Integrand size = 31, antiderivative size = 350 \[ \int \frac {\cot (d+e x)}{\sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}} \, dx=-\frac {\text {arctanh}\left (\frac {2 a+b \tan (d+e x)}{2 \sqrt {a} \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{\sqrt {a} e}-\frac {\sqrt {a-c-\sqrt {a^2+b^2-2 a c+c^2}} \text {arctanh}\left (\frac {a-c-\sqrt {a^2+b^2-2 a c+c^2}+b \tan (d+e x)}{\sqrt {2} \sqrt {a-c-\sqrt {a^2+b^2-2 a c+c^2}} \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{\sqrt {2} \sqrt {a^2+b^2-2 a c+c^2} e}+\frac {\sqrt {a-c+\sqrt {a^2+b^2-2 a c+c^2}} \text {arctanh}\left (\frac {a-c+\sqrt {a^2+b^2-2 a c+c^2}+b \tan (d+e x)}{\sqrt {2} \sqrt {a-c+\sqrt {a^2+b^2-2 a c+c^2}} \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{\sqrt {2} \sqrt {a^2+b^2-2 a c+c^2} e} \]
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Time = 0.77 (sec) , antiderivative size = 350, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.226, Rules used = {3781, 6857, 738, 212, 1050, 1044, 214} \[ \int \frac {\cot (d+e x)}{\sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}} \, dx=-\frac {\sqrt {-\sqrt {a^2-2 a c+b^2+c^2}+a-c} \text {arctanh}\left (\frac {-\sqrt {a^2-2 a c+b^2+c^2}+a+b \tan (d+e x)-c}{\sqrt {2} \sqrt {-\sqrt {a^2-2 a c+b^2+c^2}+a-c} \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{\sqrt {2} e \sqrt {a^2-2 a c+b^2+c^2}}+\frac {\sqrt {\sqrt {a^2-2 a c+b^2+c^2}+a-c} \text {arctanh}\left (\frac {\sqrt {a^2-2 a c+b^2+c^2}+a+b \tan (d+e x)-c}{\sqrt {2} \sqrt {\sqrt {a^2-2 a c+b^2+c^2}+a-c} \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{\sqrt {2} e \sqrt {a^2-2 a c+b^2+c^2}}-\frac {\text {arctanh}\left (\frac {2 a+b \tan (d+e x)}{2 \sqrt {a} \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{\sqrt {a} e} \]
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Rule 212
Rule 214
Rule 738
Rule 1044
Rule 1050
Rule 3781
Rule 6857
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {1}{x \left (1+x^2\right ) \sqrt {a+b x+c x^2}} \, dx,x,\tan (d+e x)\right )}{e} \\ & = \frac {\text {Subst}\left (\int \left (\frac {1}{x \sqrt {a+b x+c x^2}}-\frac {x}{\left (1+x^2\right ) \sqrt {a+b x+c x^2}}\right ) \, dx,x,\tan (d+e x)\right )}{e} \\ & = \frac {\text {Subst}\left (\int \frac {1}{x \sqrt {a+b x+c x^2}} \, dx,x,\tan (d+e x)\right )}{e}-\frac {\text {Subst}\left (\int \frac {x}{\left (1+x^2\right ) \sqrt {a+b x+c x^2}} \, dx,x,\tan (d+e x)\right )}{e} \\ & = -\frac {2 \text {Subst}\left (\int \frac {1}{4 a-x^2} \, dx,x,\frac {2 a+b \tan (d+e x)}{\sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{e}+\frac {\text {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,\tan (d+e x)\right )}{2 \sqrt {a^2+b^2-2 a c+c^2} e}-\frac {\text {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,\tan (d+e x)\right )}{2 \sqrt {a^2+b^2-2 a c+c^2} e} \\ & = -\frac {\text {arctanh}\left (\frac {2 a+b \tan (d+e x)}{2 \sqrt {a} \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{\sqrt {a} e}+\frac {\left (b \left (a-c-\sqrt {a^2+b^2-2 a c+c^2}\right )\right ) \text {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 \tan (d+e x)}{\sqrt {a+b \tan (d+e x)+c \tan ^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 ) \text {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 \tan (d+e x)}{\sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{\sqrt {a^2+b^2-2 a c+c^2} e} \\ & = -\frac {\text {arctanh}\left (\frac {2 a+b \tan (d+e x)}{2 \sqrt {a} \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{\sqrt {a} e}-\frac {\sqrt {a-c-\sqrt {a^2+b^2-2 a c+c^2}} \text {arctanh}\left (\frac {a-c-\sqrt {a^2+b^2-2 a c+c^2}+b \tan (d+e x)}{\sqrt {2} \sqrt {a-c-\sqrt {a^2+b^2-2 a c+c^2}} \sqrt {a+b \tan (d+e x)+c \tan ^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}} \text {arctanh}\left (\frac {a-c+\sqrt {a^2+b^2-2 a c+c^2}+b \tan (d+e x)}{\sqrt {2} \sqrt {a-c+\sqrt {a^2+b^2-2 a c+c^2}} \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{\sqrt {2} \sqrt {a^2+b^2-2 a c+c^2} e} \\ \end{align*}
Result contains complex when optimal does not.
Time = 0.44 (sec) , antiderivative size = 223, normalized size of antiderivative = 0.64 \[ \int \frac {\cot (d+e x)}{\sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}} \, dx=\frac {-\frac {2 \text {arctanh}\left (\frac {2 a+b \tan (d+e x)}{2 \sqrt {a} \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{\sqrt {a}}+\frac {\text {arctanh}\left (\frac {2 a-i b+(b-2 i c) \tan (d+e x)}{2 \sqrt {a-i b-c} \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{\sqrt {a-i b-c}}+\frac {\text {arctanh}\left (\frac {2 a+i b+(b+2 i c) \tan (d+e x)}{2 \sqrt {a+i b-c} \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{\sqrt {a+i b-c}}}{2 e} \]
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Timed out.
hanged
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Leaf count of result is larger than twice the leaf count of optimal. 10284 vs. \(2 (309) = 618\).
Time = 2.29 (sec) , antiderivative size = 20605, normalized size of antiderivative = 58.87 \[ \int \frac {\cot (d+e x)}{\sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}} \, dx=\text {Too large to display} \]
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\[ \int \frac {\cot (d+e x)}{\sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}} \, dx=\int \frac {\cot {\left (d + e x \right )}}{\sqrt {a + b \tan {\left (d + e x \right )} + c \tan ^{2}{\left (d + e x \right )}}}\, dx \]
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\[ \int \frac {\cot (d+e x)}{\sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}} \, dx=\int { \frac {\cot \left (e x + d\right )}{\sqrt {c \tan \left (e x + d\right )^{2} + b \tan \left (e x + d\right ) + a}} \,d x } \]
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\[ \int \frac {\cot (d+e x)}{\sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}} \, dx=\int { \frac {\cot \left (e x + d\right )}{\sqrt {c \tan \left (e x + d\right )^{2} + b \tan \left (e x + d\right ) + a}} \,d x } \]
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Timed out. \[ \int \frac {\cot (d+e x)}{\sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}} \, dx=\int \frac {\mathrm {cot}\left (d+e\,x\right )}{\sqrt {c\,{\mathrm {tan}\left (d+e\,x\right )}^2+b\,\mathrm {tan}\left (d+e\,x\right )+a}} \,d x \]
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