Integrand size = 24, antiderivative size = 24 \[ \int \frac {x^5}{\left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^2} \, dx=\frac {x^3}{a^3 c \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)}+\frac {x}{a^5 c^2 \sqrt {c+a^2 c x^2} \arctan (a x)}-\frac {7 \sqrt {1+a^2 x^2} \operatorname {CosIntegral}(\arctan (a x))}{4 a^6 c^2 \sqrt {c+a^2 c x^2}}+\frac {3 \sqrt {1+a^2 x^2} \operatorname {CosIntegral}(3 \arctan (a x))}{4 a^6 c^2 \sqrt {c+a^2 c x^2}}+\frac {\text {Int}\left (\frac {x}{\sqrt {c+a^2 c x^2} \arctan (a x)^2},x\right )}{a^4 c^2} \]
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Not integrable
Time = 0.61 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00, number of steps used = 0, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {x^5}{\left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^2} \, dx=\int \frac {x^5}{\left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^2} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = -\frac {\int \frac {x^3}{\left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^2} \, dx}{a^2}+\frac {\int \frac {x^3}{\left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^2} \, dx}{a^2 c} \\ & = \frac {x^3}{a^3 c \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)}-\frac {3 \int \frac {x^2}{\left (c+a^2 c x^2\right )^{5/2} \arctan (a x)} \, dx}{a^3}+\frac {\int \frac {x}{\sqrt {c+a^2 c x^2} \arctan (a x)^2} \, dx}{a^4 c^2}-\frac {\int \frac {x}{\left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^2} \, dx}{a^4 c} \\ & = \frac {x^3}{a^3 c \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)}+\frac {x}{a^5 c^2 \sqrt {c+a^2 c x^2} \arctan (a x)}+\frac {\int \frac {x}{\sqrt {c+a^2 c x^2} \arctan (a x)^2} \, dx}{a^4 c^2}-\frac {\int \frac {1}{\left (c+a^2 c x^2\right )^{3/2} \arctan (a x)} \, dx}{a^5 c}-\frac {\left (3 \sqrt {1+a^2 x^2}\right ) \int \frac {x^2}{\left (1+a^2 x^2\right )^{5/2} \arctan (a x)} \, dx}{a^3 c^2 \sqrt {c+a^2 c x^2}} \\ & = \frac {x^3}{a^3 c \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)}+\frac {x}{a^5 c^2 \sqrt {c+a^2 c x^2} \arctan (a x)}+\frac {\int \frac {x}{\sqrt {c+a^2 c x^2} \arctan (a x)^2} \, dx}{a^4 c^2}-\frac {\left (3 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \frac {\cos (x) \sin ^2(x)}{x} \, dx,x,\arctan (a x)\right )}{a^6 c^2 \sqrt {c+a^2 c x^2}}-\frac {\sqrt {1+a^2 x^2} \int \frac {1}{\left (1+a^2 x^2\right )^{3/2} \arctan (a x)} \, dx}{a^5 c^2 \sqrt {c+a^2 c x^2}} \\ & = \frac {x^3}{a^3 c \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)}+\frac {x}{a^5 c^2 \sqrt {c+a^2 c x^2} \arctan (a x)}+\frac {\int \frac {x}{\sqrt {c+a^2 c x^2} \arctan (a x)^2} \, dx}{a^4 c^2}-\frac {\sqrt {1+a^2 x^2} \text {Subst}\left (\int \frac {\cos (x)}{x} \, dx,x,\arctan (a x)\right )}{a^6 c^2 \sqrt {c+a^2 c x^2}}-\frac {\left (3 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \left (\frac {\cos (x)}{4 x}-\frac {\cos (3 x)}{4 x}\right ) \, dx,x,\arctan (a x)\right )}{a^6 c^2 \sqrt {c+a^2 c x^2}} \\ & = \frac {x^3}{a^3 c \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)}+\frac {x}{a^5 c^2 \sqrt {c+a^2 c x^2} \arctan (a x)}-\frac {\sqrt {1+a^2 x^2} \operatorname {CosIntegral}(\arctan (a x))}{a^6 c^2 \sqrt {c+a^2 c x^2}}+\frac {\int \frac {x}{\sqrt {c+a^2 c x^2} \arctan (a x)^2} \, dx}{a^4 c^2}-\frac {\left (3 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \frac {\cos (x)}{x} \, dx,x,\arctan (a x)\right )}{4 a^6 c^2 \sqrt {c+a^2 c x^2}}+\frac {\left (3 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \frac {\cos (3 x)}{x} \, dx,x,\arctan (a x)\right )}{4 a^6 c^2 \sqrt {c+a^2 c x^2}} \\ & = \frac {x^3}{a^3 c \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)}+\frac {x}{a^5 c^2 \sqrt {c+a^2 c x^2} \arctan (a x)}-\frac {7 \sqrt {1+a^2 x^2} \operatorname {CosIntegral}(\arctan (a x))}{4 a^6 c^2 \sqrt {c+a^2 c x^2}}+\frac {3 \sqrt {1+a^2 x^2} \operatorname {CosIntegral}(3 \arctan (a x))}{4 a^6 c^2 \sqrt {c+a^2 c x^2}}+\frac {\int \frac {x}{\sqrt {c+a^2 c x^2} \arctan (a x)^2} \, dx}{a^4 c^2} \\ \end{align*}
Not integrable
Time = 11.87 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.08 \[ \int \frac {x^5}{\left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^2} \, dx=\int \frac {x^5}{\left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^2} \, dx \]
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Not integrable
Time = 18.23 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.92
\[\int \frac {x^{5}}{\left (a^{2} c \,x^{2}+c \right )^{\frac {5}{2}} \arctan \left (a x \right )^{2}}d x\]
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Not integrable
Time = 0.25 (sec) , antiderivative size = 62, normalized size of antiderivative = 2.58 \[ \int \frac {x^5}{\left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^2} \, dx=\int { \frac {x^{5}}{{\left (a^{2} c x^{2} + c\right )}^{\frac {5}{2}} \arctan \left (a x\right )^{2}} \,d x } \]
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Not integrable
Time = 7.15 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00 \[ \int \frac {x^5}{\left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^2} \, dx=\int \frac {x^{5}}{\left (c \left (a^{2} x^{2} + 1\right )\right )^{\frac {5}{2}} \operatorname {atan}^{2}{\left (a x \right )}}\, dx \]
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Not integrable
Time = 0.41 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00 \[ \int \frac {x^5}{\left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^2} \, dx=\int { \frac {x^{5}}{{\left (a^{2} c x^{2} + c\right )}^{\frac {5}{2}} \arctan \left (a x\right )^{2}} \,d x } \]
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Exception generated. \[ \int \frac {x^5}{\left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^2} \, dx=\text {Exception raised: TypeError} \]
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Not integrable
Time = 0.53 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00 \[ \int \frac {x^5}{\left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^2} \, dx=\int \frac {x^5}{{\mathrm {atan}\left (a\,x\right )}^2\,{\left (c\,a^2\,x^2+c\right )}^{5/2}} \,d x \]
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