Integrand size = 24, antiderivative size = 199 \[ \int \frac {x^2 \arctan (a x)^3}{\left (c+a^2 c x^2\right )^{5/2}} \, dx=\frac {2}{27 a^3 c \left (c+a^2 c x^2\right )^{3/2}}-\frac {14}{9 a^3 c^2 \sqrt {c+a^2 c x^2}}-\frac {2 x^3 \arctan (a x)}{9 c \left (c+a^2 c x^2\right )^{3/2}}-\frac {4 x \arctan (a x)}{3 a^2 c^2 \sqrt {c+a^2 c x^2}}+\frac {x^2 \arctan (a x)^2}{3 a c \left (c+a^2 c x^2\right )^{3/2}}+\frac {2 \arctan (a x)^2}{3 a^3 c^2 \sqrt {c+a^2 c x^2}}+\frac {x^3 \arctan (a x)^3}{3 c \left (c+a^2 c x^2\right )^{3/2}} \]
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Time = 0.29 (sec) , antiderivative size = 199, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {5064, 5060, 5050, 5014, 272, 45} \[ \int \frac {x^2 \arctan (a x)^3}{\left (c+a^2 c x^2\right )^{5/2}} \, dx=-\frac {4 x \arctan (a x)}{3 a^2 c^2 \sqrt {a^2 c x^2+c}}+\frac {x^2 \arctan (a x)^2}{3 a c \left (a^2 c x^2+c\right )^{3/2}}+\frac {x^3 \arctan (a x)^3}{3 c \left (a^2 c x^2+c\right )^{3/2}}-\frac {2 x^3 \arctan (a x)}{9 c \left (a^2 c x^2+c\right )^{3/2}}+\frac {2 \arctan (a x)^2}{3 a^3 c^2 \sqrt {a^2 c x^2+c}}-\frac {14}{9 a^3 c^2 \sqrt {a^2 c x^2+c}}+\frac {2}{27 a^3 c \left (a^2 c x^2+c\right )^{3/2}} \]
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Rule 45
Rule 272
Rule 5014
Rule 5050
Rule 5060
Rule 5064
Rubi steps \begin{align*} \text {integral}& = \frac {x^3 \arctan (a x)^3}{3 c \left (c+a^2 c x^2\right )^{3/2}}-a \int \frac {x^3 \arctan (a x)^2}{\left (c+a^2 c x^2\right )^{5/2}} \, dx \\ & = -\frac {2 x^3 \arctan (a x)}{9 c \left (c+a^2 c x^2\right )^{3/2}}+\frac {x^2 \arctan (a x)^2}{3 a c \left (c+a^2 c x^2\right )^{3/2}}+\frac {x^3 \arctan (a x)^3}{3 c \left (c+a^2 c x^2\right )^{3/2}}+\frac {1}{9} (2 a) \int \frac {x^3}{\left (c+a^2 c x^2\right )^{5/2}} \, dx-\frac {2 \int \frac {x \arctan (a x)^2}{\left (c+a^2 c x^2\right )^{3/2}} \, dx}{3 a c} \\ & = -\frac {2 x^3 \arctan (a x)}{9 c \left (c+a^2 c x^2\right )^{3/2}}+\frac {x^2 \arctan (a x)^2}{3 a c \left (c+a^2 c x^2\right )^{3/2}}+\frac {2 \arctan (a x)^2}{3 a^3 c^2 \sqrt {c+a^2 c x^2}}+\frac {x^3 \arctan (a x)^3}{3 c \left (c+a^2 c x^2\right )^{3/2}}+\frac {1}{9} a \text {Subst}\left (\int \frac {x}{\left (c+a^2 c x\right )^{5/2}} \, dx,x,x^2\right )-\frac {4 \int \frac {\arctan (a x)}{\left (c+a^2 c x^2\right )^{3/2}} \, dx}{3 a^2 c} \\ & = -\frac {4}{3 a^3 c^2 \sqrt {c+a^2 c x^2}}-\frac {2 x^3 \arctan (a x)}{9 c \left (c+a^2 c x^2\right )^{3/2}}-\frac {4 x \arctan (a x)}{3 a^2 c^2 \sqrt {c+a^2 c x^2}}+\frac {x^2 \arctan (a x)^2}{3 a c \left (c+a^2 c x^2\right )^{3/2}}+\frac {2 \arctan (a x)^2}{3 a^3 c^2 \sqrt {c+a^2 c x^2}}+\frac {x^3 \arctan (a x)^3}{3 c \left (c+a^2 c x^2\right )^{3/2}}+\frac {1}{9} a \text {Subst}\left (\int \left (-\frac {1}{a^2 \left (c+a^2 c x\right )^{5/2}}+\frac {1}{a^2 c \left (c+a^2 c x\right )^{3/2}}\right ) \, dx,x,x^2\right ) \\ & = \frac {2}{27 a^3 c \left (c+a^2 c x^2\right )^{3/2}}-\frac {14}{9 a^3 c^2 \sqrt {c+a^2 c x^2}}-\frac {2 x^3 \arctan (a x)}{9 c \left (c+a^2 c x^2\right )^{3/2}}-\frac {4 x \arctan (a x)}{3 a^2 c^2 \sqrt {c+a^2 c x^2}}+\frac {x^2 \arctan (a x)^2}{3 a c \left (c+a^2 c x^2\right )^{3/2}}+\frac {2 \arctan (a x)^2}{3 a^3 c^2 \sqrt {c+a^2 c x^2}}+\frac {x^3 \arctan (a x)^3}{3 c \left (c+a^2 c x^2\right )^{3/2}} \\ \end{align*}
Time = 0.12 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.48 \[ \int \frac {x^2 \arctan (a x)^3}{\left (c+a^2 c x^2\right )^{5/2}} \, dx=\frac {\sqrt {c+a^2 c x^2} \left (-40-42 a^2 x^2-6 a x \left (6+7 a^2 x^2\right ) \arctan (a x)+9 \left (2+3 a^2 x^2\right ) \arctan (a x)^2+9 a^3 x^3 \arctan (a x)^3\right )}{27 a^3 c^3 \left (1+a^2 x^2\right )^2} \]
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Result contains complex when optimal does not.
