Optimal. Leaf size=294 \[ \frac{\sqrt{-\sqrt{a^2-2 a c+b^2+c^2}+a-c} \tanh ^{-1}\left (\frac{-\sqrt{a^2-2 a c+b^2+c^2}+a+b \tan (d+e x)-c}{\sqrt{2} \sqrt{-\sqrt{a^2-2 a c+b^2+c^2}+a-c} \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{\sqrt{2} e \sqrt{a^2-2 a c+b^2+c^2}}-\frac{\sqrt{\sqrt{a^2-2 a c+b^2+c^2}+a-c} \tanh ^{-1}\left (\frac{\sqrt{a^2-2 a c+b^2+c^2}+a+b \tan (d+e x)-c}{\sqrt{2} \sqrt{\sqrt{a^2-2 a c+b^2+c^2}+a-c} \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{\sqrt{2} e \sqrt{a^2-2 a c+b^2+c^2}} \]
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Rubi [A] time = 0.285837, antiderivative size = 294, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.129, Rules used = {3700, 1036, 1030, 208} \[ \frac{\sqrt{-\sqrt{a^2-2 a c+b^2+c^2}+a-c} \tanh ^{-1}\left (\frac{-\sqrt{a^2-2 a c+b^2+c^2}+a+b \tan (d+e x)-c}{\sqrt{2} \sqrt{-\sqrt{a^2-2 a c+b^2+c^2}+a-c} \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{\sqrt{2} e \sqrt{a^2-2 a c+b^2+c^2}}-\frac{\sqrt{\sqrt{a^2-2 a c+b^2+c^2}+a-c} \tanh ^{-1}\left (\frac{\sqrt{a^2-2 a c+b^2+c^2}+a+b \tan (d+e x)-c}{\sqrt{2} \sqrt{\sqrt{a^2-2 a c+b^2+c^2}+a-c} \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{\sqrt{2} e \sqrt{a^2-2 a c+b^2+c^2}} \]
Antiderivative was successfully verified.
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Rule 3700
Rule 1036
Rule 1030
Rule 208
Rubi steps
\begin{align*} \int \frac{\tan (d+e x)}{\sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{x}{\left (1+x^2\right ) \sqrt{a+b x+c x^2}} \, dx,x,\tan (d+e x)\right )}{e}\\ &=-\frac{\operatorname{Subst}\left (\int \frac{-b+\left (a-c-\sqrt{a^2+b^2-2 a c+c^2}\right ) x}{\left (1+x^2\right ) \sqrt{a+b x+c x^2}} \, dx,x,\tan (d+e x)\right )}{2 \sqrt{a^2+b^2-2 a c+c^2} e}+\frac{\operatorname{Subst}\left (\int \frac{-b+\left (a-c+\sqrt{a^2+b^2-2 a c+c^2}\right ) x}{\left (1+x^2\right ) \sqrt{a+b x+c x^2}} \, dx,x,\tan (d+e x)\right )}{2 \sqrt{a^2+b^2-2 a c+c^2} e}\\ &=-\frac{\left (b \left (a-c-\sqrt{a^2+b^2-2 a c+c^2}\right )\right ) \operatorname{Subst}\left (\int \frac{1}{-2 b \left (a-c-\sqrt{a^2+b^2-2 a c+c^2}\right )+b x^2} \, dx,x,\frac{a-c-\sqrt{a^2+b^2-2 a c+c^2}+b \tan (d+e x)}{\sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{\sqrt{a^2+b^2-2 a c+c^2} e}+\frac{\left (b \left (a-c+\sqrt{a^2+b^2-2 a c+c^2}\right )\right ) \operatorname{Subst}\left (\int \frac{1}{-2 b \left (a-c+\sqrt{a^2+b^2-2 a c+c^2}\right )+b x^2} \, dx,x,\frac{a-c+\sqrt{a^2+b^2-2 a c+c^2}+b \tan (d+e x)}{\sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{\sqrt{a^2+b^2-2 a c+c^2} e}\\ &=\frac{\sqrt{a-c-\sqrt{a^2+b^2-2 a c+c^2}} \tanh ^{-1}\left (\frac{a-c-\sqrt{a^2+b^2-2 a c+c^2}+b \tan (d+e x)}{\sqrt{2} \sqrt{a-c-\sqrt{a^2+b^2-2 a c+c^2}} \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{\sqrt{2} \sqrt{a^2+b^2-2 a c+c^2} e}-\frac{\sqrt{a-c+\sqrt{a^2+b^2-2 a c+c^2}} \tanh ^{-1}\left (\frac{a-c+\sqrt{a^2+b^2-2 a c+c^2}+b \tan (d+e x)}{\sqrt{2} \sqrt{a-c+\sqrt{a^2+b^2-2 a c+c^2}} \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{\sqrt{2} \sqrt{a^2+b^2-2 a c+c^2} e}\\ \end{align*}
Mathematica [C] time = 0.0930891, size = 173, normalized size = 0.59 \[ \frac{-\frac{\tanh ^{-1}\left (\frac{2 a+(b-2 i c) \tan (d+e x)-i b}{2 \sqrt{a-i b-c} \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{2 \sqrt{a-i b-c}}-\frac{\tanh ^{-1}\left (\frac{2 a+(b+2 i c) \tan (d+e x)+i b}{2 \sqrt{a+i b-c} \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{2 \sqrt{a+i b-c}}}{e} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.326, size = 9338488, normalized size = 31763.6 \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(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \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 \frac{\tan{\left (d + e x \right )}}{\sqrt{a + b \tan{\left (d + e x \right )} + c \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(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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