Integrand size = 21, antiderivative size = 438 \[ \int \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^2 \, dx=\frac {1}{12} c x \sqrt {c+a^2 c x^2}-\frac {3 c \sqrt {c+a^2 c x^2} \arctan (a x)}{4 a}-\frac {\left (c+a^2 c x^2\right )^{3/2} \arctan (a x)}{6 a}+\frac {3}{8} c x \sqrt {c+a^2 c x^2} \arctan (a x)^2+\frac {1}{4} x \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^2-\frac {3 i c^2 \sqrt {1+a^2 x^2} \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2}{4 a \sqrt {c+a^2 c x^2}}+\frac {5 c^{3/2} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {c+a^2 c x^2}}\right )}{6 a}+\frac {3 i c^2 \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )}{4 a \sqrt {c+a^2 c x^2}}-\frac {3 i c^2 \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )}{4 a \sqrt {c+a^2 c x^2}}-\frac {3 c^2 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (3,-i e^{i \arctan (a x)}\right )}{4 a \sqrt {c+a^2 c x^2}}+\frac {3 c^2 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (3,i e^{i \arctan (a x)}\right )}{4 a \sqrt {c+a^2 c x^2}} \]
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Time = 0.22 (sec) , antiderivative size = 438, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.476, Rules used = {5000, 5010, 5008, 4266, 2611, 2320, 6724, 223, 212, 201} \[ \int \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^2 \, dx=\frac {3 i c^2 \sqrt {a^2 x^2+1} \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )}{4 a \sqrt {a^2 c x^2+c}}-\frac {3 i c^2 \sqrt {a^2 x^2+1} \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )}{4 a \sqrt {a^2 c x^2+c}}-\frac {3 c^2 \sqrt {a^2 x^2+1} \operatorname {PolyLog}\left (3,-i e^{i \arctan (a x)}\right )}{4 a \sqrt {a^2 c x^2+c}}+\frac {3 c^2 \sqrt {a^2 x^2+1} \operatorname {PolyLog}\left (3,i e^{i \arctan (a x)}\right )}{4 a \sqrt {a^2 c x^2+c}}-\frac {3 i c^2 \sqrt {a^2 x^2+1} \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2}{4 a \sqrt {a^2 c x^2+c}}+\frac {3}{8} c x \arctan (a x)^2 \sqrt {a^2 c x^2+c}-\frac {3 c \arctan (a x) \sqrt {a^2 c x^2+c}}{4 a}+\frac {1}{4} x \arctan (a x)^2 \left (a^2 c x^2+c\right )^{3/2}-\frac {\arctan (a x) \left (a^2 c x^2+c\right )^{3/2}}{6 a}+\frac {5 c^{3/2} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{6 a}+\frac {1}{12} c x \sqrt {a^2 c x^2+c} \]
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Rule 201
Rule 212
Rule 223
Rule 2320
Rule 2611
Rule 4266
Rule 5000
Rule 5008
Rule 5010
Rule 6724
Rubi steps \begin{align*} \text {integral}& = -\frac {\left (c+a^2 c x^2\right )^{3/2} \arctan (a x)}{6 a}+\frac {1}{4} x \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^2+\frac {1}{6} c \int \sqrt {c+a^2 c x^2} \, dx+\frac {1}{4} (3 c) \int \sqrt {c+a^2 c x^2} \arctan (a x)^2 \, dx \\ & = \frac {1}{12} c x \sqrt {c+a^2 c x^2}-\frac {3 c \sqrt {c+a^2 c x^2} \arctan (a x)}{4 a}-\frac {\left (c+a^2 c x^2\right )^{3/2} \arctan (a x)}{6 a}+\frac {3}{8} c x \sqrt {c+a^2 c x^2} \arctan (a x)^2+\frac {1}{4} x \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^2+\frac {1}{12} c^2 \int \frac {1}{\sqrt {c+a^2 c x^2}} \, dx+\frac {1}{8} \left (3 c^2\right ) \int \frac {\arctan (a x)^2}{\sqrt {c+a^2 c x^2}} \, dx+\frac {1}{4} \left (3 c^2\right ) \int \frac {1}{\sqrt {c+a^2 c x^2}} \, dx \\ & = \frac {1}{12} c x \sqrt {c+a^2 c x^2}-\frac {3 c \sqrt {c+a^2 c x^2} \arctan (a x)}{4 a}-\frac {\left (c+a^2 c x^2\right )^{3/2} \arctan (a x)}{6 a}+\frac {3}{8} c x \sqrt {c+a^2 c x^2} \arctan (a x)^2+\frac {1}{4} x \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^2+\frac {1}{12} c^2 \text {Subst}\left (\int \frac {1}{1-a^2 c x^2} \, dx,x,\frac {x}{\sqrt {c+a^2 c x^2}}\right )+\frac {1}{4} \left (3 c^2\right ) \text {Subst}\left (\int \frac {1}{1-a^2 c x^2} \, dx,x,\frac {x}{\sqrt {c+a^2 c x^2}}\right )+\frac {\left (3 c^2 \sqrt {1+a^2 x^2}\right ) \int \frac {\arctan (a x)^2}{\sqrt {1+a^2 x^2}} \, dx}{8 \sqrt {c+a^2 c x^2}} \\ & = \frac {1}{12} c