Integrand size = 17, antiderivative size = 80 \[ \int \left (a+b \cot ^2(x)\right )^{3/2} \tan ^2(x) \, dx=(a-b)^{3/2} \arctan \left (\frac {\sqrt {a-b} \cot (x)}{\sqrt {a+b \cot ^2(x)}}\right )-b^{3/2} \text {arctanh}\left (\frac {\sqrt {b} \cot (x)}{\sqrt {a+b \cot ^2(x)}}\right )+a \sqrt {a+b \cot ^2(x)} \tan (x) \]
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Time = 0.14 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.412, Rules used = {3751, 485, 537, 223, 212, 385, 209} \[ \int \left (a+b \cot ^2(x)\right )^{3/2} \tan ^2(x) \, dx=(a-b)^{3/2} \arctan \left (\frac {\sqrt {a-b} \cot (x)}{\sqrt {a+b \cot ^2(x)}}\right )-b^{3/2} \text {arctanh}\left (\frac {\sqrt {b} \cot (x)}{\sqrt {a+b \cot ^2(x)}}\right )+a \tan (x) \sqrt {a+b \cot ^2(x)} \]
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Rule 209
Rule 212
Rule 223
Rule 385
Rule 485
Rule 537
Rule 3751
Rubi steps \begin{align*} \text {integral}& = -\text {Subst}\left (\int \frac {\left (a+b x^2\right )^{3/2}}{x^2 \left (1+x^2\right )} \, dx,x,\cot (x)\right ) \\ & = a \sqrt {a+b \cot ^2(x)} \tan (x)-\text {Subst}\left (\int \frac {-a (a-2 b)+b^2 x^2}{\left (1+x^2\right ) \sqrt {a+b x^2}} \, dx,x,\cot (x)\right ) \\ & = a \sqrt {a+b \cot ^2(x)} \tan (x)+(a-b)^2 \text {Subst}\left (\int \frac {1}{\left (1+x^2\right ) \sqrt {a+b x^2}} \, dx,x,\cot (x)\right )-b^2 \text {Subst}\left (\int \frac {1}{\sqrt {a+b x^2}} \, dx,x,\cot (x)\right ) \\ & = a \sqrt {a+b \cot ^2(x)} \tan (x)+(a-b)^2 \text {Subst}\left (\int \frac {1}{1-(-a+b) x^2} \, dx,x,\frac {\cot (x)}{\sqrt {a+b \cot ^2(x)}}\right )-b^2 \text {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {\cot (x)}{\sqrt {a+b \cot ^2(x)}}\right ) \\ & = (a-b)^{3/2} \arctan \left (\frac {\sqrt {a-b} \cot (x)}{\sqrt {a+b \cot ^2(x)}}\right )-b^{3/2} \text {arctanh}\left (\frac {\sqrt {b} \cot (x)}{\sqrt {a+b \cot ^2(x)}}\right )+a \sqrt {a+b \cot ^2(x)} \tan (x) \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(222\) vs. \(2(80)=160\).
Time = 0.82 (sec) , antiderivative size = 222, normalized size of antiderivative = 2.78 \[ \int \left (a+b \cot ^2(x)\right )^{3/2} \tan ^2(x) \, dx=\frac {\sqrt {-\left ((-a-b+(a-b) \cos (2 x)) \csc ^2(x)\right )} \left (-\sqrt {2} (a-b)^2 \sqrt {-b} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {a-b} \cos (x)}{\sqrt {-a-b+(a-b) \cos (2 x)}}\right )+\sqrt {a-b} \left (\sqrt {2} b^2 \text {arctanh}\left (\frac {\sqrt {2} \sqrt {-b} \cos (x)}{\sqrt {-a-b+(a-b) \cos (2 x)}}\right )+a \sqrt {-b} \sqrt {-a-b+(a-b) \cos (2 x)} \sec (x)\right )\right ) \sin (x)}{\sqrt {2} \sqrt {a-b} \sqrt {-b} \sqrt {-a-b+(a-b) \cos (2 x)}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(641\) vs. \(2(66)=132\).
