Optimal. Leaf size=179 \[ \frac{\sqrt{a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}{2 e}-\frac{\sqrt{a-b+c} \tanh ^{-1}\left (\frac{2 a+(b-2 c) \tan ^2(d+e x)-b}{2 \sqrt{a-b+c} \sqrt{a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}\right )}{2 e}+\frac{(b-2 c) \tanh ^{-1}\left (\frac{b+2 c \tan ^2(d+e x)}{2 \sqrt{c} \sqrt{a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}\right )}{4 \sqrt{c} e} \]
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Rubi [A] time = 0.224228, antiderivative size = 179, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.212, Rules used = {3700, 1247, 734, 843, 621, 206, 724} \[ \frac{\sqrt{a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}{2 e}-\frac{\sqrt{a-b+c} \tanh ^{-1}\left (\frac{2 a+(b-2 c) \tan ^2(d+e x)-b}{2 \sqrt{a-b+c} \sqrt{a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}\right )}{2 e}+\frac{(b-2 c) \tanh ^{-1}\left (\frac{b+2 c \tan ^2(d+e x)}{2 \sqrt{c} \sqrt{a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}\right )}{4 \sqrt{c} e} \]
Antiderivative was successfully verified.
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Rule 3700
Rule 1247
Rule 734
Rule 843
Rule 621
Rule 206
Rule 724
Rubi steps
\begin{align*} \int \tan (d+e x) \sqrt{a+b \tan ^2(d+e x)+c \tan ^4(d+e x)} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{x \sqrt{a+b x^2+c x^4}}{1+x^2} \, dx,x,\tan (d+e x)\right )}{e}\\ &=\frac{\operatorname{Subst}\left (\int \frac{\sqrt{a+b x+c x^2}}{1+x} \, dx,x,\tan ^2(d+e x)\right )}{2 e}\\ &=\frac{\sqrt{a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}{2 e}-\frac{\operatorname{Subst}\left (\int \frac{-2 a+b-(b-2 c) x}{(1+x) \sqrt{a+b x+c x^2}} \, dx,x,\tan ^2(d+e x)\right )}{4 e}\\ &=\frac{\sqrt{a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}{2 e}+\frac{(b-2 c) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+b x+c x^2}} \, dx,x,\tan ^2(d+e x)\right )}{4 e}+\frac{(a-b+c) \operatorname{Subst}\left (\int \frac{1}{(1+x) \sqrt{a+b x+c x^2}} \, dx,x,\tan ^2(d+e x)\right )}{2 e}\\ &=\frac{\sqrt{a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}{2 e}+\frac{(b-2 c) \operatorname{Subst}\left (\int \frac{1}{4 c-x^2} \, dx,x,\frac{b+2 c \tan ^2(d+e x)}{\sqrt{a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}\right )}{2 e}-\frac{(a-b+c) \operatorname{Subst}\left (\int \frac{1}{4 a-4 b+4 c-x^2} \, dx,x,\frac{2 a-b-(-b+2 c) \tan ^2(d+e x)}{\sqrt{a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}\right )}{e}\\ &=-\frac{\sqrt{a-b+c} \tanh ^{-1}\left (\frac{2 a-b+(b-2 c) \tan ^2(d+e x)}{2 \sqrt{a-b+c} \sqrt{a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}\right )}{2 e}+\frac{(b-2 c) \tanh ^{-1}\left (\frac{b+2 c \tan ^2(d+e x)}{2 \sqrt{c} \sqrt{a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}\right )}{4 \sqrt{c} e}+\frac{\sqrt{a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}{2 e}\\ \end{align*}
Mathematica [A] time = 0.278924, size = 180, normalized size = 1.01 \[ \frac{2 \sqrt{c} \sqrt{a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}-2 \sqrt{c} \sqrt{a-b+c} \tanh ^{-1}\left (\frac{2 a+(b-2 c) \tan ^2(d+e x)-b}{2 \sqrt{a-b+c} \sqrt{a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}\right )+(b-2 c) \tanh ^{-1}\left (\frac{b+2 c \tan ^2(d+e x)}{2 \sqrt{c} \sqrt{a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}\right )}{4 \sqrt{c} e} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.186, size = 289, normalized size = 1.6 \begin{align*}{\frac{1}{2\,e}\sqrt{ \left ( 1+ \left ( \tan \left ( ex+d \right ) \right ) ^{2} \right ) ^{2}c+ \left ( b-2\,c \right ) \left ( 1+ \left ( \tan \left ( ex+d \right ) \right ) ^{2} \right ) +a-b+c}}+{\frac{b}{4\,e}\ln \left ({ \left ({\frac{b}{2}}-c+c \left ( 1+ \left ( \tan \left ( ex+d \right ) \right ) ^{2} \right ) \right ){\frac{1}{\sqrt{c}}}}+\sqrt{ \left ( 1+ \left ( \tan \left ( ex+d \right ) \right ) ^{2} \right ) ^{2}c+ \left ( b-2\,c \right ) \left ( 1+ \left ( \tan \left ( ex+d \right ) \right ) ^{2} \right ) +a-b+c} \right ){\frac{1}{\sqrt{c}}}}-{\frac{1}{2\,e}\ln \left ({ \left ({\frac{b}{2}}-c+c \left ( 1+ \left ( \tan \left ( ex+d \right ) \right ) ^{2} \right ) \right ){\frac{1}{\sqrt{c}}}}+\sqrt{ \left ( 1+ \left ( \tan \left ( ex+d \right ) \right ) ^{2} \right ) ^{2}c+ \left ( b-2\,c \right ) \left ( 1+ \left ( \tan \left ( ex+d \right ) \right ) ^{2} \right ) +a-b+c} \right ) \sqrt{c}}-{\frac{1}{2\,e}\sqrt{a-b+c}\ln \left ({\frac{1}{1+ \left ( \tan \left ( ex+d \right ) \right ) ^{2}} \left ( 2\,a-2\,b+2\,c+ \left ( b-2\,c \right ) \left ( 1+ \left ( \tan \left ( ex+d \right ) \right ) ^{2} \right ) +2\,\sqrt{a-b+c}\sqrt{ \left ( 1+ \left ( \tan \left ( ex+d \right ) \right ) ^{2} \right ) ^{2}c+ \left ( b-2\,c \right ) \left ( 1+ \left ( \tan \left ( ex+d \right ) \right ) ^{2} \right ) +a-b+c} \right ) } \right ) } \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 )^{4} + b \tan \left (e x + d\right )^{2} + a} \tan \left (e x + d\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 16.1877, size = 2641, normalized size = 14.75 \begin{align*} \text{result too large to display} \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 ^{2}{\left (d + e x \right )} + c \tan ^{4}{\left (d + e x \right )}} \tan{\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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