Integrand size = 21, antiderivative size = 77 \[ \int \frac {\sqrt {\arctan (a x)}}{\left (c+a^2 c x^2\right )^2} \, dx=\frac {x \sqrt {\arctan (a x)}}{2 c^2 \left (1+a^2 x^2\right )}+\frac {\arctan (a x)^{3/2}}{3 a c^2}-\frac {\sqrt {\pi } \operatorname {FresnelS}\left (\frac {2 \sqrt {\arctan (a x)}}{\sqrt {\pi }}\right )}{8 a c^2} \]
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Time = 0.08 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {5012, 5090, 4491, 12, 3386, 3432} \[ \int \frac {\sqrt {\arctan (a x)}}{\left (c+a^2 c x^2\right )^2} \, dx=\frac {x \sqrt {\arctan (a x)}}{2 c^2 \left (a^2 x^2+1\right )}-\frac {\sqrt {\pi } \operatorname {FresnelS}\left (\frac {2 \sqrt {\arctan (a x)}}{\sqrt {\pi }}\right )}{8 a c^2}+\frac {\arctan (a x)^{3/2}}{3 a c^2} \]
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Rule 12
Rule 3386
Rule 3432
Rule 4491
Rule 5012
Rule 5090
Rubi steps \begin{align*} \text {integral}& = \frac {x \sqrt {\arctan (a x)}}{2 c^2 \left (1+a^2 x^2\right )}+\frac {\arctan (a x)^{3/2}}{3 a c^2}-\frac {1}{4} a \int \frac {x}{\left (c+a^2 c x^2\right )^2 \sqrt {\arctan (a x)}} \, dx \\ & = \frac {x \sqrt {\arctan (a x)}}{2 c^2 \left (1+a^2 x^2\right )}+\frac {\arctan (a x)^{3/2}}{3 a c^2}-\frac {\text {Subst}\left (\int \frac {\cos (x) \sin (x)}{\sqrt {x}} \, dx,x,\arctan (a x)\right )}{4 a c^2} \\ & = \frac {x \sqrt {\arctan (a x)}}{2 c^2 \left (1+a^2 x^2\right )}+\frac {\arctan (a x)^{3/2}}{3 a c^2}-\frac {\text {Subst}\left (\int \frac {\sin (2 x)}{2 \sqrt {x}} \, dx,x,\arctan (a x)\right )}{4 a c^2} \\ & = \frac {x \sqrt {\arctan (a x)}}{2 c^2 \left (1+a^2 x^2\right )}+\frac {\arctan (a x)^{3/2}}{3 a c^2}-\frac {\text {Subst}\left (\int \frac {\sin (2 x)}{\sqrt {x}} \, dx,x,\arctan (a x)\right )}{8 a c^2} \\ & = \frac {x \sqrt {\arctan (a x)}}{2 c^2 \left (1+a^2 x^2\right )}+\frac {\arctan (a x)^{3/2}}{3 a c^2}-\frac {\text {Subst}\left (\int \sin \left (2 x^2\right ) \, dx,x,\sqrt {\arctan (a x)}\right )}{4 a c^2} \\ & = \frac {x \sqrt {\arctan (a x)}}{2 c^2 \left (1+a^2 x^2\right )}+\frac {\arctan (a x)^{3/2}}{3 a c^2}-\frac {\sqrt {\pi } \operatorname {FresnelS}\left (\frac {2 \sqrt {\arctan (a x)}}{\sqrt {\pi }}\right )}{8 a c^2} \\ \end{align*}
Time = 0.30 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.88 \[ \int \frac {\sqrt {\arctan (a x)}}{\left (c+a^2 c x^2\right )^2} \, dx=\frac {4 \sqrt {\arctan (a x)} \left (\frac {3 x}{1+a^2 x^2}+\frac {2 \arctan (a x)}{a}\right )-\frac {3 \sqrt {\pi } \operatorname {FresnelS}\left (\frac {2 \sqrt {\arctan (a x)}}{\sqrt {\pi }}\right )}{a}}{24 c^2} \]
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Time = 6.53 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.74
method | result | size |
default | \(\frac {8 \arctan \left (a x \right )^{\frac {3}{2}} \sqrt {\pi }+6 \sin \left (2 \arctan \left (a x \right )\right ) \sqrt {\arctan \left (a x \right )}\, \sqrt {\pi }-3 \pi \,\operatorname {FresnelS}\left (\frac {2 \sqrt {\arctan \left (a x \right )}}{\sqrt {\pi }}\right )}{24 c^{2} a \sqrt {\pi }}\) | \(57\) |
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Exception generated. \[ \int \frac {\sqrt {\arctan (a x)}}{\left (c+a^2 c x^2\right )^2} \, dx=\text {Exception raised: TypeError} \]
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\[ \int \frac {\sqrt {\arctan (a x)}}{\left (c+a^2 c x^2\right )^2} \, dx=\frac {\int \frac {\sqrt {\operatorname {atan}{\left (a x \right )}}}{a^{4} x^{4} + 2 a^{2} x^{2} + 1}\, dx}{c^{2}} \]
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Exception generated. \[ \int \frac {\sqrt {\arctan (a x)}}{\left (c+a^2 c x^2\right )^2} \, dx=\text {Exception raised: RuntimeError} \]
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\[ \int \frac {\sqrt {\arctan (a x)}}{\left (c+a^2 c x^2\right )^2} \, dx=\int { \frac {\sqrt {\arctan \left (a x\right )}}{{\left (a^{2} c x^{2} + c\right )}^{2}} \,d x } \]
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Timed out. \[ \int \frac {\sqrt {\arctan (a x)}}{\left (c+a^2 c x^2\right )^2} \, dx=\int \frac {\sqrt {\mathrm {atan}\left (a\,x\right )}}{{\left (c\,a^2\,x^2+c\right )}^2} \,d x \]
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