Optimal. Leaf size=547 \[ -\frac {\sqrt {-\sqrt {a^2-2 a c+b^2+c^2}+a-c} \tanh ^{-1}\left (\frac {-\sqrt {a^2-2 a c+b^2+c^2}+a+b \cot (d+e x)-c}{\sqrt {2} \sqrt {-\sqrt {a^2-2 a c+b^2+c^2}+a-c} \sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}}\right )}{\sqrt {2} e \sqrt {a^2-2 a c+b^2+c^2}}+\frac {\sqrt {\sqrt {a^2-2 a c+b^2+c^2}+a-c} \tanh ^{-1}\left (\frac {\sqrt {a^2-2 a c+b^2+c^2}+a+b \cot (d+e x)-c}{\sqrt {2} \sqrt {\sqrt {a^2-2 a c+b^2+c^2}+a-c} \sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}}\right )}{\sqrt {2} e \sqrt {a^2-2 a c+b^2+c^2}}+\frac {b \left (5 b^2-12 a c\right ) \tanh ^{-1}\left (\frac {b+2 c \cot (d+e x)}{2 \sqrt {c} \sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}}\right )}{16 c^{7/2} e}-\frac {\left (-16 a c+15 b^2-10 b c \cot (d+e x)\right ) \sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}}{24 c^3 e}-\frac {b \tanh ^{-1}\left (\frac {b+2 c \cot (d+e x)}{2 \sqrt {c} \sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}}\right )}{2 c^{3/2} e}-\frac {\cot ^2(d+e x) \sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}}{3 c e}+\frac {\sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}}{c e} \]
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Rubi [A] time = 1.09, antiderivative size = 547, normalized size of antiderivative = 1.00, 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 206
Rule 208
Rule 621
Rule 640
Rule 742
Rule 779
Rule 1030
Rule 1036
Rule 3701
Rule 6725
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*}
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Mathematica [C] time = 21.38, 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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fricas [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
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 \[ \int \frac {\cot \left (e x + d\right )^{5}}{\sqrt {c \cot \left (e x + d\right )^{2} + b \cot \left (e x + d\right ) + a}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 1.07, size = 9581948, normalized size = 17517.27 \[ \text {output too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\mathrm {cot}\left (d+e\,x\right )}^5}{\sqrt {c\,{\mathrm {cot}\left (d+e\,x\right )}^2+b\,\mathrm {cot}\left (d+e\,x\right )+a}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\cot ^{5}{\left (d + e x \right )}}{\sqrt {a + b \cot {\left (d + e x \right )} + c \cot ^{2}{\left (d + e x \right )}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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