Integrand size = 35, antiderivative size = 435 \[ \int \cot ^3(d+e x) \sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)} \, dx=\frac {\sqrt {a} \text {arctanh}\left (\frac {2 a+b \tan ^2(d+e x)}{2 \sqrt {a} \sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}\right )}{2 e}-\frac {b \text {arctanh}\left (\frac {2 a+b \tan ^2(d+e x)}{2 \sqrt {a} \sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}\right )}{4 \sqrt {a} e}-\frac {\sqrt {a-b+c} \text {arctanh}\left (\frac {2 a-b+(b-2 c) \tan ^2(d+e x)}{2 \sqrt {a-b+c} \sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}\right )}{2 e}-\frac {b \text {arctanh}\left (\frac {b+2 c \tan ^2(d+e x)}{2 \sqrt {c} \sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}\right )}{4 \sqrt {c} e}+\frac {(b-2 c) \text {arctanh}\left (\frac {b+2 c \tan ^2(d+e x)}{2 \sqrt {c} \sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}\right )}{4 \sqrt {c} e}+\frac {\sqrt {c} \text {arctanh}\left (\frac {b+2 c \tan ^2(d+e x)}{2 \sqrt {c} \sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}\right )}{2 e}-\frac {\cot ^2(d+e x) \sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}{2 e} \]
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Time = 0.66 (sec) , antiderivative size = 435, normalized size of antiderivative = 1.00, number of steps used = 22, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.257, Rules used = {3781, 1265, 974, 746, 857, 635, 212, 738, 748} \[ \int \cot ^3(d+e x) \sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)} \, dx=-\frac {b \text {arctanh}\left (\frac {2 a+b \tan ^2(d+e x)}{2 \sqrt {a} \sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}\right )}{4 \sqrt {a} e}+\frac {\sqrt {a} \text {arctanh}\left (\frac {2 a+b \tan ^2(d+e x)}{2 \sqrt {a} \sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}\right )}{2 e}-\frac {\sqrt {a-b+c} \text {arctanh}\left (\frac {2 a+(b-2 c) \tan ^2(d+e x)-b}{2 \sqrt {a-b+c} \sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}\right )}{2 e}+\frac {\sqrt {c} \text {arctanh}\left (\frac {b+2 c \tan ^2(d+e x)}{2 \sqrt {c} \sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}\right )}{2 e}-\frac {b \text {arctanh}\left (\frac {b+2 c \tan ^2(d+e x)}{2 \sqrt {c} \sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}\right )}{4 \sqrt {c} e}+\frac {(b-2 c) \text {arctanh}\left (\frac {b+2 c \tan ^2(d+e x)}{2 \sqrt {c} \sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}\right )}{4 \sqrt {c} e}-\frac {\cot ^2(d+e x) \sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}{2 e} \]
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Rule 212
Rule 635
Rule 738
Rule 746
Rule 748
Rule 857
Rule 974
Rule 1265
Rule 3781
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {\sqrt {a+b x^2+c x^4}}{x^3 \left (1+x^2\right )} \, dx,x,\tan (d+e x)\right )}{e} \\ & = \frac {\text {Subst}\left (\int \frac {\sqrt {a+b x+c x^2}}{x^2 (1+x)} \, dx,x,\tan ^2(d+e x)\right )}{2 e} \\ & = \frac {\text {Subst}\left (\int \left (\frac {\sqrt {a+b x+c x^2}}{x^2}-\frac {\sqrt {a+b x+c x^2}}{x}+\frac {\sqrt {a+b x+c x^2}}{1+x}\right ) \, dx,x,\tan ^2(d+e x)\right )}{2 e} \\ & = \frac {\text {Subst}\left (\int \frac {\sqrt {a+b x+c x^2}}{x^2} \, dx,x,\tan ^2(d+e x)\right )}{2 e}-\frac {\text {Subst}\left (\int \frac {\sqrt {a+b x+c x^2}}{x} \, dx,x,\tan ^2(d+e x)\right )}{2 e}+\frac {\text {Subst}\left (\int \frac {\sqrt {a+b x+c x^2}}{1+x} \, dx,x,\tan ^2(d+e x)\right )}{2 e} \\ & = -\frac {\cot ^2(d+e x) \sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}{2 e}+\frac {\text {Subst}\left (\int \frac {-2 a-b x}{x \sqrt {a+b x+c x^2}} \, dx,x,\tan ^2(d+e x)\right )}{4 e}-\frac {\text {Subst}\left (\int \frac {-2 a+b-(b-2 c) x}{(1+x) \sqrt {a+b x+c x^2}} \, dx,x,\tan ^2(d+e x)\right )}{4 e}+\frac {\text {Subst}\left (\int \frac {b+2 c x}{x \sqrt {a+b x+c x^2}} \, dx,x,\tan ^2(d+e x)\right )}{4 e} \\ & = -\frac {\cot ^2(d+e x) \sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}{2 e}-\frac {a \text {Subst}\left (\int \frac {1}{x \sqrt {a+b x+c x^2}} \, dx,x,\tan ^2(d+e x)\right )}{2 e}-\frac {b \text {Subst}\left (\int \frac {1}{\sqrt {a+b x+c x^2}} \, dx,x,\tan ^2(d+e x)\right )}{4 e}+\frac {b \text {Subst}\left (\int \frac {1}{x \sqrt {a+b x+c x^2}} \, dx,x,\tan ^2(d+e x)\right )}{4 e}+\frac {(b-2 c) \text {Subst}\left (\int \frac {1}{\sqrt {a+b x+c x^2}} \, dx,x,\tan ^2(d+e x)\right )}{4 e}+\frac {c \text {Subst}\left (\int \frac {1}{\sqrt {a+b x+c x^2}} \, dx,x,\tan ^2(d+e x)\right )}{2 e}+\frac {(a-b+c) \text {Subst}\left (\int \frac {1}{(1+x) \sqrt {a+b x+c x^2}} \, dx,x,\tan ^2(d+e x)\right )}{2 e} \\ & = -\frac {\cot ^2(d+e x) \sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}{2 e}+\frac {a \text {Subst}\left (\int \frac {1}{4 a-x^2} \, dx,x,\frac {2 a+b \tan ^2(d+e x)}{\sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}\right )}{e}-\frac {b \text {Subst}\left (\int \frac {1}{4 a-x^2} \, dx,x,\frac {2 a+b \tan ^2(d+e x)}{\sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}\right )}{2 e}-\frac {b \text {Subst}\left (\int \frac {1}{4 c-x^2} \, dx,x,\frac {b+2 c \tan ^2(d+e x)}{\sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}\right )}{2 e}+\frac {(b-2 c) \text {Subst}\left (\int \frac {1}{4 c-x^2} \, dx,x,\frac {b+2 c \tan ^2(d+e x)}{\sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}\right )}{2 e}+\frac {c \text {Subst}\left (\int \frac {1}{4 c-x^2} \, dx,x,\frac {b+2 c \tan ^2(d+e x)}{\sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}\right )}{e}-\frac {(a-b+c) \text {Subst}\left (\int \frac {1}{4 a-4 b+4 c-x^2} \, dx,x,\frac {2 a-b-(-b+2 c) \tan ^2(d+e x)}{\sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}\right )}{e} \\ & = \frac {\sqrt {a} \text {arctanh}\left (\frac {2 a+b \tan ^2(d+e x)}{2 \sqrt {a} \sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}\right )}{2 e}-\frac {b \text {arctanh}\left (\frac {2 a+b \tan ^2(d+e x)}{2 \sqrt {a} \sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}\right )}{4 \sqrt {a} e}-\frac {\sqrt {a-b+c} \text {arctanh}\left (\frac {2 a-b+(b-2 c) \tan ^2(d+e x)}{2 \sqrt {a-b+c} \sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}\right )}{2 e}-\frac {b \text {arctanh}\left (\frac {b+2 c \tan ^2(d+e x)}{2 \sqrt {c} \sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}\right )}{4 \sqrt {c} e}+\frac {(b-2 c) \text {arctanh}\left (\frac {b+2 c \tan ^2(d+e x)}{2 \sqrt {c} \sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}\right )}{4 \sqrt {c} e}+\frac {\sqrt {c} \text {arctanh}\left (\frac {b+2 c \tan ^2(d+e x)}{2 \sqrt {c} \sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}\right )}{2 e}-\frac {\cot ^2(d+e x) \sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}{2 e} \\ \end{align*}
Time = 1.42 (sec) , antiderivative size = 187, normalized size of antiderivative = 0.43 \[ \int \cot ^3(d+e x) \sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)} \, dx=\frac {(2 a-b) \text {arctanh}\left (\frac {2 a+b \tan ^2(d+e x)}{2 \sqrt {a} \sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}\right )-2 \sqrt {a} \left (\sqrt {a-b+c} \text {arctanh}\left (\frac {2 a-b+(b-2 c) \tan ^2(d+e x)}{2 \sqrt {a-b+c} \sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}\right )+\cot ^2(d+e x) \sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}\right )}{4 \sqrt {a} e} \]
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\[\int \cot \left (e x +d \right )^{3} \sqrt {a +b \tan \left (e x +d \right )^{2}+c \tan \left (e x +d \right )^{4}}d x\]
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Time = 1.78 (sec) , antiderivative size = 1186, normalized size of antiderivative = 2.73 \[ \int \cot ^3(d+e x) \sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)} \, dx=\text {Too large to display} \]
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\[ \int \cot ^3(d+e x) \sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)} \, dx=\int \sqrt {a + b \tan ^{2}{\left (d + e x \right )} + c \tan ^{4}{\left (d + e x \right )}} \cot ^{3}{\left (d + e x \right )}\, dx \]
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\[ \int \cot ^3(d+e x) \sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)} \, dx=\int { \sqrt {c \tan \left (e x + d\right )^{4} + b \tan \left (e x + d\right )^{2} + a} \cot \left (e x + d\right )^{3} \,d x } \]
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\[ \int \cot ^3(d+e x) \sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)} \, dx=\int { \sqrt {c \tan \left (e x + d\right )^{4} + b \tan \left (e x + d\right )^{2} + a} \cot \left (e x + d\right )^{3} \,d x } \]
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Timed out. \[ \int \cot ^3(d+e x) \sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)} \, dx=\int {\mathrm {cot}\left (d+e\,x\right )}^3\,\sqrt {c\,{\mathrm {tan}\left (d+e\,x\right )}^4+b\,{\mathrm {tan}\left (d+e\,x\right )}^2+a} \,d x \]
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