Optimal. Leaf size=1189 \[ \text{result too large to display} \]
[Out]
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Rubi [A] time = 6.28863, antiderivative size = 1189, normalized size of antiderivative = 1., number of steps used = 20, number of rules used = 12, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.364, Rules used = {3701, 6725, 636, 738, 779, 621, 206, 832, 1018, 1036, 1030, 208} \[ -\frac{2 (2 a+b \cot (d+e x)) \cot ^4(d+e x)}{\left (b^2-4 a c\right ) e \sqrt{c \cot ^2(d+e x)+b \cot (d+e x)+a}}+\frac{2 b \sqrt{c \cot ^2(d+e x)+b \cot (d+e x)+a} \cot ^3(d+e x)}{c \left (b^2-4 a c\right ) e}-\frac{\left (7 b^2-16 a c\right ) \sqrt{c \cot ^2(d+e x)+b \cot (d+e x)+a} \cot ^2(d+e x)}{3 c^2 \left (b^2-4 a c\right ) e}+\frac{2 (2 a+b \cot (d+e x)) \cot ^2(d+e x)}{\left (b^2-4 a c\right ) e \sqrt{c \cot ^2(d+e x)+b \cot (d+e x)+a}}+\frac{5 b \left (7 b^2-12 a c\right ) \tanh ^{-1}\left (\frac{b+2 c \cot (d+e x)}{2 \sqrt{c} \sqrt{c \cot ^2(d+e x)+b \cot (d+e x)+a}}\right )}{16 c^{9/2} e}-\frac{3 b \tanh ^{-1}\left (\frac{b+2 c \cot (d+e x)}{2 \sqrt{c} \sqrt{c \cot ^2(d+e x)+b \cot (d+e x)+a}}\right )}{2 c^{5/2} e}+\frac{\sqrt{2 a-2 c-\sqrt{a^2-2 c a+b^2+c^2}} \sqrt{a^2-2 c a-b^2+c^2+(a-c) \sqrt{a^2-2 c a+b^2+c^2}} \tanh ^{-1}\left (\frac{b^2-\left (2 a-2 c-\sqrt{a^2-2 c a+b^2+c^2}\right ) \cot (d+e x) b-(a-c) \left (a-c+\sqrt{a^2-2 c a+b^2+c^2}\right )}{\sqrt{2} \sqrt{2 a-2 c-\sqrt{a^2-2 c a+b^2+c^2}} \sqrt{a^2-2 c a-b^2+c^2+(a-c) \sqrt{a^2-2 c a+b^2+c^2}} \sqrt{c \cot ^2(d+e x)+b \cot (d+e x)+a}}\right )}{\sqrt{2} \left (a^2-2 c a+b^2+c^2\right )^{3/2} e}-\frac{\sqrt{2 a-2 c+\sqrt{a^2-2 c a+b^2+c^2}} \sqrt{a^2-2 c a-b^2+c^2-(a-c) \sqrt{a^2-2 c a+b^2+c^2}} \tanh ^{-1}\left (\frac{b^2-\left (2 a-2 c+\sqrt{a^2-2 c a+b^2+c^2}\right ) \cot (d+e x) b-(a-c) \left (a-c-\sqrt{a^2-2 c a+b^2+c^2}\right )}{\sqrt{2} \sqrt{2 a-2 c+\sqrt{a^2-2 c a+b^2+c^2}} \sqrt{a^2-2 c a-b^2+c^2-(a-c) \sqrt{a^2-2 c a+b^2+c^2}} \sqrt{c \cot ^2(d+e x)+b \cot (d+e x)+a}}\right )}{\sqrt{2} \left (a^2-2 c a+b^2+c^2\right )^{3/2} e}+\frac{\left (3 b^2-2 c \cot (d+e x) b-8 a c\right ) \sqrt{c \cot ^2(d+e x)+b \cot (d+e x)+a}}{c^2 \left (b^2-4 a c\right ) e}-\frac{\left (105 b^4-460 a c b^2-2 c \left (35 b^2-116 a c\right ) \cot (d+e x) b+256 a^2 c^2\right ) \sqrt{c \cot ^2(d+e x)+b \cot (d+e x)+a}}{24 c^4 \left (b^2-4 a c\right ) e}-\frac{2 (2 a+b \cot (d+e x))}{\left (b^2-4 a c\right ) e \sqrt{c \cot ^2(d+e x)+b \cot (d+e x)+a}}+\frac{2 \left (a \left (b^2-2 (a-c) c\right )+b c (a+c) \cot (d+e x)\right )}{\left (b^2+(a-c)^2\right ) \left (b^2-4 a c\right ) e \sqrt{c \cot ^2(d+e x)+b \cot (d+e x)+a}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3701
