Integrand size = 24, antiderivative size = 80 \[ \int \frac {x^2 \sqrt {\arctan (a x)}}{\left (c+a^2 c x^2\right )^2} \, dx=-\frac {x \sqrt {\arctan (a x)}}{2 a^2 c^2 \left (1+a^2 x^2\right )}+\frac {\arctan (a x)^{3/2}}{3 a^3 c^2}+\frac {\sqrt {\pi } \operatorname {FresnelS}\left (\frac {2 \sqrt {\arctan (a x)}}{\sqrt {\pi }}\right )}{8 a^3 c^2} \]
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Time = 0.10 (sec) , antiderivative size = 80, 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 = {5056, 5090, 4491, 12, 3386, 3432} \[ \int \frac {x^2 \sqrt {\arctan (a x)}}{\left (c+a^2 c x^2\right )^2} \, dx=\frac {\sqrt {\pi } \operatorname {FresnelS}\left (\frac {2 \sqrt {\arctan (a x)}}{\sqrt {\pi }}\right )}{8 a^3 c^2}+\frac {\arctan (a x)^{3/2}}{3 a^3 c^2}-\frac {x \sqrt {\arctan (a x)}}{2 a^2 c^2 \left (a^2 x^2+1\right )} \]
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Rule 12
Rule 3386
Rule 3432
Rule 4491
Rule 5056
Rule 5090
Rubi steps \begin{align*} \text {integral}& = -\frac {x \sqrt {\arctan (a x)}}{2 a^2 c^2 \left (1+a^2 x^2\right )}+\frac {\arctan (a x)^{3/2}}{3 a^3 c^2}+\frac {\int \frac {x}{\left (c+a^2 c x^2\right )^2 \sqrt {\arctan (a x)}} \, dx}{4 a} \\ & = -\frac {x \sqrt {\arctan (a x)}}{2 a^2 c^2 \left (1+a^2 x^2\right )}+\frac {\arctan (a x)^{3/2}}{3 a^3 c^2}+\frac {\text {Subst}\left (\int \frac {\cos (x) \sin (x)}{\sqrt {x}} \, dx,x,\arctan (a x)\right )}{4 a^3 c^2} \\ & = -\frac {x \sqrt {\arctan (a x)}}{2 a^2 c^2 \left (1+a^2 x^2\right )}+\frac {\arctan (a x)^{3/2}}{3 a^3 c^2}+\frac {\text {Subst}\left (\int \frac {\sin (2 x)}{2 \sqrt {x}} \, dx,x,\arctan (a x)\right )}{4 a^3 c^2} \\ & = -\frac {x \sqrt {\arctan (a x)}}{2 a^2 c^2 \left (1+a^2 x^2\right )}+\frac {\arctan (a x)^{3/2}}{3 a^3 c^2}+\frac {\text {Subst}\left (\int \frac {\sin (2 x)}{\sqrt {x}} \, dx,x,\arctan (a x)\right )}{8 a^3 c^2} \\ & = -\frac {x \sqrt {\arctan (a x)}}{2 a^2 c^2 \left (1+a^2 x^2\right )}+\frac {\arctan (a x)^{3/2}}{3 a^3 c^2}+\frac {\text {Subst}\left (\int \sin \left (2 x^2\right ) \, dx,x,\sqrt {\arctan (a x)}\right )}{4 a^3 c^2} \\ & = -\frac {x \sqrt {\arctan (a x)}}{2 a^2 c^2 \left (1+a^2 x^2\right )}+\frac {\arctan (a x)^{3/2}}{3 a^3 c^2}+\frac {\sqrt {\pi } \operatorname {FresnelS}\left (\frac {2 \sqrt {\arctan (a x)}}{\sqrt {\pi }}\right )}{8 a^3 c^2} \\ \end{align*}
Time = 0.34 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.82 \[ \int \frac {x^2 \sqrt {\arctan (a x)}}{\left (c+a^2 c x^2\right )^2} \, dx=\frac {4 \sqrt {\arctan (a x)} \left (-\frac {3 a x}{1+a^2 x^2}+2 \arctan (a x)\right )+3 \sqrt {\pi } \operatorname {FresnelS}\left (\frac {2 \sqrt {\arctan (a x)}}{\sqrt {\pi }}\right )}{24 a^3 c^2} \]
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Time = 6.35 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.75
method | result | size |
default | \(\frac {3 \sqrt {\arctan \left (a x \right )}\, \sqrt {\pi }\, \operatorname {FresnelS}\left (\frac {2 \sqrt {\arctan \left (a x \right )}}{\sqrt {\pi }}\right )+8 \arctan \left (a x \right )^{2}-6 \sin \left (2 \arctan \left (a x \right )\right ) \arctan \left (a x \right )}{24 c^{2} a^{3} \sqrt {\arctan \left (a x \right )}}\) | \(60\) |
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Exception generated. \[ \int \frac {x^2 \sqrt {\arctan (a x)}}{\left (c+a^2 c x^2\right )^2} \, dx=\text {Exception raised: TypeError} \]
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\[ \int \frac {x^2 \sqrt {\arctan (a x)}}{\left (c+a^2 c x^2\right )^2} \, dx=\frac {\int \frac {x^{2} \sqrt {\operatorname {atan}{\left (a x \right )}}}{a^{4} x^{4} + 2 a^{2} x^{2} + 1}\, dx}{c^{2}} \]
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Exception generated. \[ \int \frac {x^2 \sqrt {\arctan (a x)}}{\left (c+a^2 c x^2\right )^2} \, dx=\text {Exception raised: RuntimeError} \]
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\[ \int \frac {x^2 \sqrt {\arctan (a x)}}{\left (c+a^2 c x^2\right )^2} \, dx=\int { \frac {x^{2} \sqrt {\arctan \left (a x\right )}}{{\left (a^{2} c x^{2} + c\right )}^{2}} \,d x } \]
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Timed out. \[ \int \frac {x^2 \sqrt {\arctan (a x)}}{\left (c+a^2 c x^2\right )^2} \, dx=\int \frac {x^2\,\sqrt {\mathrm {atan}\left (a\,x\right )}}{{\left (c\,a^2\,x^2+c\right )}^2} \,d x \]
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