Optimal. Leaf size=1189 \[ -\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^2-2 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 {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} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 4.47, antiderivative size = 1189, normalized size of antiderivative = 1.00, 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 = {3782, 6857,
650, 752, 793, 635, 212, 846, 1032, 1050, 1044, 214} \begin {gather*} -\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}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 212
Rule 214
Rule 635
Rule 650
Rule 752
Rule 793
Rule 846
Rule 1032
Rule 1044
Rule 1050
Rule 3782
Rule 6857
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 {\text {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 {\text {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 {\text {Subst}\left (\int \frac {x}{\left (a+b x+c x^2\right )^{3/2}} \, dx,x,\cot (d+e x)\right )}{e}+\frac {\text {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 {\text {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 {\text {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 \text {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 \text {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 \text {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) \text {Subst}\left (\int \frac {1}{\sqrt {a+b x+c x^2}} \, dx,x,\cot (d+e x)\right )}{2 c^2 e}+\frac {\text {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 {\text {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 {\text {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) \text {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 {\text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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*}
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Mathematica [C] Result contains complex when optimal does not.
time = 45.16, size = 5618, normalized size = 4.72 \begin {gather*} \text {Result too large to show} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [B] result has leaf size over 500,000. Avoiding possible recursion issues.
time = 0.51, size = 13067599, normalized size = 10990.41 \[\text {output too large to display}\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F(-1)]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \text {Hanged} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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