Integrand size = 22, antiderivative size = 308 \[ \int \frac {x^4 \arctan (a x)}{\left (c+a^2 c x^2\right )^{5/2}} \, dx=\frac {1}{9 a^5 c \left (c+a^2 c x^2\right )^{3/2}}-\frac {4}{3 a^5 c^2 \sqrt {c+a^2 c x^2}}-\frac {x^3 \arctan (a x)}{3 a^2 c \left (c+a^2 c x^2\right )^{3/2}}-\frac {x \arctan (a x)}{a^4 c^2 \sqrt {c+a^2 c x^2}}-\frac {2 i \sqrt {1+a^2 x^2} \arctan (a x) \arctan \left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{a^5 c^2 \sqrt {c+a^2 c x^2}}+\frac {i \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (2,-\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{a^5 c^2 \sqrt {c+a^2 c x^2}}-\frac {i \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (2,\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{a^5 c^2 \sqrt {c+a^2 c x^2}} \]
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Time = 0.27 (sec) , antiderivative size = 308, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.318, Rules used = {5084, 5054, 5010, 5006, 5064, 272, 45} \[ \int \frac {x^4 \arctan (a x)}{\left (c+a^2 c x^2\right )^{5/2}} \, dx=-\frac {x^3 \arctan (a x)}{3 a^2 c \left (a^2 c x^2+c\right )^{3/2}}-\frac {2 i \sqrt {a^2 x^2+1} \arctan (a x) \arctan \left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{a^5 c^2 \sqrt {a^2 c x^2+c}}+\frac {i \sqrt {a^2 x^2+1} \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a^5 c^2 \sqrt {a^2 c x^2+c}}-\frac {i \sqrt {a^2 x^2+1} \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a^5 c^2 \sqrt {a^2 c x^2+c}}-\frac {4}{3 a^5 c^2 \sqrt {a^2 c x^2+c}}+\frac {1}{9 a^5 c \left (a^2 c x^2+c\right )^{3/2}}-\frac {x \arctan (a x)}{a^4 c^2 \sqrt {a^2 c x^2+c}} \]
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Rule 45
Rule 272
Rule 5006
Rule 5010
Rule 5054
Rule 5064
Rule 5084
Rubi steps \begin{align*} \text {integral}& = -\frac {\int \frac {x^2 \arctan (a x)}{\left (c+a^2 c x^2\right )^{5/2}} \, dx}{a^2}+\frac {\int \frac {x^2 \arctan (a x)}{\left (c+a^2 c x^2\right )^{3/2}} \, dx}{a^2 c} \\ & = -\frac {1}{a^5 c^2 \sqrt {c+a^2 c x^2}}-\frac {x^3 \arctan (a x)}{3 a^2 c \left (c+a^2 c x^2\right )^{3/2}}-\frac {x \arctan (a x)}{a^4 c^2 \sqrt {c+a^2 c x^2}}+\frac {\int \frac {x^3}{\left (c+a^2 c x^2\right )^{5/2}} \, dx}{3 a}+\frac {\int \frac {\arctan (a x)}{\sqrt {c+a^2 c x^2}} \, dx}{a^4 c^2} \\ & = -\frac {1}{a^5 c^2 \sqrt {c+a^2 c x^2}}-\frac {x^3 \arctan (a x)}{3 a^2 c \left (c+a^2 c x^2\right )^{3/2}}-\frac {x \arctan (a x)}{a^4 c^2 \sqrt {c+a^2 c x^2}}+\frac {\text {Subst}\left (\int \frac {x}{\left (c+a^2 c x\right )^{5/2}} \, dx,x,x^2\right )}{6 a}+\frac {\sqrt {1+a^2 x^2} \int \frac {\arctan (a x)}{\sqrt {1+a^2 x^2}} \, dx}{a^4 c^2 \sqrt {c+a^2 c x^2}} \\ & = -\frac {1}{a^5 c^2 \sqrt {c+a^2 c x^2}}-\frac {x^3 \arctan (a x)}{3 a^2 c \left (c+a^2 c x^2\right )^{3/2}}-\frac {x \arctan (a x)}{a^4 c^2 \sqrt {c+a^2 c x^2}}-\frac {2 i \sqrt {1+a^2 x^2} \arctan (a x) \arctan \left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{a^5 c^2 \sqrt {c+a^2 c x^2}}+\frac {i \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (2,-\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{a^5 c^2 \sqrt {c+a^2 c x^2}}-\frac {i \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (2,\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{a^5 c^2 \sqrt {c+a^2 c x^2}}+\frac {\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 )}{6 a} \\ & = \frac {1}{9 a^5 c \left (c+a^2 c x^2\right )^{3/2}}-\frac {4}{3 a^5 c^2 \sqrt {c+a^2 c x^2}}-\frac {x^3 \arctan (a x)}{3 a^2 c \left (c+a^2 c x^2\right )^{3/2}}-\frac {x \arctan (a x)}{a^4 c^2 \sqrt {c+a^2 c x^2}}-\frac {2 i \sqrt {1+a^2 x^2} \arctan (a x) \arctan \left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{a^5 c^2 \sqrt {c+a^2 c x^2}}+\frac {i \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (2,-\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{a^5 c^2 \sqrt {c+a^2 c x^2}}-\frac {i \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (2,\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{a^5 c^2 \sqrt {c+a^2 c x^2}} \\ \end{align*}
