \sqrt {\pi } \sin \left (\frac {2 a}{b}\right ) \operatorname {FresnelS}\left (\frac {2 \sqrt {a+b \arcsin (c x)}}{\sqrt {b} \sqrt {\pi }}\right )}{b^{3/2} c^2}-\frac {2 x \sqrt {1-c^2 x^2}}{b c \sqrt {a+b \arcsin (c x)}} \]
[In]
[Out]
Rule 3385
Rule 3386
Rule 3387
Rule 3432
Rule 3433
Rule 4727
Rubi steps \begin{align*} \text {integral}& = -\frac {2 x \sqrt {1-c^2 x^2}}{b c \sqrt {a+b \arcsin (c x)}}+\frac {2 \text {Subst}\left (\int \frac {\cos \left (\frac {2 a}{b}-\frac {2 x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \arcsin (c x)\right )}{b^2 c^2} \\ & = -\frac {2 x \sqrt {1-c^2 x^2}}{b c \sqrt {a+b \arcsin (c x)}}+\frac {\left (2 \cos \left (\frac {2 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cos \left (\frac {2 x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \arcsin (c x)\right )}{b^2 c^2}+\frac {\left (2 \sin \left (\frac {2 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sin \left (\frac {2 x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \arcsin (c x)\right )}{b^2 c^2} \\ & = -\frac {2 x \sqrt {1-c^2 x^2}}{b c \sqrt {a+b \arcsin (c x)}}+\frac {\left (4 \cos \left (\frac {2 a}{b}\right )\right ) \text {Subst}\left (\int \cos \left (\frac {2 x^2}{b}\right ) \, dx,x,\sqrt {a+b \arcsin (c x)}\right )}{b^2 c^2}+\frac {\left (4 \sin \left (\frac {2 a}{b}\right )\right ) \text {Subst}\left (\int \sin \left (\frac {2 x^2}{b}\right ) \, dx,x,\sqrt {a+b \arcsin (c x)}\right )}{b^2 c^2} \\ & = -\frac {2 x \sqrt {1-c^2 x^2}}{b c \sqrt {a+b \arcsin (c x)}}+\frac {2 \sqrt {\pi } \cos \left (\frac {2 a}{b}\right ) \operatorname {FresnelC}\left (\frac {2 \sqrt {a+b \arcsin (c x)}}{\sqrt {b} \sqrt {\pi }}\right )}{b^{3/2} c^2}+\frac {2 \sqrt {\pi } \operatorname {FresnelS}\left (\frac {2 \sqrt {a+b \arcsin (c x)}}{\sqrt {b} \sqrt {\pi }}\right ) \sin \left (\frac {2 a}{b}\right )}{b^{3/2} c^2} \\ \end{align*}
Result contains complex when optimal does not.
Time = 0.12 (sec) , antiderivative size = 155, normalized size of antiderivative = 1.19 \[ \int \frac {x}{(a+b \arcsin (c x))^{3/2}} \, dx=\frac {i e^{-\frac {2 i a}{b}} \left (-\sqrt {2} \sqrt {-\frac {i (a+b \arcsin (c x))}{b}} \Gamma \left (\frac {1}{2},-\frac {2 i (a+b \arcsin (c x))}{b}\right )+\sqrt {2} e^{\frac {4 i a}{b}} \sqrt {\frac {i (a+b \arcsin (c x))}{b}} \Gamma \left (\frac {1}{2},\frac {2 i (a+b \arcsin (c x))}{b}\right )+2 i e^{\frac {2 i a}{b}} \sin (2 \arcsin (c x))\right )}{2 b c^2 \sqrt {a+b \arcsin (c x)}} \]
[In]
[Out]
Time = 0.07 (sec) , antiderivative size = 156, normalized size of antiderivative = 1.20
method | result | size |
default | \(\frac {2 \sqrt {-\frac {1}{b}}\, \cos \left (\frac {2 a}{b}\right ) \operatorname {FresnelC}\left (\frac {2 \sqrt {2}\, \sqrt {a +b \arcsin \left (c x \right )}}{\sqrt {\pi }\, \sqrt {-\frac {2}{b}}\, b}\right ) \sqrt {a +b \arcsin \left (c x \right )}\, \sqrt {\pi }-2 \sqrt {-\frac {1}{b}}\, \sin \left (\frac {2 a}{b}\right ) \operatorname {FresnelS}\left (\frac {2 \sqrt {2}\, \sqrt {a +b \arcsin \left (c x \right )}}{\sqrt {\pi }\, \sqrt {-\frac {2}{b}}\, b}\right ) \sqrt {a +b \arcsin \left (c x \right )}\, \sqrt {\pi }+\sin \left (-\frac {2 \left (a +b \arcsin \left (c x \right )\right )}{b}+\frac {2 a}{b}\right )}{c^{2} b \sqrt {a +b \arcsin \left (c x \right )}}\) | \(156\) |
[In]
[Out]
Exception generated. \[ \int \frac {x}{(a+b \arcsin (c x))^{3/2}} \, dx=\text {Exception raised: TypeError} \]
[In]
[Out]
\[ \int \frac {x}{(a+b \arcsin (c x))^{3/2}} \, dx=\int \frac {x}{\left (a + b \operatorname {asin}{\left (c x \right )}\right )^{\frac {3}{2}}}\, dx \]
[In]
[Out]
\[ \int \frac {x}{(a+b \arcsin (c x))^{3/2}} \, dx=\int { \frac {x}{{\left (b \arcsin \left (c x\right ) + a\right )}^{\frac {3}{2}}} \,d x } \]
[In]
[Out]
\[ \int \frac {x}{(a+b \arcsin (c x))^{3/2}} \, dx=\int { \frac {x}{{\left (b \arcsin \left (c x\right ) + a\right )}^{\frac {3}{2}}} \,d x } \]
[In]
[Out]
Timed out. \[ \int \frac {x}{(a+b \arcsin (c x))^{3/2}} \, dx=\int \frac {x}{{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^{3/2}} \,d x \]
[In]
[Out]