Integrand size = 21, antiderivative size = 39 \[ \int \frac {1}{\left (c+a^2 c x^2\right )^{3/2} \arctan (a x)} \, dx=\frac {\sqrt {1+a^2 x^2} \operatorname {CosIntegral}(\arctan (a x))}{a c \sqrt {c+a^2 c x^2}} \]
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Time = 0.06 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {5025, 5024, 3383} \[ \int \frac {1}{\left (c+a^2 c x^2\right )^{3/2} \arctan (a x)} \, dx=\frac {\sqrt {a^2 x^2+1} \operatorname {CosIntegral}(\arctan (a x))}{a c \sqrt {a^2 c x^2+c}} \]
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Rule 3383
Rule 5024
Rule 5025
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {1+a^2 x^2} \int \frac {1}{\left (1+a^2 x^2\right )^{3/2} \arctan (a x)} \, dx}{c \sqrt {c+a^2 c x^2}} \\ & = \frac {\sqrt {1+a^2 x^2} \text {Subst}\left (\int \frac {\cos (x)}{x} \, dx,x,\arctan (a x)\right )}{a c \sqrt {c+a^2 c x^2}} \\ & = \frac {\sqrt {1+a^2 x^2} \operatorname {CosIntegral}(\arctan (a x))}{a c \sqrt {c+a^2 c x^2}} \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.03 \[ \int \frac {1}{\left (c+a^2 c x^2\right )^{3/2} \arctan (a x)} \, dx=\frac {\sqrt {1+a^2 x^2} \operatorname {CosIntegral}(\arctan (a x))}{a c \sqrt {c \left (1+a^2 x^2\right )}} \]
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 3.95 (sec) , antiderivative size = 136, normalized size of antiderivative = 3.49
method | result | size |
default | \(-\frac {i \operatorname {csgn}\left (\arctan \left (a x \right )\right ) \operatorname {csgn}\left (i \arctan \left (a x \right )\right ) \pi \sqrt {c \left (a x -i\right ) \left (a x +i\right )}}{2 \sqrt {a^{2} x^{2}+1}\, a \,c^{2}}+\frac {i \operatorname {csgn}\left (i \arctan \left (a x \right )\right ) \pi \sqrt {c \left (a x -i\right ) \left (a x +i\right )}}{2 \sqrt {a^{2} x^{2}+1}\, a \,c^{2}}+\frac {\operatorname {Ci}\left (\arctan \left (a x \right )\right ) \sqrt {c \left (a x -i\right ) \left (a x +i\right )}}{\sqrt {a^{2} x^{2}+1}\, a \,c^{2}}\) | \(136\) |
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\[ \int \frac {1}{\left (c+a^2 c x^2\right )^{3/2} \arctan (a x)} \, dx=\int { \frac {1}{{\left (a^{2} c x^{2} + c\right )}^{\frac {3}{2}} \arctan \left (a x\right )} \,d x } \]
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\[ \int \frac {1}{\left (c+a^2 c x^2\right )^{3/2} \arctan (a x)} \, dx=\int \frac {1}{\left (c \left (a^{2} x^{2} + 1\right )\right )^{\frac {3}{2}} \operatorname {atan}{\left (a x \right )}}\, dx \]
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\[ \int \frac {1}{\left (c+a^2 c x^2\right )^{3/2} \arctan (a x)} \, dx=\int { \frac {1}{{\left (a^{2} c x^{2} + c\right )}^{\frac {3}{2}} \arctan \left (a x\right )} \,d x } \]
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\[ \int \frac {1}{\left (c+a^2 c x^2\right )^{3/2} \arctan (a x)} \, dx=\int { \frac {1}{{\left (a^{2} c x^{2} + c\right )}^{\frac {3}{2}} \arctan \left (a x\right )} \,d x } \]
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Timed out. \[ \int \frac {1}{\left (c+a^2 c x^2\right )^{3/2} \arctan (a x)} \, dx=\int \frac {1}{\mathrm {atan}\left (a\,x\right )\,{\left (c\,a^2\,x^2+c\right )}^{3/2}} \,d x \]
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