Integrand size = 17, antiderivative size = 85 \[ \int \sqrt {a+b \cot ^2(x)} \tan ^4(x) \, dx=-\sqrt {a-b} \arctan \left (\frac {\sqrt {a-b} \cot (x)}{\sqrt {a+b \cot ^2(x)}}\right )-\frac {(3 a-b) \sqrt {a+b \cot ^2(x)} \tan (x)}{3 a}+\frac {1}{3} \sqrt {a+b \cot ^2(x)} \tan ^3(x) \]
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Time = 0.17 (sec) , antiderivative size = 85, 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, 486, 597, 12, 385, 209} \[ \int \sqrt {a+b \cot ^2(x)} \tan ^4(x) \, dx=-\sqrt {a-b} \arctan \left (\frac {\sqrt {a-b} \cot (x)}{\sqrt {a+b \cot ^2(x)}}\right )+\frac {1}{3} \tan ^3(x) \sqrt {a+b \cot ^2(x)}-\frac {(3 a-b) \tan (x) \sqrt {a+b \cot ^2(x)}}{3 a} \]
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Rule 12
Rule 209
Rule 385
Rule 486
Rule 597
Rule 3751
Rubi steps \begin{align*} \text {integral}& = -\text {Subst}\left (\int \frac {\sqrt {a+b x^2}}{x^4 \left (1+x^2\right )} \, dx,x,\cot (x)\right ) \\ & = \frac {1}{3} \sqrt {a+b \cot ^2(x)} \tan ^3(x)-\frac {1}{3} \text {Subst}\left (\int \frac {-3 a+b-2 b x^2}{x^2 \left (1+x^2\right ) \sqrt {a+b x^2}} \, dx,x,\cot (x)\right ) \\ & = -\frac {(3 a-b) \sqrt {a+b \cot ^2(x)} \tan (x)}{3 a}+\frac {1}{3} \sqrt {a+b \cot ^2(x)} \tan ^3(x)+\frac {\text {Subst}\left (\int -\frac {3 a (a-b)}{\left (1+x^2\right ) \sqrt {a+b x^2}} \, dx,x,\cot (x)\right )}{3 a} \\ & = -\frac {(3 a-b) \sqrt {a+b \cot ^2(x)} \tan (x)}{3 a}+\frac {1}{3} \sqrt {a+b \cot ^2(x)} \tan ^3(x)+(-a+b) \text {Subst}\left (\int \frac {1}{\left (1+x^2\right ) \sqrt {a+b x^2}} \, dx,x,\cot (x)\right ) \\ & = -\frac {(3 a-b) \sqrt {a+b \cot ^2(x)} \tan (x)}{3 a}+\frac {1}{3} \sqrt {a+b \cot ^2(x)} \tan ^3(x)+(-a+b) \text {Subst}\left (\int \frac {1}{1-(-a+b) x^2} \, dx,x,\frac {\cot (x)}{\sqrt {a+b \cot ^2(x)}}\right ) \\ & = -\sqrt {a-b} \arctan \left (\frac {\sqrt {a-b} \cot (x)}{\sqrt {a+b \cot ^2(x)}}\right )-\frac {(3 a-b) \sqrt {a+b \cot ^2(x)} \tan (x)}{3 a}+\frac {1}{3} \sqrt {a+b \cot ^2(x)} \tan ^3(x) \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 1.76 (sec) , antiderivative size = 174, normalized size of antiderivative = 2.05 \[ \int \sqrt {a+b \cot ^2(x)} \tan ^4(x) \, dx=\frac {1}{3} \sqrt {a+b \cot ^2(x)} \left (1+\frac {b \cot ^2(x)}{a}\right ) \sin ^2(x) \left (-\frac {4 (a-b) \cos ^2(x) \left (a+b \cot ^2(x)\right ) \operatorname {Hypergeometric2F1}\left (2,2,\frac {3}{2},\frac {(a-b) \cos ^2(x)}{a}\right )}{a^2}+\frac {\left (a-2 b \cot ^2(x)\right ) \csc ^2(x) \left (\arcsin \left (\sqrt {\frac {(a-b) \cos ^2(x)}{a}}\right ) \sqrt {\frac {(a-b) \cos ^2(x)}{a}}+\sqrt {\frac {b \cos ^2(x)}{a}+\sin ^2(x)}\right )}{\left (a+b \cot ^2(x)\right ) \sqrt {\frac {b \cos ^2(x)}{a}+\sin ^2(x)}}\right ) \tan ^3(x) \]
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Leaf count of result is larger than twice the leaf count of optimal. \(494\) vs. \(2(71)=142\).
