Optimal. Leaf size=140 \[ a x-\frac{b \log \left (c x^2-\sqrt{2} \sqrt{c} x+1\right )}{2 \sqrt{2} \sqrt{c}}+\frac{b \log \left (c x^2+\sqrt{2} \sqrt{c} x+1\right )}{2 \sqrt{2} \sqrt{c}}+b x \tan ^{-1}\left (c x^2\right )+\frac{b \tan ^{-1}\left (1-\sqrt{2} \sqrt{c} x\right )}{\sqrt{2} \sqrt{c}}-\frac{b \tan ^{-1}\left (\sqrt{2} \sqrt{c} x+1\right )}{\sqrt{2} \sqrt{c}} \]
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Rubi [A] time = 0.104643, antiderivative size = 140, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 7, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.7, Rules used = {5027, 297, 1162, 617, 204, 1165, 628} \[ a x-\frac{b \log \left (c x^2-\sqrt{2} \sqrt{c} x+1\right )}{2 \sqrt{2} \sqrt{c}}+\frac{b \log \left (c x^2+\sqrt{2} \sqrt{c} x+1\right )}{2 \sqrt{2} \sqrt{c}}+b x \tan ^{-1}\left (c x^2\right )+\frac{b \tan ^{-1}\left (1-\sqrt{2} \sqrt{c} x\right )}{\sqrt{2} \sqrt{c}}-\frac{b \tan ^{-1}\left (\sqrt{2} \sqrt{c} x+1\right )}{\sqrt{2} \sqrt{c}} \]
Antiderivative was successfully verified.
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Rule 5027
Rule 297
Rule 1162
Rule 617
Rule 204
Rule 1165
Rule 628
Rubi steps
\begin{align*} \int \left (a+b \tan ^{-1}\left (c x^2\right )\right ) \, dx &=a x+b \int \tan ^{-1}\left (c x^2\right ) \, dx\\ &=a x+b x \tan ^{-1}\left (c x^2\right )-(2 b c) \int \frac{x^2}{1+c^2 x^4} \, dx\\ &=a x+b x \tan ^{-1}\left (c x^2\right )+b \int \frac{1-c x^2}{1+c^2 x^4} \, dx-b \int \frac{1+c x^2}{1+c^2 x^4} \, dx\\ &=a x+b x \tan ^{-1}\left (c x^2\right )-\frac{b \int \frac{1}{\frac{1}{c}-\frac{\sqrt{2} x}{\sqrt{c}}+x^2} \, dx}{2 c}-\frac{b \int \frac{1}{\frac{1}{c}+\frac{\sqrt{2} x}{\sqrt{c}}+x^2} \, dx}{2 c}-\frac{b \int \frac{\frac{\sqrt{2}}{\sqrt{c}}+2 x}{-\frac{1}{c}-\frac{\sqrt{2} x}{\sqrt{c}}-x^2} \, dx}{2 \sqrt{2} \sqrt{c}}-\frac{b \int \frac{\frac{\sqrt{2}}{\sqrt{c}}-2 x}{-\frac{1}{c}+\frac{\sqrt{2} x}{\sqrt{c}}-x^2} \, dx}{2 \sqrt{2} \sqrt{c}}\\ &=a x+b x \tan ^{-1}\left (c x^2\right )-\frac{b \log \left (1-\sqrt{2} \sqrt{c} x+c x^2\right )}{2 \sqrt{2} \sqrt{c}}+\frac{b \log \left (1+\sqrt{2} \sqrt{c} x+c x^2\right )}{2 \sqrt{2} \sqrt{c}}-\frac{b \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1-\sqrt{2} \sqrt{c} x\right )}{\sqrt{2} \sqrt{c}}+\frac{b \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1+\sqrt{2} \sqrt{c} x\right )}{\sqrt{2} \sqrt{c}}\\ &=a x+b x \tan ^{-1}\left (c x^2\right )+\frac{b \tan ^{-1}\left (1-\sqrt{2} \sqrt{c} x\right )}{\sqrt{2} \sqrt{c}}-\frac{b \tan ^{-1}\left (1+\sqrt{2} \sqrt{c} x\right )}{\sqrt{2} \sqrt{c}}-\frac{b \log \left (1-\sqrt{2} \sqrt{c} x+c x^2\right )}{2 \sqrt{2} \sqrt{c}}+\frac{b \log \left (1+\sqrt{2} \sqrt{c} x+c x^2\right )}{2 \sqrt{2} \sqrt{c}}\\ \end{align*}
