Integrand size = 22, antiderivative size = 50 \[ \int \frac {x^4}{\left (c+a^2 c x^2\right )^3 \arctan (a x)} \, dx=-\frac {\operatorname {CosIntegral}(2 \arctan (a x))}{2 a^5 c^3}+\frac {\operatorname {CosIntegral}(4 \arctan (a x))}{8 a^5 c^3}+\frac {3 \log (\arctan (a x))}{8 a^5 c^3} \]
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Time = 0.09 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {5090, 3393, 3383} \[ \int \frac {x^4}{\left (c+a^2 c x^2\right )^3 \arctan (a x)} \, dx=-\frac {\operatorname {CosIntegral}(2 \arctan (a x))}{2 a^5 c^3}+\frac {\operatorname {CosIntegral}(4 \arctan (a x))}{8 a^5 c^3}+\frac {3 \log (\arctan (a x))}{8 a^5 c^3} \]
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Rule 3383
Rule 3393
Rule 5090
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {\sin ^4(x)}{x} \, dx,x,\arctan (a x)\right )}{a^5 c^3} \\ & = \frac {\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 )}{a^5 c^3} \\ & = \frac {3 \log (\arctan (a x))}{8 a^5 c^3}+\frac {\text {Subst}\left (\int \frac {\cos (4 x)}{x} \, dx,x,\arctan (a x)\right )}{8 a^5 c^3}-\frac {\text {Subst}\left (\int \frac {\cos (2 x)}{x} \, dx,x,\arctan (a x)\right )}{2 a^5 c^3} \\ & = -\frac {\operatorname {CosIntegral}(2 \arctan (a x))}{2 a^5 c^3}+\frac {\operatorname {CosIntegral}(4 \arctan (a x))}{8 a^5 c^3}+\frac {3 \log (\arctan (a x))}{8 a^5 c^3} \\ \end{align*}
Time = 0.11 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.68 \[ \int \frac {x^4}{\left (c+a^2 c x^2\right )^3 \arctan (a x)} \, dx=\frac {-4 \operatorname {CosIntegral}(2 \arctan (a x))+\operatorname {CosIntegral}(4 \arctan (a x))+3 \log (\arctan (a x))}{8 a^5 c^3} \]
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Time = 3.51 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.70
method | result | size |
derivativedivides | \(-\frac {-3 \ln \left (\arctan \left (a x \right )\right )+4 \,\operatorname {Ci}\left (2 \arctan \left (a x \right )\right )-\operatorname {Ci}\left (4 \arctan \left (a x \right )\right )}{8 a^{5} c^{3}}\) | \(35\) |
default | \(-\frac {-3 \ln \left (\arctan \left (a x \right )\right )+4 \,\operatorname {Ci}\left (2 \arctan \left (a x \right )\right )-\operatorname {Ci}\left (4 \arctan \left (a x \right )\right )}{8 a^{5} c^{3}}\) | \(35\) |
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Result contains complex when optimal does not.
Time = 0.25 (sec) , antiderivative size = 174, normalized size of antiderivative = 3.48 \[ \int \frac {x^4}{\left (c+a^2 c x^2\right )^3 \arctan (a x)} \, dx=\frac {6 \, \log \left (\arctan \left (a x\right )\right ) + \operatorname {log\_integral}\left (\frac {a^{4} x^{4} + 4 i \, a^{3} x^{3} - 6 \, a^{2} x^{2} - 4 i \, a x + 1}{a^{4} x^{4} + 2 \, a^{2} x^{2} + 1}\right ) + \operatorname {log\_integral}\left (\frac {a^{4} x^{4} - 4 i \, a^{3} x^{3} - 6 \, a^{2} x^{2} + 4 i \, a x + 1}{a^{4} x^{4} + 2 \, a^{2} x^{2} + 1}\right ) - 4 \, \operatorname {log\_integral}\left (-\frac {a^{2} x^{2} + 2 i \, a x - 1}{a^{2} x^{2} + 1}\right ) - 4 \, \operatorname {log\_integral}\left (-\frac {a^{2} x^{2} - 2 i \, a x - 1}{a^{2} x^{2} + 1}\right )}{16 \, a^{5} c^{3}} \]
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\[ \int \frac {x^4}{\left (c+a^2 c x^2\right )^3 \arctan (a x)} \, dx=\frac {\int \frac {x^{4}}{a^{6} x^{6} \operatorname {atan}{\left (a x \right )} + 3 a^{4} x^{4} \operatorname {atan}{\left (a x \right )} + 3 a^{2} x^{2} \operatorname {atan}{\left (a x \right )} + \operatorname {atan}{\left (a x \right )}}\, dx}{c^{3}} \]
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\[ \int \frac {x^4}{\left (c+a^2 c x^2\right )^3 \arctan (a x)} \, dx=\int { \frac {x^{4}}{{\left (a^{2} c x^{2} + c\right )}^{3} \arctan \left (a x\right )} \,d x } \]
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\[ \int \frac {x^4}{\left (c+a^2 c x^2\right )^3 \arctan (a x)} \, dx=\int { \frac {x^{4}}{{\left (a^{2} c x^{2} + c\right )}^{3} \arctan \left (a x\right )} \,d x } \]
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Timed out. \[ \int \frac {x^4}{\left (c+a^2 c x^2\right )^3 \arctan (a x)} \, dx=\int \frac {x^4}{\mathrm {atan}\left (a\,x\right )\,{\left (c\,a^2\,x^2+c\right )}^3} \,d x \]
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