Rule 6725
Rule 636
Rule 738
Rule 779
Rule 621
Rule 206
Rule 832
Rule 1018
Rule 1036
Rule 1030
Rule 208
Rubi steps
\begin{align*} \int \frac{\cot ^7(d+e x)}{\left (a+b \cot (d+e x)+c \cot ^2(d+e x)\right )^{3/2}} \, dx &=-\frac{\operatorname{Subst}\left (\int \frac{x^7}{\left (1+x^2\right ) \left (a+b x+c x^2\right )^{3/2}} \, dx,x,\cot (d+e x)\right )}{e}\\ &=-\frac{\operatorname{Subst}\left (\int \left (\frac{x}{\left (a+b x+c x^2\right )^{3/2}}-\frac{x^3}{\left (a+b x+c x^2\right )^{3/2}}+\frac{x^5}{\left (a+b x+c x^2\right )^{3/2}}-\frac{x}{\left (1+x^2\right ) \left (a+b x+c x^2\right )^{3/2}}\right ) \, dx,x,\cot (d+e x)\right )}{e}\\ &=-\frac{\operatorname{Subst}\left (\int \frac{x}{\left (a+b x+c x^2\right )^{3/2}} \, dx,x,\cot (d+e x)\right )}{e}+\frac{\operatorname{Subst}\left (\int \frac{x^3}{\left (a+b x+c x^2\right )^{3/2}} \, dx,x,\cot (d+e x)\right )}{e}-\frac{\operatorname{Subst}\left (\int \frac{x^5}{\left (a+b x+c x^2\right )^{3/2}} \, dx,x,\cot (d+e x)\right )}{e}+\frac{\operatorname{Subst}\left (\int \frac{x}{\left (1+x^2\right ) \left (a+b x+c x^2\right )^{3/2}} \, dx,x,\cot (d+e x)\right )}{e}\\ &=-\frac{2 (2 a+b \cot (d+e x))}{\left (b^2-4 a c\right ) e \sqrt{a+b \cot (d+e x)+c \cot ^2(d+e x)}}+\frac{2 \cot ^2(d+e x) (2 a+b \cot (d+e x))}{\left (b^2-4 a c\right ) e \sqrt{a+b \cot (d+e x)+c \cot ^2(d+e x)}}-\frac{2 \cot ^4(d+e x) (2 a+b \cot (d+e x))}{\left (b^2-4 a c\right ) e \sqrt{a+b \cot (d+e x)+c \cot ^2(d+e x)}}+\frac{2 \left (a \left (b^2-2 (a-c) c\right )+b c (a+c) \cot (d+e x)\right )}{\left (b^2+(a-c)^2\right ) \left (b^2-4 a c\right ) e \sqrt{a+b \cot (d+e x)+c \cot ^2(d+e x)}}-\frac{2 \operatorname{Subst}\left (\int \frac{x (4 a+2 b x)}{\sqrt{a+b x+c x^2}} \, dx,x,\cot (d+e x)\right )}{\left (b^2-4 a c\right ) e}+\frac{2 \operatorname{Subst}\left (\int \frac{x^3 (8 a+4 b x)}{\sqrt{a+b x+c x^2}} \, dx,x,\cot (d+e x)\right )}{\left (b^2-4 a c\right ) e}-\frac{2 \operatorname{Subst}\left (\int \frac{-\frac{1}{2} b \left (b^2-4 a c\right )-\frac{1}{2} (a-c) \left (b^2-4 a