Integrand size = 23, antiderivative size = 60 \[ \int \frac {1}{\left (c+a^2 c x^2\right )^{3/2} \sqrt {\arctan (a x)}} \, dx=\frac {\sqrt {2 \pi } \sqrt {1+a^2 x^2} \operatorname {FresnelC}\left (\sqrt {\frac {2}{\pi }} \sqrt {\arctan (a x)}\right )}{a c \sqrt {c+a^2 c x^2}} \]
[Out]
Time = 0.06 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {5025, 5024, 3385, 3433} \[ \int \frac {1}{\left (c+a^2 c x^2\right )^{3/2} \sqrt {\arctan (a x)}} \, dx=\frac {\sqrt {2 \pi } \sqrt {a^2 x^2+1} \operatorname {FresnelC}\left (\sqrt {\frac {2}{\pi }} \sqrt {\arctan (a x)}\right )}{a c \sqrt {a^2 c x^2+c}} \]
[In]
[Out]
Rule 3385
Rule 3433
Rule 5024
Rule 5025
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {1+a^2 x^2} \int \frac {1}{\left (1+a^2 x^2\right )^{3/2} \sqrt {\arctan (a x)}} \, dx}{c \sqrt {c+a^2 c x^2}} \\ & = \frac {\sqrt {1+a^2 x^2} \text {Subst}\left (\int \frac {\cos (x)}{\sqrt {x}} \, dx,x,\arctan (a x)\right )}{a c \sqrt {c+a^2 c x^2}} \\ & = \frac {\left (2 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \cos \left (x^2\right ) \, dx,x,\sqrt {\arctan (a x)}\right )}{a c \sqrt {c+a^2 c x^2}} \\ & = \frac {\sqrt {2 \pi } \sqrt {1+a^2 x^2} \operatorname {FresnelC}\left (\sqrt {\frac {2}{\pi }} \sqrt {\arctan (a x)}\right )}{a c \sqrt {c+a^2 c x^2}} \\ \end{align*}
Result contains complex when optimal does not.
Time = 0.11 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.67 \[ \int \frac {1}{\left (c+a^2 c x^2\right )^{3/2} \sqrt {\arctan (a x)}} \, dx=-\frac {i \sqrt {1+a^2 x^2} \left (\sqrt {-i \arctan (a x)} \Gamma \left (\frac {1}{2},-i \arctan (a x)\right )-\sqrt {i \arctan (a x)} \Gamma \left (\frac {1}{2},i \arctan (a x)\right )\right )}{2 a c \sqrt {c \left (1+a^2 x^2\right )} \sqrt {\arctan (a x)}} \]
[In]
[Out]
\[\int \frac {1}{\left (a^{2} c \,x^{2}+c \right )^{\frac {3}{2}} \sqrt {\arctan \left (a x \right )}}d x\]
[In]
[Out]
Exception generated. \[ \int \frac {1}{\left (c+a^2 c x^2\right )^{3/2} \sqrt {\arctan (a x)}} \, dx=\text {Exception raised: TypeError} \]
[In]
[Out]
\[ \int \frac {1}{\left (c+a^2 c x^2\right )^{3/2} \sqrt {\arctan (a x)}} \, dx=\int \frac {1}{\left (c \left (a^{2} x^{2} + 1\right )\right )^{\frac {3}{2}} \sqrt {\operatorname {atan}{\left (a x \right )}}}\, dx \]
[In]
[Out]
Exception generated. \[ \int \frac {1}{\left (c+a^2 c x^2\right )^{3/2} \sqrt {\arctan (a x)}} \, dx=\text {Exception raised: RuntimeError} \]
[In]
[Out]
\[ \int \frac {1}{\left (c+a^2 c x^2\right )^{3/2} \sqrt {\arctan (a x)}} \, dx=\int { \frac {1}{{\left (a^{2} c x^{2} + c\right )}^{\frac {3}{2}} \sqrt {\arctan \left (a x\right )}} \,d x } \]
[In]
[Out]
Timed out. \[ \int \frac {1}{\left (c+a^2 c x^2\right )^{3/2} \sqrt {\arctan (a x)}} \, dx=\int \frac {1}{\sqrt {\mathrm {atan}\left (a\,x\right )}\,{\left (c\,a^2\,x^2+c\right )}^{3/2}} \,d x \]
[In]
[Out]