Time = 0.82 (sec) , antiderivative size = 642, normalized size of antiderivative = 8.02
method | result | size |
default | \(\frac {\sqrt {4}\, \left (2 \cos \left (x \right ) b^{\frac {7}{2}} \ln \left (4 \cos \left (x \right ) \sqrt {-a +b}\, \sqrt {-\frac {a \cos \left (x \right )^{2}-\cos \left (x \right )^{2} b -a}{\left (\cos \left (x \right )+1\right )^{2}}}-4 \cos \left (x \right ) a +4 b \cos \left (x \right )+4 \sqrt {-a +b}\, \sqrt {-\frac {a \cos \left (x \right )^{2}-\cos \left (x \right )^{2} b -a}{\left (\cos \left (x \right )+1\right )^{2}}}\right )-4 \cos \left (x \right ) b^{\frac {5}{2}} \ln \left (4 \cos \left (x \right ) \sqrt {-a +b}\, \sqrt {-\frac {a \cos \left (x \right )^{2}-\cos \left (x \right )^{2} b -a}{\left (\cos \left (x \right )+1\right )^{2}}}-4 \cos \left (x \right ) a +4 b \cos \left (x \right )+4 \sqrt {-a +b}\, \sqrt {-\frac {a \cos \left (x \right )^{2}-\cos \left (x \right )^{2} b -a}{\left (\cos \left (x \right )+1\right )^{2}}}\right ) a +2 \cos \left (x \right ) b^{\frac {3}{2}} \sqrt {-\frac {a \cos \left (x \right )^{2}-\cos \left (x \right )^{2} b -a}{\left (\cos \left (x \right )+1\right )^{2}}}\, \sqrt {-a +b}\, a +2 \cos \left (x \right ) b^{\frac {3}{2}} \ln \left (4 \cos \left (x \right ) \sqrt {-a +b}\, \sqrt {-\frac {a \cos \left (x \right )^{2}-\cos \left (x \right )^{2} b -a}{\left (\cos \left (x \right )+1\right )^{2}}}-4 \cos \left (x \right ) a +4 b \cos \left (x \right )+4 \sqrt {-a +b}\, \sqrt {-\frac {a \cos \left (x \right )^{2}-\cos \left (x \right )^{2} b -a}{\left (\cos \left (x \right )+1\right )^{2}}}\right ) a^{2}+2 b^{\frac {3}{2}} \sqrt {-\frac {a \cos \left (x \right )^{2}-\cos \left (x \right )^{2} b -a}{\left (\cos \left (x \right )+1\right )^{2}}}\, \sqrt {-a +b}\, a -\cos \left (x \right ) \ln \left (-\frac {4 \left (\sqrt {b}\, \sqrt {-\frac {a \cos \left (x \right )^{2}-\cos \left (x \right )^{2} b -a}{\left (\cos \left (x \right )+1\right )^{2}}}\, \cos \left (x \right )-\cos \left (x \right ) a +b \cos \left (x \right )+\sqrt {-\frac {a \cos \left (x \right )^{2}-\cos \left (x \right )^{2} b -a}{\left (\cos \left (x \right )+1\right )^{2}}}\, \sqrt {b}+a \right )}{\cos \left (x \right )-1}\right ) \sqrt {-a +b}\, b^{3}+\cos \left (x \right ) \ln \left (\frac {2 \sqrt {b}\, \sqrt {-\frac {a \cos \left (x \right )^{2}-\cos \left (x \right )^{2} b -a}{\left (\cos \left (x \right )+1\right )^{2}}}\, \cos \left (x \right )+2 \cos \left (x \right ) a -2 b \cos \left (x \right )+2 \sqrt {-\frac {a \cos \left (x \right )^{2}-\cos \left (x \right )^{2} b -a}{\left (\cos \left (x \right )+1\right )^{2}}}\, \sqrt {b}+2 a}{\sqrt {b}\, \left (\cos \left (x \right )+1\right )}\right ) \sqrt {-a +b}\, b^{3}\right ) \left (\cos \left (x \right )-1\right ) \left (a +b \cot \left (x \right )^{2}\right )^{\frac {3}{2}} \tan \left (x \right )}{4 b^{\frac {3}{2}} \sqrt {-a +b}\, \left (a \cos \left (x \right )^{2}-\cos \left (x \right )^{2} b -a \right ) \sqrt {-\frac {a \cos \left (x \right )^{2}-\cos \left (x \right )^{2} b -a}{\left (\cos \left (x \right )+1\right )^{2}}}}\) | \(642\) |