c\right ) x}{\left (1+x^2\right ) \sqrt{a+b x+c x^2}} \, dx,x,\cot (d+e x)\right )}{\left (b^2+(a-c)^2\right ) \left (b^2-4 a c\right ) e}\\ &=-\frac{2 (2 a+b \cot (d+e x))}{\left (b^2-4 a c\right ) e \sqrt{a+b \cot (d+e x)+c \cot ^2(d+e x)}}+\frac{2 \cot ^2(d+e x) (2 a+b \cot (d+e x))}{\left (b^2-4 a c\right ) e \sqrt{a+b \cot (d+e x)+c \cot ^2(d+e x)}}-\frac{2 \cot ^4(d+e x) (2 a+b \cot (d+e x))}{\left (b^2-4 a c\right ) e \sqrt{a+b \cot (d+e x)+c \cot ^2(d+e x)}}+\frac{2 \left (a \left (b^2-2 (a-c) c\right )+b c (a+c) \cot (d+e x)\right )}{\left (b^2+(a-c)^2\right ) \left (b^2-4 a c\right ) e \sqrt{a+b \cot (d+e x)+c \cot ^2(d+e x)}}+\frac{2 b \cot ^3(d+e x) \sqrt{a+b \cot (d+e x)+c \cot ^2(d+e x)}}{c \left (b^2-4 a c\right ) e}+\frac{\left (3 b^2-8 a c-2 b c \cot (d+e x)\right ) \sqrt{a+b \cot (d+e x)+c \cot ^2(d+e x)}}{c^2 \left (b^2-4 a c\right ) e}-\frac{(3 b) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+b x+c x^2}} \, dx,x,\cot (d+e x)\right )}{2 c^2 e}+\frac{\operatorname{Subst}\left (\int \frac{x^2 \left (-12 a b-2 \left (7 b^2-16 a c\right ) x\right )}{\sqrt{a+b x+c x^2}} \, dx,x,\cot (d+e x)\right )}{2 c \left (b^2-4 a c\right ) e}+\frac{\operatorname{Subst}\left (\int \frac{\frac{1}{2} b \left (b^2-4 a c\right ) \left (2 a-2 c+\sqrt{a^2+b^2-2 a c+c^2}\right )+\frac{1}{2} \left (b^2-4 a c\right ) \left (b^2-(a-c) \left (a-c-\sqrt{a^2+b^2-2 a c+c^2}\right )\right ) x}{\left (1+x^2\right ) \sqrt{a+b x+c x^2}} \, dx,x,\cot (d+e x)\right )}{\left (b^2-4 a c\right ) \left (a^2+b^2-2 a c+c^2\right )^{3/2} e}-\frac{\operatorname{Subst}\left (\int \frac{\frac{1}{2} b \left (b^2-4 a c\right ) \left (2 a-2 c-\sqrt{a^2+b^2-2 a c+c^2}\right )+\frac{1}{2} \left (b^2-4 a c\right ) \left (b^2-(a-c) \left (a-c+\sqrt{a^2+b^2-2 a c+c^2}\right )\right ) x}{\left (1+x^2\right ) \sqrt{a+b x+c x^2}} \, dx,x,\cot (d+e x)\right )}{\left (b^2-4 a c\right ) \left (a^2+b^2-2 a c+c^2\right )^{3/2} e}\\ &=-\frac{2 (2 a+b \cot (d+e x))}{\left (b^2-4 a c\right ) e \sqrt{a+b \cot (d+e x)+c \cot ^2(d+e x)}}+\frac{2 \cot ^2(d+e x) (2 a+b \cot (d+e x))}{\left (b^2-4 a c\right ) e \sqrt{a+b \cot (d+e x)+c \cot ^2(d+e x)}}-\frac{2 \cot ^4(d+e x) (2 a+b \cot (d+e x))}{\left (b^2-4 a c\right ) e \sqrt{a+b \cot (d+e x)+c \cot ^2(d+e x)}}+\frac{2 \left (a \left (b^2-2 (a-c) c\right )+b c (a+c) \cot (d+e x)\right )}{\left (b^2+(a-c)^2\right ) \left (b^2-4 a c\right ) e \sqrt{a+b \cot (d+e x)+c \cot ^2(d+e x)}}-\frac{\left (7 b^2-16 a c\right ) \cot ^2(d+e x) \sqrt{a+b \cot (d+e x)+c \cot ^2(d+e x)}}{3 c^2 \left (b^2-4 a c\right ) e}+\frac{2 b \cot ^3(d+e x) \sqrt{a+b \cot (d+e x)+c \cot ^2(d+e x)}}{c \left (b^2-4 a c\right ) e}+\frac{\left (3 b^2-8 a c-2 b c \cot (d+e x)\right ) \sqrt{a+b \cot (d+e x)+c \cot ^2(d+e x)}}{c^2 \left (b^2-4 a c\right ) e}-\frac{(3 b) \operatorname{Subst}\left (\int \frac{1}{4 c-x^2} \, dx,x,\frac{b+2 c \cot (d+e x)}{\sqrt{a+b \cot (d+e x)+c \cot ^2(d+e x)}}\right )}{c^2 e}+\frac{\operatorname{Subst}\left (\int \frac{x \left (4 a \left (7 b^2-16 a c\right )+b \left (35 b^2-116 a c\right ) x\right )}{\sqrt{a+b x+c x^2}} \, dx,x,\cot (d+e x)\right )}{6 c^2 \left (b^2-4 a c\right ) e}-\frac{\left (b \left (b^2-4 a c\right ) \left (2 a-2 c+\sqrt{a^2+b^2-2 a c+c^2}\right ) \left (b^2-(a-c) \left (a-c-\sqrt{a^2+b^2-2 a c+c^2}\right )\right )\right ) \operatorname{Subst}\left (\int \frac{1}{\frac{1}{2} b \left (b^2-4 a c\right )^2 \left (2 a-2 c+\sqrt{a^2+b^2-2 a c+c^2}\right ) \left (b^2-(a-c) \left (a-c-\sqrt{a^2+b^2-2 a c+c^2}\right )\right )+b x^2} \, dx,x,\frac{\frac{1}{2} \left (b^2-4 a c\right ) \left (b^2-(a-c) \left (a-c-\sqrt{a^2+b^2-2 a c+c^2}\right )\right )-\frac{1}{2} b \left (b^2-4 a c\right ) \left (2 a-2 c+\sqrt{a^2+b^2-2 a c+c^2}\right ) \cot (d+e x)}{\sqrt{a+b \cot (d+e x)+c \cot ^2(d+e x)}}\right )}{2 \left (a^2+b^2-2 a c+c^2\right )^{3/2} e}+\frac{\left (b \left (b^2-4 a c\right ) \left (2 a-2 c-\sqrt{a^2+b^2-2 a c+c^2}\right ) \left (b^2-(a-c) \left (a-c+\sqrt{a^2+b^2-2 a c+c^2}\right )\right )\right ) \operatorname{Subst}\left (\int \frac{1}{\frac{1}{2} b \left (b^2-4 a c\right )^2 \left (2 a-2 c-\sqrt{a^2+b^2-2 a c+c^2}\right ) \left (b^2-(a-c) \left (a-c+\sqrt{a^2+b^2-2 a c+c^2}\right )\right )+b x^2} \, dx,x,\frac{\frac{1}{2} \left (b^2-4 a c\right ) \left (b^2-(a-c) \left (a-c+\sqrt{a^2+b^2-2 a c+c^2}\right )\right )-\frac{1}{2} b \left (b^2-4 a c\right ) \left (2 a-2 c-\sqrt{a^2+b^2-2 a c+c^2}\right ) \cot (d+e x)}{\sqrt{a+b \cot (d+e x)+c \cot ^2(d+e x)}}\right )}{2 \left (a^2+b^2-2 a c+c^2\right )^{3/2} e}\\ &=-\frac{3 b \tanh ^{-1}\left (\frac{b+2 c \cot (d+e x)}{2 \sqrt{c} \sqrt{a+b \cot (d+e x)+c \cot ^2(d+e x)}}\right )}{2 c^{5/2} e}+\frac{\sqrt{2 a-2 c-\sqrt{a^2+b^2-2 a c+c^2}} \sqrt{a^2-b^2-2 a