Integrand size = 21, antiderivative size = 151 \[ \int \frac {\arctan (a x)^{5/2}}{\left (c+a^2 c x^2\right )^2} \, dx=-\frac {15 x \sqrt {\arctan (a x)}}{32 c^2 \left (1+a^2 x^2\right )}-\frac {5 \arctan (a x)^{3/2}}{16 a c^2}+\frac {5 \arctan (a x)^{3/2}}{8 a c^2 \left (1+a^2 x^2\right )}+\frac {x \arctan (a x)^{5/2}}{2 c^2 \left (1+a^2 x^2\right )}+\frac {\arctan (a x)^{7/2}}{7 a c^2}+\frac {15 \sqrt {\pi } \operatorname {FresnelS}\left (\frac {2 \sqrt {\arctan (a x)}}{\sqrt {\pi }}\right )}{128 a c^2} \]
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Time = 0.14 (sec) , antiderivative size = 151, 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 = {5012, 5050, 5090, 4491, 12, 3386, 3432} \[ \int \frac {\arctan (a x)^{5/2}}{\left (c+a^2 c x^2\right )^2} \, dx=\frac {x \arctan (a x)^{5/2}}{2 c^2 \left (a^2 x^2+1\right )}+\frac {5 \arctan (a x)^{3/2}}{8 a c^2 \left (a^2 x^2+1\right )}-\frac {15 x \sqrt {\arctan (a x)}}{32 c^2 \left (a^2 x^2+1\right )}+\frac {15 \sqrt {\pi } \operatorname {FresnelS}\left (\frac {2 \sqrt {\arctan (a x)}}{\sqrt {\pi }}\right )}{128 a c^2}+\frac {\arctan (a x)^{7/2}}{7 a c^2}-\frac {5 \arctan (a x)^{3/2}}{16 a c^2} \]
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Rule 12
Rule 3386
Rule 3432
Rule 4491
Rule 5012
Rule 5050
Rule 5090
Rubi steps \begin{align*} \text {integral}& = \frac {x \arctan (a x)^{5/2}}{2 c^2 \left (1+a^2 x^2\right )}+\frac {\arctan (a x)^{7/2}}{7 a c^2}-\frac {1}{4} (5 a) \int \frac {x \arctan (a x)^{3/2}}{\left (c+a^2 c x^2\right )^2} \, dx \\ & = \frac {5 \arctan (a x)^{3/2}}{8 a c^2 \left (1+a^2 x^2\right )}+\frac {x \arctan (a x)^{5/2}}{2 c^2 \left (1+a^2 x^2\right )}+\frac {\arctan (a x)^{7/2}}{7 a c^2}-\frac {15}{16} \int \frac {\sqrt {\arctan (a x)}}{\left (c+a^2 c x^2\right )^2} \, dx \\ & = -\frac {15 x \sqrt {\arctan (a x)}}{32 c^2 \left (1+a^2 x^2\right )}-\frac {5 \arctan (a x)^{3/2}}{16 a c^2}+\frac {5 \arctan (a x)^{3/2}}{8 a c^2 \left (1+a^2 x^2\right )}+\frac {x \arctan (a x)^{5/2}}{2 c^2 \left (1+a^2 x^2\right )}+\frac {\arctan (a x)^{7/2}}{7 a c^2}+\frac {1}{64} (15 a) \int \frac {x}{\left (c+a^2 c x^2\right )^2 \sqrt {\arctan (a x)}} \, dx \\ & = -\frac {15 x \sqrt {\arctan (a x)}}{32 c^2 \left (1+a^2 x^2\right )}-\frac {5 \arctan (a x)^{3/2}}{16 a c^2}+\frac {5 \arctan (a x)^{3/2}}{8 a c^2 \left (1+a^2 x^2\right )}+\frac {x \arctan (a x)^{5/2}}{2 c^2 \left (1+a^2 x^2\right )}+\frac {\arctan (a x)^{7/2}}{7 a c^2}+\frac {15 \text {Subst}\left (\int \frac {\cos (x) \sin (x)}{\sqrt {x}} \, dx,x,\arctan (a x)\right )}{64 a c^2} \\ & = -\frac {15 x \sqrt {\arctan (a x)}}{32 c^2 \left (1+a^2 x^2\right )}-\frac {5 \arctan (a x)^{3/2}}{16 a c^2}+\frac {5 \arctan (a x)^{3/2}}{8 a c^2 \left (1+a^2 x^2\right )}+\frac {x \arctan (a x)^{5/2}}{2 c^2 \left (1+a^2 x^2\right )}+\frac {\arctan (a x)^{7/2}}{7 a c^2}+\frac {15 \text {Subst}\left (\int \frac {\sin (2 x)}{2 \sqrt {x}} \, dx,x,\arctan (a x)\right )}{64 a c^2} \\ & = -\frac {15 x \sqrt {\arctan (a x)}}{32 c^2 \left (1+a^2 x^2\right )}-\frac {5 \arctan (a x)^{3/2}}{16 a c^2}+\frac {5 \arctan (a x)^{3/2}}{8 a c^2 \left (1+a^2 x^2\right )}+\frac {x \arctan (a x)^{5/2}}{2 c^2 \left (1+a^2 x^2\right )}+\frac {\arctan (a x)^{7/2}}{7 a c^2}+\frac {15 \text {Subst}\left (\int \frac {\sin (2 x)}{\sqrt {x}} \, dx,x,\arctan (a x)\right )}{128 a c^2} \\ & = -\frac {15 x \sqrt {\arctan (a x)}}{32 c^2 \left (1+a^2 x^2\right )}-\frac {5 \arctan (a x)^{3/2}}{16 a c^2}+\frac {5 \arctan (a x)^{3/2}}{8 a c^2 \left (1+a^2 x^2\right )}+\frac {x \arctan (a x)^{5/2}}{2 c^2 \left (1+a^2 x^2\right )}+\frac {\arctan (a x)^{7/2}}{7 a c^2}+\frac {15 \text {Subst}\left (\int \sin \left (2 x^2\right ) \, dx,x,\sqrt {\arctan (a x)}\right )}{64 a c^2} \\ & = -\frac {15 x \sqrt {\arctan (a x)}}{32 c^2 \left (1+a^2 x^2\right )}-\frac {5 \arctan (a x)^{3/2}}{16 a c^2}+\frac {5 \arctan (a x)^{3/2}}{8 a c^2 \left (1+a^2 x^2\right )}+\frac {x \arctan (a x)^{5/2}}{2 c^2 \left (1+a^2 x^2\right )}+\frac {\arctan (a x)^{7/2}}{7 a c^2}+\frac {15 \sqrt {\pi } \operatorname {FresnelS}\left (\frac {2 \sqrt {\arctan (a x)}}{\sqrt {\pi }}\right )}{128 a c^2} \\ \end{align*}
Time = 0.31 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.72 \[ \int \frac {\arctan (a x)^{5/2}}{\left (c+a^2 c x^2\right )^2} \, dx=\frac {4 \sqrt {\arctan (a x)} \left (-105 a x-70 \left (-1+a^2 x^2\right ) \arctan (a x)+112 a x \arctan (a x)^2+32 \left (1+a^2 x^2\right ) \arctan (a x)^3\right )+105 \sqrt {\pi } \left (1+a^2 x^2\right ) \operatorname {FresnelS}\left (\frac {2 \sqrt {\arctan (a x)}}{\sqrt {\pi }}\right )}{896 c^2 \left (a+a^3 x^2\right )} \]
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Time = 25.05 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.62
method | result | size |
default | \(\frac {128 \arctan \left (a x \right )^{\frac {7}{2}} \sqrt {\pi }+224 \arctan \left (a x \right )^{\frac {5}{2}} \sqrt {\pi }\, \sin \left (2 \arctan \left (a x \right )\right )+280 \arctan \left (a x \right )^{\frac {3}{2}} \sqrt {\pi }\, \cos \left (2 \arctan \left (a x \right )\right )+105 \pi \,\operatorname {FresnelS}\left (\frac {2 \sqrt {\arctan \left (a x \right )}}{\sqrt {\pi }}\right )-210 \sqrt {\arctan \left (a x \right )}\, \sqrt {\pi }\, \sin \left (2 \arctan \left (a x \right )\right )}{896 c^{2} a \sqrt {\pi }}\) | \(93\) |
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Exception generated. \[ \int \frac {\arctan (a x)^{5/2}}{\left (c+a^2 c x^2\right )^2} \, dx=\text {Exception raised: TypeError} \]
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\[ \int \frac {\arctan (a x)^{5/2}}{\left (c+a^2 c x^2\right )^2} \, dx=\frac {\int \frac {\operatorname {atan}^{\frac {5}{2}}{\left (a x \right )}}{a^{4} x^{4} + 2 a^{2} x^{2} + 1}\, dx}{c^{2}} \]
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Exception generated. \[ \int \frac {\arctan (a x)^{5/2}}{\left (c+a^2 c x^2\right )^2} \, dx=\text {Exception raised: RuntimeError} \]
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\[ \int \frac {\arctan (a x)^{5/2}}{\left (c+a^2 c x^2\right )^2} \, dx=\int { \frac {\arctan \left (a x\right )^{\frac {5}{2}}}{{\left (a^{2} c x^{2} + c\right )}^{2}} \,d x } \]
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Timed out. \[ \int \frac {\arctan (a x)^{5/2}}{\left (c+a^2 c x^2\right )^2} \, dx=\int \frac {{\mathrm {atan}\left (a\,x\right )}^{5/2}}{{\left (c\,a^2\,x^2+c\right )}^2} \,d x \]
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