Optimal. Leaf size=1007 \[ -\frac{\left (5 b^2-12 a c\right ) \sqrt{c \tan ^2(d+e x)+b \tan (d+e x)+a} \cot ^2(d+e x)}{2 a^2 \left (b^2-4 a c\right ) e}+\frac{2 \left (b^2+c \tan (d+e x) b-2 a c\right ) \cot ^2(d+e x)}{a \left (b^2-4 a c\right ) e \sqrt{c \tan ^2(d+e x)+b \tan (d+e x)+a}}+\frac{b \left (15 b^2-52 a c\right ) \sqrt{c \tan ^2(d+e x)+b \tan (d+e x)+a} \cot (d+e x)}{4 a^3 \left (b^2-4 a c\right ) e}-\frac{3 \left (5 b^2-4 a c\right ) \tanh ^{-1}\left (\frac{2 a+b \tan (d+e x)}{2 \sqrt{a} \sqrt{c \tan ^2(d+e x)+b \tan (d+e x)+a}}\right )}{8 a^{7/2} e}+\frac{\tanh ^{-1}\left (\frac{2 a+b \tan (d+e x)}{2 \sqrt{a} \sqrt{c \tan ^2(d+e x)+b \tan (d+e x)+a}}\right )}{a^{3/2} e}+\frac{\sqrt{2 a-2 c-\sqrt{a^2-2 c a+b^2+c^2}} \sqrt{a^2-2 c a-b^2+c^2+(a-c) \sqrt{a^2-2 c a+b^2+c^2}} \tanh ^{-1}\left (\frac{b^2-\left (2 a-2 c-\sqrt{a^2-2 c a+b^2+c^2}\right ) \tan (d+e x) b-(a-c) \left (a-c+\sqrt{a^2-2 c a+b^2+c^2}\right )}{\sqrt{2} \sqrt{2 a-2 c-\sqrt{a^2-2 c a+b^2+c^2}} \sqrt{a^2-2 c a-b^2+c^2+(a-c) \sqrt{a^2-2 c a+b^2+c^2}} \sqrt{c \tan ^2(d+e x)+b \tan (d+e x)+a}}\right )}{\sqrt{2} \left (a^2-2 c a+b^2+c^2\right )^{3/2} e}-\frac{\sqrt{2 a-2 c+\sqrt{a^2-2 c a+b^2+c^2}} \sqrt{a^2-2 c a-b^2+c^2-(a-c) \sqrt{a^2-2 c a+b^2+c^2}} \tanh ^{-1}\left (\frac{b^2-\left (2 a-2 c+\sqrt{a^2-2 c a+b^2+c^2}\right ) \tan (d+e x) b-(a-c) \left (a-c-\sqrt{a^2-2 c a+b^2+c^2}\right )}{\sqrt{2} \sqrt{2 a-2 c+\sqrt{a^2-2 c a+b^2+c^2}} \sqrt{a^2-2 c a-b^2+c^2-(a-c) \sqrt{a^2-2 c a+b^2+c^2}} \sqrt{c \tan ^2(d+e x)+b \tan (d+e x)+a}}\right )}{\sqrt{2} \left (a^2-2 c a+b^2+c^2\right )^{3/2} e}-\frac{2 \left (b^2+c \tan (d+e x) b-2 a c\right )}{a \left (b^2-4 a c\right ) e \sqrt{c \tan ^2(d+e x)+b \tan (d+e x)+a}}+\frac{2 \left (a \left (b^2-2 (a-c) c\right )+b c (a+c) \tan (d+e x)\right )}{\left (b^2+(a-c)^2\right ) \left (b^2-4 a c\right ) e \sqrt{c \tan ^2(d+e x)+b \tan (d+e x)+a}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 4.92768, antiderivative size = 1007, normalized size of antiderivative = 1., number of steps used = 18, number of rules used = 12, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.364, Rules used = {3700, 6725, 740, 834, 806, 724, 206, 12, 1018, 1036, 1030, 208} \[ -\frac{\left (5 b^2-12 a c\right ) \sqrt{c \tan ^2(d+e x)+b \tan (d+e x)+a} \cot ^2(d+e x)}{2 a^2 \left (b^2-4 a c\right ) e}+\frac{2 \left (b^2+c \tan (d+e x) b-2 a c\right ) \cot ^2(d+e x)}{a \left (b^2-4 a c\right ) e \sqrt{c \tan ^2(d+e x)+b \tan (d+e x)+a}}+\frac{b \left (15 b^2-52 a c\right ) \sqrt{c \tan ^2(d+e x)+b \tan (d+e x)+a} \cot (d+e x)}{4 a^3 \left (b^2-4 a c\right ) e}-\frac{3 \left (5 b^2-4 a c\right ) \tanh ^{-1}\left (\frac{2 a+b \tan (d+e x)}{2 \sqrt{a} \sqrt{c \tan ^2(d+e x)+b \tan (d+e x)+a}}\right )}{8 a^{7/2} e}+\frac{\tanh ^{-1}\left (\frac{2 a+b \tan (d+e x)}{2 \sqrt{a} \sqrt{c \tan ^2(d+e x)+b \tan (d+e x)+a}}\right )}{a^{3/2} e}+\frac{\sqrt{2 a-2 c-\sqrt{a^2-2 c a+b^2+c^2}} \sqrt{a^2-2 c a-b^2+c^2+(a-c) \sqrt{a^2-2 c a+b^2+c^2}} \tanh ^{-1}\left (\frac{b^2-\left (2 a-2 c-\sqrt{a^2-2 c a+b^2+c^2}\right ) \tan (d+e x) b-(a-c) \left (a-c+\sqrt{a^2-2 c a+b^2+c^2}\right )}{\sqrt{2} \sqrt{2 a-2 c-\sqrt{a^2-2 c a+b^2+c^2}} \sqrt{a^2-2 c a-b^2+c^2+(a-c) \sqrt{a^2-2 c a+b^2+c^2}} \sqrt{c \tan ^2(d+e x)+b \tan (d+e x)+a}}\right )}{\sqrt{2} \left (a^2-2 c a+b^2+c^2\right )^{3/2} e}-\frac{\sqrt{2 a-2 c+\sqrt{a^2-2 c a+b^2+c^2}} \sqrt{a^2-2 c a-b^2+c^2-(a-c) \sqrt{a^2-2 c a+b^2+c^2}} \tanh ^{-1}\left (\frac{b^2-\left (2 a-2 c+\sqrt{a^2-2 c a+b^2+c^2}\right ) \tan (d+e x) b-(a-c) \left (a-c-\sqrt{a^2-2 c a+b^2+c^2}\right )}{\sqrt{2} \sqrt{2 a-2 c+\sqrt{a^2-2 c a+b^2+c^2}} \sqrt{a^2-2 c a-b^2+c^2-(a-c) \sqrt{a^2-2 c a+b^2+c^2}} \sqrt{c \tan ^2(d+e x)+b \tan (d+e x)+a}}\right )}{\sqrt{2} \left (a^2-2 c a+b^2+c^2\right )^{3/2} e}-\frac{2 \left (b^2+c \tan (d+e x) b-2 a c\right )}{a \left (b^2-4 a c\right ) e \sqrt{c \tan ^2(d+e x)+b \tan (d+e x)+a}}+\frac{2 \left (a \left (b^2-2 (a-c) c\right )+b c (a+c) \tan (d+e x)\right )}{\left (b^2+(a-c)^2\right ) \left (b^2-4 a c\right ) e \sqrt{c \tan ^2(d+e x)+b \tan (d+e x)+a}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3700
Rule 6725
Rule 740
Rule 834
Rule 806
Rule 724
Rule 206
Rule 12
Rule 1018
Rule 1036
Rule 1030
Rule 208
Rubi steps
\begin{align*} \int \frac{\cot ^3(d+e x)}{\left (a+b \tan (d+e x)+c \tan ^2(d+e x)\right )^{3/2}} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1}{x^3 \left (1+x^2\right ) \left (a+b x+c x^2\right )^{3/2}} \, dx,x,\tan (d+e x)\right )}{e}\\ &=\frac{\operatorname{Subst}\left (\int \left (\frac{1}{x^3 \left (a+b x+c x^2\right )^{3/2}}-\frac{1}{x \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,\tan (d+e x)\right )}{e}\\ &=\frac{\operatorname{Subst}\left (\int \frac{1}{x^3 \left (a+b x+c x^2\right )^{3/2}} \, dx,x,\tan (d+e x)\right )}{e}-\frac{\operatorname{Subst}\left (\int \frac{1}{x \left (a+b x+c x^2\right )^{3/2}} \, dx,x,\tan (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,\tan (d+e x)\right )}{e}\\ &=-\frac{2 \left (b^2-2 a c+b c \tan (d+e x)\right )}{a \left (b^2-4 a c\right ) e \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}+\frac{2 \cot ^2(d+e x) \left (b^2-2 a c+b c \tan (d+e x)\right )}{a \left (b^2-4 a c\right ) e \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}+\frac{2 \left (a \left (b^2-2 (a-c) c\right )+b c (a+c) \tan (d+e x)\right )}{\left (b^2+(a-c)^2\right ) \left (b^2-4 a c\right ) e \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}+\frac{2 \operatorname{Subst}\left (\int \frac{-\frac{b^2}{2}+2 a c}{x \sqrt{a+b x+c x^2}} \, dx,x,\tan (d+e x)\right )}{a \left (b^2-4 a c\right ) e}-\frac{2 \operatorname{Subst}\left (\int \frac{\frac{1}{2} \left (-5 b^2+12 a c\right )-2 b c x}{x^3 \sqrt{a+b x+c x^2}} \, dx,x,\tan (d+e x)\right )}{a \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,\tan (d+e x)\right )}{\left (b^2+(a-c)^2\right ) \left (b^2-4 a c\right ) e}\\ &=-\frac{2 \left (b^2-2 a c+b c \tan (d+e x)\right )}{a \left (b^2-4 a c\right ) e \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}+\frac{2 \cot ^2(d+e x) \left (b^2-2 a c+b c \tan (d+e x)\right )}{a \left (b^2-4 a c\right ) e \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}+\frac{2 \left (a \left (b^2-2 (a-c) c\right )+b c (a+c) \tan (d+e x)\right )}{\left (b^2+(a-c)^2\right ) \left (b^2-4 a c\right ) e \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}-\frac{\left (5 b^2-12 a c\right ) \cot ^2(d+e x) \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}{2 a^2 \left (b^2-4 a c\right ) e}-\frac{\operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+b x+c x^2}} \, dx,x,\tan (d+e x)\right )}{a e}+\frac{\operatorname{Subst}\left (\int \frac{-\frac{1}{4} b \left (15 b^2-52 a c\right )-\frac{1}{2} c \left (5 b^2-12 a c\right ) x}{x^2 \sqrt{a+b x+c x^2}} \, dx,x,\tan (d+e x)\right )}{a^2 \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,\tan (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,\tan (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 \left (b^2-2 a c+b c \tan (d+e x)\right )}{a \left (b^2-4 a c\right ) e \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}+\frac{2 \cot ^2(d+e x) \left (b^2-2 a c+b c \tan (d+e x)\right )}{a \left (b^2-4 a c\right ) e \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}+\frac{2 \left (a \left (b^2-2 (a-c) c\right )+b c (a+c) \tan (d+e x)\right )}{\left (b^2+(a-c)^2\right ) \left (b^2-4 a c\right ) e \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}+\frac{b \left (15 b^2-52 a c\right ) \cot (d+e x) \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}{4 a^3 \left (b^2-4 a c\right ) e}-\frac{\left (5 b^2-12 a c\right ) \cot ^2(d+e x) \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}{2 a^2 \left (b^2-4 a c\right ) 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 )}{a e}+\frac{\left (3 \left (5 b^2-4 a c\right )\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+b x+c x^2}} \, dx,x,\tan (d+e x)\right )}{8 a^3 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 ) \tan (d+e x)}{\sqrt{a+b \tan (d+e x)+c \tan ^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 ) \tan (d+e x)}{\sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{2 \left (a^2+b^2-2 a c+c^2\right )^{3/2} 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 )}{a^{3/2} 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 ) \tan (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 \tan (d+e x)+c \tan ^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 ) \tan (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 \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{\sqrt{2} \left (a^2+b^2-2 a c+c^2\right )^{3/2} e}-\frac{2 \left (b^2-2 a c+b c \tan (d+e x)\right )}{a \left (b^2-4 a c\right ) e \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}+\frac{2 \cot ^2(d+e x) \left (b^2-2 a c+b c \tan (d+e x)\right )}{a \left (b^2-4 a c\right ) e \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}+\frac{2 \left (a \left (b^2-2 (a-c) c\right )+b c (a+c) \tan (d+e x)\right )}{\left (b^2+(a-c)^2\right ) \left (b^2-4 a c\right ) e \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}+\frac{b \left (15 b^2-52 a c\right ) \cot (d+e x) \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}{4 a^3 \left (b^2-4 a c\right ) e}-\frac{\left (5 b^2-12 a c\right ) \cot ^2(d+e x) \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}{2 a^2 \left (b^2-4 a c\right ) e}-\frac{\left (3 \left (5 b^2-4 a c\right )\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^3 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 )}{a^{3/2} e}-\frac{3 \left (5 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^{7/2} 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 ) \tan (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 \tan (d+e x)+c \tan ^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 ) \tan (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 \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{\sqrt{2} \left (a^2+b^2-2 a c+c^2\right )^{3/2} e}-\frac{2 \left (b^2-2 a c+b c \tan (d+e x)\right )}{a \left (b^2-4 a c\right ) e \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}+\frac{2 \cot ^2(d+e x) \left (b^2-2 a c+b c \tan (d+e x)\right )}{a \left (b^2-4 a c\right ) e \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}+\frac{2 \left (a \left (b^2-2 (a-c) c\right )+b c (a+c) \tan (d+e x)\right )}{\left (b^2+(a-c)^2\right ) \left (b^2-4 a c\right ) e \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}+\frac{b \left (15 b^2-52 a c\right ) \cot (d+e x) \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}{4 a^3 \left (b^2-4 a c\right ) e}-\frac{\left (5 b^2-12 a c\right ) \cot ^2(d+e x) \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}{2 a^2 \left (b^2-4 a c\right ) e}\\ \end{align*}
Mathematica [C] time = 6.17195, size = 786, normalized size = 0.78 \[ \frac{-\frac{2 \left (2 a c-\frac{b^2}{2}\right ) \tanh ^{-1}\left (\frac{2 a+b \tan (d+e x)}{2 \sqrt{a} \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{a^{3/2} \left (b^2-4 a c\right )}-\frac{2 \left (-\frac{\frac{\left (a c \left (5 b^2-12 a c\right )-\frac{1}{4} b^2 \left (15 b^2-52 a c\right )\right ) \tanh ^{-1}\left (\frac{2 a+b \tan (d+e x)}{2 \sqrt{a} \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{2 a^{3/2}}+\frac{b \left (15 b^2-52 a c\right ) \cot (d+e x) \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}{4 a}}{2 a}-\frac{\left (12 a c-5 b^2\right ) \cot ^2(d+e x) \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}{4 a}\right )}{a \left (b^2-4 a c\right )}-\frac{2 \left (c (-a b-b c) \tan (d+e x)-a \left (-2 a c+b^2+2 c^2\right )\right )}{\left (b^2-4 a c\right ) \left ((c-a)^2+b^2\right ) \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}+\frac{2 \left (2 a c-b^2-b c \tan (d+e x)\right )}{a \left (b^2-4 a c\right ) \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}-\frac{2 \left (-\frac{4 \sqrt{a+i b-c} \left (-\frac{1}{4} (a-c) \left (b^2-4 a c\right )+\frac{1}{4} i b \left (b^2-4 a c\right )\right ) \tanh ^{-1}\left (\frac{2 a-(-b-2 i c) \tan (d+e x)+i b}{2 \sqrt{a+i b-c} \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{4 a+4 i b-4 c}-\frac{4 \sqrt{a-i b-c} \left (-\frac{1}{4} (a-c) \left (b^2-4 a c\right )-\frac{1}{4} i b \left (b^2-4 a c\right )\right ) \tanh ^{-1}\left (\frac{2 a-(-b+2 i c) \tan (d+e x)-i b}{2 \sqrt{a-i b-c} \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{4 a-4 i b-4 c}\right )}{\left (b^2-4 a c\right ) \left ((c-a)^2+b^2\right )}-\frac{2 \cot ^2(d+e x) \left (2 a c-b^2-b c \tan (d+e x)\right )}{a \left (b^2-4 a c\right ) \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}}{e} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 180., size = 0, normalized size = 0. \begin{align*} \int{ \left ( \cot \left ( ex+d \right ) \right ) ^{3} \left ( a+b\tan \left ( ex+d \right ) +c \left ( \tan \left ( ex+d \right ) \right ) ^{2} \right ) ^{-{\frac{3}{2}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\cot ^{3}{\left (d + e x \right )}}{\left (a + b \tan{\left (d + e x \right )} + c \tan ^{2}{\left (d + e x \right )}\right )^{\frac{3}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\cot \left (e x + d\right )^{3}}{{\left (c \tan \left (e x + d\right )^{2} + b \tan \left (e x + d\right ) + a\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]