Integrand size = 22, antiderivative size = 22 \[ \int \frac {1}{x^3 \left (c+a^2 c x^2\right )^3 \arctan (a x)^3} \, dx=-\frac {1}{2 a c^3 x^3 \arctan (a x)^2}+\frac {a}{c^3 x \arctan (a x)^2}-\frac {a^3 x}{2 c^3 \left (1+a^2 x^2\right )^2 \arctan (a x)^2}-\frac {a^3 x}{c^3 \left (1+a^2 x^2\right ) \arctan (a x)^2}-\frac {2 a^2}{c^3 \left (1+a^2 x^2\right )^2 \arctan (a x)}+\frac {3 a^2}{2 c^3 \left (1+a^2 x^2\right ) \arctan (a x)}-\frac {a^2 \left (1-a^2 x^2\right )}{c^3 \left (1+a^2 x^2\right ) \arctan (a x)}-\frac {5 a^2 \text {Si}(2 \arctan (a x))}{2 c^3}-\frac {a^2 \text {Si}(4 \arctan (a x))}{c^3}-\frac {3 \text {Int}\left (\frac {1}{x^4 \arctan (a x)^2},x\right )}{2 a c^3}+\frac {a \text {Int}\left (\frac {1}{x^2 \arctan (a x)^2},x\right )}{c^3} \]
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Not integrable
Time = 0.90 (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 = {} \[ \int \frac {1}{x^3 \left (c+a^2 c x^2\right )^3 \arctan (a x)^3} \, dx=\int \frac {1}{x^3 \left (c+a^2 c x^2\right )^3 \arctan (a x)^3} \, dx \]
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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)^3} \, dx\right )+\frac {\int \frac {1}{x^3 \left (c+a^2 c x^2\right )^2 \arctan (a x)^3} \, dx}{c} \\ & = a^4 \int \frac {x}{\left (c+a^2 c x^2\right )^3 \arctan (a x)^3} \, dx+\frac {\int \frac {1}{x^3 \left (c+a^2 c x^2\right ) \arctan (a x)^3} \, dx}{c^2}-2 \frac {a^2 \int \frac {1}{x \left (c+a^2 c x^2\right )^2 \arctan (a x)^3} \, dx}{c} \\ & = -\frac {1}{2 a c^3 x^3 \arctan (a x)^2}-\frac {a^3 x}{2 c^3 \left (1+a^2 x^2\right )^2 \arctan (a x)^2}+\frac {1}{2} a^3 \int \frac {1}{\left (c+a^2 c x^2\right )^3 \arctan (a x)^2} \, dx-\frac {1}{2} \left (3 a^5\right ) \int \frac {x^2}{\left (c+a^2 c x^2\right )^3 \arctan (a x)^2} \, dx-\frac {3 \int \frac {1}{x^4 \arctan (a x)^2} \, dx}{2 a c^3}-2 \left (\frac {a^2 \int \frac {1}{x \left (c+a^2 c x^2\right ) \arctan (a x)^3} \, dx}{c^2}-\frac {a^4 \int \frac {x}{\left (c+a^2 c x^2\right )^2 \arctan (a x)^3} \, dx}{c}\right ) \\ & = -\frac {1}{2 a c^3 x^3 \arctan (a x)^2}-\frac {a^3 x}{2 c^3 \left (1+a^2 x^2\right )^2 \arctan (a x)^2}-\frac {a^2}{2 c^3 \left (1+a^2 x^2\right )^2 \arctan (a x)}+\frac {1}{2} \left (3 a^3\right ) \int \frac {1}{\left (c+a^2 c x^2\right )^3 \arctan (a x)^2} \, dx-\left (2 a^4\right ) \int \frac {x}{\left (c+a^2 c x^2\right )^3 \arctan (a x)} \, dx-\frac {3 \int \frac {1}{x^4 \arctan (a x)^2} \, dx}{2 a c^3}-\frac {\left (3 a^3\right ) \int \frac {1}{\left (c+a^2 c x^2\right )^2 \arctan (a x)^2} \, dx}{2 c}-2 \left (-\frac {a}{2 c^3 x \arctan (a x)^2}+\frac {a^3 x}{2 c^3 \left (1+a^2 x^2\right ) \arctan (a x)^2}+\frac {a^2 \left (1-a^2 x^2\right )}{2 c^3 \left (1+a^2 x^2\right ) \arctan (a x)}-\frac {a \int \frac {1}{x^2 \arctan (a x)^2} \, dx}{2 c^3}+\frac {\left (2 a^4\right ) \int \frac {x}{\left (c+a^2 c x^2\right )^2 \arctan (a x)} \, dx}{c}\right ) \\ & = -\frac {1}{2 a c^3 x^3 \arctan (a x)^2}-\frac {a^3 x}{2 c^3 \left (1+a^2 x^2\right )^2 \arctan (a x)^2}-\frac {2 a^2}{c^3 \left (1+a^2 x^2\right )^2 \arctan (a x)}+\frac {3 a^2}{2 c^3 \left (1+a^2 x^2\right ) \arctan (a x)}-\left (6 a^4\right ) \int \frac {x}{\left (c+a^2 c x^2\right )^3 \arctan (a x)} \, dx-\frac {3 \int \frac {1}{x^4 \arctan (a x)^2} \, dx}{2 a c^3}-2 \left (-\frac {a}{2 c^3 x \arctan (a x)^2}+\frac {a^3 x}{2 c^3 \left (1+a^2 x^2\right ) \arctan (a x)^2}+\frac {a^2 \left (1-a^2 x^2\right )}{2 c^3 \left (1+a^2 x^2\right ) \arctan (a x)}-\frac {a \int \frac {1}{x^2 \arctan (a x)^2} \, dx}{2 c^3}+\frac {\left (2 a^2\right ) \text {Subst}\left (\int \frac {\cos (x) \sin (x)}{x} \, dx,x,\arctan (a x)\right )}{c^3}\right )-\frac {\left (2 a^2\right ) \text {Subst}\left (\int \frac {\cos ^3(x) \sin (x)}{x} \, dx,x,\arctan (a x)\right )}{c^3}+\frac {\left (3 a^4\right ) \int \frac {x}{\left (c+a^2 c x^2\right )^2 \arctan (a x)} \, dx}{c} \\ & = -\frac {1}{2 a c^3 x^3 \arctan (a x)^2}-\frac {a^3 x}{2 c^3 \left (1+a^2 x^2\right )^2 \arctan (a x)^2}-\frac {2 a^2}{c^3 \left (1+a^2 x^2\right )^2 \arctan (a x)}+\frac {3 a^2}{2 c^3 \left (1+a^2 x^2\right ) \arctan (a x)}-\frac {3 \int \frac {1}{x^4 \arctan (a x)^2} \, dx}{2 a c^3}-2 \left (-\frac {a}{2 c^3 x \arctan (a x)^2}+\frac {a^3 x}{2 c^3 \left (1+a^2 x^2\right ) \arctan (a x)^2}+\frac {a^2 \left (1-a^2 x^2\right )}{2 c^3 \left (1+a^2 x^2\right ) \arctan (a x)}-\frac {a \int \frac {1}{x^2 \arctan (a x)^2} \, dx}{2 c^3}+\frac {\left (2 a^2\right ) \text {Subst}\left (\int \frac {\sin (2 x)}{2 x} \, dx,x,\arctan (a x)\right )}{c^3}\right )-\frac {\left (2 a^2\right ) \text {Subst}\left (\int \left (\frac {\sin (2 x)}{4 x}+\frac {\sin (4 x)}{8 x}\right ) \, dx,x,\arctan (a x)\right )}{c^3}+\frac {\left (3 a^2\right ) \text {Subst}\left (\int \frac {\cos (x) \sin (x)}{x} \, dx,x,\arctan (a x)\right )}{c^3}-\frac {\left (6 a^2\right ) \text {Subst}\left (\int \frac {\cos ^3(x) \sin (x)}{x} \, dx,x,\arctan (a x)\right )}{c^3} \\ & = -\frac {1}{2 a c^3 x^3 \arctan (a x)^2}-\frac {a^3 x}{2 c^3 \left (1+a^2 x^2\right )^2 \arctan (a x)^2}-\frac {2 a^2}{c^3 \left (1+a^2 x^2\right )^2 \arctan (a x)}+\frac {3 a^2}{2 c^3 \left (1+a^2 x^2\right ) \arctan (a x)}-\frac {3 \int \frac {1}{x^4 \arctan (a x)^2} \, dx}{2 a c^3}-\frac {a^2 \text {Subst}\left (\int \frac {\sin (4 x)}{x} \, dx,x,\arctan (a x)\right )}{4 c^3}-\frac {a^2 \text {Subst}\left (\int \frac {\sin (2 x)}{x} \, dx,x,\arctan (a x)\right )}{2 c^3}-2 \left (-\frac {a}{2 c^3 x \arctan (a x)^2}+\frac {a^3 x}{2 c^3 \left (1+a^2 x^2\right ) \arctan (a x)^2}+\frac {a^2 \left (1-a^2 x^2\right )}{2 c^3 \left (1+a^2 x^2\right ) \arctan (a x)}-\frac {a \int \frac {1}{x^2 \arctan (a x)^2} \, dx}{2 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 \frac {\sin (2 x)}{2 x} \, dx,x,\arctan (a x)\right )}{c^3}-\frac {\left (6 a^2\right ) \text {Subst}\left (\int \left (\frac {\sin (2 x)}{4 x}+\frac {\sin (4 x)}{8 x}\right ) \, dx,x,\arctan (a x)\right )}{c^3} \\ & = -\frac {1}{2 a c^3 x^3 \arctan (a x)^2}-\frac {a^3 x}{2 c^3 \left (1+a^2 x^2\right )^2 \arctan (a x)^2}-\frac {2 a^2}{c^3 \left (1+a^2 x^2\right )^2 \arctan (a x)}+\frac {3 a^2}{2 c^3 \left (1+a^2 x^2\right ) \arctan (a x)}-\frac {a^2 \text {Si}(2 \arctan (a x))}{2 c^3}-\frac {a^2 \text {Si}(4 \arctan (a x))}{4 c^3}-\frac {3 \int \frac {1}{x^4 \arctan (a x)^2} \, dx}{2 a c^3}-2 \left (-\frac {a}{2 c^3 x \arctan (a x)^2}+\frac {a^3 x}{2 c^3 \left (1+a^2 x^2\right ) \arctan (a x)^2}+\frac {a^2 \left (1-a^2 x^2\right )}{2 c^3 \left (1+a^2 x^2\right ) \arctan (a