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Time = 0.76 (sec) , antiderivative size = 543, normalized size of antiderivative = 6.79 \[ \int \left (a+b \cot ^2(x)\right )^{3/2} \tan ^2(x) \, dx=\left [\frac {1}{4} \, {\left (-a + b\right )}^{\frac {3}{2}} \log \left (-\frac {a^{2} \tan \left (x\right )^{4} - 2 \, {\left (3 \, a^{2} - 4 \, a b\right )} \tan \left (x\right )^{2} + a^{2} - 8 \, a b + 8 \, b^{2} + 4 \, {\left (a \tan \left (x\right )^{3} - {\left (a - 2 \, b\right )} \tan \left (x\right )\right )} \sqrt {-a + b} \sqrt {\frac {a \tan \left (x\right )^{2} + b}{\tan \left (x\right )^{2}}}}{\tan \left (x\right )^{4} + 2 \, \tan \left (x\right )^{2} + 1}\right ) + \frac {1}{2} \, b^{\frac {3}{2}} \log \left (\frac {a \tan \left (x\right )^{2} - 2 \, \sqrt {b} \sqrt {\frac {a \tan \left (x\right )^{2} + b}{\tan \left (x\right )^{2}}} \tan \left (x\right ) + 2 \, b}{\tan \left (x\right )^{2}}\right ) + a \sqrt {\frac {a \tan \left (x\right )^{2} + b}{\tan \left (x\right )^{2}}} \tan \left (x\right ), \sqrt {-b} b \arctan \left (\frac {\sqrt {-b} \sqrt {\frac {a \tan \left (x\right )^{2} + b}{\tan \left (x\right )^{2}}} \tan \left (x\right )}{b}\right ) + \frac {1}{4} \, {\left (-a + b\right )}^{\frac {3}{2}} \log \left (-\frac {a^{2} \tan \left (x\right )^{4} - 2 \, {\left (3 \, a^{2} - 4 \, a b\right )} \tan \left (x\right )^{2} + a^{2} - 8 \, a b + 8 \, b^{2} + 4 \, {\left (a \tan \left (x\right )^{3} - {\left (a - 2 \, b\right )} \tan \left (x\right )\right )} \sqrt {-a + b} \sqrt {\frac {a \tan \left (x\right )^{2} + b}{\tan \left (x\right )^{2}}}}{\tan \left (x\right )^{4} + 2 \, \tan \left (x\right )^{2} + 1}\right ) + a \sqrt {\frac {a \tan \left (x\right )^{2} + b}{\tan \left (x\right )^{2}}} \tan \left (x\right ), \frac {1}{2} \, {\left (a - b\right )}^{\frac {3}{2}} \arctan \left (\frac {2 \, \sqrt {a - b} \sqrt {\frac {a \tan \left (x\right )^{2} + b}{\tan \left (x\right )^{2}}} \tan \left (x\right )}{a \tan \left (x\right )^{2} - a + 2 \, b}\right ) + \frac {1}{2} \, b^{\frac {3}{2}} \log \left (\frac {a \tan \left (x\right )^{2} - 2 \, \sqrt {b} \sqrt {\frac {a \tan \left (x\right )^{2} + b}{\tan \left (x\right )^{2}}} \tan \left (x\right ) + 2 \, b}{\tan \left (x\right )^{2}}\right ) + a \sqrt {\frac {a \tan \left (x\right )^{2} + b}{\tan \left (x\right )^{2}}} \tan \left (x\right ), \frac {1}{2} \, {\left (a - b\right )}^{\frac {3}{2}} \arctan \left (\frac {2 \, \sqrt {a - b} \sqrt {\frac {a \tan \left (x\right )^{2} + b}{\tan \left (x\right )^{2}}} \tan \left (x\right )}{a \tan \left (x\right )^{2} - a + 2 \, b}\right ) + \sqrt {-b} b \arctan \left (\frac {\sqrt {-b} \sqrt {\frac {a \tan \left (x\right )^{2} + b}{\tan \left (x\right )^{2}}} \tan \left (x\right )}{b}\right ) + a \sqrt {\frac {a \tan \left (x\right )^{2} + b}{\tan \left (x\right )^{2}}} \tan \left (x\right )\right ] \]
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\[ \int \left (a+b \cot ^2(x)\right )^{3/2} \tan ^2(x) \, dx=\int \left (a + b \cot ^{2}{\left (x \right )}\right )^{\frac {3}{2}} \tan ^{2}{\left (x \right )}\, dx \]
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\[ \int \left (a+b \cot ^2(x)\right )^{3/2} \tan ^2(x) \, dx=\int { {\left (b \cot \left (x\right )^{2} + a\right )}^{\frac {3}{2}} \tan \left (x\right )^{2} \,d x } \]
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Leaf count of result is larger than twice the leaf count of optimal. 625 vs. \(2 (66) = 132\).