c+c^2+(a-c) \sqrt{a^2+b^2-2 a c+c^2}} \tanh ^{-1}\left (\frac{b^2-(a-c) \left (a-c+\sqrt{a^2+b^2-2 a c+c^2}\right )-b \left (2 a-2 c-\sqrt{a^2+b^2-2 a c+c^2}\right ) \cot (d+e x)}{\sqrt{2} \sqrt{2 a-2 c-\sqrt{a^2+b^2-2 a c+c^2}} \sqrt{a^2-b^2-2 a c+c^2+(a-c) \sqrt{a^2+b^2-2 a c+c^2}} \sqrt{a+b \cot (d+e x)+c \cot ^2(d+e x)}}\right )}{\sqrt{2} \left (a^2+b^2-2 a c+c^2\right )^{3/2} e}-\frac{\sqrt{2 a-2 c+\sqrt{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 )} \tanh ^{-1}\left (\frac{b^2-(a-c) \left (a-c-\sqrt{a^2+b^2-2 a c+c^2}\right )-b \left (2 a-2 c+\sqrt{a^2+b^2-2 a c+c^2}\right ) \cot (d+e x)}{\sqrt{2} \sqrt{2 a-2 c+\sqrt{a^2+b^2-2 a c+c^2}} \sqrt{a^2-b^2-2 a c+c^2-(a-c) \sqrt{a^2+b^2-2 a c+c^2}} \sqrt{a+b \cot (d+e x)+c \cot ^2(d+e x)}}\right )}{\sqrt{2} \left (a^2+b^2-2 a c+c^2\right )^{3/2} e}-\frac{2 (2 a+b \cot (d+e x))}{\left (b^2-4 a c\right ) e \sqrt{a+b \cot (d+e x)+c \cot ^2(d+e x)}}+\frac{2 \cot ^2(d+e x) (2 a+b \cot (d+e x))}{\left (b^2-4 a c\right ) e \sqrt{a+b \cot (d+e x)+c \cot ^2(d+e x)}}-\frac{2 \cot ^4(d+e x) (2 a+b \cot (d+e x))}{\left (b^2-4 a c\right ) e \sqrt{a+b \cot (d+e x)+c \cot ^2(d+e x)}}+\frac{2 \left (a \left (b^2-2 (a-c) c\right )+b c (a+c) \cot (d+e x)\right )}{\left (b^2+(a-c)^2\right ) \left (b^2-4 a c\right ) e \sqrt{a+b \cot (d+e x)+c \cot ^2(d+e x)}}-\frac{\left (7 b^2-16 a c\right ) \cot ^2(d+e x) \sqrt{a+b \cot (d+e x)+c \cot ^2(d+e x)}}{3 c^2 \left (b^2-4 a c\right ) e}+\frac{2 b \cot ^3(d+e x) \sqrt{a+b \cot (d+e x)+c \cot ^2(d+e x)}}{c \left (b^2-4 a c\right ) e}+\frac{\left (3 b^2-8 a c-2 b c \cot (d+e x)\right ) \sqrt{a+b \cot (d+e x)+c \cot ^2(d+e x)}}{c^2 \left (b^2-4 a c\right ) e}-\frac{\left (105 b^4-460 a b^2 c+256 a^2 c^2-2 b c \left (35 b^2-116 a c\right ) \cot (d+e x)\right ) \sqrt{a+b \cot (d+e x)+c \cot ^2(d+e x)}}{24 c^4 \left (b^2-4 a c\right ) e}+\frac{\left (5 b \left (7 b^2-12 a c\right )\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+b x+c x^2}} \, dx,x,\cot (d+e x)\right )}{16 c^4 e}\\ &=-\frac{3 b \tanh ^{-1}\left (\frac{b+2 c \cot (d+e x)}{2 \sqrt{c} \sqrt{a+b \cot (d+e x)+c \cot ^2(d+e x)}}\right )}{2 c^{5/2} e}+\frac{\sqrt{2 a-2 c-\sqrt{a^2+b^2-2 a c+c^2}} \sqrt{a^2-b^2-2 a c+c^2+(a-c) \sqrt{a^2+b^2-2 a c+c^2}} \tanh ^{-1}\left (\frac{b^2-(a-c) \left (a-c+\sqrt{a^2+b^2-2 a c+c^2}\right )-b \left (2 a-2 c-\sqrt{a^2+b^2-2 a c+c^2}\right ) \cot (d+e