Time = 4.79 (sec) , antiderivative size = 308, normalized size of antiderivative = 1.55
method | result | size |
default | \(\frac {\left (9 i \arctan \left (a x \right )^{2}+9 \arctan \left (a x \right )^{3}-2 i-6 \arctan \left (a x \right )\right ) \left (a^{3} x^{3}-3 i a^{2} x^{2}-3 a x +i\right ) \sqrt {c \left (a x -i\right ) \left (a x +i\right )}}{216 \left (a^{2} x^{2}+1\right )^{2} a^{3} c^{3}}+\frac {\left (\arctan \left (a x \right )^{3}-6 \arctan \left (a x \right )+3 i \arctan \left (a x \right )^{2}-6 i\right ) \left (a x -i\right ) \sqrt {c \left (a x -i\right ) \left (a x +i\right )}}{8 c^{3} a^{3} \left (a^{2} x^{2}+1\right )}+\frac {\sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (a x +i\right ) \left (\arctan \left (a x \right )^{3}-6 \arctan \left (a x \right )-3 i \arctan \left (a x \right )^{2}+6 i\right )}{8 c^{3} a^{3} \left (a^{2} x^{2}+1\right )}+\frac {\left (-9 i \arctan \left (a x \right )^{2}+9 \arctan \left (a x \right )^{3}+2 i-6 \arctan \left (a x \right )\right ) \sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (a^{3} x^{3}+3 i a^{2} x^{2}-3 a x -i\right )}{216 \left (a^{4} x^{4}+2 a^{2} x^{2}+1\right ) a^{3} c^{3}}\) | \(308\) |
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none
Time = 0.27 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.53 \[ \int \frac {x^2 \arctan (a x)^3}{\left (c+a^2 c x^2\right )^{5/2}} \, dx=\frac {{\left (9 \, a^{3} x^{3} \arctan \left (a x\right )^{3} - 42 \, a^{2} x^{2} + 9 \, {\left (3 \, a^{2} x^{2} + 2\right )} \arctan \left (a x\right )^{2} - 6 \, {\left (7 \, a^{3} x^{3} + 6 \, a x\right )} \arctan \left (a x\right ) - 40\right )} \sqrt {a^{2} c x^{2} + c}}{27 \, {\left (a^{7} c^{3} x^{4} + 2 \, a^{5} c^{3} x^{2} + a^{3} c^{3}\right )}} \]
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\[ \int \frac {x^2 \arctan (a x)^3}{\left (c+a^2 c x^2\right )^{5/2}} \, dx=\int \frac {x^{2} \operatorname {atan}^{3}{\left (a x \right )}}{\left (c \left (a^{2} x^{2} + 1\right )\right )^{\frac {5}{2}}}\, dx \]
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\[ \int \frac {x^2 \arctan (a x)^3}{\left (c+a^2 c x^2\right )^{5/2}} \, dx=\int { \frac {x^{2} \arctan \left (a x\right )^{3}}{{\left (a^{2} c x^{2} + c\right )}^{\frac {5}{2}}} \,d x } \]
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\[ \int \frac {x^2 \arctan (a x)^3}{\left (c+a^2 c x^2\right )^{5/2}} \, dx=\int { \frac {x^{2} \arctan \left (a x\right )^{3}}{{\left (a^{2} c x^{2} + c\right )}^{\frac {5}{2}}} \,d x } \]
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Timed out. \[ \int \frac {x^2 \arctan (a x)^3}{\left (c+a^2 c x^2\right )^{5/2}} \, dx=\int \frac {x^2\,{\mathrm {atan}\left (a\,x\right )}^3}{{\left (c\,a^2\,x^2+c\right )}^{5/2}} \,d x \]
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