Time = 0.40 (sec) , antiderivative size = 177, normalized size of antiderivative = 0.57 \[ \int \frac {x^4 \arctan (a x)}{\left (c+a^2 c x^2\right )^{5/2}} \, dx=\frac {\sqrt {c \left (1+a^2 x^2\right )} \left (-\frac {45}{\sqrt {1+a^2 x^2}}-\frac {45 a x \arctan (a x)}{\sqrt {1+a^2 x^2}}+\cos (3 \arctan (a x))+36 \arctan (a x) \left (\log \left (1-i e^{i \arctan (a x)}\right )-\log \left (1+i e^{i \arctan (a x)}\right )\right )+36 i \left (\operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-\operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )\right )+3 \arctan (a x) \sin (3 \arctan (a x))\right )}{36 a^5 c^3 \sqrt {1+a^2 x^2}} \]
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Time = 0.88 (sec) , antiderivative size = 389, normalized size of antiderivative = 1.26
method | result | size |
default | \(-\frac {\left (i+3 \arctan \left (a x \right )\right ) \left (a^{3} x^{3}-3 i a^{2} x^{2}-3 a x +i\right ) \sqrt {c \left (a x -i\right ) \left (a x +i\right )}}{72 \left (a^{2} x^{2}+1\right )^{2} a^{5} c^{3}}-\frac {5 \left (\arctan \left (a x \right )+i\right ) \left (a x -i\right ) \sqrt {c \left (a x -i\right ) \left (a x +i\right )}}{8 c^{3} a^{5} \left (a^{2} x^{2}+1\right )}-\frac {5 \sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (a x +i\right ) \left (\arctan \left (a x \right )-i\right )}{8 c^{3} a^{5} \left (a^{2} x^{2}+1\right )}-\frac {\left (-i+3 \arctan \left (a x \right )\right ) \sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (a^{3} x^{3}+3 i a^{2} x^{2}-3 a x -i\right )}{72 \left (a^{4} x^{4}+2 a^{2} x^{2}+1\right ) a^{5} c^{3}}-\frac {\left (\arctan \left (a x \right ) \ln \left (1+\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-\arctan \left (a x \right ) \ln \left (1-\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-i \operatorname {dilog}\left (1+\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )+i \operatorname {dilog}\left (1-\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )\right ) \sqrt {c \left (a x -i\right ) \left (a x +i\right )}}{\sqrt {a^{2} x^{2}+1}\, c^{3} a^{5}}\) | \(389\) |
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\[ \int \frac {x^4 \arctan (a x)}{\left (c+a^2 c x^2\right )^{5/2}} \, dx=\int { \frac {x^{4} \arctan \left (a x\right )}{{\left (a^{2} c x^{2} + c\right )}^{\frac {5}{2}}} \,d x } \]
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\[ \int \frac {x^4 \arctan (a x)}{\left (c+a^2 c x^2\right )^{5/2}} \, dx=\int \frac {x^{4} \operatorname {atan}{\left (a x \right )}}{\left (c \left (a^{2} x^{2} + 1\right )\right )^{\frac {5}{2}}}\, dx \]
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\[ \int \frac {x^4 \arctan (a x)}{\left (c+a^2 c x^2\right )^{5/2}} \, dx=\int { \frac {x^{4} \arctan \left (a x\right )}{{\left (a^{2} c x^{2} + c\right )}^{\frac {5}{2}}} \,d x } \]
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\[ \int \frac {x^4 \arctan (a x)}{\left (c+a^2 c x^2\right )^{5/2}} \, dx=\int { \frac {x^{4} \arctan \left (a x\right )}{{\left (a^{2} c x^{2} + c\right )}^{\frac {5}{2}}} \,d x } \]
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Timed out. \[ \int \frac {x^4 \arctan (a x)}{\left (c+a^2 c x^2\right )^{5/2}} \, dx=\int \frac {x^4\,\mathrm {atan}\left (a\,x\right )}{{\left (c\,a^2\,x^2+c\right )}^{5/2}} \,d x \]
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