Optimal. Leaf size=547 \[ -\frac{\sqrt{-\sqrt{a^2-2 a c+b^2+c^2}+a-c} \tanh ^{-1}\left (\frac{-\sqrt{a^2-2 a c+b^2+c^2}+a+b \cot (d+e x)-c}{\sqrt{2} \sqrt{-\sqrt{a^2-2 a c+b^2+c^2}+a-c} \sqrt{a+b \cot (d+e x)+c \cot ^2(d+e x)}}\right )}{\sqrt{2} e \sqrt{a^2-2 a c+b^2+c^2}}+\frac{\sqrt{\sqrt{a^2-2 a c+b^2+c^2}+a-c} \tanh ^{-1}\left (\frac{\sqrt{a^2-2 a c+b^2+c^2}+a+b \cot (d+e x)-c}{\sqrt{2} \sqrt{\sqrt{a^2-2 a c+b^2+c^2}+a-c} \sqrt{a+b \cot (d+e x)+c \cot ^2(d+e x)}}\right )}{\sqrt{2} e \sqrt{a^2-2 a c+b^2+c^2}}-\frac{\left (-16 a c+15 b^2-10 b c \cot (d+e x)\right ) \sqrt{a+b \cot (d+e x)+c \cot ^2(d+e x)}}{24 c^3 e}+\frac{b \left (5 b^2-12 a c\right ) \tanh ^{-1}\left (\frac{b+2 c \cot (d+e x)}{2 \sqrt{c} \sqrt{a+b \cot (d+e x)+c \cot ^2(d+e x)}}\right )}{16 c^{7/2} e}-\frac{b \tanh ^{-1}\left (\frac{b+2 c \cot (d+e x)}{2 \sqrt{c} \sqrt{a+b \cot (d+e x)+c \cot ^2(d+e x)}}\right )}{2 c^{3/2} e}-\frac{\cot ^2(d+e x) \sqrt{a+b \cot (d+e x)+c \cot ^2(d+e x)}}{3 c e}+\frac{\sqrt{a+b \cot (d+e x)+c \cot ^2(d+e x)}}{c e} \]
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Rubi [A] time = 1.08568, antiderivative size = 547, normalized size of antiderivative = 1., number of steps used = 15, number of rules used = 10, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.303, Rules used = {3701, 6725, 640, 621, 206, 742, 779, 1036, 1030, 208} \[ -\frac{\sqrt{-\sqrt{a^2-2 a c+b^2+c^2}+a-c} \tanh ^{-1}\left (\frac{-\sqrt{a^2-2 a c+b^2+c^2}+a+b \cot (d+e x)-c}{\sqrt{2} \sqrt{-\sqrt{a^2-2 a c+b^2+c^2}+a-c} \sqrt{a+b \cot (d+e x)+c \cot ^2(d+e x)}}\right )}{\sqrt{2} e \sqrt{a^2-2 a c+b^2+c^2}}+\frac{\sqrt{\sqrt{a^2-2 a c+b^2+c^2}+a-c} \tanh ^{-1}\left (\frac{\sqrt{a^2-2 a c+b^2+c^2}+a+b \cot (d+e x)-c}{\sqrt{2} \sqrt{\sqrt{a^2-2 a c+b^2+c^2}+a-c} \sqrt{a+b \cot (d+e x)+c \cot ^2(d+e x)}}\right )}{\sqrt{2} e \sqrt{a^2-2 a c+b^2+c^2}}-\frac{\left (-16 a c+15 b^2-10 b c \cot (d+e x)\right ) \sqrt{a+b \cot (d+e x)+c \cot ^2(d+e x)}}{24 c^3 e}+\frac{b \left (5 b^2-12 a c\right ) \tanh ^{-1}\left (\frac{b+2 c \cot (d+e x)}{2 \sqrt{c} \sqrt{a+b \cot (d+e x)+c \cot ^2(d+e x)}}\right )}{16 c^{7/2} e}-\frac{b \tanh ^{-1}\left (\frac{b+2 c \cot (d+e x)}{2 \sqrt{c} \sqrt{a+b \cot (d+e x)+c \cot ^2(d+e x)}}\right )}{2 c^{3/2} e}-\frac{\cot ^2(d+e x) \sqrt{a+b \cot (d+e x)+c \cot ^2(d+e x)}}{3 c e}+\frac{\sqrt{a+b \cot (d+e x)+c \cot ^2(d+e x)}}{c e} \]
Antiderivative was successfully verified.
