Integrand size = 17, antiderivative size = 94 \[ \int \frac {\cot ^2(x)}{\left (a+b \cot ^2(x)\right )^{5/2}} \, dx=\frac {\arctan \left (\frac {\sqrt {a-b} \cot (x)}{\sqrt {a+b \cot ^2(x)}}\right )}{(a-b)^{5/2}}-\frac {\cot (x)}{3 (a-b) \left (a+b \cot ^2(x)\right )^{3/2}}-\frac {(2 a+b) \cot (x)}{3 a (a-b)^2 \sqrt {a+b \cot ^2(x)}} \]
[Out]
Time = 0.15 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.353, Rules used = {3751, 482, 541, 12, 385, 209} \[ \int \frac {\cot ^2(x)}{\left (a+b \cot ^2(x)\right )^{5/2}} \, dx=\frac {\arctan \left (\frac {\sqrt {a-b} \cot (x)}{\sqrt {a+b \cot ^2(x)}}\right )}{(a-b)^{5/2}}-\frac {(2 a+b) \cot (x)}{3 a (a-b)^2 \sqrt {a+b \cot ^2(x)}}-\frac {\cot (x)}{3 (a-b) \left (a+b \cot ^2(x)\right )^{3/2}} \]
[In]
[Out]
Rule 12
Rule 209
Rule 385
Rule 482
Rule 541
Rule 3751
Rubi steps \begin{align*} \text {integral}& = -\text {Subst}\left (\int \frac {x^2}{\left (1+x^2\right ) \left (a+b x^2\right )^{5/2}} \, dx,x,\cot (x)\right ) \\ & = -\frac {\cot (x)}{3 (a-b) \left (a+b \cot ^2(x)\right )^{3/2}}+\frac {\text {Subst}\left (\int \frac {1-2 x^2}{\left (1+x^2\right ) \left (a+b x^2\right )^{3/2}} \, dx,x,\cot (x)\right )}{3 (a-b)} \\ & = -\frac {\cot (x)}{3 (a-b) \left (a+b \cot ^2(x)\right )^{3/2}}-\frac {(2 a+b) \cot (x)}{3 a (a-b)^2 \sqrt {a+b \cot ^2(x)}}+\frac {\text {Subst}\left (\int \frac {3 a}{\left (1+x^2\right ) \sqrt {a+b x^2}} \, dx,x,\cot (x)\right )}{3 a (a-b)^2} \\ & = -\frac {\cot (x)}{3 (a-b) \left (a+b \cot ^2(x)\right )^{3/2}}-\frac {(2 a+b) \cot (x)}{3 a (a-b)^2 \sqrt {a+b \cot ^2(x)}}+\frac {\text {Subst}\left (\int \frac {1}{\left (1+x^2\right ) \sqrt {a+b x^2}} \, dx,x,\cot (x)\right )}{(a-b)^2} \\ & = -\frac {\cot (x)}{3 (a-b) \left (a+b \cot ^2(x)\right )^{3/2}}-\frac {(2 a+b) \cot (x)}{3 a (a-b)^2 \sqrt {a+b \cot ^2(x)}}+\frac {\text {Subst}\left (\int \frac {1}{1-(-a+b) x^2} \, dx,x,\frac {\cot (x)}{\sqrt {a+b \cot ^2(x)}}\right )}{(a-b)^2} \\ & = \frac {\arctan \left (\frac {\sqrt {a-b} \cot (x)}{\sqrt {a+b \cot ^2(x)}}\right )}{(a-b)^{5/2}}-\frac {\cot (x)}{3 (a-b) \left (a+b \cot ^2(x)\right )^{3/2}}-\frac {(2 a+b) \cot (x)}{3 a (a-b)^2 \sqrt {a+b \cot ^2(x)}} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 6.34 (sec) , antiderivative size = 197, normalized size of antiderivative = 2.10 \[ \int \frac {\cot ^2(x)}{\left (a+b \cot ^2(x)\right )^{5/2}} \, dx=\frac {\cos (x) \left (-12 (a-b)^3 \cos ^3(x) \cot (x) \left (a+b \cot ^2(x)\right ) \operatorname {Hypergeometric2F1}\left (2,2,\frac {9}{2},\frac {(a-b) \cos ^2(x)}{a}\right )-\frac {35 a \left (5 a+2 b \cot ^2(x)\right ) \sin (x) \left (a \left ((a-4 b) \csc ^2(x)-3 a \sec ^2(x)\right ) \sqrt {\frac {(a-b) \cos ^2(x) \left (a+b \cot ^2(x)\right ) \sin ^2(x)}{a^2}}+3 \arcsin \left (\sqrt {\frac {(a-b) \cos ^2(x)}{a}}\right ) (b \cot (x)+a \tan (x))^2\right )}{\sqrt {\frac {(a-b) \cos ^2(x) \left (a+b \cot ^2(x)\right ) \sin ^2(x)}{a^2}}}\right )}{315 a^3 (a-b)^2 \left (a+b \cot ^2(x)\right )^{3/2}} \]
[In]
[Out]
Time = 0.06 (sec) , antiderivative size = 161, normalized size of antiderivative = 1.71
