Integrand size = 14, antiderivative size = 144 \[ \int \frac {1}{(a+b \arcsin (c+d x))^{3/2}} \, dx=-\frac {2 \sqrt {1-(c+d x)^2}}{b d \sqrt {a+b \arcsin (c+d x)}}-\frac {2 \sqrt {2 \pi } \cos \left (\frac {a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b}}\right )}{b^{3/2} d}+\frac {2 \sqrt {2 \pi } \operatorname {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b}}\right ) \sin \left (\frac {a}{b}\right )}{b^{3/2} d} \]
[Out]
Time = 0.17 (sec) , antiderivative size = 144, 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 = {4887, 4717, 4809, 3387, 3386, 3432, 3385, 3433} \[ \int \frac {1}{(a+b \arcsin (c+d x))^{3/2}} \, dx=\frac {2 \sqrt {2 \pi } \sin \left (\frac {a}{b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b}}\right )}{b^{3/2} d}-\frac {2 \sqrt {2 \pi } \cos \left (\frac {a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b}}\right )}{b^{3/2} d}-\frac {2 \sqrt {1-(c+d x)^2}}{b d \sqrt {a+b \arcsin (c+d x)}} \]
[In]
[Out]
Rule 3385
Rule 3386
Rule 3387
Rule 3432
Rule 3433
Rule 4717
Rule 4809
Rule 4887
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {1}{(a+b \arcsin (x))^{3/2}} \, dx,x,c+d x\right )}{d} \\ & = -\frac {2 \sqrt {1-(c+d x)^2}}{b d \sqrt {a+b \arcsin (c+d x)}}-\frac {2 \text {Subst}\left (\int \frac {x}{\sqrt {1-x^2} \sqrt {a+b \arcsin (x)}} \, dx,x,c+d x\right )}{b d} \\ & = -\frac {2 \sqrt {1-(c+d x)^2}}{b d \sqrt {a+b \arcsin (c+d x)}}+\frac {2 \text {Subst}\left (\int \frac {\sin \left (\frac {a}{b}-\frac {x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \arcsin (c+d x)\right )}{b^2 d} \\ & = -\frac {2 \sqrt {1-(c+d x)^2}}{b d \sqrt {a+b \arcsin (c+d x)}}-\frac {\left (2 \cos \left (\frac {a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sin \left (\frac {x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \arcsin (c+d x)\right )}{b^2 d}+\frac {\left (2 \sin \left (\frac {a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cos \left (\frac {x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \arcsin (c+d x)\right )}{b^2 d} \\ & = -\frac {2 \sqrt {1-(c+d x)^2}}{b d \sqrt {a+b \arcsin (c+d x)}}-\frac {\left (4 \cos \left (\frac {a}{b}\right )\right ) \text {Subst}\left (\int \sin \left (\frac {x^2}{b}\right ) \, dx,x,\sqrt {a+b \arcsin (c+d x)}\right )}{b^2 d}+\frac {\left (4 \sin \left (\frac {a}{b}\right )\right ) \text {Subst}\left (\int \cos \left (\frac {x^2}{b}\right ) \, dx,x,\sqrt {a+b \arcsin (c+d x)}\right )}{b^2 d} \\ & = -\frac {2 \sqrt {1-(c+d x)^2}}{b d \sqrt {a+b \arcsin (c+d x)}}-\frac {2 \sqrt {2 \pi } \cos \left (\frac {a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b}}\right )}{b^{3/2} d}+\frac {2 \sqrt {2 \pi } \operatorname {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b}}\right ) \sin \left (\frac {a}{b}\right )}{b^{3/2} d} \\ \end{align*}
Result contains complex when optimal does not.
Time = 0.23 (sec) , antiderivative size = 185, normalized size of antiderivative = 1.28 \[ \int \frac {1}{(a+b \arcsin (c+d x))^{3/2}} \, dx=\frac {e^{-\frac {i (a+b \arcsin (c+d x))}{b}} \left (e^{i \arcsin (c+d x)} \sqrt {-\frac {i (a+b \arcsin (c+d x))}{b}} \Gamma \left (\frac {1}{2},-\frac {i (a+b \arcsin (c+d x))}{b}\right )+e^{\frac {i a}{b}} \left (-1-e^{2 i \arcsin (c+d x)}+e^{\frac {i (a+b \arcsin (c+d x))}{b}} \sqrt {\frac {i (a+b \arcsin (c+d x))}{b}} \Gamma \left (\frac {1}{2},\frac {i (a+b \arcsin (c+d x))}{b}\right )\right )\right )}{b d \sqrt {a+b \arcsin (c+d x)}} \]
[In]
[Out]
Time = 0.72 (sec) , antiderivative size = 170, normalized size of antiderivative = 1.18
method | result | size |
default | \(-\frac {2 \left (-\sqrt {a +b \arcsin \left (d x +c \right )}\, \cos \left (\frac {a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {2}\, \sqrt {a +b \arcsin \left (d x +c \right )}}{\sqrt {\pi }\, \sqrt {-\frac {1}{b}}\, b}\right ) \sqrt {-\frac {1}{b}}\, \sqrt {\pi }\, \sqrt {2}-\sqrt {a +b \arcsin \left (d x +c \right )}\, \sin \left (\frac {a}{b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {2}\, \sqrt {a +b \arcsin \left (d x +c \right )}}{\sqrt {\pi }\, \sqrt {-\frac {1}{b}}\, b}\right ) \sqrt {-\frac {1}{b}}\, \sqrt {\pi }\, \sqrt {2}+\cos \left (-\frac {a +b \arcsin \left (d x +c \right )}{b}+\frac {a}{b}\right )\right )}{d b \sqrt {a +b \arcsin \left (d x +c \right )}}\) | \(170\) |
[In]
[Out]
Exception generated. \[ \int \frac {1}{(a+b \arcsin (c+d x))^{3/2}} \, dx=\text {Exception raised: TypeError} \]
[In]
[Out]
\[ \int \frac {1}{(a+b \arcsin (c+d x))^{3/2}} \, dx=\int \frac {1}{\left (a + b \operatorname {asin}{\left (c + d x \right )}\right )^{\frac {3}{2}}}\, dx \]
[In]
[Out]
\[ \int \frac {1}{(a+b \arcsin (c+d x))^{3/2}} \, dx=\int { \frac {1}{{\left (b \arcsin \left (d x + c\right ) + a\right )}^{\frac {3}{2}}} \,d x } \]
[In]
[Out]
\[ \int \frac {1}{(a+b \arcsin (c+d x))^{3/2}} \, dx=\int { \frac {1}{{\left (b \arcsin \left (d x + c\right ) + a\right )}^{\frac {3}{2}}} \,d x } \]
[In]
[Out]
Timed out. \[ \int \frac {1}{(a+b \arcsin (c+d x))^{3/2}} \, dx=\int \frac {1}{{\left (a+b\,\mathrm {asin}\left (c+d\,x\right )\right )}^{3/2}} \,d x \]
[In]
[Out]