Optimal. Leaf size=865 \[ -\frac{2 (2 a+b \cot (d+e x)) \cot ^2(d+e x)}{\left (b^2-4 a c\right ) e \sqrt{c \cot ^2(d+e x)+b \cot (d+e x)+a}}+\frac{3 b \tanh ^{-1}\left (\frac{b+2 c \cot (d+e x)}{2 \sqrt{c} \sqrt{c \cot ^2(d+e x)+b \cot (d+e x)+a}}\right )}{2 c^{5/2} e}-\frac{\sqrt{2 a-2 c-\sqrt{a^2-2 c a+b^2+c^2}} \sqrt{a^2-2 c a-b^2+c^2+(a-c) \sqrt{a^2-2 c a+b^2+c^2}} \tanh ^{-1}\left (\frac{b^2-\left (2 a-2 c-\sqrt{a^2-2 c a+b^2+c^2}\right ) \cot (d+e x) b-(a-c) \left (a-c+\sqrt{a^2-2 c a+b^2+c^2}\right )}{\sqrt{2} \sqrt{2 a-2 c-\sqrt{a^2-2 c a+b^2+c^2}} \sqrt{a^2-2 c a-b^2+c^2+(a-c) \sqrt{a^2-2 c a+b^2+c^2}} \sqrt{c \cot ^2(d+e x)+b \cot (d+e x)+a}}\right )}{\sqrt{2} \left (a^2-2 c a+b^2+c^2\right )^{3/2} e}+\frac{\sqrt{2 a-2 c+\sqrt{a^2-2 c a+b^2+c^2}} \sqrt{a^2-2 c a-b^2+c^2-(a-c) \sqrt{a^2-2 c a+b^2+c^2}} \tanh ^{-1}\left (\frac{b^2-\left (2 a-2 c+\sqrt{a^2-2 c a+b^2+c^2}\right ) \cot (d+e x) b-(a-c) \left (a-c-\sqrt{a^2-2 c a+b^2+c^2}\right )}{\sqrt{2} \sqrt{2 a-2 c+\sqrt{a^2-2 c a+b^2+c^2}} \sqrt{a^2-2 c a-b^2+c^2-(a-c) \sqrt{a^2-2 c a+b^2+c^2}} \sqrt{c \cot ^2(d+e x)+b \cot (d+e x)+a}}\right )}{\sqrt{2} \left (a^2-2 c a+b^2+c^2\right )^{3/2} e}-\frac{\left (3 b^2-2 c \cot (d+e x) b-8 a c\right ) \sqrt{c \cot ^2(d+e x)+b \cot (d+e x)+a}}{c^2 \left (b^2-4 a c\right ) e}+\frac{2 (2 a+b \cot (d+e x))}{\left (b^2-4 a c\right ) e \sqrt{c \cot ^2(d+e x)+b \cot (d+e x)+a}}-\frac{2 \left (a \left (b^2-2 (a-c) c\right )+b c (a+c) \cot (d+e x)\right )}{\left (b^2+(a-c)^2\right ) \left (b^2-4 a c\right ) e \sqrt{c \cot ^2(d+e x)+b \cot (d+e x)+a}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 4.77397, antiderivative size = 865, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 11, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {3701, 6725, 636, 738, 779, 621, 206, 1018, 1036, 1030, 208} \[ -\frac{2 (2 a+b \cot (d+e x)) \cot ^2(d+e x)}{\left (b^2-4 a c\right ) e \sqrt{c \cot ^2(d+e x)+b \cot (d+e x)+a}}+\frac{3 b \tanh ^{-1}\left (\frac{b+2 c \cot (d+e x)}{2 \sqrt{c} \sqrt{c \cot ^2(d+e x)+b \cot (d+e x)+a}}\right )}{2 c^{5/2} e}-\frac{\sqrt{2 a-2 c-\sqrt{a^2-2 c a+b^2+c^2}} \sqrt{a^2-2 c a-b^2+c^2+(a-c) \sqrt{a^2-2 c a+b^2+c^2}} \tanh ^{-1}\left (\frac{b^2-\left (2 a-2 c-\sqrt{a^2-2 c a+b^2+c^2}\right ) \cot (d+e x) b-(a-c) \left (a-c+\sqrt{a^2-2 c a+b^2+c^2}\right )}{\sqrt{2} \sqrt{2 a-2 c-\sqrt{a^2-2 c a+b^2+c^2}} \sqrt{a^2-2 c a-b^2+c^2+(a-c) \sqrt{a^2-2 c a+b^2+c^2}} \sqrt{c \cot ^2(d+e x)+b \cot (d+e x)+a}}\right )}{\sqrt{2} \left (a^2-2 c a+b^2+c^2\right )^{3/2} e}+\frac{\sqrt{2 a-2 c+\sqrt{a^2-2 c a+b^2+c^2}} \sqrt{a^2-2 c a-b^2+c^2-(a-c) \sqrt{a^2-2 c a+b^2+c^2}} \tanh ^{-1}\left (\frac{b^2-\left (2 a-2 c+\sqrt{a^2-2 c a+b^2+c^2}\right ) \cot (d+e x) b-(a-c) \left (a-c-\sqrt{a^2-2 c a+b^2+c^2}\right )}{\sqrt{2} \sqrt{2 a-2 c+\sqrt{a^2-2 c a+b^2+c^2}} \sqrt{a^2-2 c a-b^2+c^2-(a-c) \sqrt{a^2-2 c a+b^2+c^2}} \sqrt{c \cot ^2(d+e x)+b \cot (d+e x)+a}}\right )}{\sqrt{2} \left (a^2-2 c a+b^2+c^2\right )^{3/2} e}-\frac{\left (3 b^2-2 c \cot (d+e x) b-8 a c\right ) \sqrt{c \cot ^2(d+e x)+b \cot (d+e x)+a}}{c^2 \left (b^2-4 a c\right ) e}+\frac{2 (2 a+b \cot (d+e x))}{\left (b^2-4 a c\right ) e \sqrt{c \cot ^2(d+e x)+b \cot (d+e x)+a}}-\frac{2 \left (a \left (b^2-2 (a-c) c\right )+b c (a+c) \cot (d+e x)\right )}{\left (b^2+(a-c)^2\right ) \left (b^2-4 a c\right ) e \sqrt{c \cot ^2(d+e x)+b \cot (d+e x)+a}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3701
