Integrand size = 21, antiderivative size = 57 \[ \int \frac {1}{\left (c+a^2 c x^2\right )^2 \arctan (a x)^{3/2}} \, dx=-\frac {2}{a c^2 \left (1+a^2 x^2\right ) \sqrt {\arctan (a x)}}-\frac {2 \sqrt {\pi } \operatorname {FresnelS}\left (\frac {2 \sqrt {\arctan (a x)}}{\sqrt {\pi }}\right )}{a c^2} \]
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Time = 0.08 (sec) , antiderivative size = 57, 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 = {5022, 5090, 4491, 12, 3386, 3432} \[ \int \frac {1}{\left (c+a^2 c x^2\right )^2 \arctan (a x)^{3/2}} \, dx=-\frac {2}{a c^2 \left (a^2 x^2+1\right ) \sqrt {\arctan (a x)}}-\frac {2 \sqrt {\pi } \operatorname {FresnelS}\left (\frac {2 \sqrt {\arctan (a x)}}{\sqrt {\pi }}\right )}{a c^2} \]
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Rule 12
Rule 3386
Rule 3432
Rule 4491
Rule 5022
Rule 5090
Rubi steps \begin{align*} \text {integral}& = -\frac {2}{a c^2 \left (1+a^2 x^2\right ) \sqrt {\arctan (a x)}}-(4 a) \int \frac {x}{\left (c+a^2 c x^2\right )^2 \sqrt {\arctan (a x)}} \, dx \\ & = -\frac {2}{a c^2 \left (1+a^2 x^2\right ) \sqrt {\arctan (a x)}}-\frac {4 \text {Subst}\left (\int \frac {\cos (x) \sin (x)}{\sqrt {x}} \, dx,x,\arctan (a x)\right )}{a c^2} \\ & = -\frac {2}{a c^2 \left (1+a^2 x^2\right ) \sqrt {\arctan (a x)}}-\frac {4 \text {Subst}\left (\int \frac {\sin (2 x)}{2 \sqrt {x}} \, dx,x,\arctan (a x)\right )}{a c^2} \\ & = -\frac {2}{a c^2 \left (1+a^2 x^2\right ) \sqrt {\arctan (a x)}}-\frac {2 \text {Subst}\left (\int \frac {\sin (2 x)}{\sqrt {x}} \, dx,x,\arctan (a x)\right )}{a c^2} \\ & = -\frac {2}{a c^2 \left (1+a^2 x^2\right ) \sqrt {\arctan (a x)}}-\frac {4 \text {Subst}\left (\int \sin \left (2 x^2\right ) \, dx,x,\sqrt {\arctan (a x)}\right )}{a c^2} \\ & = -\frac {2}{a c^2 \left (1+a^2 x^2\right ) \sqrt {\arctan (a x)}}-\frac {2 \sqrt {\pi } \operatorname {FresnelS}\left (\frac {2 \sqrt {\arctan (a x)}}{\sqrt {\pi }}\right )}{a c^2} \\ \end{align*}
Time = 0.28 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.91 \[ \int \frac {1}{\left (c+a^2 c x^2\right )^2 \arctan (a x)^{3/2}} \, dx=\frac {-\frac {2}{\left (1+a^2 x^2\right ) \sqrt {\arctan (a x)}}-2 \sqrt {\pi } \operatorname {FresnelS}\left (\frac {2 \sqrt {\arctan (a x)}}{\sqrt {\pi }}\right )}{a c^2} \]
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Time = 2.06 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.82
method | result | size |
default | \(-\frac {2 \sqrt {\arctan \left (a x \right )}\, \sqrt {\pi }\, \operatorname {FresnelS}\left (\frac {2 \sqrt {\arctan \left (a x \right )}}{\sqrt {\pi }}\right )+\cos \left (2 \arctan \left (a x \right )\right )+1}{c^{2} a \sqrt {\arctan \left (a x \right )}}\) | \(47\) |
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Exception generated. \[ \int \frac {1}{\left (c+a^2 c x^2\right )^2 \arctan (a x)^{3/2}} \, dx=\text {Exception raised: TypeError} \]
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\[ \int \frac {1}{\left (c+a^2 c x^2\right )^2 \arctan (a x)^{3/2}} \, dx=\frac {\int \frac {1}{a^{4} x^{4} \operatorname {atan}^{\frac {3}{2}}{\left (a x \right )} + 2 a^{2} x^{2} \operatorname {atan}^{\frac {3}{2}}{\left (a x \right )} + \operatorname {atan}^{\frac {3}{2}}{\left (a x \right )}}\, dx}{c^{2}} \]
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Exception generated. \[ \int \frac {1}{\left (c+a^2 c x^2\right )^2 \arctan (a x)^{3/2}} \, dx=\text {Exception raised: RuntimeError} \]
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\[ \int \frac {1}{\left (c+a^2 c x^2\right )^2 \arctan (a x)^{3/2}} \, dx=\int { \frac {1}{{\left (a^{2} c x^{2} + c\right )}^{2} \arctan \left (a x\right )^{\frac {3}{2}}} \,d x } \]
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Timed out. \[ \int \frac {1}{\left (c+a^2 c x^2\right )^2 \arctan (a x)^{3/2}} \, dx=\int \frac {1}{{\mathrm {atan}\left (a\,x\right )}^{3/2}\,{\left (c\,a^2\,x^2+c\right )}^2} \,d x \]
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