Integrand size = 22, antiderivative size = 87 \[ \int \frac {x}{\left (c+a^2 c x^2\right )^{5/2} \arctan (a x)} \, dx=\frac {\sqrt {1+a^2 x^2} \text {Si}(\arctan (a x))}{4 a^2 c^2 \sqrt {c+a^2 c x^2}}+\frac {\sqrt {1+a^2 x^2} \text {Si}(3 \arctan (a x))}{4 a^2 c^2 \sqrt {c+a^2 c x^2}} \]
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Time = 0.14 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {5091, 5090, 4491, 3380} \[ \int \frac {x}{\left (c+a^2 c x^2\right )^{5/2} \arctan (a x)} \, dx=\frac {\sqrt {a^2 x^2+1} \text {Si}(\arctan (a x))}{4 a^2 c^2 \sqrt {a^2 c x^2+c}}+\frac {\sqrt {a^2 x^2+1} \text {Si}(3 \arctan (a x))}{4 a^2 c^2 \sqrt {a^2 c x^2+c}} \]
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Rule 3380
Rule 4491
Rule 5090
Rule 5091
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {1+a^2 x^2} \int \frac {x}{\left (1+a^2 x^2\right )^{5/2} \arctan (a x)} \, dx}{c^2 \sqrt {c+a^2 c x^2}} \\ & = \frac {\sqrt {1+a^2 x^2} \text {Subst}\left (\int \frac {\cos ^2(x) \sin (x)}{x} \, dx,x,\arctan (a x)\right )}{a^2 c^2 \sqrt {c+a^2 c x^2}} \\ & = \frac {\sqrt {1+a^2 x^2} \text {Subst}\left (\int \left (\frac {\sin (x)}{4 x}+\frac {\sin (3 x)}{4 x}\right ) \, dx,x,\arctan (a x)\right )}{a^2 c^2 \sqrt {c+a^2 c x^2}} \\ & = \frac {\sqrt {1+a^2 x^2} \text {Subst}\left (\int \frac {\sin (x)}{x} \, dx,x,\arctan (a x)\right )}{4 a^2 c^2 \sqrt {c+a^2 c x^2}}+\frac {\sqrt {1+a^2 x^2} \text {Subst}\left (\int \frac {\sin (3 x)}{x} \, dx,x,\arctan (a x)\right )}{4 a^2 c^2 \sqrt {c+a^2 c x^2}} \\ & = \frac {\sqrt {1+a^2 x^2} \text {Si}(\arctan (a x))}{4 a^2 c^2 \sqrt {c+a^2 c x^2}}+\frac {\sqrt {1+a^2 x^2} \text {Si}(3 \arctan (a x))}{4 a^2 c^2 \sqrt {c+a^2 c x^2}} \\ \end{align*}
Time = 0.14 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.59 \[ \int \frac {x}{\left (c+a^2 c x^2\right )^{5/2} \arctan (a x)} \, dx=\frac {\left (1+a^2 x^2\right )^{3/2} (\text {Si}(\arctan (a x))+\text {Si}(3 \arctan (a x)))}{4 a^2 c \left (c \left (1+a^2 x^2\right )\right )^{3/2}} \]
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 3.38 (sec) , antiderivative size = 125, normalized size of antiderivative = 1.44
method | result | size |
default | \(-\frac {\operatorname {csgn}\left (\arctan \left (a x \right )\right ) \pi \sqrt {c \left (a x -i\right ) \left (a x +i\right )}}{4 \sqrt {a^{2} x^{2}+1}\, a^{2} c^{3}}+\frac {\operatorname {Si}\left (3 \arctan \left (a x \right )\right ) \sqrt {c \left (a x -i\right ) \left (a x +i\right )}}{4 \sqrt {a^{2} x^{2}+1}\, a^{2} c^{3}}+\frac {\operatorname {Si}\left (\arctan \left (a x \right )\right ) \sqrt {c \left (a x -i\right ) \left (a x +i\right )}}{4 \sqrt {a^{2} x^{2}+1}\, a^{2} c^{3}}\) | \(125\) |
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\[ \int \frac {x}{\left (c+a^2 c x^2\right )^{5/2} \arctan (a x)} \, dx=\int { \frac {x}{{\left (a^{2} c x^{2} + c\right )}^{\frac {5}{2}} \arctan \left (a x\right )} \,d x } \]
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\[ \int \frac {x}{\left (c+a^2 c x^2\right )^{5/2} \arctan (a x)} \, dx=\int \frac {x}{\left (c \left (a^{2} x^{2} + 1\right )\right )^{\frac {5}{2}} \operatorname {atan}{\left (a x \right )}}\, dx \]
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\[ \int \frac {x}{\left (c+a^2 c x^2\right )^{5/2} \arctan (a x)} \, dx=\int { \frac {x}{{\left (a^{2} c x^{2} + c\right )}^{\frac {5}{2}} \arctan \left (a x\right )} \,d x } \]
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Exception generated. \[ \int \frac {x}{\left (c+a^2 c x^2\right )^{5/2} \arctan (a x)} \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int \frac {x}{\left (c+a^2 c x^2\right )^{5/2} \arctan (a x)} \, dx=\int \frac {x}{\mathrm {atan}\left (a\,x\right )\,{\left (c\,a^2\,x^2+c\right )}^{5/2}} \,d x \]
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