Optimal. Leaf size=602 \[ -\frac {\sqrt {-a \left (\sqrt {a^2-2 a c+b^2+c^2}+2 c\right )+c \left (\sqrt {a^2-2 a c+b^2+c^2}+c\right )+a^2+b^2} \tan ^{-1}\left (\frac {-b \sqrt {a^2-2 a c+b^2+c^2} \cot (d+e x)+(a-c) \left (-\sqrt {a^2-2 a c+b^2+c^2}+a-c\right )+b^2}{\sqrt {2} \sqrt [4]{a^2-2 a c+b^2+c^2} \sqrt {-a \left (\sqrt {a^2-2 a c+b^2+c^2}+2 c\right )+c \left (\sqrt {a^2-2 a c+b^2+c^2}+c\right )+a^2+b^2} \sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}}\right )}{\sqrt {2} e \sqrt [4]{a^2-2 a c+b^2+c^2}}+\frac {\sqrt {-a \left (2 c-\sqrt {a^2-2 a c+b^2+c^2}\right )+c \left (c-\sqrt {a^2-2 a c+b^2+c^2}\right )+a^2+b^2} \tanh ^{-1}\left (\frac {b \sqrt {a^2-2 a c+b^2+c^2} \cot (d+e x)+(a-c) \left (\sqrt {a^2-2 a c+b^2+c^2}+a-c\right )+b^2}{\sqrt {2} \sqrt [4]{a^2-2 a c+b^2+c^2} \sqrt {-a \left (2 c-\sqrt {a^2-2 a c+b^2+c^2}\right )+c \left (c-\sqrt {a^2-2 a c+b^2+c^2}\right )+a^2+b^2} \sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}}\right )}{\sqrt {2} e \sqrt [4]{a^2-2 a c+b^2+c^2}}-\frac {\sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}}{e}-\frac {b \tanh ^{-1}\left (\frac {b+2 c \cot (d+e x)}{2 \sqrt {c} \sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}}\right )}{2 \sqrt {c} e} \]
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Rubi [A] time = 23.18, antiderivative size = 602, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 9, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.290, Rules used = {3701, 1021, 1078, 621, 206, 1036, 1030, 208, 205} \[ -\frac {\sqrt {-a \left (\sqrt {a^2-2 a c+b^2+c^2}+2 c\right )+c \left (\sqrt {a^2-2 a c+b^2+c^2}+c\right )+a^2+b^2} \tan ^{-1}\left (\frac {-b \sqrt {a^2-2 a c+b^2+c^2} \cot (d+e x)+(a-c) \left (-\sqrt {a^2-2 a c+b^2+c^2}+a-c\right )+b^2}{\sqrt {2} \sqrt [4]{a^2-2 a c+b^2+c^2} \sqrt {-a \left (\sqrt {a^2-2 a c+b^2+c^2}+2 c\right )+c \left (\sqrt {a^2-2 a c+b^2+c^2}+c\right )+a^2+b^2} \sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}}\right )}{\sqrt {2} e \sqrt [4]{a^2-2 a c+b^2+c^2}}+\frac {\sqrt {-a \left (2 c-\sqrt {a^2-2 a c+b^2+c^2}\right )+c \left (c-\sqrt {a^2-2 a c+b^2+c^2}\right )+a^2+b^2} \tanh ^{-1}\left (\frac {b \sqrt {a^2-2 a c+b^2+c^2} \cot (d+e x)+(a-c) \left (\sqrt {a^2-2 a c+b^2+c^2}+a-c\right )+b^2}{\sqrt {2} \sqrt [4]{a^2-2 a c+b^2+c^2} \sqrt {-a \left (2 c-\sqrt {a^2-2 a c+b^2+c^2}\right )+c \left (c-\sqrt {a^2-2 a c+b^2+c^2}\right )+a^2+b^2} \sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}}\right )}{\sqrt {2} e \sqrt [4]{a^2-2 a c+b^2+c^2}}-\frac {\sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}}{e}-\frac {b \tanh ^{-1}\left (\frac {b+2 c \cot (d+e x)}{2 \sqrt {c} \sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}}\right )}{2 \sqrt {c} e} \]
Antiderivative was successfully verified.
