Integrand size = 22, antiderivative size = 104 \[ \int \frac {x}{\left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^3} \, dx=-\frac {x}{2 a c \sqrt {c+a^2 c x^2} \arctan (a x)^2}-\frac {1}{2 a^2 c \sqrt {c+a^2 c x^2} \arctan (a x)}-\frac {\sqrt {1+a^2 x^2} \text {Si}(\arctan (a x))}{2 a^2 c \sqrt {c+a^2 c x^2}} \]
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Time = 0.19 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {5062, 5022, 5091, 5090, 3380} \[ \int \frac {x}{\left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^3} \, dx=-\frac {\sqrt {a^2 x^2+1} \text {Si}(\arctan (a x))}{2 a^2 c \sqrt {a^2 c x^2+c}}-\frac {x}{2 a c \arctan (a x)^2 \sqrt {a^2 c x^2+c}}-\frac {1}{2 a^2 c \arctan (a x) \sqrt {a^2 c x^2+c}} \]
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Rule 3380
Rule 5022
Rule 5062
Rule 5090
Rule 5091
Rubi steps \begin{align*} \text {integral}& = -\frac {x}{2 a c \sqrt {c+a^2 c x^2} \arctan (a x)^2}+\frac {\int \frac {1}{\left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^2} \, dx}{2 a} \\ & = -\frac {x}{2 a c \sqrt {c+a^2 c x^2} \arctan (a x)^2}-\frac {1}{2 a^2 c \sqrt {c+a^2 c x^2} \arctan (a x)}-\frac {1}{2} \int \frac {x}{\left (c+a^2 c x^2\right )^{3/2} \arctan (a x)} \, dx \\ & = -\frac {x}{2 a c \sqrt {c+a^2 c x^2} \arctan (a x)^2}-\frac {1}{2 a^2 c \sqrt {c+a^2 c x^2} \arctan (a x)}-\frac {\sqrt {1+a^2 x^2} \int \frac {x}{\left (1+a^2 x^2\right )^{3/2} \arctan (a x)} \, dx}{2 c \sqrt {c+a^2 c x^2}} \\ & = -\frac {x}{2 a c \sqrt {c+a^2 c x^2} \arctan (a x)^2}-\frac {1}{2 a^2 c \sqrt {c+a^2 c x^2} \arctan (a x)}-\frac {\sqrt {1+a^2 x^2} \text {Subst}\left (\int \frac {\sin (x)}{x} \, dx,x,\arctan (a x)\right )}{2 a^2 c \sqrt {c+a^2 c x^2}} \\ & = -\frac {x}{2 a c \sqrt {c+a^2 c x^2} \arctan (a x)^2}-\frac {1}{2 a^2 c \sqrt {c+a^2 c x^2} \arctan (a x)}-\frac {\sqrt {1+a^2 x^2} \text {Si}(\arctan (a x))}{2 a^2 c \sqrt {c+a^2 c x^2}} \\ \end{align*}
Time = 0.19 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.61 \[ \int \frac {x}{\left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^3} \, dx=-\frac {a x+\arctan (a x)+\sqrt {1+a^2 x^2} \arctan (a x)^2 \text {Si}(\arctan (a x))}{2 a^2 c \sqrt {c+a^2 c x^2} \arctan (a x)^2} \]
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Result contains complex when optimal does not.
Time = 9.25 (sec) , antiderivative size = 156, normalized size of antiderivative = 1.50
method | result | size |
default | \(-\frac {i \left (\arctan \left (a x \right )^{2} \operatorname {Ei}_{1}\left (-i \arctan \left (a x \right )\right ) a^{2} x^{2}-\arctan \left (a x \right )^{2} \operatorname {Ei}_{1}\left (i \arctan \left (a x \right )\right ) a^{2} x^{2}+\operatorname {Ei}_{1}\left (-i \arctan \left (a x \right )\right ) \arctan \left (a x \right )^{2}-\operatorname {Ei}_{1}\left (i \arctan \left (a x \right )\right ) \arctan \left (a x \right )^{2}-2 i \sqrt {a^{2} x^{2}+1}\, a x -2 i \sqrt {a^{2} x^{2}+1}\, \arctan \left (a x \right )\right ) \sqrt {c \left (a x -i\right ) \left (a x +i\right )}}{4 \left (a^{2} x^{2}+1\right )^{\frac {3}{2}} \arctan \left (a x \right )^{2} a^{2} c^{2}}\) | \(156\) |
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\[ \int \frac {x}{\left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^3} \, dx=\int { \frac {x}{{\left (a^{2} c x^{2} + c\right )}^{\frac {3}{2}} \arctan \left (a x\right )^{3}} \,d x } \]
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\[ \int \frac {x}{\left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^3} \, dx=\int \frac {x}{\left (c \left (a^{2} x^{2} + 1\right )\right )^{\frac {3}{2}} \operatorname {atan}^{3}{\left (a x \right )}}\, dx \]
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\[ \int \frac {x}{\left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^3} \, dx=\int { \frac {x}{{\left (a^{2} c x^{2} + c\right )}^{\frac {3}{2}} \arctan \left (a x\right )^{3}} \,d x } \]
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Exception generated. \[ \int \frac {x}{\left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^3} \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int \frac {x}{\left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^3} \, dx=\int \frac {x}{{\mathrm {atan}\left (a\,x\right )}^3\,{\left (c\,a^2\,x^2+c\right )}^{3/2}} \,d x \]
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