Integrand size = 22, antiderivative size = 387 \[ \int x \left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^2 \, dx=\frac {5 c^2 \sqrt {c+a^2 c x^2}}{56 a^2}+\frac {5 c \left (c+a^2 c x^2\right )^{3/2}}{252 a^2}+\frac {\left (c+a^2 c x^2\right )^{5/2}}{105 a^2}-\frac {5 c^2 x \sqrt {c+a^2 c x^2} \arctan (a x)}{56 a}-\frac {5 c x \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)}{84 a}-\frac {x \left (c+a^2 c x^2\right )^{5/2} \arctan (a x)}{21 a}+\frac {\left (c+a^2 c x^2\right )^{7/2} \arctan (a x)^2}{7 a^2 c}+\frac {5 i c^3 \sqrt {1+a^2 x^2} \arctan (a x) \arctan \left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{28 a^2 \sqrt {c+a^2 c x^2}}-\frac {5 i c^3 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (2,-\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{56 a^2 \sqrt {c+a^2 c x^2}}+\frac {5 i c^3 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (2,\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{56 a^2 \sqrt {c+a^2 c x^2}} \]
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Time = 0.21 (sec) , antiderivative size = 387, 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 = {5050, 4998, 5010, 5006} \[ \int x \left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^2 \, dx=\frac {5 i c^3 \sqrt {a^2 x^2+1} \arctan (a x) \arctan \left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{28 a^2 \sqrt {a^2 c x^2+c}}-\frac {5 c^2 x \arctan (a x) \sqrt {a^2 c x^2+c}}{56 a}+\frac {\arctan (a x)^2 \left (a^2 c x^2+c\right )^{7/2}}{7 a^2 c}-\frac {x \arctan (a x) \left (a^2 c x^2+c\right )^{5/2}}{21 a}-\frac {5 c x \arctan (a x) \left (a^2 c x^2+c\right )^{3/2}}{84 a}-\frac {5 i c^3 \sqrt {a^2 x^2+1} \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{56 a^2 \sqrt {a^2 c x^2+c}}+\frac {5 i c^3 \sqrt {a^2 x^2+1} \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{56 a^2 \sqrt {a^2 c x^2+c}}+\frac {5 c^2 \sqrt {a^2 c x^2+c}}{56 a^2}+\frac {\left (a^2 c x^2+c\right )^{5/2}}{105 a^2}+\frac {5 c \left (a^2 c x^2+c\right )^{3/2}}{252 a^2} \]
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Rule 4998
Rule 5006
Rule 5010
Rule 5050
Rubi steps \begin{align*} \text {integral}& = \frac {\left (c+a^2 c x^2\right )^{7/2} \arctan (a x)^2}{7 a^2 c}-\frac {2 \int \left (c+a^2 c x^2\right )^{5/2} \arctan (a x) \, dx}{7 a} \\ & = \frac {\left (c+a^2 c x^2\right )^{5/2}}{105 a^2}-\frac {x \left (c+a^2 c x^2\right )^{5/2} \arctan (a x)}{21 a}+\frac {\left (c+a^2 c x^2\right )^{7/2} \arctan (a x)^2}{7 a^2 c}-\frac {(5 c) \int \left (c+a^2 c x^2\right )^{3/2} \arctan (a x) \, dx}{21 a} \\ & = \frac {5 c \left (c+a^2 c x^2\right )^{3/2}}{252 a^2}+\frac {\left (c+a^2 c x^2\right )^{5/2}}{105 a^2}-\frac {5 c x \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)}{84 a}-\frac {x \left (c+a^2 c x^2\right )^{5/2} \arctan (a x)}{21 a}+\frac {\left (c+a^2 c x^2\right )^{7/2} \arctan (a x)^2}{7 a^2 c}-\frac {\left (5 c^2\right ) \int \sqrt {c+a^2 c x^2} \arctan (a x) \, dx}{28 a} \\ & = \frac {5 c^2 \sqrt {c+a^2 c x^2}}{56 a^2}+\frac {5 c \left (c+a^2 c x^2\right )^{3/2}}{252 a^2}+\frac {\left (c+a^2 c x^2\right )^{5/2}}{105 a^2}-\frac {5 c^2 x \sqrt {c+a^2 c x^2} \arctan (a x)}{56 a}-\frac {5 c x \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)}{84 a}-\frac {x \left (c+a^2 c x^2\right )^{5/2} \arctan (a x)}{21 a}+\frac {\left (c+a^2 c x^2\right )^{7/2} \arctan (a x)^2}{7 a^2 c}-\frac {\left (5 c^3\right ) \int \frac {\arctan (a x)}{\sqrt {c+a^2 c x^2}} \, dx}{56 a} \\ & = \frac {5 c^2 \sqrt {c+a^2 c x^2}}{56 a^2}+\frac {5 c \left (c+a^2 c x^2\right )^{3/2}}{252 a^2}+\frac {\left (c+a^2 c x^2\right )^{5/2}}{105 a^2}-\frac {5 c^2 x \sqrt {c+a^2 c x^2} \arctan (a x)}{56 a}-\frac {5 c x \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)}{84 a}-\frac {x \left (c+a^2 c x^2\right )^{5/2} \arctan (a x)}{21 a}+\frac {\left (c+a^2 c x^2\right )^{7/2} \arctan (a x)^2}{7 a^2 c}-\frac {\left (5 c^3 \sqrt {1+a^2 x^2}\right ) \int \frac {\arctan (a x)}{\sqrt {1+a^2 x^2}} \, dx}{56 a \sqrt {c+a^2 c x^2}} \\ & = \frac {5 c^2 \sqrt {c+a^2 c x^2}}{56 a^2}+\frac {5 c \left (c+a^2 c x^2\right )^{3/2}}{252 a^2}+\frac {\left (c+a^2 c x^2\right )^{5/2}}{105 a^2}-\frac {5 c^2 x \sqrt {c+a^2 c x^2} \arctan (a x)}{56 a}-\frac {5 c x \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)}{84 a}-\frac {x \left (c+a^2 c x^2\right )^{5/2} \arctan (a x)}{21 a}+\frac {\left (c+a^2 c x^2\right )^{7/2} \arctan (a x)^2}{7 a^2 c}+\frac {5 i c^3 \sqrt {1+a^2 x^2} \arctan (a x) \arctan \left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{28 a^2 \sqrt {c+a^2 c x^2}}-\frac {5 i c^3 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (2,-\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{56 a^2 \sqrt {c+a^2 c x^2}}+\frac {5 i c^3 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (2,\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{56 a^2 \sqrt {c+a^2 c x^2}} \\ \end{align*}
Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(1087\) vs. \(2(387)=774\).
Time = 7.75 (sec) , antiderivative size = 1087, normalized size of antiderivative = 2.81 \[ \int x \left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^2 \, dx=\frac {c^2 \left (1+a^2 x^2\right ) \sqrt {c \left (1+a^2 x^2\right )} \left (2+4 \arctan (a x)^2+2 \cos (2 \arctan (a x))-\frac {3 \arctan (a x) \log \left (1-i e^{i \arctan (a x)}\right )}{\sqrt {1+a^2 x^2}}-\arctan (a x) \cos (3 \arctan (a x)) \log \left (1-i e^{i \arctan (a x)}\right )+\frac {3 \arctan (a x) \log \left (1+i e^{i \arctan (a x)}\right )}{\sqrt {1+a^2 x^2}}+\arctan (a x) \cos (3 \arctan (a x)) \log \left (1+i e^{i \arctan (a x)}\right )-\frac {4 i \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )}{\left (1+a^2 x^2\right )^{3/2}}+\frac {4 i \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )}{\left (1+a^2 x^2\right )^{3/2}}-2 \arctan (a x) \sin (2 \arctan (a x))\right )}{12 a^2}-\frac {c^2 \left (1+a^2 x^2\right )^2 \sqrt {c \left (1+a^2 x^2\right )} \left (50-32 \arctan (a x)^2+72 \cos (2 \arctan (a x))+160 \arctan (a x)^2 \cos (2 \arctan (a x))+22 \cos (4 \arctan (a x))-\frac {110 \arctan (a x) \log \left (1-i e^{i \arctan (a x)}\right )}{\sqrt {1+a^2 x^2}}-55 \arctan (a x) \cos (3 \arctan (a x)) \log \left (1-i e^{i \arctan (a x)}\right )-11 \arctan (a x) \cos (5 \arctan (a x)) \log \left (1-i e^{i \arctan (a x)}\right )+\frac {110 \arctan (a x) \log \left (1+i e^{i \arctan (a x)}\right )}{\sqrt {1+a^2 x^2}}+55 \arctan (a x) \cos (3 \arctan (a x)) \log \left (1+i e^{i \arctan (a x)}\right )+11 \arctan (a x) \cos (5 \arctan (a x)) \log \left (1+i e^{i \arctan (a x)}\right )-\frac {176 i \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )}{\left (1+a^2 x^2\right )^{5/2}}+\frac {176 i \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )}{\left (1+a^2 x^2\right )^{5/2}}+4 \arctan (a x) \sin (2 \arctan (a x))-22 \arctan (a x) \sin (4 \arctan (a x))\right )}{480 a^2}+\frac {c^2 \left (1+a^2 x^2\right )^3 \sqrt {c \left (1+a^2 x^2\right )} \left (4116+10944 \arctan (a x)^2+6262 \cos (2 \arctan (a x))-5376 \arctan (a x)^2 \cos (2 \arctan (a x))+2764 \cos (4 \arctan (a x))+6720 \arctan (a x)^2 \cos (4 \arctan (a x))+618 \cos (6 \arctan (a x))-\frac {10815 \arctan (a x) \log \left (1-i e^{i \arctan (a x)}\right )}{\sqrt {1+a^2 x^2}}-6489 \arctan (a x) \cos (3 \arctan (a x)) \log \left (1-i e^{i \arctan (a x)}\right )-2163 \arctan (a x) \cos (5 \arctan (a x)) \log \left (1-i e^{i \arctan (a x)}\right )-309 \arctan (a x) \cos (7 \arctan (a x)) \log \left (1-i e^{i \arctan (a x)}\right )+\frac {10815 \arctan (a x) \log \left (1+i e^{i \arctan (a x)}\right )}{\sqrt {1+a^2 x^2}}+6489 \arctan (a x) \cos (3 \arctan (a x)) \log \left (1+i e^{i \arctan (a x)}\right )+2163 \arctan (a x) \cos (5 \arctan (a x)) \log \left (1+i e^{i \arctan (a x)}\right )+309 \arctan (a x) \cos (7 \arctan (a x)) \log \left (1+i e^{i \arctan (a x)}\right )-\frac {19776 i \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )}{\left (1+a^2 x^2\right )^{7/2}}+\frac {19776 i \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )}{\left (1+a^2 x^2\right )^{7/2}}-1266 \arctan (a x) \sin (2 \arctan (a x))+360 \arctan (a x) \sin (4 \arctan (a x))-618 \arctan (a x) \sin (6 \arctan (a x))\right )}{161280 a^2} \]
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Time = 3.84 (sec) , antiderivative size = 275, normalized size of antiderivative = 0.71
method | result | size |
default | \(\frac {c^{2} \sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (360 a^{6} x^{6} \arctan \left (a x \right )^{2}-120 \arctan \left (a x \right ) a^{5} x^{5}+1080 a^{4} \arctan \left (a x \right )^{2} x^{4}+24 a^{4} x^{4}-390 \arctan \left (a x \right ) x^{3} a^{3}+1080 x^{2} \arctan \left (a x \right )^{2} a^{2}+98 a^{2} x^{2}-495 x \arctan \left (a x \right ) a +360 \arctan \left (a x \right )^{2}+299\right )}{2520 a^{2}}+\frac {5 c^{2} \sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (\arctan \left (a x \right ) \ln \left (1+\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-\arctan \left (a x \right ) \ln \left (1-\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-i \operatorname {dilog}\left (1+\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )+i \operatorname {dilog}\left (1-\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )\right )}{56 a^{2} \sqrt {a^{2} x^{2}+1}}\) | \(275\) |
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\[ \int x \left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^2 \, dx=\int { {\left (a^{2} c x^{2} + c\right )}^{\frac {5}{2}} x \arctan \left (a x\right )^{2} \,d x } \]
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\[ \int x \left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^2 \, dx=\int x \left (c \left (a^{2} x^{2} + 1\right )\right )^{\frac {5}{2}} \operatorname {atan}^{2}{\left (a x \right )}\, dx \]
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\[ \int x \left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^2 \, dx=\int { {\left (a^{2} c x^{2} + c\right )}^{\frac {5}{2}} x \arctan \left (a x\right )^{2} \,d x } \]
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Exception generated. \[ \int x \left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^2 \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int x \left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^2 \, dx=\int x\,{\mathrm {atan}\left (a\,x\right )}^2\,{\left (c\,a^2\,x^2+c\right )}^{5/2} \,d x \]
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