Integrand size = 22, antiderivative size = 22 \[ \int \frac {1}{x^3 \left (c+a^2 c x^2\right )^3 \arctan (a x)^2} \, dx=-\frac {1}{a c^3 x^3 \arctan (a x)}+\frac {2 a}{c^3 x \arctan (a x)}-\frac {a^3 x}{c^3 \left (1+a^2 x^2\right )^2 \arctan (a x)}-\frac {2 a^3 x}{c^3 \left (1+a^2 x^2\right ) \arctan (a x)}+\frac {5 a^2 \operatorname {CosIntegral}(2 \arctan (a x))}{2 c^3}+\frac {a^2 \operatorname {CosIntegral}(4 \arctan (a x))}{2 c^3}-\frac {3 \text {Int}\left (\frac {1}{x^4 \arctan (a x)},x\right )}{a c^3}+\frac {2 a \text {Int}\left (\frac {1}{x^2 \arctan (a x)},x\right )}{c^3} \]
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Not integrable
Time = 0.88 (sec) , antiderivative size = 22, 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 = {} \[ \int \frac {1}{x^3 \left (c+a^2 c x^2\right )^3 \arctan (a x)^2} \, dx=\int \frac {1}{x^3 \left (c+a^2 c x^2\right )^3 \arctan (a x)^2} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = -\left (a^2 \int \frac {1}{x \left (c+a^2 c x^2\right )^3 \arctan (a x)^2} \, dx\right )+\frac {\int \frac {1}{x^3 \left (c+a^2 c x^2\right )^2 \arctan (a x)^2} \, dx}{c} \\ & = a^4 \int \frac {x}{\left (c+a^2 c x^2\right )^3 \arctan (a x)^2} \, dx+\frac {\int \frac {1}{x^3 \left (c+a^2 c x^2\right ) \arctan (a x)^2} \, dx}{c^2}-2 \frac {a^2 \int \frac {1}{x \left (c+a^2 c x^2\right )^2 \arctan (a x)^2} \, dx}{c} \\ & = -\frac {1}{a c^3 x^3 \arctan (a x)}-\frac {a^3 x}{c^3 \left (1+a^2 x^2\right )^2 \arctan (a x)}+a^3 \int \frac {1}{\left (c+a^2 c x^2\right )^3 \arctan (a x)} \, dx-\left (3 a^5\right ) \int \frac {x^2}{\left (c+a^2 c x^2\right )^3 \arctan (a x)} \, dx-\frac {3 \int \frac {1}{x^4 \arctan (a x)} \, dx}{a c^3}-2 \left (\frac {a^2 \int \frac {1}{x \left (c+a^2 c x^2\right ) \arctan (a x)^2} \, dx}{c^2}-\frac {a^4 \int \frac {x}{\left (c+a^2 c x^2\right )^2 \arctan (a x)^2} \, dx}{c}\right ) \\ & = -\frac {1}{a c^3 x^3 \arctan (a x)}-\frac {a^3 x}{c^3 \left (1+a^2 x^2\right )^2 \arctan (a x)}-\frac {3 \int \frac {1}{x^4 \arctan (a x)} \, dx}{a c^3}+\frac {a^2 \text {Subst}\left (\int \frac {\cos ^4(x)}{x} \, dx,x,\arctan (a x)\right )}{c^3}-\frac {\left (3 a^2\right ) \text {Subst}\left (\int \frac {\cos ^2(x) \sin ^2(x)}{x} \, dx,x,\arctan (a x)\right )}{c^3}-2 \left (-\frac {a}{c^3 x \arctan (a x)}+\frac {a^3 x}{c^3 \left (1+a^2 x^2\right ) \arctan (a x)}-\frac {a \int \frac {1}{x^2 \arctan (a x)} \, dx}{c^3}-\frac {a^3 \int \frac {1}{\left (c+a^2 c x^2\right )^2 \arctan (a x)} \, dx}{c}+\frac {a^5 \int \frac {x^2}{\left (c+a^2 c x^2\right )^2 \arctan (a x)} \, dx}{c}\right ) \\ & = -\frac {1}{a c^3 x^3 \arctan (a x)}-\frac {a^3 x}{c^3 \left (1+a^2 x^2\right )^2 \arctan (a x)}-\frac {3 \int \frac {1}{x^4 \arctan (a x)} \, dx}{a c^3}+\frac {a^2 \text {Subst}\left (\int \left (\frac {3}{8 x}+\frac {\cos (2 x)}{2 x}+\frac {\cos (4 x)}{8 x}\right ) \, dx,x,\arctan (a x)\right )}{c^3}-2 \left (-\frac {a}{c^3 x \arctan (a x)}+\frac {a^3 x}{c^3 \left (1+a^2 x^2\right ) \arctan (a x)}-\frac {a \int \frac {1}{x^2 \arctan (a x)} \, dx}{c^3}-\frac {a^2 \text {Subst}\left (\int \frac {\cos ^2(x)}{x} \, dx,x,\arctan (a x)\right )}{c^3}+\frac {a^2 \text {Subst}\left (\int \frac {\sin ^2(x)}{x} \, dx,x,\arctan (a x)\right )}{c^3}\right )-\frac {\left (3 a^2\right ) \text {Subst}\left (\int \left (\frac {1}{8 x}-\frac {\cos (4 x)}{8 x}\right ) \, dx,x,\arctan (a x)\right )}{c^3} \\ & = -\frac {1}{a c^3 x^3 \arctan (a x)}-\frac {a^3 x}{c^3 \left (1+a^2 x^2\right )^2 \arctan (a x)}-\frac {3 \int \frac {1}{x^4 \arctan (a x)} \, dx}{a c^3}+\frac {a^2 \text {Subst}\left (\int \frac {\cos (4 x)}{x} \, dx,x,\arctan (a x)\right )}{8 c^3}+\frac {\left (3 a^2\right ) \text {Subst}\left (\int \frac {\cos (4 x)}{x} \, dx,x,\arctan (a x)\right )}{8 c^3}+\frac {a^2 \text {Subst}\left (\int \frac {\cos (2 x)}{x} \, dx,x,\arctan (a x)\right )}{2 c^3}-2 \left (-\frac {a}{c^3 x \arctan (a x)}+\frac {a^3 x}{c^3 \left (1+a^2 x^2\right ) \arctan (a x)}-\frac {a \int \frac {1}{x^2 \arctan (a x)} \, dx}{c^3}+\frac {a^2 \text {Subst}\left (\int \left (\frac {1}{2 x}-\frac {\cos (2 x)}{2 x}\right ) \, dx,x,\arctan (a x)\right )}{c^3}-\frac {a^2 \text {Subst}\left (\int \left (\frac {1}{2 x}+\frac {\cos (2 x)}{2 x}\right ) \, dx,x,\arctan (a x)\right )}{c^3}\right ) \\ & = -\frac {1}{a c^3 x^3 \arctan (a x)}-\frac {a^3 x}{c^3 \left (1+a^2 x^2\right )^2 \arctan (a x)}+\frac {a^2 \operatorname {CosIntegral}(2 \arctan (a x))}{2 c^3}+\frac {a^2 \operatorname {CosIntegral}(4 \arctan (a x))}{2 c^3}-\frac {3 \int \frac {1}{x^4 \arctan (a x)} \, dx}{a c^3}-2 \left (-\frac {a}{c^3 x \arctan (a x)}+\frac {a^3 x}{c^3 \left (1+a^2 x^2\right ) \arctan (a x)}-\frac {a \int \frac {1}{x^2 \arctan (a x)} \, dx}{c^3}-2 \frac {a^2 \text {Subst}\left (\int \frac {\cos (2 x)}{x} \, dx,x,\arctan (a x)\right )}{2 c^3}\right ) \\ & = -\frac {1}{a c^3 x^3 \arctan (a x)}-\frac {a^3 x}{c^3 \left (1+a^2 x^2\right )^2 \arctan (a x)}+\frac {a^2 \operatorname {CosIntegral}(2 \arctan (a x))}{2 c^3}+\frac {a^2 \operatorname {CosIntegral}(4 \arctan (a x))}{2 c^3}-\frac {3 \int \frac {1}{x^4 \arctan (a x)} \, dx}{a c^3}-2 \left (-\frac {a}{c^3 x \arctan (a x)}+\frac {a^3 x}{c^3 \left (1+a^2 x^2\right ) \arctan (a x)}-\frac {a^2 \operatorname {CosIntegral}(2 \arctan (a x))}{c^3}-\frac {a \int \frac {1}{x^2 \arctan (a x)} \, dx}{c^3}\right ) \\ \end{align*}
Not integrable
Time = 2.70 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int \frac {1}{x^3 \left (c+a^2 c x^2\right )^3 \arctan (a x)^2} \, dx=\int \frac {1}{x^3 \left (c+a^2 c x^2\right )^3 \arctan (a x)^2} \, dx \]
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Not integrable
Time = 20.69 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00
\[\int \frac {1}{x^{3} \left (a^{2} c \,x^{2}+c \right )^{3} \arctan \left (a x \right )^{2}}d x\]
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Not integrable
Time = 0.25 (sec) , antiderivative size = 51, normalized size of antiderivative = 2.32 \[ \int \frac {1}{x^3 \left (c+a^2 c x^2\right )^3 \arctan (a x)^2} \, dx=\int { \frac {1}{{\left (a^{2} c x^{2} + c\right )}^{3} x^{3} \arctan \left (a x\right )^{2}} \,d x } \]
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Not integrable
Time = 1.76 (sec) , antiderivative size = 60, normalized size of antiderivative = 2.73 \[ \int \frac {1}{x^3 \left (c+a^2 c x^2\right )^3 \arctan (a x)^2} \, dx=\frac {\int \frac {1}{a^{6} x^{9} \operatorname {atan}^{2}{\left (a x \right )} + 3 a^{4} x^{7} \operatorname {atan}^{2}{\left (a x \right )} + 3 a^{2} x^{5} \operatorname {atan}^{2}{\left (a x \right )} + x^{3} \operatorname {atan}^{2}{\left (a x \right )}}\, dx}{c^{3}} \]
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Not integrable
Time = 0.31 (sec) , antiderivative size = 139, normalized size of antiderivative = 6.32 \[ \int \frac {1}{x^3 \left (c+a^2 c x^2\right )^3 \arctan (a x)^2} \, dx=\int { \frac {1}{{\left (a^{2} c x^{2} + c\right )}^{3} x^{3} \arctan \left (a x\right )^{2}} \,d x } \]
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Not integrable
Time = 127.82 (sec) , antiderivative size = 3, normalized size of antiderivative = 0.14 \[ \int \frac {1}{x^3 \left (c+a^2 c x^2\right )^3 \arctan (a x)^2} \, dx=\int { \frac {1}{{\left (a^{2} c x^{2} + c\right )}^{3} x^{3} \arctan \left (a x\right )^{2}} \,d x } \]
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Not integrable
Time = 0.42 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int \frac {1}{x^3 \left (c+a^2 c x^2\right )^3 \arctan (a x)^2} \, dx=\int \frac {1}{x^3\,{\mathrm {atan}\left (a\,x\right )}^2\,{\left (c\,a^2\,x^2+c\right )}^3} \,d x \]
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