Optimal. Leaf size=435 \[ -\frac{b \tanh ^{-1}\left (\frac{2 a+b \tan ^2(d+e x)}{2 \sqrt{a} \sqrt{a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}\right )}{4 \sqrt{a} e}+\frac{\sqrt{a} \tanh ^{-1}\left (\frac{2 a+b \tan ^2(d+e x)}{2 \sqrt{a} \sqrt{a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}\right )}{2 e}-\frac{\sqrt{a-b+c} \tanh ^{-1}\left (\frac{2 a+(b-2 c) \tan ^2(d+e x)-b}{2 \sqrt{a-b+c} \sqrt{a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}\right )}{2 e}+\frac{\sqrt{c} \tanh ^{-1}\left (\frac{b+2 c \tan ^2(d+e x)}{2 \sqrt{c} \sqrt{a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}\right )}{2 e}-\frac{b \tanh ^{-1}\left (\frac{b+2 c \tan ^2(d+e x)}{2 \sqrt{c} \sqrt{a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}\right )}{4 \sqrt{c} e}+\frac{(b-2 c) \tanh ^{-1}\left (\frac{b+2 c \tan ^2(d+e x)}{2 \sqrt{c} \sqrt{a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}\right )}{4 \sqrt{c} e}-\frac{\cot ^2(d+e x) \sqrt{a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}{2 e} \]
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Rubi [A] time = 0.544183, antiderivative size = 435, normalized size of antiderivative = 1., number of steps used = 22, number of rules used = 9, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.257, Rules used = {3700, 1251, 960, 732, 843, 621, 206, 724, 734} \[ -\frac{b \tanh ^{-1}\left (\frac{2 a+b \tan ^2(d+e x)}{2 \sqrt{a} \sqrt{a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}\right )}{4 \sqrt{a} e}+\frac{\sqrt{a} \tanh ^{-1}\left (\frac{2 a+b \tan ^2(d+e x)}{2 \sqrt{a} \sqrt{a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}\right )}{2 e}-\frac{\sqrt{a-b+c} \tanh ^{-1}\left (\frac{2 a+(b-2 c) \tan ^2(d+e x)-b}{2 \sqrt{a-b+c} \sqrt{a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}\right )}{2 e}+\frac{\sqrt{c} \tanh ^{-1}\left (\frac{b+2 c \tan ^2(d+e x)}{2 \sqrt{c} \sqrt{a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}\right )}{2 e}-\frac{b \tanh ^{-1}\left (\frac{b+2 c \tan ^2(d+e x)}{2 \sqrt{c} \sqrt{a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}\right )}{4 \sqrt{c} e}+\frac{(b-2 c) \tanh ^{-1}\left (\frac{b+2 c \tan ^2(d+e x)}{2 \sqrt{c} \sqrt{a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}\right )}{4 \sqrt{c} e}-\frac{\cot ^2(d+e x) \sqrt{a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}{2 e} \]
Antiderivative was successfully verified.
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Rule 3700
Rule 1251
Rule 960
Rule 732
Rule 843
Rule 621
Rule 206
Rule 724
Rule 734
Rubi steps
\begin{align*} \int \cot ^3(d+e x) \sqrt{a+b \tan ^2(d+e x)+c \tan ^4(d+e x)} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\sqrt{a+b x^2+c x^4}}{x^3 \left (1+x^2\right )} \, dx,x,\tan (d+e x)\right )}{e}\\ &=\frac{\operatorname{Subst}\left (\int \frac{\sqrt{a+b x+c x^2}}{x^2 (1+x)} \, dx,x,\tan ^2(d+e x)\right )}{2 e}\\ &=\frac{\operatorname{Subst}\left (\int \left (\frac{\sqrt{a+b x+c x^2}}{x^2}-\frac{\sqrt{a+b x+c x^2}}{x}+\frac{\sqrt{a+b x+c x^2}}{1+x}\right ) \, dx,x,\tan ^2(d+e x)\right )}{2 e}\\ &=\frac{\operatorname{Subst}\left (\int \frac{\sqrt{a+b x+c x^2}}{x^2} \, dx,x,\tan ^2(d+e x)\right )}{2 e}-\frac{\operatorname{Subst}\left (\int \frac{\sqrt{a+b x+c x^2}}{x} \, dx,x,\tan ^2(d+e x)\right )}{2 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{\cot ^2(d+e x) \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 x}{x \sqrt{a+b x+c x^2}} \, dx,x,\tan ^2(d+e x)\right )}{4 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{\operatorname{Subst}\left (\int \frac{b+2 c x}{x \sqrt{a+b x+c x^2}} \, dx,x,\tan ^2(d+e x)\right )}{4 e}\\ &=-\frac{\cot ^2(d+e x) \sqrt{a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}{2 e}-\frac{a \operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+b x+c x^2}} \, dx,x,\tan ^2(d+e x)\right )}{2 e}-\frac{b \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+b x+c x^2}} \, dx,x,\tan ^2(d+e x)\right )}{4 e}+\frac{b \operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+b x+c x^2}} \, dx,x,\tan ^2(d+e x)\right )}{4 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{c \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+b x+c x^2}} \, dx,x,\tan ^2(d+e x)\right )}{2 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{\cot ^2(d+e x) \sqrt{a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}{2 e}+\frac{a \operatorname{Subst}\left (\int \frac{1}{4 a-x^2} \, dx,x,\frac{2 a+b \tan ^2(d+e x)}{\sqrt{a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}\right )}{e}-\frac{b \operatorname{Subst}\left (\int \frac{1}{4 a-x^2} \, dx,x,\frac{2 a+b \tan ^2(d+e x)}{\sqrt{a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}\right )}{2 e}-\frac{b \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{(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{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 )}{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} \tanh ^{-1}\left (\frac{2 a+b \tan ^2(d+e x)}{2 \sqrt{a} \sqrt{a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}\right )}{2 e}-\frac{b \tanh ^{-1}\left (\frac{2 a+b \tan ^2(d+e x)}{2 \sqrt{a} \sqrt{a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}\right )}{4 \sqrt{a} 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 \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{(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{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 )}{2 e}-\frac{\cot ^2(d+e x) \sqrt{a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}{2 e}\\ \end{align*}
Mathematica [A] time = 1.15561, size = 187, normalized size = 0.43 \[ \frac{(2 a-b) \tanh ^{-1}\left (\frac{2 a+b \tan ^2(d+e x)}{2 \sqrt{a} \sqrt{a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}\right )-2 \sqrt{a} \left (\sqrt{a-b+c} \tanh ^{-1}\left (\frac{2 a+(b-2 c) \tan ^2(d+e x)-b}{2 \sqrt{a-b+c} \sqrt{a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}\right )+\cot ^2(d+e x) \sqrt{a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}\right )}{4 \sqrt{a} e} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.509, size = 0, normalized size = 0. \begin{align*} \int \left ( \cot \left ( ex+d \right ) \right ) ^{3}\sqrt{a+b \left ( \tan \left ( ex+d \right ) \right ) ^{2}+c \left ( \tan \left ( ex+d \right ) \right ) ^{4}}\, dx \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} \cot \left (e x + d\right )^{3}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 19.0857, size = 2934, normalized size = 6.74 \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 )}} \cot ^{3}{\left (d + e x \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [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} \cot \left (e x + d\right )^{3}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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