Integrand size = 22, antiderivative size = 138 \[ \int \frac {x}{\left (c+a^2 c x^2\right )^2 \arctan (a x)^{3/2}} \, dx=-\frac {2 x}{a c^2 \left (1+a^2 x^2\right ) \sqrt {\arctan (a x)}}+\frac {4 \sqrt {\arctan (a x)}}{a^2 c^2}-\frac {8 \sqrt {\arctan (a x)}}{a^2 c^2 \left (1+a^2 x^2\right )}+\frac {4 \left (1-a^2 x^2\right ) \sqrt {\arctan (a x)}}{a^2 c^2 \left (1+a^2 x^2\right )}+\frac {2 \sqrt {\pi } \operatorname {FresnelC}\left (\frac {2 \sqrt {\arctan (a x)}}{\sqrt {\pi }}\right )}{a^2 c^2} \]
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Time = 0.12 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {5052, 5050, 5024, 3393, 3385, 3433} \[ \int \frac {x}{\left (c+a^2 c x^2\right )^2 \arctan (a x)^{3/2}} \, dx=\frac {2 \sqrt {\pi } \operatorname {FresnelC}\left (\frac {2 \sqrt {\arctan (a x)}}{\sqrt {\pi }}\right )}{a^2 c^2}-\frac {2 x}{a c^2 \left (a^2 x^2+1\right ) \sqrt {\arctan (a x)}}+\frac {4 \left (1-a^2 x^2\right ) \sqrt {\arctan (a x)}}{a^2 c^2 \left (a^2 x^2+1\right )}-\frac {8 \sqrt {\arctan (a x)}}{a^2 c^2 \left (a^2 x^2+1\right )}+\frac {4 \sqrt {\arctan (a x)}}{a^2 c^2} \]
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Rule 3385
Rule 3393
Rule 3433
Rule 5024
Rule 5050
Rule 5052
Rubi steps \begin{align*} \text {integral}& = -\frac {2 x}{a c^2 \left (1+a^2 x^2\right ) \sqrt {\arctan (a x)}}+\frac {4 \left (1-a^2 x^2\right ) \sqrt {\arctan (a x)}}{a^2 c^2 \left (1+a^2 x^2\right )}+16 \int \frac {x \sqrt {\arctan (a x)}}{\left (c+a^2 c x^2\right )^2} \, dx \\ & = -\frac {2 x}{a c^2 \left (1+a^2 x^2\right ) \sqrt {\arctan (a x)}}-\frac {8 \sqrt {\arctan (a x)}}{a^2 c^2 \left (1+a^2 x^2\right )}+\frac {4 \left (1-a^2 x^2\right ) \sqrt {\arctan (a x)}}{a^2 c^2 \left (1+a^2 x^2\right )}+\frac {4 \int \frac {1}{\left (c+a^2 c x^2\right )^2 \sqrt {\arctan (a x)}} \, dx}{a} \\ & = -\frac {2 x}{a c^2 \left (1+a^2 x^2\right ) \sqrt {\arctan (a x)}}-\frac {8 \sqrt {\arctan (a x)}}{a^2 c^2 \left (1+a^2 x^2\right )}+\frac {4 \left (1-a^2 x^2\right ) \sqrt {\arctan (a x)}}{a^2 c^2 \left (1+a^2 x^2\right )}+\frac {4 \text {Subst}\left (\int \frac {\cos ^2(x)}{\sqrt {x}} \, dx,x,\arctan (a x)\right )}{a^2 c^2} \\ & = -\frac {2 x}{a c^2 \left (1+a^2 x^2\right ) \sqrt {\arctan (a x)}}-\frac {8 \sqrt {\arctan (a x)}}{a^2 c^2 \left (1+a^2 x^2\right )}+\frac {4 \left (1-a^2 x^2\right ) \sqrt {\arctan (a x)}}{a^2 c^2 \left (1+a^2 x^2\right )}+\frac {4 \text {Subst}\left (\int \left (\frac {1}{2 \sqrt {x}}+\frac {\cos (2 x)}{2 \sqrt {x}}\right ) \, dx,x,\arctan (a x)\right )}{a^2 c^2} \\ & = -\frac {2 x}{a c^2 \left (1+a^2 x^2\right ) \sqrt {\arctan (a x)}}+\frac {4 \sqrt {\arctan (a x)}}{a^2 c^2}-\frac {8 \sqrt {\arctan (a x)}}{a^2 c^2 \left (1+a^2 x^2\right )}+\frac {4 \left (1-a^2 x^2\right ) \sqrt {\arctan (a x)}}{a^2 c^2 \left (1+a^2 x^2\right )}+\frac {2 \text {Subst}\left (\int \frac {\cos (2 x)}{\sqrt {x}} \, dx,x,\arctan (a x)\right )}{a^2 c^2} \\ & = -\frac {2 x}{a c^2 \left (1+a^2 x^2\right ) \sqrt {\arctan (a x)}}+\frac {4 \sqrt {\arctan (a x)}}{a^2 c^2}-\frac {8 \sqrt {\arctan (a x)}}{a^2 c^2 \left (1+a^2 x^2\right )}+\frac {4 \left (1-a^2 x^2\right ) \sqrt {\arctan (a x)}}{a^2 c^2 \left (1+a^2 x^2\right )}+\frac {4 \text {Subst}\left (\int \cos \left (2 x^2\right ) \, dx,x,\sqrt {\arctan (a x)}\right )}{a^2 c^2} \\ & = -\frac {2 x}{a c^2 \left (1+a^2 x^2\right ) \sqrt {\arctan (a x)}}+\frac {4 \sqrt {\arctan (a x)}}{a^2 c^2}-\frac {8 \sqrt {\arctan (a x)}}{a^2 c^2 \left (1+a^2 x^2\right )}+\frac {4 \left (1-a^2 x^2\right ) \sqrt {\arctan (a x)}}{a^2 c^2 \left (1+a^2 x^2\right )}+\frac {2 \sqrt {\pi } \operatorname {FresnelC}\left (\frac {2 \sqrt {\arctan (a x)}}{\sqrt {\pi }}\right )}{a^2 c^2} \\ \end{align*}
Result contains complex when optimal does not.
Time = 0.17 (sec) , antiderivative size = 158, normalized size of antiderivative = 1.14 \[ \int \frac {x}{\left (c+a^2 c x^2\right )^2 \arctan (a x)^{3/2}} \, dx=\frac {-8 a x+4 \sqrt {\pi } \left (1+a^2 x^2\right ) \sqrt {\arctan (a x)} \operatorname {FresnelC}\left (\frac {2 \sqrt {\arctan (a x)}}{\sqrt {\pi }}\right )-i \sqrt {2} \left (1+a^2 x^2\right ) \sqrt {-i \arctan (a x)} \Gamma \left (\frac {1}{2},-2 i \arctan (a x)\right )+i \sqrt {2} \left (1+a^2 x^2\right ) \sqrt {i \arctan (a x)} \Gamma \left (\frac {1}{2},2 i \arctan (a x)\right )}{4 a^2 c^2 \left (1+a^2 x^2\right ) \sqrt {\arctan (a x)}} \]
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Time = 1.74 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.34
method | result | size |
default | \(\frac {2 \,\operatorname {FresnelC}\left (\frac {2 \sqrt {\arctan \left (a x \right )}}{\sqrt {\pi }}\right ) \sqrt {\arctan \left (a x \right )}\, \sqrt {\pi }-\sin \left (2 \arctan \left (a x \right )\right )}{c^{2} a^{2} \sqrt {\arctan \left (a x \right )}}\) | \(47\) |
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Exception generated. \[ \int \frac {x}{\left (c+a^2 c x^2\right )^2 \arctan (a x)^{3/2}} \, dx=\text {Exception raised: TypeError} \]
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\[ \int \frac {x}{\left (c+a^2 c x^2\right )^2 \arctan (a x)^{3/2}} \, dx=\frac {\int \frac {x}{a^{4} x^{4} \operatorname {atan}^{\frac {3}{2}}{\left (a x \right )} + 2 a^{2} x^{2} \operatorname {atan}^{\frac {3}{2}}{\left (a x \right )} + \operatorname {atan}^{\frac {3}{2}}{\left (a x \right )}}\, dx}{c^{2}} \]
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Exception generated. \[ \int \frac {x}{\left (c+a^2 c x^2\right )^2 \arctan (a x)^{3/2}} \, dx=\text {Exception raised: RuntimeError} \]
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Timed out. \[ \int \frac {x}{\left (c+a^2 c x^2\right )^2 \arctan (a x)^{3/2}} \, dx=\text {Timed out} \]
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Timed out. \[ \int \frac {x}{\left (c+a^2 c x^2\right )^2 \arctan (a x)^{3/2}} \, dx=\int \frac {x}{{\mathrm {atan}\left (a\,x\right )}^{3/2}\,{\left (c\,a^2\,x^2+c\right )}^2} \,d x \]
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