Optimal. Leaf size=340 \[ \frac {i c \sqrt {a^2 x^2+1} \tan ^{-1}(a x) \text {Li}_2\left (-i e^{i \tan ^{-1}(a x)}\right )}{a \sqrt {a^2 c x^2+c}}-\frac {i c \sqrt {a^2 x^2+1} \tan ^{-1}(a x) \text {Li}_2\left (i e^{i \tan ^{-1}(a x)}\right )}{a \sqrt {a^2 c x^2+c}}-\frac {c \sqrt {a^2 x^2+1} \text {Li}_3\left (-i e^{i \tan ^{-1}(a x)}\right )}{a \sqrt {a^2 c x^2+c}}+\frac {c \sqrt {a^2 x^2+1} \text {Li}_3\left (i e^{i \tan ^{-1}(a x)}\right )}{a \sqrt {a^2 c x^2+c}}-\frac {i c \sqrt {a^2 x^2+1} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^2}{a \sqrt {a^2 c x^2+c}}+\frac {1}{2} x \sqrt {a^2 c x^2+c} \tan ^{-1}(a x)^2-\frac {\sqrt {a^2 c x^2+c} \tan ^{-1}(a x)}{a}+\frac {\sqrt {c} \tanh ^{-1}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{a} \]
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Rubi [A] time = 0.22, antiderivative size = 340, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 9, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {4880, 4890, 4888, 4181, 2531, 2282, 6589, 217, 206} \[ \frac {i c \sqrt {a^2 x^2+1} \tan ^{-1}(a x) \text {PolyLog}\left (2,-i e^{i \tan ^{-1}(a x)}\right )}{a \sqrt {a^2 c x^2+c}}-\frac {i c \sqrt {a^2 x^2+1} \tan ^{-1}(a x) \text {PolyLog}\left (2,i e^{i \tan ^{-1}(a x)}\right )}{a \sqrt {a^2 c x^2+c}}-\frac {c \sqrt {a^2 x^2+1} \text {PolyLog}\left (3,-i e^{i \tan ^{-1}(a x)}\right )}{a \sqrt {a^2 c x^2+c}}+\frac {c \sqrt {a^2 x^2+1} \text {PolyLog}\left (3,i e^{i \tan ^{-1}(a x)}\right )}{a \sqrt {a^2 c x^2+c}}-\frac {i c \sqrt {a^2 x^2+1} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^2}{a \sqrt {a^2 c x^2+c}}+\frac {1}{2} x \sqrt {a^2 c x^2+c} \tan ^{-1}(a x)^2-\frac {\sqrt {a^2 c x^2+c} \tan ^{-1}(a x)}{a}+\frac {\sqrt {c} \tanh ^{-1}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{a} \]
Antiderivative was successfully verified.
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Rule 206
Rule 217
Rule 2282
Rule 2531
Rule 4181
Rule 4880
Rule 4888
Rule 4890
Rule 6589
Rubi steps
\begin {align*} \int \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2 \, dx &=-\frac {\sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{a}+\frac {1}{2} x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac {1}{2} c \int \frac {\tan ^{-1}(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} \tan ^{-1}(a x)}{a}+\frac {1}{2} x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2+c \operatorname {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 {\tan ^{-1}(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} \tan ^{-1}(a x)}{a}+\frac {1}{2} x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac {\sqrt {c} \tanh ^{-1}\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 ) \operatorname {Subst}\left (\int x^2 \sec (x) \, dx,x,\tan ^{-1}(a x)\right )}{2 a \sqrt {c+a^2 c x^2}}\\ &=-\frac {\sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{a}+\frac {1}{2} x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2-\frac {i c \sqrt {1+a^2 x^2} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^2}{a \sqrt {c+a^2 c x^2}}+\frac {\sqrt {c} \tanh ^{-1}\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 ) \operatorname {Subst}\left (\int x \log \left (1-i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{a \sqrt {c+a^2 c x^2}}+\frac {\left (c \sqrt {1+a^2 x^2}\right ) \operatorname {Subst}\left (\int x \log \left (1+i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{a \sqrt {c+a^2 c x^2}}\\ &=-\frac {\sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{a}+\frac {1}{2} x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2-\frac {i c \sqrt {1+a^2 x^2} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^2}{a \sqrt {c+a^2 c x^2}}+\frac {\sqrt {c} \tanh ^{-1}\left (\frac {a \sqrt {c} x}{\sqrt {c+a^2 c x^2}}\right )}{a}+\frac {i c \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \text {Li}_2\left (-i e^{i \tan ^{-1}(a x)}\right )}{a \sqrt {c+a^2 c x^2}}-\frac {i c \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \text {Li}_2\left (i e^{i \tan ^{-1}(a x)}\right )}{a \sqrt {c+a^2 c x^2}}-\frac {\left (i c \sqrt {1+a^2 x^2}\right ) \operatorname {Subst}\left (\int \text {Li}_2\left (-i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{a \sqrt {c+a^2 c x^2}}+\frac {\left (i c \sqrt {1+a^2 x^2}\right ) \operatorname {Subst}\left (\int \text {Li}_2\left (i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{a \sqrt {c+a^2 c x^2}}\\ &=-\frac {\sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{a}+\frac {1}{2} x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2-\frac {i c \sqrt {1+a^2 x^2} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^2}{a \sqrt {c+a^2 c x^2}}+\frac {\sqrt {c} \tanh ^{-1}\left (\frac {a \sqrt {c} x}{\sqrt {c+a^2 c x^2}}\right )}{a}+\frac {i c \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \text {Li}_2\left (-i e^{i \tan ^{-1}(a x)}\right )}{a \sqrt {c+a^2 c x^2}}-\frac {i c \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \text {Li}_2\left (i e^{i \tan ^{-1}(a x)}\right )}{a \sqrt {c+a^2 c x^2}}-\frac {\left (c \sqrt {1+a^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_2(-i x)}{x} \, dx,x,e^{i \tan ^{-1}(a x)}\right )}{a \sqrt {c+a^2 c x^2}}+\frac {\left (c \sqrt {1+a^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_2(i x)}{x} \, dx,x,e^{i \tan ^{-1}(a x)}\right )}{a \sqrt {c+a^2 c x^2}}\\ &=-\frac {\sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{a}+\frac {1}{2} x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2-\frac {i c \sqrt {1+a^2 x^2} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^2}{a \sqrt {c+a^2 c x^2}}+\frac {\sqrt {c} \tanh ^{-1}\left (\frac {a \sqrt {c} x}{\sqrt {c+a^2 c x^2}}\right )}{a}+\frac {i c \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \text {Li}_2\left (-i e^{i \tan ^{-1}(a x)}\right )}{a \sqrt {c+a^2 c x^2}}-\frac {i c \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \text {Li}_2\left (i e^{i \tan ^{-1}(a x)}\right )}{a \sqrt {c+a^2 c x^2}}-\frac {c \sqrt {1+a^2 x^2} \text {Li}_3\left (-i e^{i \tan ^{-1}(a x)}\right )}{a \sqrt {c+a^2 c x^2}}+\frac {c \sqrt {1+a^2 x^2} \text {Li}_3\left (i e^{i \tan ^{-1}(a x)}\right )}{a \sqrt {c+a^2 c x^2}}\\ \end {align*}
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Mathematica [A] time = 0.42, size = 201, normalized size = 0.59 \[ \frac {\sqrt {c \left (a^2 x^2+1\right )} \left (a x \sqrt {a^2 x^2+1} \tan ^{-1}(a x)^2-2 \sqrt {a^2 x^2+1} \tan ^{-1}(a x)+2 \tanh ^{-1}\left (\frac {a x}{\sqrt {a^2 x^2+1}}\right )+2 i \tan ^{-1}(a x) \text {Li}_2\left (-i e^{i \tan ^{-1}(a x)}\right )-2 i \tan ^{-1}(a x) \text {Li}_2\left (i e^{i \tan ^{-1}(a x)}\right )-2 \text {Li}_3\left (-i e^{i \tan ^{-1}(a x)}\right )+2 \text {Li}_3\left (i e^{i \tan ^{-1}(a x)}\right )-2 i \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^2\right )}{2 a \sqrt {a^2 x^2+1}} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.53, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\sqrt {a^{2} c x^{2} + c} \arctan \left (a x\right )^{2}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.84, size = 268, normalized size = 0.79 \[ \frac {\sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \arctan \left (a x \right ) \left (\arctan \left (a x \right ) x a -2\right )}{2 a}-\frac {i \sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (i \arctan \left (a x \right )^{2} \ln \left (1-\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-i \arctan \left (a x \right )^{2} \ln \left (1+\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )+2 \arctan \left (a x \right ) \polylog \left (2, \frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )+2 i \polylog \left (3, \frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-2 \arctan \left (a x \right ) \polylog \left (2, -\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-2 i \polylog \left (3, -\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )+4 \arctan \left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )\right )}{2 a \sqrt {a^{2} x^{2}+1}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {a^{2} c x^{2} + c} \arctan \left (a x\right )^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int {\mathrm {atan}\left (a\,x\right )}^2\,\sqrt {c\,a^2\,x^2+c} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {c \left (a^{2} x^{2} + 1\right )} \operatorname {atan}^{2}{\left (a x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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