Integrand size = 21, antiderivative size = 89 \[ \int \frac {1}{\left (c+a^2 c x^2\right )^3 \sqrt {\arctan (a x)}} \, dx=\frac {3 \sqrt {\arctan (a x)}}{4 a c^3}+\frac {\sqrt {\frac {\pi }{2}} \operatorname {FresnelC}\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\arctan (a x)}\right )}{8 a c^3}+\frac {\sqrt {\pi } \operatorname {FresnelC}\left (\frac {2 \sqrt {\arctan (a x)}}{\sqrt {\pi }}\right )}{2 a c^3} \]
[Out]
Time = 0.07 (sec) , antiderivative size = 89, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {5024, 3393, 3385, 3433} \[ \int \frac {1}{\left (c+a^2 c x^2\right )^3 \sqrt {\arctan (a x)}} \, dx=\frac {\sqrt {\frac {\pi }{2}} \operatorname {FresnelC}\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\arctan (a x)}\right )}{8 a c^3}+\frac {\sqrt {\pi } \operatorname {FresnelC}\left (\frac {2 \sqrt {\arctan (a x)}}{\sqrt {\pi }}\right )}{2 a c^3}+\frac {3 \sqrt {\arctan (a x)}}{4 a c^3} \]
[In]
[Out]
Rule 3385
Rule 3393
Rule 3433
Rule 5024
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {\cos ^4(x)}{\sqrt {x}} \, dx,x,\arctan (a x)\right )}{a c^3} \\ & = \frac {\text {Subst}\left (\int \left (\frac {3}{8 \sqrt {x}}+\frac {\cos (2 x)}{2 \sqrt {x}}+\frac {\cos (4 x)}{8 \sqrt {x}}\right ) \, dx,x,\arctan (a x)\right )}{a c^3} \\ & = \frac {3 \sqrt {\arctan (a x)}}{4 a c^3}+\frac {\text {Subst}\left (\int \frac {\cos (4 x)}{\sqrt {x}} \, dx,x,\arctan (a x)\right )}{8 a c^3}+\frac {\text {Subst}\left (\int \frac {\cos (2 x)}{\sqrt {x}} \, dx,x,\arctan (a x)\right )}{2 a c^3} \\ & = \frac {3 \sqrt {\arctan (a x)}}{4 a c^3}+\frac {\text {Subst}\left (\int \cos \left (4 x^2\right ) \, dx,x,\sqrt {\arctan (a x)}\right )}{4 a c^3}+\frac {\text {Subst}\left (\int \cos \left (2 x^2\right ) \, dx,x,\sqrt {\arctan (a x)}\right )}{a c^3} \\ & = \frac {3 \sqrt {\arctan (a x)}}{4 a c^3}+\frac {\sqrt {\frac {\pi }{2}} \operatorname {FresnelC}\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\arctan (a x)}\right )}{8 a c^3}+\frac {\sqrt {\pi } \operatorname {FresnelC}\left (\frac {2 \sqrt {\arctan (a x)}}{\sqrt {\pi }}\right )}{2 a c^3} \\ \end{align*}
Result contains complex when optimal does not.
Time = 0.49 (sec) , antiderivative size = 229, normalized size of antiderivative = 2.57 \[ \int \frac {1}{\left (c+a^2 c x^2\right )^3 \sqrt {\arctan (a x)}} \, dx=\frac {-6 \sqrt {2 \pi } \sqrt {\arctan (a x)^2} \operatorname {FresnelC}\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\arctan (a x)}\right )+48 \sqrt {\pi } \sqrt {\arctan (a x)^2} \operatorname {FresnelC}\left (\frac {2 \sqrt {\arctan (a x)}}{\sqrt {\pi }}\right )+\sqrt {\arctan (a x)} \left (192 \sqrt {\arctan (a x)^2}-20 \sqrt {2} \sqrt {i \arctan (a x)} \Gamma \left (\frac {1}{2},-2 i \arctan (a x)\right )-20 \sqrt {2} \sqrt {-i \arctan (a x)} \Gamma \left (\frac {1}{2},2 i \arctan (a x)\right )-11 \sqrt {i \arctan (a x)} \Gamma \left (\frac {1}{2},-4 i \arctan (a x)\right )-11 \sqrt {-i \arctan (a x)} \Gamma \left (\frac {1}{2},4 i \arctan (a x)\right )\right )}{256 a c^3 \sqrt {\arctan (a x)^2}} \]
[In]
[Out]
Time = 1.01 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.66
method | result | size |
default | \(\frac {\pi \sqrt {2}\, \operatorname {FresnelC}\left (\frac {2 \sqrt {2}\, \sqrt {\arctan \left (a x \right )}}{\sqrt {\pi }}\right )+8 \pi \,\operatorname {FresnelC}\left (\frac {2 \sqrt {\arctan \left (a x \right )}}{\sqrt {\pi }}\right )+12 \sqrt {\arctan \left (a x \right )}\, \sqrt {\pi }}{16 c^{3} a \sqrt {\pi }}\) | \(59\) |
[In]
[Out]
Exception generated. \[ \int \frac {1}{\left (c+a^2 c x^2\right )^3 \sqrt {\arctan (a x)}} \, dx=\text {Exception raised: TypeError} \]
[In]
[Out]
\[ \int \frac {1}{\left (c+a^2 c x^2\right )^3 \sqrt {\arctan (a x)}} \, dx=\frac {\int \frac {1}{a^{6} x^{6} \sqrt {\operatorname {atan}{\left (a x \right )}} + 3 a^{4} x^{4} \sqrt {\operatorname {atan}{\left (a x \right )}} + 3 a^{2} x^{2} \sqrt {\operatorname {atan}{\left (a x \right )}} + \sqrt {\operatorname {atan}{\left (a x \right )}}}\, dx}{c^{3}} \]
[In]
[Out]
Exception generated. \[ \int \frac {1}{\left (c+a^2 c x^2\right )^3 \sqrt {\arctan (a x)}} \, dx=\text {Exception raised: RuntimeError} \]
[In]
[Out]
\[ \int \frac {1}{\left (c+a^2 c x^2\right )^3 \sqrt {\arctan (a x)}} \, dx=\int { \frac {1}{{\left (a^{2} c x^{2} + c\right )}^{3} \sqrt {\arctan \left (a x\right )}} \,d x } \]
[In]
[Out]
Timed out. \[ \int \frac {1}{\left (c+a^2 c x^2\right )^3 \sqrt {\arctan (a x)}} \, dx=\int \frac {1}{\sqrt {\mathrm {atan}\left (a\,x\right )}\,{\left (c\,a^2\,x^2+c\right )}^3} \,d x \]
[In]
[Out]