x)}{\sqrt{2} \sqrt{2 a-2 c-\sqrt{a^2+b^2-2 a c+c^2}} \sqrt{a^2-b^2-2 a c+c^2+(a-c) \sqrt{a^2+b^2-2 a c+c^2}} \sqrt{a+b \cot (d+e x)+c \cot ^2(d+e x)}}\right )}{\sqrt{2} \left (a^2+b^2-2 a c+c^2\right )^{3/2} e}-\frac{\sqrt{2 a-2 c+\sqrt{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 )} \tanh ^{-1}\left (\frac{b^2-(a-c) \left (a-c-\sqrt{a^2+b^2-2 a c+c^2}\right )-b \left (2 a-2 c+\sqrt{a^2+b^2-2 a c+c^2}\right ) \cot (d+e x)}{\sqrt{2} \sqrt{2 a-2 c+\sqrt{a^2+b^2-2 a c+c^2}} \sqrt{a^2-b^2-2 a c+c^2-(a-c) \sqrt{a^2+b^2-2 a c+c^2}} \sqrt{a+b \cot (d+e x)+c \cot ^2(d+e x)}}\right )}{\sqrt{2} \left (a^2+b^2-2 a c+c^2\right )^{3/2} e}-\frac{2 (2 a+b \cot (d+e x))}{\left (b^2-4 a c\right ) e \sqrt{a+b \cot (d+e x)+c \cot ^2(d+e x)}}+\frac{2 \cot ^2(d+e x) (2 a+b \cot (d+e x))}{\left (b^2-4 a c\right ) e \sqrt{a+b \cot (d+e x)+c \cot ^2(d+e x)}}-\frac{2 \cot ^4(d+e x) (2 a+b \cot (d+e x))}{\left (b^2-4 a c\right ) e \sqrt{a+b \cot (d+e x)+c \cot ^2(d+e x)}}+\frac{2 \left (a \left (b^2-2 (a-c) c\right )+b c (a+c) \cot (d+e x)\right )}{\left (b^2+(a-c)^2\right ) \left (b^2-4 a c\right ) e \sqrt{a+b \cot (d+e x)+c \cot ^2(d+e x)}}-\frac{\left (7 b^2-16 a c\right ) \cot ^2(d+e x) \sqrt{a+b \cot (d+e x)+c \cot ^2(d+e x)}}{3 c^2 \left (b^2-4 a c\right ) e}+\frac{2 b \cot ^3(d+e x) \sqrt{a+b \cot (d+e x)+c \cot ^2(d+e x)}}{c \left (b^2-4 a c\right ) e}+\frac{\left (3 b^2-8 a c-2 b c \cot (d+e x)\right ) \sqrt{a+b \cot (d+e x)+c \cot ^2(d+e x)}}{c^2 \left (b^2-4 a c\right ) e}-\frac{\left (105 b^4-460 a b^2 c+256 a^2 c^2-2 b c \left (35 b^2-116 a c\right ) \cot (d+e x)\right ) \sqrt{a+b \cot (d+e x)+c \cot ^2(d+e x)}}{24 c^4 \left (b^2-4 a c\right ) e}+\frac{\left (5 b \left (7 b^2-12 a c\right )\right ) \operatorname{Subst}\left (\int \frac{1}{4 c-x^2} \, dx,x,\frac{b+2 c \cot (d+e x)}{\sqrt{a+b \cot (d+e x)+c \cot ^2(d+e x)}}\right )}{8 c^4 e}\\ &=-\frac{3 b \tanh ^{-1}\left (\frac{b+2 c \cot (d+e x)}{2 \sqrt{c} \sqrt{a+b \cot (d+e x)+c \cot ^2(d+e x)}}\right )}{2 c^{5/2} e}+\frac{5 b \left (7 b^2-12 a c\right ) \tanh ^{-1}\left (\frac{b+2 c \cot (d+e x)}{2 \sqrt{c} \sqrt{a+b \cot (d+e x)+c \cot ^2(d+e x)}}\right )}{16 c^{9/2} e}+\frac{\sqrt{2 a-2 c-\sqrt{a^2+b^2-2 a c+c^2}} \sqrt{a^2-b^2-2 a c+c^2+(a-c) \sqrt{a^2+b^2-2 a c+c^2}} \tanh ^{-1}\left (\frac{b^2-(a-c) \left (a-c+\sqrt{a^2+b^2-2 a c+c^2}\right )-b \left (2 a-2 