Optimal. Leaf size=676 \[ \frac{\sqrt{-a \left (2 c-\sqrt{a^2-2 a c+b^2+c^2}\right )+c \left (c-\sqrt{a^2-2 a c+b^2+c^2}\right )+a^2+b^2} \tan ^{-1}\left (\frac{b \sqrt{a^2-2 a c+b^2+c^2}-\left (-a \left (2 c-\sqrt{a^2-2 a c+b^2+c^2}\right )+c \left (c-\sqrt{a^2-2 a c+b^2+c^2}\right )+a^2+b^2\right ) \tan (d+e x)}{\sqrt{2} \sqrt [4]{a^2-2 a c+b^2+c^2} \sqrt{-a \left (2 c-\sqrt{a^2-2 a c+b^2+c^2}\right )+c \left (c-\sqrt{a^2-2 a c+b^2+c^2}\right )+a^2+b^2} \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{\sqrt{2} e \sqrt [4]{a^2-2 a c+b^2+c^2}}+\frac{\sqrt{-a \left (\sqrt{a^2-2 a c+b^2+c^2}+2 c\right )+c \left (\sqrt{a^2-2 a c+b^2+c^2}+c\right )+a^2+b^2} \tanh ^{-1}\left (\frac{\left (-a \left (\sqrt{a^2-2 a c+b^2+c^2}+2 c\right )+c \left (\sqrt{a^2-2 a c+b^2+c^2}+c\right )+a^2+b^2\right ) \tan (d+e x)+b \sqrt{a^2-2 a c+b^2+c^2}}{\sqrt{2} \sqrt [4]{a^2-2 a c+b^2+c^2} \sqrt{-a \left (\sqrt{a^2-2 a c+b^2+c^2}+2 c\right )+c \left (\sqrt{a^2-2 a c+b^2+c^2}+c\right )+a^2+b^2} \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{\sqrt{2} e \sqrt [4]{a^2-2 a c+b^2+c^2}}-\frac{\left (b^2-4 c (a-2 c)\right ) \tanh ^{-1}\left (\frac{b+2 c \tan (d+e x)}{2 \sqrt{c} \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{8 c^{3/2} e}+\frac{(b+2 c \tan (d+e x)) \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}{4 c e} \]
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Rubi [A] time = 26.5713, antiderivative size = 676, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 9, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273, Rules used = {3700, 1071, 1078, 621, 206, 1036, 1030, 208, 205} \[ \frac{\sqrt{-a \left (2 c-\sqrt{a^2-2 a c+b^2+c^2}\right )+c \left (c-\sqrt{a^2-2 a c+b^2+c^2}\right )+a^2+b^2} \tan ^{-1}\left (\frac{b \sqrt{a^2-2 a c+b^2+c^2}-\left (-a \left (2 c-\sqrt{a^2-2 a c+b^2+c^2}\right )+c \left (c-\sqrt{a^2-2 a c+b^2+c^2}\right )+a^2+b^2\right ) \tan (d+e x)}{\sqrt{2} \sqrt [4]{a^2-2 a c+b^2+c^2} \sqrt{-a \left (2 c-\sqrt{a^2-2 a c+b^2+c^2}\right )+c \left (c-\sqrt{a^2-2 a c+b^2+c^2}\right )+a^2+b^2} \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{\sqrt{2} e \sqrt [4]{a^2-2 a c+b^2+c^2}}+\frac{\sqrt{-a \left (\sqrt{a^2-2 a c+b^2+c^2}+2 c\right )+c \left (\sqrt{a^2-2 a c+b^2+c^2}+c\right )+a^2+b^2} \tanh ^{-1}\left (\frac{\left (-a \left (\sqrt{a^2-2 a c+b^2+c^2}+2 c\right )+c \left (\sqrt{a^2-2 a c+b^2+c^2}+c\right )+a^2+b^2\right ) \tan (d+e x)+b \sqrt{a^2-2 a c+b^2+c^2}}{\sqrt{2} \sqrt [4]{a^2-2 a c+b^2+c^2} \sqrt{-a \left (\sqrt{a^2-2 a c+b^2+c^2}+2 c\right )+c \left (\sqrt{a^2-2 a c+b^2+c^2}+c\right )+a^2+b^2} \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{\sqrt{2} e \sqrt [4]{a^2-2 a c+b^2+c^2}}-\frac{\left (b^2-4 c (a-2 c)\right ) \tanh ^{-1}\left (\frac{b+2 c \tan (d+e x)}{2 \sqrt{c} \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{8 c^{3/2} e}+\frac{(b+2 c \tan (d+e x)) \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}{4 c e} \]
Antiderivative was successfully verified.
