Integrand size = 19, antiderivative size = 298 \[ \int \left (c+a^2 c x^2\right )^{3/2} \arctan (a x) \, dx=-\frac {3 c \sqrt {c+a^2 c x^2}}{8 a}-\frac {\left (c+a^2 c x^2\right )^{3/2}}{12 a}+\frac {3}{8} c x \sqrt {c+a^2 c x^2} \arctan (a x)+\frac {1}{4} x \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)-\frac {3 i c^2 \sqrt {1+a^2 x^2} \arctan (a x) \arctan \left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{4 a \sqrt {c+a^2 c x^2}}+\frac {3 i c^2 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (2,-\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{8 a \sqrt {c+a^2 c x^2}}-\frac {3 i c^2 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (2,\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{8 a \sqrt {c+a^2 c x^2}} \]
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Time = 0.10 (sec) , antiderivative size = 298, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {4998, 5010, 5006} \[ \int \left (c+a^2 c x^2\right )^{3/2} \arctan (a x) \, dx=-\frac {3 i c^2 \sqrt {a^2 x^2+1} \arctan (a x) \arctan \left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{4 a \sqrt {a^2 c x^2+c}}+\frac {3}{8} c x \arctan (a x) \sqrt {a^2 c x^2+c}+\frac {1}{4} x \arctan (a x) \left (a^2 c x^2+c\right )^{3/2}+\frac {3 i c^2 \sqrt {a^2 x^2+1} \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{8 a \sqrt {a^2 c x^2+c}}-\frac {3 i c^2 \sqrt {a^2 x^2+1} \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{8 a \sqrt {a^2 c x^2+c}}-\frac {3 c \sqrt {a^2 c x^2+c}}{8 a}-\frac {\left (a^2 c x^2+c\right )^{3/2}}{12 a} \]
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Rule 4998
Rule 5006
Rule 5010
Rubi steps \begin{align*} \text {integral}& = -\frac {\left (c+a^2 c x^2\right )^{3/2}}{12 a}+\frac {1}{4} x \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)+\frac {1}{4} (3 c) \int \sqrt {c+a^2 c x^2} \arctan (a x) \, dx \\ & = -\frac {3 c \sqrt {c+a^2 c x^2}}{8 a}-\frac {\left (c+a^2 c x^2\right )^{3/2}}{12 a}+\frac {3}{8} c x \sqrt {c+a^2 c x^2} \arctan (a x)+\frac {1}{4} x \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)+\frac {1}{8} \left (3 c^2\right ) \int \frac {\arctan (a x)}{\sqrt {c+a^2 c x^2}} \, dx \\ & = -\frac {3 c \sqrt {c+a^2 c x^2}}{8 a}-\frac {\left (c+a^2 c x^2\right )^{3/2}}{12 a}+\frac {3}{8} c x \sqrt {c+a^2 c x^2} \arctan (a x)+\frac {1}{4} x \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)+\frac {\left (3 c^2 \sqrt {1+a^2 x^2}\right ) \int \frac {\arctan (a x)}{\sqrt {1+a^2 x^2}} \, dx}{8 \sqrt {c+a^2 c x^2}} \\ & = -\frac {3 c \sqrt {c+a^2 c x^2}}{8 a}-\frac {\left (c+a^2 c x^2\right )^{3/2}}{12 a}+\frac {3}{8} c x \sqrt {c+a^2 c x^2} \arctan (a x)+\frac {1}{4} x \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)-\frac {3 i c^2 \sqrt {1+a^2 x^2} \arctan (a x) \arctan \left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{4 a \sqrt {c+a^2 c x^2}}+\frac {3 i c^2 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (2,-\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{8 a \sqrt {c+a^2 c x^2}}-\frac {3 i c^2 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (2,\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{8 a \sqrt {c+a^2 c x^2}} \\ \end{align*}
Time = 2.48 (sec) , antiderivative size = 351, normalized size of antiderivative = 1.18 \[ \int \left (c+a^2 c x^2\right )^{3/2} \arctan (a x) \, dx=\frac {c \sqrt {c+a^2 c x^2} \left (2 \left (1+a^2 x^2\right )^{3/2}+96 \sqrt {1+a^2 x^2} (-1+a x \arctan (a x))+6 \left (1+a^2 x^2\right )^2 \cos (3 \arctan (a x))+96 \arctan (a x) \left (\log \left (1-i e^{i \arctan (a x)}\right )-\log \left (1+i e^{i \arctan (a x)}\right )\right )+72 i \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-72 i \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-3 \left (1+a^2 x^2\right )^2 \arctan (a x) \left (-\frac {14 a x}{\sqrt {1+a^2 x^2}}+3 \log \left (1-i e^{i \arctan (a x)}\right )+4 \cos (2 \arctan (a x)) \left (\log \left (1-i e^{i \arctan (a x)}\right )-\log \left (1+i e^{i \arctan (a x)}\right )\right )+\cos (4 \arctan (a x)) \left (\log \left (1-i e^{i \arctan (a x)}\right )-\log \left (1+i e^{i \arctan (a x)}\right )\right )-3 \log \left (1+i e^{i \arctan (a x)}\right )+2 \sin (3 \arctan (a x))\right )\right )}{192 a \sqrt {1+a^2 x^2}} \]
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Time = 0.41 (sec) , antiderivative size = 201, normalized size of antiderivative = 0.67
method | result | size |
default | \(\frac {c \sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (6 \arctan \left (a x \right ) x^{3} a^{3}-2 a^{2} x^{2}+15 x \arctan \left (a x \right ) a -11\right )}{24 a}-\frac {3 c \sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (\arctan \left (a x \right ) \ln \left (1+\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-\arctan \left (a x \right ) \ln \left (1-\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-i \operatorname {dilog}\left (1+\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )+i \operatorname {dilog}\left (1-\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )\right )}{8 a \sqrt {a^{2} x^{2}+1}}\) | \(201\) |
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\[ \int \left (c+a^2 c x^2\right )^{3/2} \arctan (a x) \, dx=\int { {\left (a^{2} c x^{2} + c\right )}^{\frac {3}{2}} \arctan \left (a x\right ) \,d x } \]
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\[ \int \left (c+a^2 c x^2\right )^{3/2} \arctan (a x) \, dx=\int \left (c \left (a^{2} x^{2} + 1\right )\right )^{\frac {3}{2}} \operatorname {atan}{\left (a x \right )}\, dx \]
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\[ \int \left (c+a^2 c x^2\right )^{3/2} \arctan (a x) \, dx=\int { {\left (a^{2} c x^{2} + c\right )}^{\frac {3}{2}} \arctan \left (a x\right ) \,d x } \]
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Exception generated. \[ \int \left (c+a^2 c x^2\right )^{3/2} \arctan (a x) \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int \left (c+a^2 c x^2\right )^{3/2} \arctan (a x) \, dx=\int \mathrm {atan}\left (a\,x\right )\,{\left (c\,a^2\,x^2+c\right )}^{3/2} \,d x \]
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