Mathematica [A] time = 0.042094, size = 107, normalized size = 0.76 \[ a x+b x \tan ^{-1}\left (c x^2\right )-\frac{b \left (\log \left (c x^2-\sqrt{2} \sqrt{c} x+1\right )-\log \left (c x^2+\sqrt{2} \sqrt{c} x+1\right )-2 \tan ^{-1}\left (1-\sqrt{2} \sqrt{c} x\right )+2 \tan ^{-1}\left (\sqrt{2} \sqrt{c} x+1\right )\right )}{2 \sqrt{2} \sqrt{c}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.023, size = 125, normalized size = 0.9 \begin{align*} ax+bx\arctan \left ( c{x}^{2} \right ) -{\frac{b\sqrt{2}}{4\,c}\ln \left ({ \left ({x}^{2}-\sqrt [4]{{c}^{-2}}x\sqrt{2}+\sqrt{{c}^{-2}} \right ) \left ({x}^{2}+\sqrt [4]{{c}^{-2}}x\sqrt{2}+\sqrt{{c}^{-2}} \right ) ^{-1}} \right ){\frac{1}{\sqrt [4]{{c}^{-2}}}}}-{\frac{b\sqrt{2}}{2\,c}\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{{c}^{-2}}}}}+1 \right ){\frac{1}{\sqrt [4]{{c}^{-2}}}}}-{\frac{b\sqrt{2}}{2\,c}\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{{c}^{-2}}}}}-1 \right ){\frac{1}{\sqrt [4]{{c}^{-2}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.51946, size = 348, normalized size = 2.49 \begin{align*} \frac{1}{4} \,{\left (c{\left (\frac{\sqrt{2} \log \left (\sqrt{c^{2}} x^{2} + \sqrt{2}{\left (c^{2}\right )}^{\frac{1}{4}} x + 1\right )}{{\left (c^{2}\right )}^{\frac{3}{4}}} - \frac{\sqrt{2} \log \left (\sqrt{c^{2}} x^{2} - \sqrt{2}{\left (c^{2}\right )}^{\frac{1}{4}} x + 1\right )}{{\left (c^{2}\right )}^{\frac{3}{4}}} - \frac{\sqrt{2} \log \left (\frac{2 \, \sqrt{c^{2}} x - \sqrt{2} \sqrt{-\sqrt{c^{2}}} + \sqrt{2}{\left (c^{2}\right )}^{\frac{1}{4}}}{2 \, \sqrt{c^{2}} x + \sqrt{2} \sqrt{-\sqrt{c^{2}}} + \sqrt{2}{\left (c^{2}\right )}^{\frac{1}{4}}}\right )}{\sqrt{c^{2}} \sqrt{-\sqrt{c^{2}}}} - \frac{\sqrt{2} \log \left (\frac{2 \, \sqrt{c^{2}} x - \sqrt{2} \sqrt{-\sqrt{c^{2}}} - \sqrt{2}{\left (c^{2}\right )}^{\frac{1}{4}}}{2 \, \sqrt{c^{2}} x + \sqrt{2} \sqrt{-\sqrt{c^{2}}} - \sqrt{2}{\left (c^{2}\right )}^{\frac{1}{4}}}\right )}{\sqrt{c^{2}} \sqrt{-\sqrt{c^{2}}}}\right )} + 4 \, x \arctan \left (c x^{2}\right )\right )} b + a x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.82601, size = 768, normalized size = 5.49 \begin{align*} b x \arctan \left (c x^{2}\right ) + a x + \sqrt{2} \left (\frac{b^{4}}{c^{2}}\right )^{\frac{1}{4}} \arctan \left (-\frac{\sqrt{2} \left (\frac{b^{4}}{c^{2}}\right )^{\frac{1}{4}} b^{3} c x + b^{4} - \sqrt{2} \sqrt{b^{6} x^{2} + \sqrt{2} \left (\frac{b^{4}}{c^{2}}\right )^{\frac{3}{4}} b^{3} c x + \sqrt{\frac{b^{4}}{c^{2}}} b^{4}} \left (\frac{b^{4}}{c^{2}}\right )^{\frac{1}{4}} c}{b^{4}}\right ) + \sqrt{2} \left (\frac{b^{4}}{c^{2}}\right )^{\frac{1}{4}} \arctan \left (-\frac{\sqrt{2} \left (\frac{b^{4}}{c^{2}}\right )^{\frac{1}{4}} b^{3} c x - b^{4} - \sqrt{2} \sqrt{b^{6} x^{2} - \sqrt{2} \left (\frac{b^{4}}{c^{2}}\right )^{\frac{3}{4}} b^{3} c x + \sqrt{\frac{b^{4}}{c^{2}}} b^{4}} \left (\frac{b^{4}}{c^{2}}\right )^{\frac{1}{4}} c}{b^{4}}\right ) + \frac{1}{4} \, \sqrt{2} \left (\frac{b^{4}}{c^{2}}\right )^{\frac{1}{4}} \log \left (b^{6} x^{2} + \sqrt{2} \left (\frac{b^{4}}{c^{2}}\right )^{\frac{3}{4}} b^{3} c x + \sqrt{\frac{b^{4}}{c^{2}}} b^{4}\right ) - \frac{1}{4} \, \sqrt{2} \left (\frac{b^{4}}{c^{2}}\right )^{\frac{1}{4}} \log \left (b^{6} x^{2} - \sqrt{2} \left (\frac{b^{4}}{c^{2}}\right )^{\frac{3}{4}} b^{3} c x + \sqrt{\frac{b^{4}}{c^{2}}} b^{4}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 15.0671, size = 146, normalized size = 1.04 \begin{align*} a x + b \left (\begin{cases} - \frac{\left (-1\right )^{\frac{3}{4}} c^{3} \left (\frac{1}{c^{2}}\right )^{\frac{7}{4}} \log{\left (x^{2} + i \sqrt{\frac{1}{c^{2}}} \right )}}{2} + \left (-1\right )^{\frac{3}{4}} c \left (\frac{1}{c^{2}}\right )^{\frac{3}{4}} \log{\left (x - \sqrt [4]{-1} \sqrt [4]{\frac{1}{c^{2}}} \right )} - \left (-1\right )^{\frac{3}{4}} c \left (\frac{1}{c^{2}}\right )^{\frac{3}{4}} \operatorname{atan}{\left (\frac{\left (-1\right )^{\frac{3}{4}} x}{\sqrt [4]{\frac{1}{c^{2}}}} \right )} + x \operatorname{atan}{\left (c x^{2} \right )} + \frac{\sqrt [4]{-1} \operatorname{atan}{\left (c x^{2} \right )}}{c^{4} \left (\frac{1}{c^{2}}\right )^{\frac{7}{4}}} & \text{for}\: c \neq 0 \\0 & \text{otherwise} \end{cases}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.14414, size = 201, normalized size = 1.44 \begin{align*} -\frac{1}{4} \,{\left (c{\left (\frac{2 \, \sqrt{2} \sqrt{{\left | c \right |}} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (2 \, x + \frac{\sqrt{2}}{\sqrt{{\left | c \right |}}}\right )} \sqrt{{\left | c \right |}}\right )}{c^{2}} + \frac{2 \, \sqrt{2} \sqrt{{\left | c \right |}} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (2 \, x - \frac{\sqrt{2}}{\sqrt{{\left | c \right |}}}\right )} \sqrt{{\left | c \right |}}\right )}{c^{2}} - \frac{\sqrt{2} \sqrt{{\left | c \right |}} \log \left (x^{2} + \frac{\sqrt{2} x}{\sqrt{{\left | c \right |}}} + \frac{1}{{\left | c \right |}}\right )}{c^{2}} + \frac{\sqrt{2} \sqrt{{\left | c \right |}} \log \left (x^{2} - \frac{\sqrt{2} x}{\sqrt{{\left | c \right |}}} + \frac{1}{{\left | c \right |}}\right )}{c^{2}}\right )} - 4 \, x \arctan \left (c x^{2}\right )\right )} b + a x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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