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Rule 3700
Rule 1071
Rule 1078
Rule 621
Rule 206
Rule 1036
Rule 1030
Rule 208
Rule 205
Rubi steps
\begin{align*} \int \tan ^2(d+e x) \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{x^2 \sqrt{a+b x+c x^2}}{1+x^2} \, dx,x,\tan (d+e x)\right )}{e}\\ &=\frac{(b+2 c \tan (d+e x)) \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}{4 c e}-\frac{\operatorname{Subst}\left (\int \frac{\frac{1}{4} \left (b^2+4 a c\right )+2 b c x+\frac{1}{4} \left (b^2-4 (a-2 c) c\right ) x^2}{\left (1+x^2\right ) \sqrt{a+b x+c x^2}} \, dx,x,\tan (d+e x)\right )}{2 c e}\\ &=\frac{(b+2 c \tan (d+e x)) \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}{4 c e}-\frac{\operatorname{Subst}\left (\int \frac{\frac{1}{4} \left (b^2+4 a c\right )+\frac{1}{4} \left (-b^2+4 (a-2 c) c\right )+2 b c x}{\left (1+x^2\right ) \sqrt{a+b x+c x^2}} \, dx,x,\tan (d+e x)\right )}{2 c e}-\frac{\left (b^2-4 (a-2 c) c\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+b x+c x^2}} \, dx,x,\tan (d+e x)\right )}{8 c e}\\ &=\frac{(b+2 c \tan (d+e x)) \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}{4 c e}-\frac{\left (b^2-4 (a-2 c) c\right ) \operatorname{Subst}\left (\int \frac{1}{4 c-x^2} \, dx,x,\frac{b+2 c \tan (d+e x)}{\sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{4 c e}+\frac{\operatorname{Subst}\left (\int \frac{-2 c \left (a^2+b^2+c \left (c-\sqrt{a^2+b^2-2 a c+c^2}\right )-a \left (2 c-\sqrt{a^2+b^2-2 a c+c^2}\right )\right )-2 b c \sqrt{a^2+b^2-2 a c+c^2} x}{\left (1+x^2\right ) \sqrt{a+b x+c x^2}} \, dx,x,\tan (d+e x)\right )}{4 c \sqrt{a^2+b^2-2 a c+c^2} e}-\frac{\operatorname{Subst}\left (\int \frac{-2 c \left (a^2+b^2+c \left (c+\sqrt{a^2+b^2-2 a c+c^2}\right )-a \left (2 c+\sqrt{a^2+b^2-2 a c+c^2}\right )\right )+2 b c \sqrt{a^2+b^2-2 a c+c^2} x}{\left (1+x^2\right ) \sqrt{a+b x+c x^2}} \, dx,x,\tan (d+e x)\right )}{4 c \sqrt{a^2+b^2-2 a c+c^2} e}\\ &=-\frac{\left (b^2-4 (a-2 c) c\right ) \tanh ^{-1}\left (\frac{b+2 c \tan (d+e x)}{2 \sqrt{c} \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{8 c^{3/2} e}+\frac{(b+2 c \tan (d+e x)) \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}{4 c e}-\frac{\left (2 b c \left (a^2+b^2+c \left (c-\sqrt{a^2+b^2-2 a c+c^2}\right )-a \left (2 c-\sqrt{a^2+b^2-2 a c+c^2}\right )\right )\right ) \operatorname{Subst}\left (\int \frac{1}{8 b c^2 \sqrt{a^2+b^2-2 a c+c^2} \left (a^2+b^2+c \left (c-\sqrt{a^2+b^2-2 a c+c^2}\right )-a \left (2 c-\sqrt{a^2+b^2-2 a c+c^2}\right )\right )+b x^2} \, dx,x,\frac{-2 b c \sqrt{a^2+b^2-2 a c+c^2}+2 c \left (a^2+b^2+c \left (c-\sqrt{a^2+b^2-2 a c+c^2}\right )-a \left (2 c-\sqrt{a^2+b^2-2 a c+c^2}\right )\right ) \tan (d+e x)}{\sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{e}-\frac{\left (2 b c \left (a^2+b^2+c \left (c+\sqrt{a^2+b^2-2 a c+c^2}\right )-a \left (2 c+\sqrt{a^2+b^2-2 a c+c^2}\right )\right )\right ) \operatorname{Subst}\left (\int \frac{1}{-8 b c^2 \sqrt{a^2+b^2-2 a c+c^2} \left (a^2+b^2+c \left (c+\sqrt{a^2+b^2-2 a c+c^2}\right )-a \left (2 c+\sqrt{a^2+b^2-2 a c+c^2}\right )\right )+b x^2} \, dx,x,\frac{2 b c \sqrt{a^2+b^2-2 a c+c^2}+2 c \left (a^2+b^2+c \left (c+\sqrt{a^2+b^2-2 a c+c^2}\right )-a \left (2 c+\sqrt{a^2+b^2-2 a c+c^2}\right )\right ) \tan (d+e x)}{\sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{e}\\ &=\frac{\sqrt{a^2+b^2+c \left (c-\sqrt{a^2+b^2-2 a c+c^2}\right )-a \left (2 c-\sqrt{a^2+b^2-2 a c+c^2}\right )} \tan ^{-1}\left (\frac{b \sqrt{a^2+b^2-2 a c+c^2}-\left (a^2+b^2+c \left (c-\sqrt{a^2+b^2-2 a c+c^2}\right )-a \left (2 c-\sqrt{a^2+b^2-2 a c+c^2}\right )\right ) \tan (d+e x)}{\sqrt{2} \sqrt [4]{a^2+b^2-2 a c+c^2} \sqrt{a^2+b^2+c \left (c-\sqrt{a^2+b^2-2 a c+c^2}\right )-a \left (2 c-\sqrt{a^2+b^2-2 a c+c^2}\right )} \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{\sqrt{2} \sqrt [4]{a^2+b^2-2 a c+c^2} e}-\frac{\left (b^2-4 (a-2 c) c\right ) \tanh ^{-1}\left (\frac{b+2 c \tan (d+e x)}{2 \sqrt{c} \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{8 c^{3/2} e}+\frac{\sqrt{a^2+b^2+c \left (c+\sqrt{a^2+b^2-2 a c+c^2}\right )-a \left (2 c+\sqrt{a^2+b^2-2 a c+c^2}\right )} \tanh ^{-1}\left (\frac{b \sqrt{a^2+b^2-2 a c+c^2}+\left (a^2+b^2+c \left (c+\sqrt{a^2+b^2-2 a c+c^2}\right )-a \left (2 c+\sqrt{a^2+b^2-2 a c+c^2}\right )\right ) \tan (d+e x)}{\sqrt{2} \sqrt [4]{a^2+b^2-2 a c+c^2} \sqrt{a^2+b^2+c \left (c+\sqrt{a^2+b^2-2 a c+c^2}\right )-a \left (2 c+\sqrt{a^2+b^2-2 a c+c^2}\right )} \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{\sqrt{2} \sqrt [4]{a^2+b^2-2 a c+c^2} e}+\frac{(b+2 c \tan (d+e x)) \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}{4 c e}\\ \end{align*}
Mathematica [C] time = 0.53491, size = 405, normalized size = 0.6 \[ \frac{-\frac{\left (b^2-4 a c\right ) \tanh ^{-1}\left (\frac{b+2 c \tan (d+e x)}{2 \sqrt{c} \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{8 c^{3/2}}+\frac{(b+2 c \tan (d+e x)) \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}{4 c}+\frac{1}{4} i \left (2 \sqrt{a-i b-c} \tanh ^{-1}\left (\frac{2 a+(b-2 i c) \tan (d+e x)-i b}{2 \sqrt{a-i b-c} \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )-\frac{(b-2 i c) \tanh ^{-1}\left (\frac{b+2 c \tan (d+e x)}{2 \sqrt{c} \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{\sqrt{c}}\right )-\frac{1}{4} i \left (2 \sqrt{a+i b-c} \tanh ^{-1}\left (\frac{2 a+(b+2 i c) \tan (d+e x)+i b}{2 \sqrt{a+i b-c} \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )-\frac{(b+2 i c) \tanh ^{-1}\left (\frac{b+2 c \tan (d+e x)}{2 \sqrt{c} \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{\sqrt{c}}\right )}{e} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.45, size = 17247074, normalized size = 25513.4 \begin{align*} \text{output too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{c \tan \left (e x + d\right )^{2} + b \tan \left (e x + d\right ) + a} \tan \left (e x + d\right )^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{a + b \tan{\left (d + e x \right )} + c \tan ^{2}{\left (d + e x \right )}} \tan ^{2}{\left (d + e x \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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