x \sqrt {c+a^2 c x^2}-\frac {3 c \sqrt {c+a^2 c x^2} \arctan (a x)}{4 a}-\frac {\left (c+a^2 c x^2\right )^{3/2} \arctan (a x)}{6 a}+\frac {3}{8} c x \sqrt {c+a^2 c x^2} \arctan (a x)^2+\frac {1}{4} x \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^2+\frac {5 c^{3/2} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {c+a^2 c x^2}}\right )}{6 a}+\frac {\left (3 c^2 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int x^2 \sec (x) \, dx,x,\arctan (a x)\right )}{8 a \sqrt {c+a^2 c x^2}} \\ & = \frac {1}{12} c x \sqrt {c+a^2 c x^2}-\frac {3 c \sqrt {c+a^2 c x^2} \arctan (a x)}{4 a}-\frac {\left (c+a^2 c x^2\right )^{3/2} \arctan (a x)}{6 a}+\frac {3}{8} c x \sqrt {c+a^2 c x^2} \arctan (a x)^2+\frac {1}{4} x \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^2-\frac {3 i c^2 \sqrt {1+a^2 x^2} \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2}{4 a \sqrt {c+a^2 c x^2}}+\frac {5 c^{3/2} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {c+a^2 c x^2}}\right )}{6 a}-\frac {\left (3 c^2 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int x \log \left (1-i e^{i x}\right ) \, dx,x,\arctan (a x)\right )}{4 a \sqrt {c+a^2 c x^2}}+\frac {\left (3 c^2 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int x \log \left (1+i e^{i x}\right ) \, dx,x,\arctan (a x)\right )}{4 a \sqrt {c+a^2 c x^2}} \\ & = \frac {1}{12} c x \sqrt {c+a^2 c x^2}-\frac {3 c \sqrt {c+a^2 c x^2} \arctan (a x)}{4 a}-\frac {\left (c+a^2 c x^2\right )^{3/2} \arctan (a x)}{6 a}+\frac {3}{8} c x \sqrt {c+a^2 c x^2} \arctan (a x)^2+\frac {1}{4} x \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^2-\frac {3 i c^2 \sqrt {1+a^2 x^2} \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2}{4 a \sqrt {c+a^2 c x^2}}+\frac {5 c^{3/2} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {c+a^2 c x^2}}\right )}{6 a}+\frac {3 i c^2 \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )}{4 a \sqrt {c+a^2 c x^2}}-\frac {3 i c^2 \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )}{4 a \sqrt {c+a^2 c x^2}}-\frac {\left (3 i c^2 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \operatorname {PolyLog}\left (2,-i e^{i x}\right ) \, dx,x,\arctan (a x)\right )}{4 a \sqrt {c+a^2 c x^2}}+\frac {\left (3 i c^2 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \operatorname {PolyLog}\left (2,i e^{i x}\right ) \, dx,x,\arctan (a x)\right )}{4 a \sqrt {c+a^2 c x^2}} \\ & = \frac {1}{12} c x \sqrt {c+a^2 c x^2}-\frac {3 c \sqrt {c+a^2 c x^2} \arctan (a x)}{4 a}-\frac {\left (c+a^2 c x^2\right )^{3/2} \arctan (a x)}{6 a}+\frac {3}{8} c x \sqrt {c+a^2 c x^2} \arctan (a x)^2+\frac {1}{4} x \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^2-\frac {3 i c^2 \sqrt {1+a^2 x^2} \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2}{4 a \sqrt {c+a^2 c x^2}}+\frac {5 c^{3/2} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {c+a^2 c x^2}}\right )}{6 a}+\frac {3 i c^2 \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )}{4 a \sqrt {c+a^2 c x^2}}-\frac {3 i c^2 \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )}{4 a \sqrt {c+a^2 c x^2}}-\frac {\left (3 c^2 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(2,-i x)}{x} \, dx,x,e^{i \arctan (a x)}\right )}{4 a \sqrt {c+a^2 c x^2}}+\frac {\left (3 c^2 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(2,i x)}{x} \, dx,x,e^{i \arctan (a x)}\right )}{4 a \sqrt {c+a^2 c x^2}} \\ & = \frac {1}{12} c x \sqrt {c+a^2 c x^2}-\frac {3 c \sqrt {c+a^2 c x^2} \arctan (a x)}{4 a}-\frac {\left (c+a^2 c x^2\right )^{3/2} \arctan (a x)}{6 a}+\frac {3}{8} c x \sqrt {c+a^2 c x^2} \arctan (a x)^2+\frac {1}{4} x \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^2-\frac {3 i c^2 \sqrt {1+a^2 x^2} \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2}{4 a \sqrt {c+a^2 c x^2}}+\frac {5 c^{3/2} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {c+a^2 c x^2}}\right )}{6 a}+\frac {3 i c^2 \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )}{4 a \sqrt {c+a^2 c x^2}}-\frac {3 i c^2 \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )}{4 a \sqrt {c+a^2 c x^2}}-\frac {3 c^2 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (3,-i e^{i \arctan (a x)}\right )}{4 a \sqrt {c+a^2 c x^2}}+\frac {3 c^2 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (3,i e^{i \arctan (a x)}\right )}{4 a \sqrt {c+a^2 c x^2}} \\ \end{align*}