Rule 6725
Rule 636
Rule 738
Rule 779
Rule 621
Rule 206
Rule 1018
Rule 1036
Rule 1030
Rule 208
Rubi steps
\begin{align*} \int \frac{\cot ^5(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^5}{\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}{\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}{\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 \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{-\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 \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 (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{\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 \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 (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{\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 \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 (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}\\ \end{align*}
Mathematica [C] time = 26.505, size = 737, normalized size = 0.85 \[ \frac{\sqrt{\frac{a \cos (2 (d+e x))-a-b \sin (2 (d+e x))-c \cos (2 (d+e x))-c}{\cos (2 (d+e x))-1}} \left (-\frac{2 \left (8 a^2 b^2 c-2 a^2 b^3 \sin (2 (d+e x))-2 a^3 b^2-5 a^2 b c^2 \sin (2 (d+e x))+6 a^3 b c \sin (2 (d+e x))-a^4 b \sin (2 (d+e x))-4 a^3 c^2+4 a^4 c+5 a b^3 c \sin (2 (d+e x))-2 a b^4+b^5 (-\sin (2 (d+e x)))\right )}{c (a-c) (a-i b-c) (a+i b-c) \left (4 a c-b^2\right ) (a \cos (2 (d+e x))-a-b \sin (2 (d+e x))-c \cos (2 (d+e x))-c)}-\frac{15 a^2 b^2 c-3 a^3 b^2-16 a^3 c^2+12 a^2 c^3+8 a^4 c-7 a b^2 c^2-3 a b^4-4 a c^4+b^2 c^3+b^4 c}{c^2 (a-c) (a-i b-c) (a+i b-c) \left (4 a c-b^2\right )}\right )}{e}-\frac{\tan (d+e x) \sqrt{a+b \cot (d+e x)+c \cot ^2(d+e x)} \left (-3 b \left (a^2-2 a c+b^2+c^2\right ) \tanh ^{-1}\left (\frac{b \tan (d+e x)+2 c}{2 \sqrt{c} \sqrt{\tan (d+e x) (a \tan (d+e x)+b)+c}}\right )+\frac{i c^{5/2} (a+i b-c) \tan ^{-1}\left (\frac{(b+2 i a) \tan (d+e x)+i b+2 c}{2 \sqrt{a-i b-c} \sqrt{\tan (d+e x) (a \tan (d+e x)+b)+c}}\right )}{\sqrt{a-i b-c}}+\frac{c^{5/2} (-a+i b+c) \tanh ^{-1}\left (\frac{(2 a+i b) \tan (d+e x)+b+2 i c}{2 \sqrt{a+i b-c} \sqrt{\tan (d+e x) (a \tan (d+e x)+b)+c}}\right )}{\sqrt{a+i b-c}}\right )}{2 c^{5/2} e \left (a^2-2 a c+b^2+c^2\right ) \sqrt{\tan (d+e x) (a \tan (d+e x)+b)+c}} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.05, size = 13067692, normalized size = 15107.2 \begin{align*} \text{output too large to display} \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(-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]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\cot \left (e x + d\right )^{5}}{{\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.
[In]
[Out]