Integrand size = 19, antiderivative size = 65 \[ \int \frac {1}{\left (c+a^2 c x^2\right )^2 \arctan (a x)^3} \, dx=-\frac {1}{2 a c^2 \left (1+a^2 x^2\right ) \arctan (a x)^2}+\frac {x}{c^2 \left (1+a^2 x^2\right ) \arctan (a x)}-\frac {\operatorname {CosIntegral}(2 \arctan (a x))}{a c^2} \]
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Time = 0.18 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.316, Rules used = {5022, 5088, 5090, 3393, 3383, 5024} \[ \int \frac {1}{\left (c+a^2 c x^2\right )^2 \arctan (a x)^3} \, dx=\frac {x}{c^2 \left (a^2 x^2+1\right ) \arctan (a x)}-\frac {1}{2 a c^2 \left (a^2 x^2+1\right ) \arctan (a x)^2}-\frac {\operatorname {CosIntegral}(2 \arctan (a x))}{a c^2} \]
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Rule 3383
Rule 3393
Rule 5022
Rule 5024
Rule 5088
Rule 5090
Rubi steps \begin{align*} \text {integral}& = -\frac {1}{2 a c^2 \left (1+a^2 x^2\right ) \arctan (a x)^2}-a \int \frac {x}{\left (c+a^2 c x^2\right )^2 \arctan (a x)^2} \, dx \\ & = -\frac {1}{2 a c^2 \left (1+a^2 x^2\right ) \arctan (a x)^2}+\frac {x}{c^2 \left (1+a^2 x^2\right ) \arctan (a x)}+a^2 \int \frac {x^2}{\left (c+a^2 c x^2\right )^2 \arctan (a x)} \, dx-\int \frac {1}{\left (c+a^2 c x^2\right )^2 \arctan (a x)} \, dx \\ & = -\frac {1}{2 a c^2 \left (1+a^2 x^2\right ) \arctan (a x)^2}+\frac {x}{c^2 \left (1+a^2 x^2\right ) \arctan (a x)}-\frac {\text {Subst}\left (\int \frac {\cos ^2(x)}{x} \, dx,x,\arctan (a x)\right )}{a c^2}+\frac {\text {Subst}\left (\int \frac {\sin ^2(x)}{x} \, dx,x,\arctan (a x)\right )}{a c^2} \\ & = -\frac {1}{2 a c^2 \left (1+a^2 x^2\right ) \arctan (a x)^2}+\frac {x}{c^2 \left (1+a^2 x^2\right ) \arctan (a x)}+\frac {\text {Subst}\left (\int \left (\frac {1}{2 x}-\frac {\cos (2 x)}{2 x}\right ) \, dx,x,\arctan (a x)\right )}{a c^2}-\frac {\text {Subst}\left (\int \left (\frac {1}{2 x}+\frac {\cos (2 x)}{2 x}\right ) \, dx,x,\arctan (a x)\right )}{a c^2} \\ & = -\frac {1}{2 a c^2 \left (1+a^2 x^2\right ) \arctan (a x)^2}+\frac {x}{c^2 \left (1+a^2 x^2\right ) \arctan (a x)}-2 \frac {\text {Subst}\left (\int \frac {\cos (2 x)}{x} \, dx,x,\arctan (a x)\right )}{2 a c^2} \\ & = -\frac {1}{2 a c^2 \left (1+a^2 x^2\right ) \arctan (a x)^2}+\frac {x}{c^2 \left (1+a^2 x^2\right ) \arctan (a x)}-\frac {\operatorname {CosIntegral}(2 \arctan (a x))}{a c^2} \\ \end{align*}
Time = 0.07 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.89 \[ \int \frac {1}{\left (c+a^2 c x^2\right )^2 \arctan (a x)^3} \, dx=\frac {-1+2 a x \arctan (a x)-2 \left (1+a^2 x^2\right ) \arctan (a x)^2 \operatorname {CosIntegral}(2 \arctan (a x))}{2 c^2 \left (a+a^3 x^2\right ) \arctan (a x)^2} \]
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Time = 8.88 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.80
method | result | size |
derivativedivides | \(-\frac {4 \,\operatorname {Ci}\left (2 \arctan \left (a x \right )\right ) \arctan \left (a x \right )^{2}-2 \sin \left (2 \arctan \left (a x \right )\right ) \arctan \left (a x \right )+\cos \left (2 \arctan \left (a x \right )\right )+1}{4 a \,c^{2} \arctan \left (a x \right )^{2}}\) | \(52\) |
default | \(-\frac {4 \,\operatorname {Ci}\left (2 \arctan \left (a x \right )\right ) \arctan \left (a x \right )^{2}-2 \sin \left (2 \arctan \left (a x \right )\right ) \arctan \left (a x \right )+\cos \left (2 \arctan \left (a x \right )\right )+1}{4 a \,c^{2} \arctan \left (a x \right )^{2}}\) | \(52\) |
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Result contains complex when optimal does not.
Time = 0.25 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.88 \[ \int \frac {1}{\left (c+a^2 c x^2\right )^2 \arctan (a x)^3} \, dx=-\frac {{\left (a^{2} x^{2} + 1\right )} \arctan \left (a x\right )^{2} \operatorname {log\_integral}\left (-\frac {a^{2} x^{2} + 2 i \, a x - 1}{a^{2} x^{2} + 1}\right ) + {\left (a^{2} x^{2} + 1\right )} \arctan \left (a x\right )^{2} \operatorname {log\_integral}\left (-\frac {a^{2} x^{2} - 2 i \, a x - 1}{a^{2} x^{2} + 1}\right ) - 2 \, a x \arctan \left (a x\right ) + 1}{2 \, {\left (a^{3} c^{2} x^{2} + a c^{2}\right )} \arctan \left (a x\right )^{2}} \]
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\[ \int \frac {1}{\left (c+a^2 c x^2\right )^2 \arctan (a x)^3} \, dx=\frac {\int \frac {1}{a^{4} x^{4} \operatorname {atan}^{3}{\left (a x \right )} + 2 a^{2} x^{2} \operatorname {atan}^{3}{\left (a x \right )} + \operatorname {atan}^{3}{\left (a x \right )}}\, dx}{c^{2}} \]
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\[ \int \frac {1}{\left (c+a^2 c x^2\right )^2 \arctan (a x)^3} \, dx=\int { \frac {1}{{\left (a^{2} c x^{2} + c\right )}^{2} \arctan \left (a x\right )^{3}} \,d x } \]
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\[ \int \frac {1}{\left (c+a^2 c x^2\right )^2 \arctan (a x)^3} \, dx=\int { \frac {1}{{\left (a^{2} c x^{2} + c\right )}^{2} \arctan \left (a x\right )^{3}} \,d x } \]
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Timed out. \[ \int \frac {1}{\left (c+a^2 c x^2\right )^2 \arctan (a x)^3} \, dx=\int \frac {1}{{\mathrm {atan}\left (a\,x\right )}^3\,{\left (c\,a^2\,x^2+c\right )}^2} \,d x \]
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