x)}+\frac {a^2 \text {Si}(2 \arctan (a x))}{c^3}-\frac {a \int \frac {1}{x^2 \arctan (a x)^2} \, dx}{2 c^3}\right )-\frac {\left (3 a^2\right ) \text {Subst}\left (\int \frac {\sin (4 x)}{x} \, dx,x,\arctan (a x)\right )}{4 c^3} \\ & = -\frac {1}{2 a c^3 x^3 \arctan (a x)^2}-\frac {a^3 x}{2 c^3 \left (1+a^2 x^2\right )^2 \arctan (a x)^2}-\frac {2 a^2}{c^3 \left (1+a^2 x^2\right )^2 \arctan (a x)}+\frac {3 a^2}{2 c^3 \left (1+a^2 x^2\right ) \arctan (a x)}-\frac {a^2 \text {Si}(2 \arctan (a x))}{2 c^3}-\frac {a^2 \text {Si}(4 \arctan (a x))}{c^3}-\frac {3 \int \frac {1}{x^4 \arctan (a x)^2} \, dx}{2 a c^3}-2 \left (-\frac {a}{2 c^3 x \arctan (a x)^2}+\frac {a^3 x}{2 c^3 \left (1+a^2 x^2\right ) \arctan (a x)^2}+\frac {a^2 \left (1-a^2 x^2\right )}{2 c^3 \left (1+a^2 x^2\right ) \arctan (a x)}+\frac {a^2 \text {Si}(2 \arctan (a x))}{c^3}-\frac {a \int \frac {1}{x^2 \arctan (a x)^2} \, dx}{2 c^3}\right ) \\ \end{align*}
Not integrable
Time = 3.92 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int \frac {1}{x^3 \left (c+a^2 c x^2\right )^3 \arctan (a x)^3} \, dx=\int \frac {1}{x^3 \left (c+a^2 c x^2\right )^3 \arctan (a x)^3} \, dx \]
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Not integrable
Time = 84.94 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00
\[\int \frac {1}{x^{3} \left (a^{2} c \,x^{2}+c \right )^{3} \arctan \left (a x \right )^{3}}d x\]
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Not integrable
Time = 0.24 (sec) , antiderivative size = 51, normalized size of antiderivative = 2.32 \[ \int \frac {1}{x^3 \left (c+a^2 c x^2\right )^3 \arctan (a x)^3} \, dx=\int { \frac {1}{{\left (a^{2} c x^{2} + c\right )}^{3} x^{3} \arctan \left (a x\right )^{3}} \,d x } \]
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Not integrable
Time = 2.09 (sec) , antiderivative size = 60, normalized size of antiderivative = 2.73 \[ \int \frac {1}{x^3 \left (c+a^2 c x^2\right )^3 \arctan (a x)^3} \, dx=\frac {\int \frac {1}{a^{6} x^{9} \operatorname {atan}^{3}{\left (a x \right )} + 3 a^{4} x^{7} \operatorname {atan}^{3}{\left (a x \right )} + 3 a^{2} x^{5} \operatorname {atan}^{3}{\left (a x \right )} + x^{3} \operatorname {atan}^{3}{\left (a x \right )}}\, dx}{c^{3}} \]
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Not integrable
Time = 0.40 (sec) , antiderivative size = 174, normalized size of antiderivative = 7.91 \[ \int \frac {1}{x^3 \left (c+a^2 c x^2\right )^3 \arctan (a x)^3} \, dx=\int { \frac {1}{{\left (a^{2} c x^{2} + c\right )}^{3} x^{3} \arctan \left (a x\right )^{3}} \,d x } \]
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Not integrable
Time = 184.69 (sec) , antiderivative size = 3, normalized size of antiderivative = 0.14 \[ \int \frac {1}{x^3 \left (c+a^2 c x^2\right )^3 \arctan (a x)^3} \, dx=\int { \frac {1}{{\left (a^{2} c x^{2} + c\right )}^{3} x^{3} \arctan \left (a x\right )^{3}} \,d x } \]
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Not integrable
Time = 0.44 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int \frac {1}{x^3 \left (c+a^2 c x^2\right )^3 \arctan (a x)^3} \, dx=\int \frac {1}{x^3\,{\mathrm {atan}\left (a\,x\right )}^3\,{\left (c\,a^2\,x^2+c\right )}^3} \,d x \]
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