Integrand size = 22, antiderivative size = 107 \[ \int \frac {x^3 \arctan (a x)}{\left (c+a^2 c x^2\right )^{3/2}} \, dx=-\frac {x}{a^3 c \sqrt {c+a^2 c x^2}}+\frac {\arctan (a x)}{a^4 c \sqrt {c+a^2 c x^2}}+\frac {\sqrt {c+a^2 c x^2} \arctan (a x)}{a^4 c^2}-\frac {\text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {c+a^2 c x^2}}\right )}{a^4 c^{3/2}} \]
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Time = 0.14 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {5084, 5050, 223, 212, 197} \[ \int \frac {x^3 \arctan (a x)}{\left (c+a^2 c x^2\right )^{3/2}} \, dx=\frac {\arctan (a x) \sqrt {a^2 c x^2+c}}{a^4 c^2}+\frac {\arctan (a x)}{a^4 c \sqrt {a^2 c x^2+c}}-\frac {\text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{a^4 c^{3/2}}-\frac {x}{a^3 c \sqrt {a^2 c x^2+c}} \]
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Rule 197
Rule 212
Rule 223
Rule 5050
Rule 5084
Rubi steps \begin{align*} \text {integral}& = -\frac {\int \frac {x \arctan (a x)}{\left (c+a^2 c x^2\right )^{3/2}} \, dx}{a^2}+\frac {\int \frac {x \arctan (a x)}{\sqrt {c+a^2 c x^2}} \, dx}{a^2 c} \\ & = \frac {\arctan (a x)}{a^4 c \sqrt {c+a^2 c x^2}}+\frac {\sqrt {c+a^2 c x^2} \arctan (a x)}{a^4 c^2}-\frac {\int \frac {1}{\left (c+a^2 c x^2\right )^{3/2}} \, dx}{a^3}-\frac {\int \frac {1}{\sqrt {c+a^2 c x^2}} \, dx}{a^3 c} \\ & = -\frac {x}{a^3 c \sqrt {c+a^2 c x^2}}+\frac {\arctan (a x)}{a^4 c \sqrt {c+a^2 c x^2}}+\frac {\sqrt {c+a^2 c x^2} \arctan (a x)}{a^4 c^2}-\frac {\text {Subst}\left (\int \frac {1}{1-a^2 c x^2} \, dx,x,\frac {x}{\sqrt {c+a^2 c x^2}}\right )}{a^3 c} \\ & = -\frac {x}{a^3 c \sqrt {c+a^2 c x^2}}+\frac {\arctan (a x)}{a^4 c \sqrt {c+a^2 c x^2}}+\frac {\sqrt {c+a^2 c x^2} \arctan (a x)}{a^4 c^2}-\frac {\text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {c+a^2 c x^2}}\right )}{a^4 c^{3/2}} \\ \end{align*}
Time = 0.12 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.00 \[ \int \frac {x^3 \arctan (a x)}{\left (c+a^2 c x^2\right )^{3/2}} \, dx=\frac {-a x \sqrt {c+a^2 c x^2}+\left (2+a^2 x^2\right ) \sqrt {c+a^2 c x^2} \arctan (a x)-\sqrt {c} \left (1+a^2 x^2\right ) \log \left (a c x+\sqrt {c} \sqrt {c+a^2 c x^2}\right )}{a^4 c^2 \left (1+a^2 x^2\right )} \]
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Result contains complex when optimal does not.
Time = 0.41 (sec) , antiderivative size = 218, normalized size of antiderivative = 2.04
method | result | size |
default | \(\frac {\left (\arctan \left (a x \right ) \sqrt {a^{2} x^{2}+1}\, a^{2} x^{2}-\ln \left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}+i\right ) a^{2} x^{2}+\ln \left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}-i\right ) a^{2} x^{2}-\sqrt {a^{2} x^{2}+1}\, a x +2 \arctan \left (a x \right ) \sqrt {a^{2} x^{2}+1}-\ln \left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}+i\right )+\ln \left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}-i\right )\right ) \sqrt {a^{2} x^{2}+1}\, \sqrt {c \left (a x -i\right ) \left (a x +i\right )}}{a^{4} c^{2} \left (a^{4} x^{4}+2 a^{2} x^{2}+1\right )}\) | \(218\) |
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Time = 0.27 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.95 \[ \int \frac {x^3 \arctan (a x)}{\left (c+a^2 c x^2\right )^{3/2}} \, dx=\frac {{\left (a^{2} x^{2} + 1\right )} \sqrt {c} \log \left (-2 \, a^{2} c x^{2} + 2 \, \sqrt {a^{2} c x^{2} + c} a \sqrt {c} x - c\right ) - 2 \, \sqrt {a^{2} c x^{2} + c} {\left (a x - {\left (a^{2} x^{2} + 2\right )} \arctan \left (a x\right )\right )}}{2 \, {\left (a^{6} c^{2} x^{2} + a^{4} c^{2}\right )}} \]
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\[ \int \frac {x^3 \arctan (a x)}{\left (c+a^2 c x^2\right )^{3/2}} \, dx=\int \frac {x^{3} \operatorname {atan}{\left (a x \right )}}{\left (c \left (a^{2} x^{2} + 1\right )\right )^{\frac {3}{2}}}\, dx \]
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\[ \int \frac {x^3 \arctan (a x)}{\left (c+a^2 c x^2\right )^{3/2}} \, dx=\int { \frac {x^{3} \arctan \left (a x\right )}{{\left (a^{2} c x^{2} + c\right )}^{\frac {3}{2}}} \,d x } \]
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Exception generated. \[ \int \frac {x^3 \arctan (a x)}{\left (c+a^2 c x^2\right )^{3/2}} \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int \frac {x^3 \arctan (a x)}{\left (c+a^2 c x^2\right )^{3/2}} \, dx=\int \frac {x^3\,\mathrm {atan}\left (a\,x\right )}{{\left (c\,a^2\,x^2+c\right )}^{3/2}} \,d x \]
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