Integrand size = 24, antiderivative size = 129 \[ \int \frac {x}{\left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^{5/2}} \, dx=-\frac {2 x}{3 a c \sqrt {c+a^2 c x^2} \arctan (a x)^{3/2}}-\frac {4}{3 a^2 c \sqrt {c+a^2 c x^2} \sqrt {\arctan (a x)}}-\frac {4 \sqrt {2 \pi } \sqrt {1+a^2 x^2} \operatorname {FresnelS}\left (\sqrt {\frac {2}{\pi }} \sqrt {\arctan (a x)}\right )}{3 a^2 c \sqrt {c+a^2 c x^2}} \]
[Out]
Time = 0.20 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {5062, 5022, 5091, 5090, 3386, 3432} \[ \int \frac {x}{\left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^{5/2}} \, dx=-\frac {4 \sqrt {2 \pi } \sqrt {a^2 x^2+1} \operatorname {FresnelS}\left (\sqrt {\frac {2}{\pi }} \sqrt {\arctan (a x)}\right )}{3 a^2 c \sqrt {a^2 c x^2+c}}-\frac {2 x}{3 a c \arctan (a x)^{3/2} \sqrt {a^2 c x^2+c}}-\frac {4}{3 a^2 c \sqrt {\arctan (a x)} \sqrt {a^2 c x^2+c}} \]
[In]
[Out]
Rule 3386
Rule 3432
Rule 5022
Rule 5062
Rule 5090
Rule 5091
Rubi steps \begin{align*} \text {integral}& = -\frac {2 x}{3 a c \sqrt {c+a^2 c x^2} \arctan (a x)^{3/2}}+\frac {2 \int \frac {1}{\left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^{3/2}} \, dx}{3 a} \\ & = -\frac {2 x}{3 a c \sqrt {c+a^2 c x^2} \arctan (a x)^{3/2}}-\frac {4}{3 a^2 c \sqrt {c+a^2 c x^2} \sqrt {\arctan (a x)}}-\frac {4}{3} \int \frac {x}{\left (c+a^2 c x^2\right )^{3/2} \sqrt {\arctan (a x)}} \, dx \\ & = -\frac {2 x}{3 a c \sqrt {c+a^2 c x^2} \arctan (a x)^{3/2}}-\frac {4}{3 a^2 c \sqrt {c+a^2 c x^2} \sqrt {\arctan (a x)}}-\frac {\left (4 \sqrt {1+a^2 x^2}\right ) \int \frac {x}{\left (1+a^2 x^2\right )^{3/2} \sqrt {\arctan (a x)}} \, dx}{3 c \sqrt {c+a^2 c x^2}} \\ & = -\frac {2 x}{3 a c \sqrt {c+a^2 c x^2} \arctan (a x)^{3/2}}-\frac {4}{3 a^2 c \sqrt {c+a^2 c x^2} \sqrt {\arctan (a x)}}-\frac {\left (4 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \frac {\sin (x)}{\sqrt {x}} \, dx,x,\arctan (a x)\right )}{3 a^2 c \sqrt {c+a^2 c x^2}} \\ & = -\frac {2 x}{3 a c \sqrt {c+a^2 c x^2} \arctan (a x)^{3/2}}-\frac {4}{3 a^2 c \sqrt {c+a^2 c x^2} \sqrt {\arctan (a x)}}-\frac {\left (8 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \sin \left (x^2\right ) \, dx,x,\sqrt {\arctan (a x)}\right )}{3 a^2 c \sqrt {c+a^2 c x^2}} \\ & = -\frac {2 x}{3 a c \sqrt {c+a^2 c x^2} \arctan (a x)^{3/2}}-\frac {4}{3 a^2 c \sqrt {c+a^2 c x^2} \sqrt {\arctan (a x)}}-\frac {4 \sqrt {2 \pi } \sqrt {1+a^2 x^2} \operatorname {FresnelS}\left (\sqrt {\frac {2}{\pi }} \sqrt {\arctan (a x)}\right )}{3 a^2 c \sqrt {c+a^2 c x^2}} \\ \end{align*}
Result contains complex when optimal does not.
Time = 0.18 (sec) , antiderivative size = 124, normalized size of antiderivative = 0.96 \[ \int \frac {x}{\left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^{5/2}} \, dx=-\frac {2 \left (a x+2 \arctan (a x)-i \sqrt {1+a^2 x^2} (-i \arctan (a x))^{3/2} \Gamma \left (\frac {1}{2},-i \arctan (a x)\right )+i \sqrt {1+a^2 x^2} (i \arctan (a x))^{3/2} \Gamma \left (\frac {1}{2},i \arctan (a x)\right )\right )}{3 a^2 c \sqrt {c+a^2 c x^2} \arctan (a x)^{3/2}} \]
[In]
[Out]
\[\int \frac {x}{\left (a^{2} c \,x^{2}+c \right )^{\frac {3}{2}} \arctan \left (a x \right )^{\frac {5}{2}}}d x\]
[In]
[Out]
Exception generated. \[ \int \frac {x}{\left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^{5/2}} \, dx=\text {Exception raised: TypeError} \]
[In]
[Out]
\[ \int \frac {x}{\left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^{5/2}} \, dx=\int \frac {x}{\left (c \left (a^{2} x^{2} + 1\right )\right )^{\frac {3}{2}} \operatorname {atan}^{\frac {5}{2}}{\left (a x \right )}}\, dx \]
[In]
[Out]
Exception generated. \[ \int \frac {x}{\left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^{5/2}} \, dx=\text {Exception raised: RuntimeError} \]
[In]
[Out]
Exception generated. \[ \int \frac {x}{\left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^{5/2}} \, dx=\text {Exception raised: TypeError} \]
[In]
[Out]
Timed out. \[ \int \frac {x}{\left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^{5/2}} \, dx=\int \frac {x}{{\mathrm {atan}\left (a\,x\right )}^{5/2}\,{\left (c\,a^2\,x^2+c\right )}^{3/2}} \,d x \]
[In]
[Out]