Integrand size = 33, antiderivative size = 747 \[ \int \cot ^3(d+e x) \sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)} \, dx=\frac {\sqrt {a^2+b^2+c \left (c+\sqrt {a^2+b^2-2 a c+c^2}\right )-a \left (2 c+\sqrt {a^2+b^2-2 a c+c^2}\right )} \arctan \left (\frac {b^2+(a-c) \left (a-c-\sqrt {a^2+b^2-2 a c+c^2}\right )-b \sqrt {a^2+b^2-2 a c+c^2} \cot (d+e x)}{\sqrt {2} \sqrt [4]{a^2+b^2-2 a c+c^2} \sqrt {a^2+b^2+c \left (c+\sqrt {a^2+b^2-2 a c+c^2}\right )-a \left (2 c+\sqrt {a^2+b^2-2 a c+c^2}\right )} \sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}}\right )}{\sqrt {2} \sqrt [4]{a^2+b^2-2 a c+c^2} e}+\frac {b \text {arctanh}\left (\frac {b+2 c \cot (d+e x)}{2 \sqrt {c} \sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}}\right )}{2 \sqrt {c} e}-\frac {b \left (b^2-4 a c\right ) \text {arctanh}\left (\frac {b+2 c \cot (d+e x)}{2 \sqrt {c} \sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}}\right )}{16 c^{5/2} e}-\frac {\sqrt {a^2+b^2+c \left (c-\sqrt {a^2+b^2-2 a c+c^2}\right )-a \left (2 c-\sqrt {a^2+b^2-2 a c+c^2}\right )} \text {arctanh}\left (\frac {b^2+(a-c) \left (a-c+\sqrt {a^2+b^2-2 a c+c^2}\right )+b \sqrt {a^2+b^2-2 a c+c^2} \cot (d+e x)}{\sqrt {2} \sqrt [4]{a^2+b^2-2 a c+c^2} \sqrt {a^2+b^2+c \left (c-\sqrt {a^2+b^2-2 a c+c^2}\right )-a \left (2 c-\sqrt {a^2+b^2-2 a c+c^2}\right )} \sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}}\right )}{\sqrt {2} \sqrt [4]{a^2+b^2-2 a c+c^2} e}+\frac {\sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}}{e}+\frac {b (b+2 c \cot (d+e x)) \sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}}{8 c^2 e}-\frac {\left (a+b \cot (d+e x)+c \cot ^2(d+e x)\right )^{3/2}}{3 c e} \]
[Out]
Time = 24.33 (sec) , antiderivative size = 747, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {3782, 6857, 654, 626, 635, 212, 1035, 1092, 1050, 1044, 214, 211} \[ \int \cot ^3(d+e x) \sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)} \, dx=\frac {\sqrt {-a \left (\sqrt {a^2-2 a c+b^2+c^2}+2 c\right )+c \left (\sqrt {a^2-2 a c+b^2+c^2}+c\right )+a^2+b^2} \arctan \left (\frac {-b \sqrt {a^2-2 a c+b^2+c^2} \cot (d+e x)+(a-c) \left (-\sqrt {a^2-2 a c+b^2+c^2}+a-c\right )+b^2}{\sqrt {2} \sqrt [4]{a^2-2 a c+b^2+c^2} \sqrt {-a \left (\sqrt {a^2-2 a c+b^2+c^2}+2 c\right )+c \left (\sqrt {a^2-2 a c+b^2+c^2}+c\right )+a^2+b^2} \sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}}\right )}{\sqrt {2} e \sqrt [4]{a^2-2 a c+b^2+c^2}}-\frac {\sqrt {-a \left (2 c-\sqrt {a^2-2 a c+b^2+c^2}\right )+c \left (c-\sqrt {a^2-2 a c+b^2+c^2}\right )+a^2+b^2} \text {arctanh}\left (\frac {b \sqrt {a^2-2 a c+b^2+c^2} \cot (d+e x)+(a-c) \left (\sqrt {a^2-2 a c+b^2+c^2}+a-c\right )+b^2}{\sqrt {2} \sqrt [4]{a^2-2 a c+b^2+c^2} \sqrt {-a \left (2 c-\sqrt {a^2-2 a c+b^2+c^2}\right )+c \left (c-\sqrt {a^2-2 a c+b^2+c^2}\right )+a^2+b^2} \sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}}\right )}{\sqrt {2} e \sqrt [4]{a^2-2 a c+b^2+c^2}}-\frac {b \left (b^2-4 a c\right ) \text {arctanh}\left (\frac {b+2 c \cot (d+e x)}{2 \sqrt {c} \sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}}\right )}{16 c^{5/2} e}+\frac {b \text {arctanh}\left (\frac {b+2 c \cot (d+e x)}{2 \sqrt {c} \sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}}\right )}{2 \sqrt {c} e}+\frac {b (b+2 c \cot (d+e x)) \sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}}{8 c^2 e}-\frac {\left (a+b \cot (d+e x)+c \cot ^2(d+e x)\right )^{3/2}}{3 c e}+\frac {\sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}}{e} \]
[In]
[Out]
Rule 211
Rule 212
Rule 214
Rule 626
Rule 635
Rule 654
Rule 1035
Rule 1044
Rule 1050
Rule 1092
Rule 3782
Rule 6857
Rubi steps \begin{align*} \text {integral}& = -\frac {\text {Subst}\left (\int \frac {x^3 \sqrt {a+b x+c x^2}}{1+x^2} \, dx,x,\cot (d+e x)\right )}{e} \\ & = -\frac {\text {Subst}\left (\int \left (x \sqrt {a+b x+c x^2}-\frac {x \sqrt {a+b x+c x^2}}{1+x^2}\right ) \, dx,x,\cot (d+e x)\right )}{e} \\ & = -\frac {\text {Subst}\left (\int x \sqrt {a+b x+c x^2} \, dx,x,\cot (d+e x)\right )}{e}+\frac {\text {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 {\left (a+b \cot (d+e x)+c \cot ^2(d+e x)\right )^{3/2}}{3 c e}-\frac {\text {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 {b \text {Subst}\left (\int \sqrt {a+b x+c x^2} \, dx,x,\cot (d+e x)\right )}{2 c e} \\ & = \frac {\sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}}{e}+\frac {b (b+2 c \cot (d+e x)) \sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}}{8 c^2 e}-\frac {\left (a+b \cot (d+e x)+c \cot ^2(d+e x)\right )^{3/2}}{3 c e}-\frac {\text {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 \text {Subst}\left (\int \frac {1}{\sqrt {a+b x+c x^2}} \, dx,x,\cot (d+e x)\right )}{2 e}-\frac {\left (b \left (b^2-4 a c\right )\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a+b x+c x^2}} \, dx,x,\cot (d+e x)\right )}{16 c^2 e} \\ & = \frac {\sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}}{e}+\frac {b (b+2 c \cot (d+e x)) \sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}}{8 c^2 e}-\frac {\left (a+b \cot (d+e x)+c \cot ^2(d+e x)\right )^{3/2}}{3 c e}+\frac {b \text {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 {\left (b \left (b^2-4 a c\right )\right ) \text {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^2 e}+\frac {\text {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 {\text {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 \text {arctanh}\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 {b \left (b^2-4 a c\right ) \text {arctanh}\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^{5/2} e}+\frac {\sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}}{e}+\frac {b (b+2 c \cot (d+e x)) \sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}}{8 c^2 e}-\frac {\left (a+b \cot (d+e x)+c \cot ^2(d+e x)\right )^{3/2}}{3 c 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 ) \text {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 ) \text {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 )} \arctan \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 \text {arctanh}\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 {b \left (b^2-4 a c\right ) \text {arctanh}\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^{5/2} 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 )} \text {arctanh}\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}+\frac {b (b+2 c \cot (d+e x)) \sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}}{8 c^2 e}-\frac {\left (a+b \cot (d+e x)+c \cot ^2(d+e x)\right )^{3/2}}{3 c e} \\ \end{align*}
Result contains complex when optimal does not.
Time = 6.30 (sec) , antiderivative size = 381, normalized size of antiderivative = 0.51 \[ \int \cot ^3(d+e x) \sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)} \, dx=\frac {\sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)} \tan (d+e x) \left (-3 b \left (b^2-4 c (a+2 c)\right ) \text {arctanh}\left (\frac {2 c+b \tan (d+e x)}{2 \sqrt {c} \sqrt {c+b \tan (d+e x)+a \tan ^2(d+e x)}}\right )+2 \sqrt {c} \left (12 i \sqrt {a+i b-c} c^2 \arctan \left (\frac {i b-2 c+(2 i a-b) \tan (d+e x)}{2 \sqrt {a+i b-c} \sqrt {c+b \tan (d+e x)+a \tan ^2(d+e x)}}\right )+12 i \sqrt {a-i b-c} c^2 \arctan \left (\frac {i b+2 c+(2 i a+b) \tan (d+e x)}{2 \sqrt {a-i b-c} \sqrt {c+b \tan (d+e x)+a \tan ^2(d+e x)}}\right )+\cot (d+e x) \left (3 b^2-8 a c+24 c^2-2 b c \cot (d+e x)-8 c^2 \cot ^2(d+e x)\right ) \sqrt {c+b \tan (d+e x)+a \tan ^2(d+e x)}\right )\right )}{48 c^{5/2} e \sqrt {c+b \tan (d+e x)+a \tan ^2(d+e x)}} \]
[In]
[Out]
result has leaf size over 500,000. Avoiding possible recursion issues.
Time = 3.96 (sec) , antiderivative size = 17768080, normalized size of antiderivative = 23785.92
\[\text {output too large to display}\]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 5923 vs. \(2 (670) = 1340\).
Time = 2.41 (sec) , antiderivative size = 11897, normalized size of antiderivative = 15.93 \[ \int \cot ^3(d+e x) \sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)} \, dx=\text {Too large to display} \]
[In]
[Out]
\[ \int \cot ^3(d+e x) \sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)} \, dx=\int \sqrt {a + b \cot {\left (d + e x \right )} + c \cot ^{2}{\left (d + e x \right )}} \cot ^{3}{\left (d + e x \right )}\, dx \]
[In]
[Out]
\[ \int \cot ^3(d+e x) \sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)} \, dx=\int { \sqrt {c \cot \left (e x + d\right )^{2} + b \cot \left (e x + d\right ) + a} \cot \left (e x + d\right )^{3} \,d x } \]
[In]
[Out]
\[ \int \cot ^3(d+e x) \sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)} \, dx=\int { \sqrt {c \cot \left (e x + d\right )^{2} + b \cot \left (e x + d\right ) + a} \cot \left (e x + d\right )^{3} \,d x } \]
[In]
[Out]
Timed out. \[ \int \cot ^3(d+e x) \sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)} \, dx=\int {\mathrm {cot}\left (d+e\,x\right )}^3\,\sqrt {c\,{\mathrm {cot}\left (d+e\,x\right )}^2+b\,\mathrm {cot}\left (d+e\,x\right )+a} \,d x \]
[In]
[Out]