Integrand size = 24, antiderivative size = 87 \[ \int \frac {x^2}{\left (c+a^2 c x^2\right )^{5/2} \arctan (a x)} \, dx=\frac {\sqrt {1+a^2 x^2} \operatorname {CosIntegral}(\arctan (a x))}{4 a^3 c^2 \sqrt {c+a^2 c x^2}}-\frac {\sqrt {1+a^2 x^2} \operatorname {CosIntegral}(3 \arctan (a x))}{4 a^3 c^2 \sqrt {c+a^2 c x^2}} \]
[Out]
Time = 0.20 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {5091, 5090, 4491, 3383} \[ \int \frac {x^2}{\left (c+a^2 c x^2\right )^{5/2} \arctan (a x)} \, dx=\frac {\sqrt {a^2 x^2+1} \operatorname {CosIntegral}(\arctan (a x))}{4 a^3 c^2 \sqrt {a^2 c x^2+c}}-\frac {\sqrt {a^2 x^2+1} \operatorname {CosIntegral}(3 \arctan (a x))}{4 a^3 c^2 \sqrt {a^2 c x^2+c}} \]
[In]
[Out]
Rule 3383
Rule 4491
Rule 5090
Rule 5091
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {1+a^2 x^2} \int \frac {x^2}{\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 (x) \sin ^2(x)}{x} \, dx,x,\arctan (a x)\right )}{a^3 c^2 \sqrt {c+a^2 c x^2}} \\ & = \frac {\sqrt {1+a^2 x^2} \text {Subst}\left (\int \left (\frac {\cos (x)}{4 x}-\frac {\cos (3 x)}{4 x}\right ) \, dx,x,\arctan (a x)\right )}{a^3 c^2 \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 )}{4 a^3 c^2 \sqrt {c+a^2 c x^2}}-\frac {\sqrt {1+a^2 x^2} \text {Subst}\left (\int \frac {\cos (3 x)}{x} \, dx,x,\arctan (a x)\right )}{4 a^3 c^2 \sqrt {c+a^2 c x^2}} \\ & = \frac {\sqrt {1+a^2 x^2} \operatorname {CosIntegral}(\arctan (a x))}{4 a^3 c^2 \sqrt {c+a^2 c x^2}}-\frac {\sqrt {1+a^2 x^2} \operatorname {CosIntegral}(3 \arctan (a x))}{4 a^3 c^2 \sqrt {c+a^2 c x^2}} \\ \end{align*}
Time = 0.12 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.61 \[ \int \frac {x^2}{\left (c+a^2 c x^2\right )^{5/2} \arctan (a x)} \, dx=\frac {\sqrt {1+a^2 x^2} (\operatorname {CosIntegral}(\arctan (a x))-\operatorname {CosIntegral}(3 \arctan (a x)))}{4 a^3 c^2 \sqrt {c \left (1+a^2 x^2\right )}} \]
[In]
[Out]
Result contains complex when optimal does not.
Time = 8.89 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.97
method | result | size |
default | \(-\frac {\operatorname {Ci}\left (3 \arctan \left (a x \right )\right ) \sqrt {c \left (a x -i\right ) \left (a x +i\right )}}{4 \sqrt {a^{2} x^{2}+1}\, a^{3} c^{3}}+\frac {\operatorname {Ci}\left (\arctan \left (a x \right )\right ) \sqrt {c \left (a x -i\right ) \left (a x +i\right )}}{4 \sqrt {a^{2} x^{2}+1}\, a^{3} c^{3}}\) | \(84\) |
[In]
[Out]
\[ \int \frac {x^2}{\left (c+a^2 c x^2\right )^{5/2} \arctan (a x)} \, dx=\int { \frac {x^{2}}{{\left (a^{2} c x^{2} + c\right )}^{\frac {5}{2}} \arctan \left (a x\right )} \,d x } \]
[In]
[Out]
\[ \int \frac {x^2}{\left (c+a^2 c x^2\right )^{5/2} \arctan (a x)} \, dx=\int \frac {x^{2}}{\left (c \left (a^{2} x^{2} + 1\right )\right )^{\frac {5}{2}} \operatorname {atan}{\left (a x \right )}}\, dx \]
[In]
[Out]
\[ \int \frac {x^2}{\left (c+a^2 c x^2\right )^{5/2} \arctan (a x)} \, dx=\int { \frac {x^{2}}{{\left (a^{2} c x^{2} + c\right )}^{\frac {5}{2}} \arctan \left (a x\right )} \,d x } \]
[In]
[Out]
\[ \int \frac {x^2}{\left (c+a^2 c x^2\right )^{5/2} \arctan (a x)} \, dx=\int { \frac {x^{2}}{{\left (a^{2} c x^{2} + c\right )}^{\frac {5}{2}} \arctan \left (a x\right )} \,d x } \]
[In]
[Out]
Timed out. \[ \int \frac {x^2}{\left (c+a^2 c x^2\right )^{5/2} \arctan (a x)} \, dx=\int \frac {x^2}{\mathrm {atan}\left (a\,x\right )\,{\left (c\,a^2\,x^2+c\right )}^{5/2}} \,d x \]
[In]
[Out]