Integrand size = 21, antiderivative size = 340 \[ \int \sqrt {c+a^2 c x^2} \arctan (a x)^2 \, dx=-\frac {\sqrt {c+a^2 c x^2} \arctan (a x)}{a}+\frac {1}{2} x \sqrt {c+a^2 c x^2} \arctan (a x)^2-\frac {i c \sqrt {1+a^2 x^2} \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2}{a \sqrt {c+a^2 c x^2}}+\frac {\sqrt {c} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {c+a^2 c x^2}}\right )}{a}+\frac {i c \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )}{a \sqrt {c+a^2 c x^2}}-\frac {i c \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )}{a \sqrt {c+a^2 c x^2}}-\frac {c \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (3,-i e^{i \arctan (a x)}\right )}{a \sqrt {c+a^2 c x^2}}+\frac {c \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (3,i e^{i \arctan (a x)}\right )}{a \sqrt {c+a^2 c x^2}} \]
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Time = 0.16 (sec) , antiderivative size = 340, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {5000, 5010, 5008, 4266, 2611, 2320, 6724, 223, 212} \[ \int \sqrt {c+a^2 c x^2} \arctan (a x)^2 \, dx=\frac {i c \sqrt {a^2 x^2+1} \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )}{a \sqrt {a^2 c x^2+c}}-\frac {i c \sqrt {a^2 x^2+1} \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )}{a \sqrt {a^2 c x^2+c}}-\frac {c \sqrt {a^2 x^2+1} \operatorname {PolyLog}\left (3,-i e^{i \arctan (a x)}\right )}{a \sqrt {a^2 c x^2+c}}+\frac {c \sqrt {a^2 x^2+1} \operatorname {PolyLog}\left (3,i e^{i \arctan (a x)}\right )}{a \sqrt {a^2 c x^2+c}}-\frac {i c \sqrt {a^2 x^2+1} \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2}{a \sqrt {a^2 c x^2+c}}+\frac {1}{2} x \arctan (a x)^2 \sqrt {a^2 c x^2+c}-\frac {\arctan (a x) \sqrt {a^2 c x^2+c}}{a}+\frac {\sqrt {c} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{a} \]
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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 {\sqrt {c+a^2 c x^2} \arctan (a x)}{a}+\frac {1}{2} x \sqrt {c+a^2 c x^2} \arctan (a x)^2+\frac {1}{2} c \int \frac {\arctan (a x)^2}{\sqrt {c+a^2 c x^2}} \, dx+c \int \frac {1}{\sqrt {c+a^2 c x^2}} \, dx \\ & = -\frac {\sqrt {c+a^2 c x^2} \arctan (a x)}{a}+\frac {1}{2} x \sqrt {c+a^2 c x^2} \arctan (a x)^2+c \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 (c \sqrt {1+a^2 x^2}\right ) \int \frac {\arctan (a x)^2}{\sqrt {1+a^2 x^2}} \, dx}{2 \sqrt {c+a^2 c x^2}} \\ & = -\frac {\sqrt {c+a^2 c x^2} \arctan (a x)}{a}+\frac {1}{2} x \sqrt {c+a^2 c x^2} \arctan (a x)^2+\frac {\sqrt {c} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {c+a^2 c x^2}}\right )}{a}+\frac {\left (c \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int x^2 \sec (x) \, dx,x,\arctan (a x)\right )}{2 a \sqrt {c+a^2 c x^2}} \\ & = -\frac {\sqrt {c+a^2 c x^2} \arctan (a x)}{a}+\frac {1}{2} x \sqrt {c+a^2 c x^2} \arctan (a x)^2-\frac {i c \sqrt {1+a^2 x^2} \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2}{a \sqrt {c+a^2 c x^2}}+\frac {\sqrt {c} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {c+a^2 c x^2}}\right )}{a}-\frac {\left (c \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 )}{a \sqrt {c+a^2 c x^2}}+\frac {\left (c \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 )}{a \sqrt {c+a^2 c x^2}} \\ & = -\frac {\sqrt {c+a^2 c x^2} \arctan (a x)}{a}+\frac {1}{2} x \sqrt {c+a^2 c x^2} \arctan (a x)^2-\frac {i c \sqrt {1+a^2 x^2} \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2}{a \sqrt {c+a^2 c x^2}}+\frac {\sqrt {c} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {c+a^2 c x^2}}\right )}{a}+\frac {i c \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )}{a \sqrt {c+a^2 c x^2}}-\frac {i c \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )}{a \sqrt {c+a^2 c x^2}}-\frac {\left (i c \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 )}{a \sqrt {c+a^2 c x^2}}+\frac {\left (i c \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 )}{a \sqrt {c+a^2 c x^2}} \\ & = -\frac {\sqrt {c+a^2 c x^2} \arctan (a x)}{a}+\frac {1}{2} x \sqrt {c+a^2 c x^2} \arctan (a x)^2-\frac {i c \sqrt {1+a^2 x^2} \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2}{a \sqrt {c+a^2 c x^2}}+\frac {\sqrt {c} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {c+a^2 c x^2}}\right )}{a}+\frac {i c \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )}{a \sqrt {c+a^2 c x^2}}-\frac {i c \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )}{a \sqrt {c+a^2 c x^2}}-\frac {\left (c \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 )}{a \sqrt {c+a^2 c x^2}}+\frac {\left (c \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 )}{a \sqrt {c+a^2 c x^2}} \\ & = -\frac {\sqrt {c+a^2 c x^2} \arctan (a x)}{a}+\frac {1}{2} x \sqrt {c+a^2 c x^2} \arctan (a x)^2-\frac {i c \sqrt {1+a^2 x^2} \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2}{a \sqrt {c+a^2 c x^2}}+\frac {\sqrt {c} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {c+a^2 c x^2}}\right )}{a}+\frac {i c \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )}{a \sqrt {c+a^2 c x^2}}-\frac {i c \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )}{a \sqrt {c+a^2 c x^2}}-\frac {c \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (3,-i e^{i \arctan (a x)}\right )}{a \sqrt {c+a^2 c x^2}}+\frac {c \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (3,i e^{i \arctan (a x)}\right )}{a \sqrt {c+a^2 c x^2}} \\ \end{align*}
Time = 0.35 (sec) , antiderivative size = 201, normalized size of antiderivative = 0.59 \[ \int \sqrt {c+a^2 c x^2} \arctan (a x)^2 \, dx=\frac {\sqrt {c \left (1+a^2 x^2\right )} \left (-2 \sqrt {1+a^2 x^2} \arctan (a x)+a x \sqrt {1+a^2 x^2} \arctan (a x)^2-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2+2 \text {arctanh}\left (\frac {a x}{\sqrt {1+a^2 x^2}}\right )+2 i \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-2 i \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-2 \operatorname {PolyLog}\left (3,-i e^{i \arctan (a x)}\right )+2 \operatorname {PolyLog}\left (3,i e^{i \arctan (a x)}\right )\right )}{2 a \sqrt {1+a^2 x^2}} \]
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Time = 1.29 (sec) , antiderivative size = 268, normalized size of antiderivative = 0.79
method | result | size |
default | \(\frac {\sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \arctan \left (a x \right ) \left (x \arctan \left (a x \right ) a -2\right )}{2 a}+\frac {i \sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (i \arctan \left (a x \right )^{2} \ln \left (1+\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-i \arctan \left (a x \right )^{2} \ln \left (1-\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )+2 \arctan \left (a x \right ) \operatorname {polylog}\left (2, -\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-2 \arctan \left (a x \right ) \operatorname {polylog}\left (2, \frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )+2 i \operatorname {polylog}\left (3, -\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-2 i \operatorname {polylog}\left (3, \frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-4 \arctan \left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )\right )}{2 a \sqrt {a^{2} x^{2}+1}}\) | \(268\) |
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\[ \int \sqrt {c+a^2 c x^2} \arctan (a x)^2 \, dx=\int { \sqrt {a^{2} c x^{2} + c} \arctan \left (a x\right )^{2} \,d x } \]
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\[ \int \sqrt {c+a^2 c x^2} \arctan (a x)^2 \, dx=\int \sqrt {c \left (a^{2} x^{2} + 1\right )} \operatorname {atan}^{2}{\left (a x \right )}\, dx \]
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\[ \int \sqrt {c+a^2 c x^2} \arctan (a x)^2 \, dx=\int { \sqrt {a^{2} c x^{2} + c} \arctan \left (a x\right )^{2} \,d x } \]
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Exception generated. \[ \int \sqrt {c+a^2 c x^2} \arctan (a x)^2 \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int \sqrt {c+a^2 c x^2} \arctan (a x)^2 \, dx=\int {\mathrm {atan}\left (a\,x\right )}^2\,\sqrt {c\,a^2\,x^2+c} \,d x \]
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