Optimal. Leaf size=500 \[ \frac{\sqrt{-\sqrt{a^2-2 a c+b^2+c^2}+a-c} \tanh ^{-1}\left (\frac{-\sqrt{a^2-2 a c+b^2+c^2}+a+b \tan (d+e x)-c}{\sqrt{2} \sqrt{-\sqrt{a^2-2 a c+b^2+c^2}+a-c} \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{\sqrt{2} e \sqrt{a^2-2 a c+b^2+c^2}}-\frac{\sqrt{\sqrt{a^2-2 a c+b^2+c^2}+a-c} \tanh ^{-1}\left (\frac{\sqrt{a^2-2 a c+b^2+c^2}+a+b \tan (d+e x)-c}{\sqrt{2} \sqrt{\sqrt{a^2-2 a c+b^2+c^2}+a-c} \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{\sqrt{2} e \sqrt{a^2-2 a c+b^2+c^2}}-\frac{\left (3 b^2-4 a c\right ) \tanh ^{-1}\left (\frac{2 a+b \tan (d+e x)}{2 \sqrt{a} \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{8 a^{5/2} e}+\frac{3 b \cot (d+e x) \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}{4 a^2 e}+\frac{\tanh ^{-1}\left (\frac{2 a+b \tan (d+e x)}{2 \sqrt{a} \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{\sqrt{a} e}-\frac{\cot ^2(d+e x) \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}{2 a e} \]
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Rubi [A] time = 0.799917, antiderivative size = 500, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 9, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273, Rules used = {3700, 6725, 744, 806, 724, 206, 1036, 1030, 208} \[ \frac{\sqrt{-\sqrt{a^2-2 a c+b^2+c^2}+a-c} \tanh ^{-1}\left (\frac{-\sqrt{a^2-2 a c+b^2+c^2}+a+b \tan (d+e x)-c}{\sqrt{2} \sqrt{-\sqrt{a^2-2 a c+b^2+c^2}+a-c} \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{\sqrt{2} e \sqrt{a^2-2 a c+b^2+c^2}}-\frac{\sqrt{\sqrt{a^2-2 a c+b^2+c^2}+a-c} \tanh ^{-1}\left (\frac{\sqrt{a^2-2 a c+b^2+c^2}+a+b \tan (d+e x)-c}{\sqrt{2} \sqrt{\sqrt{a^2-2 a c+b^2+c^2}+a-c} \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{\sqrt{2} e \sqrt{a^2-2 a c+b^2+c^2}}-\frac{\left (3 b^2-4 a c\right ) \tanh ^{-1}\left (\frac{2 a+b \tan (d+e x)}{2 \sqrt{a} \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{8 a^{5/2} e}+\frac{3 b \cot (d+e x) \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}{4 a^2 e}+\frac{\tanh ^{-1}\left (\frac{2 a+b \tan (d+e x)}{2 \sqrt{a} \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{\sqrt{a} e}-\frac{\cot ^2(d+e x) \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}{2 a e} \]
Antiderivative was successfully verified.
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Rule 3700
Rule 6725
Rule 744
Rule 806
Rule 724
Rule 206
Rule 1036
Rule 1030
Rule 208
Rubi steps
\begin{align*} \int \frac{\cot ^3(d+e x)}{\sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1}{x^3 \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 \left (\frac{1}{x^3 \sqrt{a+b x+c x^2}}-\frac{1}{x \sqrt{a+b x+c x^2}}+\frac{x}{\left (1+x^2\right ) \sqrt{a+b x+c x^2}}\right ) \, dx,x,\tan (d+e x)\right )}{e}\\ &=\frac{\operatorname{Subst}\left (\int \frac{1}{x^3 \sqrt{a+b x+c x^2}} \, dx,x,\tan (d+e x)\right )}{e}-\frac{\operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+b x+c x^2}} \, dx,x,\tan (d+e x)\right )}{e}+\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{\cot ^2(d+e x) \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}{2 a e}+\frac{2 \operatorname{Subst}\left (\int \frac{1}{4 a-x^2} \, dx,x,\frac{2 a+b \tan (d+e x)}{\sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{e}-\frac{\operatorname{Subst}\left (\int \frac{\frac{3 b}{2}+c x}{x^2 \sqrt{a+b x+c x^2}} \, dx,x,\tan (d+e x)\right )}{2 a 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{\tanh ^{-1}\left (\frac{2 a+b \tan (d+e x)}{2 \sqrt{a} \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{\sqrt{a} e}+\frac{3 b \cot (d+e x) \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}{4 a^2 e}-\frac{\cot ^2(d+e x) \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}{2 a e}+\frac{\left (3 b^2-4 a c\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+b x+c x^2}} \, dx,x,\tan (d+e x)\right )}{8 a^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{\tanh ^{-1}\left (\frac{2 a+b \tan (d+e x)}{2 \sqrt{a} \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{\sqrt{a} 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}+\frac{3 b \cot (d+e x) \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}{4 a^2 e}-\frac{\cot ^2(d+e x) \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}{2 a e}-\frac{\left (3 b^2-4 a c\right ) \operatorname{Subst}\left (\int \frac{1}{4 a-x^2} \, dx,x,\frac{2 a+b \tan (d+e x)}{\sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{4 a^2 e}\\ &=\frac{\tanh ^{-1}\left (\frac{2 a+b \tan (d+e x)}{2 \sqrt{a} \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{\sqrt{a} e}-\frac{\left (3 b^2-4 a c\right ) \tanh ^{-1}\left (\frac{2 a+b \tan (d+e x)}{2 \sqrt{a} \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{8 a^{5/2} 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}+\frac{3 b \cot (d+e x) \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}{4 a^2 e}-\frac{\cot ^2(d+e x) \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}{2 a e}\\ \end{align*}
Mathematica [C] time = 5.38261, size = 315, normalized size = 0.63 \[ \frac{\frac{\left (8 a^2+4 a c-3 b^2\right ) \tanh ^{-1}\left (\frac{2 a+b \tan (d+e x)}{2 \sqrt{a} \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{2 a^{5/2}}+\frac{3 b \cot (d+e x) \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}{a^2}-\frac{2 \tanh ^{-1}\left (\frac{2 a+(b-2 i c) \tan (d+e x)-i b}{2 \sqrt{a-i b-c} \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{\sqrt{a-i b-c}}-\frac{2 \tanh ^{-1}\left (\frac{2 a+(b+2 i c) \tan (d+e x)+i b}{2 \sqrt{a+i b-c} \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{\sqrt{a+i b-c}}-\frac{2 \cot ^2(d+e x) \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}{a}}{4 e} \]
Antiderivative was successfully verified.
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Maple [B] time = 45.891, size = 1981914, normalized size = 3963.8 \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(-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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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{\cot ^{3}{\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] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\cot \left (e x + d\right )^{3}}{\sqrt{c \tan \left (e x + d\right )^{2} + b \tan \left (e x + d\right ) + a}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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