Time = 18.70 (sec) , antiderivative size = 625, normalized size of antiderivative = 7.81 \[ \int \left (a+b \cot ^2(x)\right )^{3/2} \tan ^2(x) \, dx=-\frac {1}{2} \, {\left (\frac {2 \, \sqrt {-a + b} b^{2} \arctan \left (\frac {{\left (\sqrt {-a + b} \cos \left (x\right ) - \sqrt {-a \cos \left (x\right )^{2} + b \cos \left (x\right )^{2} + a}\right )}^{2} + a - 2 \, b}{2 \, \sqrt {a b - b^{2}}}\right )}{\sqrt {a b - b^{2}}} - {\left (a - b\right )} \sqrt {-a + b} \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 )}^{2}\right ) + \frac {4 \, a^{2} \sqrt {-a + b}}{{\left (\sqrt {-a + b} \cos \left (x\right ) - \sqrt {-a \cos \left (x\right )^{2} + b \cos \left (x\right )^{2} + a}\right )}^{2} - a}\right )} \mathrm {sgn}\left (\sin \left (x\right )\right ) - \frac {{\left (2 \, a \sqrt {-a + b} b^{2} \arctan \left (\frac {\sqrt {-a + b} \sqrt {b}}{\sqrt {a b - b^{2}}}\right ) - 2 \, a b^{\frac {5}{2}} \arctan \left (\frac {\sqrt {-a + b} \sqrt {b}}{\sqrt {a b - b^{2}}}\right ) - 2 \, \sqrt {-a + b} b^{3} \arctan \left (\frac {\sqrt {-a + b} \sqrt {b}}{\sqrt {a b - b^{2}}}\right ) + 2 \, b^{\frac {7}{2}} \arctan \left (\frac {\sqrt {-a + b} \sqrt {b}}{\sqrt {a b - b^{2}}}\right ) + \sqrt {a b - b^{2}} a^{2} \sqrt {-a + b} \log \left (-a - 2 \, \sqrt {-a + b} \sqrt {b} + 2 \, b\right ) - \sqrt {a b - b^{2}} a^{2} \sqrt {b} \log \left (-a - 2 \, \sqrt {-a + b} \sqrt {b} + 2 \, b\right ) - 2 \, \sqrt {a b - b^{2}} a \sqrt {-a + b} b \log \left (-a - 2 \, \sqrt {-a + b} \sqrt {b} + 2 \, b\right ) + 2 \, \sqrt {a b - b^{2}} a b^{\frac {3}{2}} \log \left (-a - 2 \, \sqrt {-a + b} \sqrt {b} + 2 \, b\right ) + \sqrt {a b - b^{2}} \sqrt {-a + b} b^{2} \log \left (-a - 2 \, \sqrt {-a + b} \sqrt {b} + 2 \, b\right ) - \sqrt {a b - b^{2}} b^{\frac {5}{2}} \log \left (-a - 2 \, \sqrt {-a + b} \sqrt {b} + 2 \, b\right ) + 2 \, \sqrt {a b - b^{2}} a^{2} \sqrt {-a + b}\right )} \mathrm {sgn}\left (\sin \left (x\right )\right )}{2 \, {\left (\sqrt {a b - b^{2}} a + \sqrt {a b - b^{2}} \sqrt {-a + b} \sqrt {b} - \sqrt {a b - b^{2}} b\right )}} \]
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Timed out. \[ \int \left (a+b \cot ^2(x)\right )^{3/2} \tan ^2(x) \, dx=\int {\mathrm {tan}\left (x\right )}^2\,{\left (b\,{\mathrm {cot}\left (x\right )}^2+a\right )}^{3/2} \,d x \]
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