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Rule 3701
Rule 6725
Rule 640
Rule 621
Rule 206
Rule 742
Rule 779
Rule 1036
Rule 1030
Rule 208
Rubi steps
\begin{align*} \int \frac{\cot ^5(d+e x)}{\sqrt{a+b \cot (d+e x)+c \cot ^2(d+e x)}} \, dx &=-\frac{\operatorname{Subst}\left (\int \frac{x^5}{\left (1+x^2\right ) \sqrt{a+b x+c x^2}} \, dx,x,\cot (d+e x)\right )}{e}\\ &=-\frac{\operatorname{Subst}\left (\int \left (-\frac{x}{\sqrt{a+b x+c x^2}}+\frac{x^3}{\sqrt{a+b x+c x^2}}+\frac{x}{\left (1+x^2\right ) \sqrt{a+b x+c x^2}}\right ) \, dx,x,\cot (d+e x)\right )}{e}\\ &=\frac{\operatorname{Subst}\left (\int \frac{x}{\sqrt{a+b x+c x^2}} \, dx,x,\cot (d+e x)\right )}{e}-\frac{\operatorname{Subst}\left (\int \frac{x^3}{\sqrt{a+b x+c x^2}} \, dx,x,\cot (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,\cot (d+e x)\right )}{e}\\ &=\frac{\sqrt{a+b \cot (d+e x)+c \cot ^2(d+e x)}}{c e}-\frac{\cot ^2(d+e x) \sqrt{a+b \cot (d+e x)+c \cot ^2(d+e x)}}{3 c e}-\frac{\operatorname{Subst}\left (\int \frac{x \left (-2 a-\frac{5 b x}{2}\right )}{\sqrt{a+b x+c x^2}} \, dx,x,\cot (d+e x)\right )}{3 c e}-\frac{b \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+b x+c x^2}} \, dx,x,\cot (d+e x)\right )}{2 c 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,\cot (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,\cot (d+e x)\right )}{2 \sqrt{a^2+b^2-2 a c+c^2} e}\\ &=\frac{\sqrt{a+b \cot (d+e x)+c \cot ^2(d+e x)}}{c e}-\frac{\cot ^2(d+e x) \sqrt{a+b \cot (d+e x)+c \cot ^2(d+e x)}}{3 c e}-\frac{\left (15 b^2-16 a c-10 b c \cot (d+e x)\right ) \sqrt{a+b \cot (d+e x)+c \cot ^2(d+e x)}}{24 c^3 e}-\frac{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 e}+\frac{\left (b \left (5 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^3 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 \cot (d+e x)}{\sqrt{a+b \cot (d+e x)+c \cot ^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 \cot (d+e x)}{\sqrt{a+b \cot (d+e x)+c \cot ^2(d+e x)}}\right )}{\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 \cot (d+e x)}{\sqrt{2} \sqrt{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} \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 \cot (d+e x)}{\sqrt{2} \sqrt{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} \sqrt{a^2+b^2-2 a c+c^2} e}-\frac{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^{3/2} e}+\frac{\sqrt{a+b \cot (d+e x)+c \cot ^2(d+e x)}}{c e}-\frac{\cot ^2(d+e x) \sqrt{a+b \cot (d+e x)+c \cot ^2(d+e x)}}{3 c e}-\frac{\left (15 b^2-16 a c-10 b c \cot (d+e x)\right ) \sqrt{a+b \cot (d+e x)+c \cot ^2(d+e x)}}{24 c^3 e}+\frac{\left (b \left (5 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^3 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 \cot (d+e x)}{\sqrt{2} \sqrt{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} \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 \cot (d+e x)}{\sqrt{2} \sqrt{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} \sqrt{a^2+b^2-2 a c+c^2} e}-\frac{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^{3/2} e}+\frac{b \left (5 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^{7/2} e}+\frac{\sqrt{a+b \cot (d+e x)+c \cot ^2(d+e x)}}{c e}-\frac{\cot ^2(d+e x) \sqrt{a+b \cot (d+e x)+c \cot ^2(d+e x)}}{3 c e}-\frac{\left (15 b^2-16 a c-10 b c \cot (d+e x)\right ) \sqrt{a+b \cot (d+e x)+c \cot ^2(d+e x)}}{24 c^3 e}\\ \end{align*}
Mathematica [C] time = 21.286, size = 416, normalized size = 0.76 \[ \frac{\sqrt{\frac{a \cos (2 (d+e x))-a-b \sin (2 (d+e x))-c \cos (2 (d+e x))-c}{\cos (2 (d+e x))-1}} \left (\frac{16 a c-15 b^2+32 c^2}{24 c^3}+\frac{5 b \cot (d+e x)}{12 c^2}-\frac{\csc ^2(d+e x)}{3 c}\right )}{e}+\frac{\tan (d+e x) \sqrt{a+b \cot (d+e x)+c \cot ^2(d+e x)} \left (b \left (5 b^2-4 c (3 a+2 c)\right ) \tanh ^{-1}\left (\frac{b \tan (d+e x)+2 c}{2 \sqrt{c} \sqrt{\tan (d+e x) (a \tan (d+e x)+b)+c}}\right )-\frac{8 i c^{7/2} \tan ^{-1}\left (\frac{(b+2 i a) \tan (d+e x)+i b+2 c}{2 \sqrt{a-i b-c} \sqrt{\tan (d+e x) (a \tan (d+e x)+b)+c}}\right )}{\sqrt{a-i b-c}}+\frac{8 c^{7/2} \tanh ^{-1}\left (\frac{(2 a+i b) \tan (d+e x)+b+2 i c}{2 \sqrt{a+i b-c} \sqrt{\tan (d+e x) (a \tan (d+e x)+b)+c}}\right )}{\sqrt{a+i b-c}}\right )}{16 c^{7/2} e \sqrt{\tan (d+e x) (a \tan (d+e x)+b)+c}} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.725, size = 9581948, normalized size = 17517.3 \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 )^{5}}{\sqrt{c \cot \left (e x + d\right )^{2} + b \cot \left (e x + d\right ) + a}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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