Integrand size = 21, antiderivative size = 174 \[ \int \frac {1}{\left (c+a^2 c x^2\right )^2 \arctan (a x)^{5/2}} \, dx=-\frac {2}{3 a c^2 \left (1+a^2 x^2\right ) \arctan (a x)^{3/2}}+\frac {8 x}{3 c^2 \left (1+a^2 x^2\right ) \sqrt {\arctan (a x)}}-\frac {16 \sqrt {\arctan (a x)}}{3 a c^2}+\frac {32 \sqrt {\arctan (a x)}}{3 a c^2 \left (1+a^2 x^2\right )}-\frac {16 \left (1-a^2 x^2\right ) \sqrt {\arctan (a x)}}{3 a c^2 \left (1+a^2 x^2\right )}-\frac {8 \sqrt {\pi } \operatorname {FresnelC}\left (\frac {2 \sqrt {\arctan (a x)}}{\sqrt {\pi }}\right )}{3 a c^2} \]
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Time = 0.14 (sec) , antiderivative size = 174, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {5022, 5052, 5050, 5024, 3393, 3385, 3433} \[ \int \frac {1}{\left (c+a^2 c x^2\right )^2 \arctan (a x)^{5/2}} \, dx=\frac {8 x}{3 c^2 \left (a^2 x^2+1\right ) \sqrt {\arctan (a x)}}-\frac {16 \left (1-a^2 x^2\right ) \sqrt {\arctan (a x)}}{3 a c^2 \left (a^2 x^2+1\right )}+\frac {32 \sqrt {\arctan (a x)}}{3 a c^2 \left (a^2 x^2+1\right )}-\frac {2}{3 a c^2 \left (a^2 x^2+1\right ) \arctan (a x)^{3/2}}-\frac {8 \sqrt {\pi } \operatorname {FresnelC}\left (\frac {2 \sqrt {\arctan (a x)}}{\sqrt {\pi }}\right )}{3 a c^2}-\frac {16 \sqrt {\arctan (a x)}}{3 a c^2} \]
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Rule 3385
Rule 3393
Rule 3433
Rule 5022
Rule 5024
Rule 5050
Rule 5052
Rubi steps \begin{align*} \text {integral}& = -\frac {2}{3 a c^2 \left (1+a^2 x^2\right ) \arctan (a x)^{3/2}}-\frac {1}{3} (4 a) \int \frac {x}{\left (c+a^2 c x^2\right )^2 \arctan (a x)^{3/2}} \, dx \\ & = -\frac {2}{3 a c^2 \left (1+a^2 x^2\right ) \arctan (a x)^{3/2}}+\frac {8 x}{3 c^2 \left (1+a^2 x^2\right ) \sqrt {\arctan (a x)}}-\frac {16 \left (1-a^2 x^2\right ) \sqrt {\arctan (a x)}}{3 a c^2 \left (1+a^2 x^2\right )}-\frac {1}{3} (64 a) \int \frac {x \sqrt {\arctan (a x)}}{\left (c+a^2 c x^2\right )^2} \, dx \\ & = -\frac {2}{3 a c^2 \left (1+a^2 x^2\right ) \arctan (a x)^{3/2}}+\frac {8 x}{3 c^2 \left (1+a^2 x^2\right ) \sqrt {\arctan (a x)}}+\frac {32 \sqrt {\arctan (a x)}}{3 a c^2 \left (1+a^2 x^2\right )}-\frac {16 \left (1-a^2 x^2\right ) \sqrt {\arctan (a x)}}{3 a c^2 \left (1+a^2 x^2\right )}-\frac {16}{3} \int \frac {1}{\left (c+a^2 c x^2\right )^2 \sqrt {\arctan (a x)}} \, dx \\ & = -\frac {2}{3 a c^2 \left (1+a^2 x^2\right ) \arctan (a x)^{3/2}}+\frac {8 x}{3 c^2 \left (1+a^2 x^2\right ) \sqrt {\arctan (a x)}}+\frac {32 \sqrt {\arctan (a x)}}{3 a c^2 \left (1+a^2 x^2\right )}-\frac {16 \left (1-a^2 x^2\right ) \sqrt {\arctan (a x)}}{3 a c^2 \left (1+a^2 x^2\right )}-\frac {16 \text {Subst}\left (\int \frac {\cos ^2(x)}{\sqrt {x}} \, dx,x,\arctan (a x)\right )}{3 a c^2} \\ & = -\frac {2}{3 a c^2 \left (1+a^2 x^2\right ) \arctan (a x)^{3/2}}+\frac {8 x}{3 c^2 \left (1+a^2 x^2\right ) \sqrt {\arctan (a x)}}+\frac {32 \sqrt {\arctan (a x)}}{3 a c^2 \left (1+a^2 x^2\right )}-\frac {16 \left (1-a^2 x^2\right ) \sqrt {\arctan (a x)}}{3 a c^2 \left (1+a^2 x^2\right )}-\frac {16 \text {Subst}\left (\int \left (\frac {1}{2 \sqrt {x}}+\frac {\cos (2 x)}{2 \sqrt {x}}\right ) \, dx,x,\arctan (a x)\right )}{3 a c^2} \\ & = -\frac {2}{3 a c^2 \left (1+a^2 x^2\right ) \arctan (a x)^{3/2}}+\frac {8 x}{3 c^2 \left (1+a^2 x^2\right ) \sqrt {\arctan (a x)}}-\frac {16 \sqrt {\arctan (a x)}}{3 a c^2}+\frac {32 \sqrt {\arctan (a x)}}{3 a c^2 \left (1+a^2 x^2\right )}-\frac {16 \left (1-a^2 x^2\right ) \sqrt {\arctan (a x)}}{3 a c^2 \left (1+a^2 x^2\right )}-\frac {8 \text {Subst}\left (\int \frac {\cos (2 x)}{\sqrt {x}} \, dx,x,\arctan (a x)\right )}{3 a c^2} \\ & = -\frac {2}{3 a c^2 \left (1+a^2 x^2\right ) \arctan (a x)^{3/2}}+\frac {8 x}{3 c^2 \left (1+a^2 x^2\right ) \sqrt {\arctan (a x)}}-\frac {16 \sqrt {\arctan (a x)}}{3 a c^2}+\frac {32 \sqrt {\arctan (a x)}}{3 a c^2 \left (1+a^2 x^2\right )}-\frac {16 \left (1-a^2 x^2\right ) \sqrt {\arctan (a x)}}{3 a c^2 \left (1+a^2 x^2\right )}-\frac {16 \text {Subst}\left (\int \cos \left (2 x^2\right ) \, dx,x,\sqrt {\arctan (a x)}\right )}{3 a c^2} \\ & = -\frac {2}{3 a c^2 \left (1+a^2 x^2\right ) \arctan (a x)^{3/2}}+\frac {8 x}{3 c^2 \left (1+a^2 x^2\right ) \sqrt {\arctan (a x)}}-\frac {16 \sqrt {\arctan (a x)}}{3 a c^2}+\frac {32 \sqrt {\arctan (a x)}}{3 a c^2 \left (1+a^2 x^2\right )}-\frac {16 \left (1-a^2 x^2\right ) \sqrt {\arctan (a x)}}{3 a c^2 \left (1+a^2 x^2\right )}-\frac {8 \sqrt {\pi } \operatorname {FresnelC}\left (\frac {2 \sqrt {\arctan (a x)}}{\sqrt {\pi }}\right )}{3 a c^2} \\ \end{align*}
Result contains complex when optimal does not.
Time = 0.51 (sec) , antiderivative size = 170, normalized size of antiderivative = 0.98 \[ \int \frac {1}{\left (c+a^2 c x^2\right )^2 \arctan (a x)^{5/2}} \, dx=\frac {-2+8 a x \arctan (a x)-4 \sqrt {\pi } \left (1+a^2 x^2\right ) \arctan (a x)^{3/2} \operatorname {FresnelC}\left (\frac {2 \sqrt {\arctan (a x)}}{\sqrt {\pi }}\right )+\sqrt {2} \left (1+a^2 x^2\right ) \sqrt {i \arctan (a x)} \sqrt {\arctan (a x)^2} \Gamma \left (\frac {1}{2},-2 i \arctan (a x)\right )+\frac {\sqrt {2} \left (1+a^2 x^2\right ) \arctan (a x)^2 \Gamma \left (\frac {1}{2},2 i \arctan (a x)\right )}{\sqrt {i \arctan (a x)}}}{3 c^2 \left (a+a^3 x^2\right ) \arctan (a x)^{3/2}} \]
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Time = 0.94 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.36
method | result | size |
default | \(\frac {-8 \,\operatorname {FresnelC}\left (\frac {2 \sqrt {\arctan \left (a x \right )}}{\sqrt {\pi }}\right ) \sqrt {\pi }\, \arctan \left (a x \right )^{\frac {3}{2}}+4 \sin \left (2 \arctan \left (a x \right )\right ) \arctan \left (a x \right )-\cos \left (2 \arctan \left (a x \right )\right )-1}{3 c^{2} a \arctan \left (a x \right )^{\frac {3}{2}}}\) | \(62\) |
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Exception generated. \[ \int \frac {1}{\left (c+a^2 c x^2\right )^2 \arctan (a x)^{5/2}} \, dx=\text {Exception raised: TypeError} \]
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\[ \int \frac {1}{\left (c+a^2 c x^2\right )^2 \arctan (a x)^{5/2}} \, dx=\frac {\int \frac {1}{a^{4} x^{4} \operatorname {atan}^{\frac {5}{2}}{\left (a x \right )} + 2 a^{2} x^{2} \operatorname {atan}^{\frac {5}{2}}{\left (a x \right )} + \operatorname {atan}^{\frac {5}{2}}{\left (a x \right )}}\, dx}{c^{2}} \]
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Exception generated. \[ \int \frac {1}{\left (c+a^2 c x^2\right )^2 \arctan (a x)^{5/2}} \, dx=\text {Exception raised: RuntimeError} \]
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\[ \int \frac {1}{\left (c+a^2 c x^2\right )^2 \arctan (a x)^{5/2}} \, dx=\int { \frac {1}{{\left (a^{2} c x^{2} + c\right )}^{2} \arctan \left (a x\right )^{\frac {5}{2}}} \,d x } \]
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Timed out. \[ \int \frac {1}{\left (c+a^2 c x^2\right )^2 \arctan (a x)^{5/2}} \, dx=\int \frac {1}{{\mathrm {atan}\left (a\,x\right )}^{5/2}\,{\left (c\,a^2\,x^2+c\right )}^2} \,d x \]
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