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Rule 205
Rule 206
Rule 208
Rule 621
Rule 1021
Rule 1030
Rule 1036
Rule 1078
Rule 3701
Rubi steps
\begin {align*} \int \cot (d+e x) \sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)} \, dx &=-\frac {\operatorname {Subst}\left (\int \frac {x \sqrt {a+b x+c x^2}}{1+x^2} \, dx,x,\cot (d+e x)\right )}{e}\\ &=-\frac {\sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}}{e}+\frac {\operatorname {Subst}\left (\int \frac {\frac {b}{2}-(a-c) x-\frac {b x^2}{2}}{\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)}}{e}+\frac {\operatorname {Subst}\left (\int \frac {b+(-a+c) x}{\left (1+x^2\right ) \sqrt {a+b x+c x^2}} \, dx,x,\cot (d+e x)\right )}{e}-\frac {b \operatorname {Subst}\left (\int \frac {1}{\sqrt {a+b x+c x^2}} \, dx,x,\cot (d+e x)\right )}{2 e}\\ &=-\frac {\sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}}{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 )}{e}-\frac {\operatorname {Subst}\left (\int \frac {-b \sqrt {a^2+b^2-2 a c+c^2}+\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 )}{2 \sqrt {a^2+b^2-2 a c+c^2} e}+\frac {\operatorname {Subst}\left (\int \frac {b \sqrt {a^2+b^2-2 a c+c^2}+\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 )}{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 \sqrt {c} e}-\frac {\sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}}{e}+\frac {\left (b \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}{2 b \sqrt {a^2+b^2-2 a c+c^2} \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 {-b^2-(a-c) \left (a-c-\sqrt {a^2+b^2-2 a c+c^2}\right )+b \sqrt {a^2+b^2-2 a c+c^2} \cot (d+e x)}{\sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}}\right )}{e}+\frac {\left (b \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}{-2 b \sqrt {a^2+b^2-2 a c+c^2} \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 {-b^2-(a-c) \left (a-c+\sqrt {a^2+b^2-2 a c+c^2}\right )-b \sqrt {a^2+b^2-2 a c+c^2} \cot (d+e x)}{\sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}}\right )}{e}\\ &=-\frac {\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 )} \tan ^{-1}\left (\frac {b^2+(a-c) \left (a-c-\sqrt {a^2+b^2-2 a c+c^2}\right )-b \sqrt {a^2+b^2-2 a c+c^2} \cot (d+e x)}{\sqrt {2} \sqrt [4]{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 )} \sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}}\right )}{\sqrt {2} \sqrt [4]{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 \sqrt {c} e}+\frac {\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 \sqrt {a^2+b^2-2 a c+c^2} \cot (d+e x)}{\sqrt {2} \sqrt [4]{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 )} \sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}}\right )}{\sqrt {2} \sqrt [4]{a^2+b^2-2 a c+c^2} e}-\frac {\sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}}{e}\\ \end {align*}
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Mathematica [C] time = 23.79, size = 350, normalized size = 0.58 \[ -\frac {\sqrt {\frac {a \cos (2 (d+e x))-a-b \sin (2 (d+e x))-c \cos (2 (d+e x))-c}{\cos (2 (d+e x))-1}}}{e}+\frac {\tan (d+e x) \sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)} \left (-i \sqrt {c} \sqrt {a-i b-c} \tan ^{-1}\left (\frac {(b+2 i a) \tan (d+e x)+i b+2 c}{2 \sqrt {a-i b-c} \sqrt {\tan (d+e x) (a \tan (d+e x)+b)+c}}\right )+\sqrt {c} \sqrt {a+i b-c} \tanh ^{-1}\left (\frac {(2 a+i b) \tan (d+e x)+b+2 i c}{2 \sqrt {a+i b-c} \sqrt {\tan (d+e x) (a \tan (d+e x)+b)+c}}\right )-b \tanh ^{-1}\left (\frac {b \tan (d+e x)+2 c}{2 \sqrt {c} \sqrt {\tan (d+e x) (a \tan (d+e x)+b)+c}}\right )\right )}{2 \sqrt {c} e \sqrt {a \tan ^2(d+e x)+b \tan (d+e x)+c}} \]
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 \[ \int \sqrt {c \cot \left (e x + d\right )^{2} + b \cot \left (e x + d\right ) + a} \cot \left (e x + d\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.92, size = 21948746, normalized size = 36459.71 \[ \text {output too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {c \cot \left (e x + d\right )^{2} + b \cot \left (e x + d\right ) + a} \cot \left (e x + d\right )\,{d x} \]
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 \[ \int \mathrm {cot}\left (d+e\,x\right )\,\sqrt {c\,{\mathrm {cot}\left (d+e\,x\right )}^2+b\,\mathrm {cot}\left (d+e\,x\right )+a} \,d x \]
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 \[ \int \sqrt {a + b \cot {\left (d + e x \right )} + c \cot ^{2}{\left (d + e x \right )}} \cot {\left (d + e x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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