method | result | size |
derivativedivides | \(-\frac {\cot \left (x \right )}{3 a \left (a +b \cot \left (x \right )^{2}\right )^{\frac {3}{2}}}-\frac {2 \cot \left (x \right )}{3 a^{2} \sqrt {a +b \cot \left (x \right )^{2}}}-\frac {b \left (\frac {\cot \left (x \right )}{3 a \left (a +b \cot \left (x \right )^{2}\right )^{\frac {3}{2}}}+\frac {2 \cot \left (x \right )}{3 a^{2} \sqrt {a +b \cot \left (x \right )^{2}}}\right )}{a -b}-\frac {b \cot \left (x \right )}{\left (a -b \right )^{2} a \sqrt {a +b \cot \left (x \right )^{2}}}+\frac {\sqrt {b^{4} \left (a -b \right )}\, \arctan \left (\frac {b^{2} \left (a -b \right ) \cot \left (x \right )}{\sqrt {b^{4} \left (a -b \right )}\, \sqrt {a +b \cot \left (x \right )^{2}}}\right )}{\left (a -b \right )^{3} b^{2}}\) | \(161\) |
default | \(-\frac {\cot \left (x \right )}{3 a \left (a +b \cot \left (x \right )^{2}\right )^{\frac {3}{2}}}-\frac {2 \cot \left (x \right )}{3 a^{2} \sqrt {a +b \cot \left (x \right )^{2}}}-\frac {b \left (\frac {\cot \left (x \right )}{3 a \left (a +b \cot \left (x \right )^{2}\right )^{\frac {3}{2}}}+\frac {2 \cot \left (x \right )}{3 a^{2} \sqrt {a +b \cot \left (x \right )^{2}}}\right )}{a -b}-\frac {b \cot \left (x \right )}{\left (a -b \right )^{2} a \sqrt {a +b \cot \left (x \right )^{2}}}+\frac {\sqrt {b^{4} \left (a -b \right )}\, \arctan \left (\frac {b^{2} \left (a -b \right ) \cot \left (x \right )}{\sqrt {b^{4} \left (a -b \right )}\, \sqrt {a +b \cot \left (x \right )^{2}}}\right )}{\left (a -b \right )^{3} b^{2}}\) | \(161\) |
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 339 vs. \(2 (80) = 160\).
Time = 0.39 (sec) , antiderivative size = 720, normalized size of antiderivative = 7.66 \[ \int \frac {\cot ^2(x)}{\left (a+b \cot ^2(x)\right )^{5/2}} \, dx=\left [-\frac {3 \, {\left (a^{3} + 2 \, a^{2} b + a b^{2} + {\left (a^{3} - 2 \, a^{2} b + a b^{2}\right )} \cos \left (2 \, x\right )^{2} - 2 \, {\left (a^{3} - a b^{2}\right )} \cos \left (2 \, x\right )\right )} \sqrt {-a + b} \log \left (-2 \, {\left (a^{2} - 2 \, a b + b^{2}\right )} \cos \left (2 \, x\right )^{2} + 2 \, {\left ({\left (a - b\right )} \cos \left (2 \, x\right ) - b\right )} \sqrt {-a + b} \sqrt {\frac {{\left (a - b\right )} \cos \left (2 \, x\right ) - a - b}{\cos \left (2 \, x\right ) - 1}} \sin \left (2 \, x\right ) + a^{2} - 2 \, b^{2} + 4 \, {\left (a b - b^{2}\right )} \cos \left (2 \, x\right )\right ) + 4 \, {\left (3 \, a^{3} - a^{2} b - a b^{2} - b^{3} - {\left (3 \, a^{3} - 5 \, a^{2} b + a b^{2} + b^{3}\right )} \cos \left (2 \, x\right )\right )} \sqrt {\frac {{\left (a - b\right )} \cos \left (2 \, x\right ) - a - b}{\cos \left (2 \, x\right ) - 1}} \sin \left (2 \, x\right )}{12 \, {\left (a^{6} - a^{5} b - 2 \, a^{4} b^{2} + 2 \, a^{3} b^{3} + a^{2} b^{4} - a b^{5} + {\left (a^{6} - 5 \, a^{5} b + 10 \, a^{4} b^{2} - 10 \, a^{3} b^{3} + 5 \, a^{2} b^{4} - a b^{5}\right )} \cos \left (2 \, x\right )^{2} - 2 \, {\left (a^{6} - 3 \, a^{5} b + 2 \, a^{4} b^{2} + 2 \, a^{3} b^{3} - 3 \, a^{2} b^{4} + a