c-\sqrt{a^2+b^2-2 a c+c^2}\right ) \cot (d+e x)}{\sqrt{2} \sqrt{2 a-2 c-\sqrt{a^2+b^2-2 a c+c^2}} \sqrt{a^2-b^2-2 a c+c^2+(a-c) \sqrt{a^2+b^2-2 a c+c^2}} \sqrt{a+b \cot (d+e x)+c \cot ^2(d+e x)}}\right )}{\sqrt{2} \left (a^2+b^2-2 a c+c^2\right )^{3/2} e}-\frac{\sqrt{2 a-2 c+\sqrt{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 )} \tanh ^{-1}\left (\frac{b^2-(a-c) \left (a-c-\sqrt{a^2+b^2-2 a c+c^2}\right )-b \left (2 a-2 c+\sqrt{a^2+b^2-2 a c+c^2}\right ) \cot (d+e x)}{\sqrt{2} \sqrt{2 a-2 c+\sqrt{a^2+b^2-2 a c+c^2}} \sqrt{a^2-b^2-2 a c+c^2-(a-c) \sqrt{a^2+b^2-2 a c+c^2}} \sqrt{a+b \cot (d+e x)+c \cot ^2(d+e x)}}\right )}{\sqrt{2} \left (a^2+b^2-2 a c+c^2\right )^{3/2} e}-\frac{2 (2 a+b \cot (d+e x))}{\left (b^2-4 a c\right ) e \sqrt{a+b \cot (d+e x)+c \cot ^2(d+e x)}}+\frac{2 \cot ^2(d+e x) (2 a+b \cot (d+e x))}{\left (b^2-4 a c\right ) e \sqrt{a+b \cot (d+e x)+c \cot ^2(d+e x)}}-\frac{2 \cot ^4(d+e x) (2 a+b \cot (d+e x))}{\left (b^2-4 a c\right ) e \sqrt{a+b \cot (d+e x)+c \cot ^2(d+e x)}}+\frac{2 \left (a \left (b^2-2 (a-c) c\right )+b c (a+c) \cot (d+e x)\right )}{\left (b^2+(a-c)^2\right ) \left (b^2-4 a c\right ) e \sqrt{a+b \cot (d+e x)+c \cot ^2(d+e x)}}-\frac{\left (7 b^2-16 a c\right ) \cot ^2(d+e x) \sqrt{a+b \cot (d+e x)+c \cot ^2(d+e x)}}{3 c^2 \left (b^2-4 a c\right ) e}+\frac{2 b \cot ^3(d+e x) \sqrt{a+b \cot (d+e x)+c \cot ^2(d+e x)}}{c \left (b^2-4 a c\right ) e}+\frac{\left (3 b^2-8 a c-2 b c \cot (d+e x)\right ) \sqrt{a+b \cot (d+e x)+c \cot ^2(d+e x)}}{c^2 \left (b^2-4 a c\right ) e}-\frac{\left (105 b^4-460 a b^2 c+256 a^2 c^2-2 b c \left (35 b^2-116 a c\right ) \cot (d+e x)\right ) \sqrt{a+b \cot (d+e x)+c \cot ^2(d+e x)}}{24 c^4 \left (b^2-4 a c\right ) e}\\ \end{align*}
Mathematica [C] time = 38.5354, size = 5618, normalized size = 4.72 \[ \text{Result too large to show} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.173, size = 13068424, normalized size = 10991.1 \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(-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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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\cot \left (e x + d\right )^{7}}{{\left (c \cot \left (e x + d\right )^{2} + b \cot \left (e x + d\right ) + a\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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