Time = 0.83 (sec) , antiderivative size = 439, normalized size of antiderivative = 1.00 \[ \int \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^2 \, dx=\frac {c \sqrt {c+a^2 c x^2} \left (2 a x \sqrt {1+a^2 x^2}+2 a^3 x^3 \sqrt {1+a^2 x^2}-94 \sqrt {1+a^2 x^2} \arctan (a x)+2 a^2 x^2 \sqrt {1+a^2 x^2} \arctan (a x)+69 a x \sqrt {1+a^2 x^2} \arctan (a x)^2+21 a^3 x^3 \sqrt {1+a^2 x^2} \arctan (a x)^2-72 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2+80 \text {arctanh}\left (\frac {a x}{\sqrt {1+a^2 x^2}}\right )+6 \arctan (a x) \cos (3 \arctan (a x))+12 a^2 x^2 \arctan (a x) \cos (3 \arctan (a x))+6 a^4 x^4 \arctan (a x) \cos (3 \arctan (a x))+72 i \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-72 i \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-72 \operatorname {PolyLog}\left (3,-i e^{i \arctan (a x)}\right )+72 \operatorname {PolyLog}\left (3,i e^{i \arctan (a x)}\right )+2 \sin (3 \arctan (a x))+4 a^2 x^2 \sin (3 \arctan (a x))+2 a^4 x^4 \sin (3 \arctan (a x))-3 \arctan (a x)^2 \sin (3 \arctan (a x))-6 a^2 x^2 \arctan (a x)^2 \sin (3 \arctan (a x))-3 a^4 x^4 \arctan (a x)^2 \sin (3 \arctan (a x))\right )}{96 a \sqrt {1+a^2 x^2}} \]
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Time = 1.28 (sec) , antiderivative size = 304, normalized size of antiderivative = 0.69
method | result | size |
default | \(\frac {c \sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (6 a^{3} \arctan \left (a x \right )^{2} x^{3}-4 a^{2} \arctan \left (a x \right ) x^{2}+15 a \arctan \left (a x \right )^{2} x +2 a x -22 \arctan \left (a x \right )\right )}{24 a}+\frac {i c \sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (9 i \arctan \left (a x \right )^{2} \ln \left (1+\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-9 i \arctan \left (a x \right )^{2} \ln \left (1-\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )+18 \arctan \left (a x \right ) \operatorname {polylog}\left (2, -\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-18 \arctan \left (a x \right ) \operatorname {polylog}\left (2, \frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )+18 i \operatorname {polylog}\left (3, -\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-18 i \operatorname {polylog}\left (3, \frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-40 \arctan \left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )\right )}{24 a \sqrt {a^{2} x^{2}+1}}\) | \(304\) |
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\[ \int \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^2 \, dx=\int { {\left (a^{2} c x^{2} + c\right )}^{\frac {3}{2}} \arctan \left (a x\right )^{2} \,d x } \]
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\[ \int \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^2 \, dx=\int \left (c \left (a^{2} x^{2} + 1\right )\right )^{\frac {3}{2}} \operatorname {atan}^{2}{\left (a x \right )}\, dx \]
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\[ \int \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^2 \, dx=\int { {\left (a^{2} c x^{2} + c\right )}^{\frac {3}{2}} \arctan \left (a x\right )^{2} \,d x } \]
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Exception generated. \[ \int \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^2 \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^2 \, dx=\int {\mathrm {atan}\left (a\,x\right )}^2\,{\left (c\,a^2\,x^2+c\right )}^{3/2} \,d x \]
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