b^{5}\right )} \cos \left (2 \, x\right )\right )}}, \frac {3 \, {\left (a^{3} + 2 \, a^{2} b + a b^{2} + {\left (a^{3} - 2 \, a^{2} b + a b^{2}\right )} \cos \left (2 \, x\right )^{2} - 2 \, {\left (a^{3} - a b^{2}\right )} \cos \left (2 \, x\right )\right )} \sqrt {a - b} \arctan \left (-\frac {\sqrt {a - b} \sqrt {\frac {{\left (a - b\right )} \cos \left (2 \, x\right ) - a - b}{\cos \left (2 \, x\right ) - 1}} \sin \left (2 \, x\right )}{{\left (a - b\right )} \cos \left (2 \, x\right ) - b}\right ) - 2 \, {\left (3 \, a^{3} - a^{2} b - a b^{2} - b^{3} - {\left (3 \, a^{3} - 5 \, a^{2} b + a b^{2} + b^{3}\right )} \cos \left (2 \, x\right )\right )} \sqrt {\frac {{\left (a - b\right )} \cos \left (2 \, x\right ) - a - b}{\cos \left (2 \, x\right ) - 1}} \sin \left (2 \, x\right )}{6 \, {\left (a^{6} - a^{5} b - 2 \, a^{4} b^{2} + 2 \, a^{3} b^{3} + a^{2} b^{4} - a b^{5} + {\left (a^{6} - 5 \, a^{5} b + 10 \, a^{4} b^{2} - 10 \, a^{3} b^{3} + 5 \, a^{2} b^{4} - a b^{5}\right )} \cos \left (2 \, x\right )^{2} - 2 \, {\left (a^{6} - 3 \, a^{5} b + 2 \, a^{4} b^{2} + 2 \, a^{3} b^{3} - 3 \, a^{2} b^{4} + a b^{5}\right )} \cos \left (2 \, x\right )\right )}}\right ] \]
[In]
[Out]
\[ \int \frac {\cot ^2(x)}{\left (a+b \cot ^2(x)\right )^{5/2}} \, dx=\int \frac {\cot ^{2}{\left (x \right )}}{\left (a + b \cot ^{2}{\left (x \right )}\right )^{\frac {5}{2}}}\, dx \]
[In]
[Out]
Exception generated. \[ \int \frac {\cot ^2(x)}{\left (a+b \cot ^2(x)\right )^{5/2}} \, dx=\text {Exception raised: ValueError} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 281 vs. \(2 (80) = 160\).
Time = 0.34 (sec) , antiderivative size = 281, normalized size of antiderivative = 2.99 \[ \int \frac {\cot ^2(x)}{\left (a+b \cot ^2(x)\right )^{5/2}} \, dx=\frac {{\left (3 \, a \sqrt {b} \log \left ({\left | -\sqrt {-a + b} + \sqrt {b} \right |}\right ) + 2 \, a \sqrt {-a + b} + \sqrt {-a + b} b\right )} \mathrm {sgn}\left (\sin \left (x\right )\right )}{3 \, {\left (a^{3} \sqrt {-a + b} \sqrt {b} - 2 \, a^{2} \sqrt {-a + b} b^{\frac {3}{2}} + a \sqrt {-a + b} b^{\frac {5}{2}}\right )}} - \frac {\frac {{\left (\frac {{\left (3 \, a^{3} - 5 \, a^{2} b + a b^{2} + b^{3}\right )} \cos \left (x\right )^{2}}{a^{4} - 3 \, a^{3} b + 3 \, a^{2} b^{2} - a b^{3}} - \frac {3 \, {\left (a^{3} - a^{2} b\right )}}{a^{4} - 3 \, a^{3} b + 3 \, a^{2} b^{2} - a b^{3}}\right )} \cos \left (x\right )}{{\left (a \cos \left (x\right )^{2} - b \cos \left (x\right )^{2} - a\right )} \sqrt {-a \cos \left (x\right )^{2} + b \cos \left (x\right )^{2} + a}} + \frac {3 \, \log \left ({\left | -\sqrt {-a + b} \cos \left (x\right ) + \sqrt {-a \cos \left (x\right )^{2} + b \cos \left (x\right )^{2} + a} \right |}\right )}{{\left (a^{2} - 2 \, a b + b^{2}\right )} \sqrt {-a + b}}}{3 \, \mathrm {sgn}\left (\sin \left (x\right )\right )} \]
[In]
[Out]
Timed out. \[ \int \frac {\cot ^2(x)}{\left (a+b \cot ^2(x)\right )^{5/2}} \, dx=\int \frac {{\mathrm {cot}\left (x\right )}^2}{{\left (b\,{\mathrm {cot}\left (x\right )}^2+a\right )}^{5/2}} \,d x \]
[In]
[Out]