Time = 1.51 (sec) , antiderivative size = 495, normalized size of antiderivative = 5.82
method | result | size |
default | \(-\frac {\sqrt {4}\, \left (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}}}\, a \cos \left (x \right )^{3}-\cos \left (x \right )^{3} \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}\, b +3 \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 ) \cos \left (x \right )^{3} a^{2}-3 \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 ) \cos \left (x \right )^{3} a b +4 \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} \sqrt {-a +b}\, a -\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 {-a +b}\, b -\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 )-\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 \right ) \sqrt {a +b \cot \left (x \right )^{2}}\, \tan \left (x \right ) \sec \left (x \right )^{2}}{6 a \sqrt {-a +b}\, \left (\cos \left (x \right )+1\right ) \sqrt {-\frac {a \cos \left (x \right )^{2}-\cos \left (x \right )^{2} b -a}{\left (\cos \left (x \right )+1\right )^{2}}}}\) | \(495\) |
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none
Time = 0.33 (sec) , antiderivative size = 239, normalized size of antiderivative = 2.81 \[ \int \sqrt {a+b \cot ^2(x)} \tan ^4(x) \, dx=\left [\frac {3 \, a \sqrt {-a + b} \log \left (-\frac {a^{2} \tan \left (x\right )^{4} - 2 \, {\left (3 \, a^{2} - 4 \, a b\right )} \tan \left (x\right )^{2} + a^{2} - 8 \, a b + 8 \, b^{2} + 4 \, {\left (a \tan \left (x\right )^{3} - {\left (a - 2 \, b\right )} \tan \left (x\right )\right )} \sqrt {-a + b} \sqrt {\frac {a \tan \left (x\right )^{2} + b}{\tan \left (x\right )^{2}}}}{\tan \left (x\right )^{4} + 2 \, \tan \left (x\right )^{2} + 1}\right ) + 4 \, {\left (a \tan \left (x\right )^{3} - {\left (3 \, a - b\right )} \tan \left (x\right )\right )} \sqrt {\frac {a \tan \left (x\right )^{2} + b}{\tan \left (x\right )^{2}}}}{12 \, a}, -\frac {3 \, \sqrt {a - b} a \arctan \left (\frac {2 \, \sqrt {a - b} \sqrt {\frac {a \tan \left (x\right )^{2} + b}{\tan \left (x\right )^{2}}} \tan \left (x\right )}{a \tan \left (x\right )^{2} - a + 2 \, b}\right ) - 2 \, {\left (a \tan \left (x\right )^{3} - {\left (3 \, a - b\right )} \tan \left (x\right )\right )} \sqrt {\frac {a \tan \left (x\right )^{2} + b}{\tan \left (x\right )^{2}}}}{6 \, a}\right ] \]
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\[ \int \sqrt {a+b \cot ^2(x)} \tan ^4(x) \, dx=\int \sqrt {a + b \cot ^{2}{\left (x \right )}} \tan ^{4}{\left (x \right )}\, dx \]
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\[ \int \sqrt {a+b \cot ^2(x)} \tan ^4(x) \, dx=\int { \sqrt {b \cot \left (x\right )^{2} + a} \tan \left (x\right )^{4} \,d x } \]
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Leaf count of result is larger than twice the leaf count of optimal. 476 vs. \(2 (71) = 142\).
Time = 0.33 (sec) , antiderivative size = 476, normalized size of antiderivative = 5.60 \[ \int \sqrt {a+b \cot ^2(x)} \tan ^4(x) \, dx=-\frac {1}{6} \, {\left (3 \, \sqrt {-a + b} \log \left ({\left (\sqrt {-a + b} \cos \left (x\right ) - \sqrt {-a \cos \left (x\right )^{2} + b \cos \left (x\right )^{2} + a}\right )}^{2}\right ) - \frac {4 \, {\left (3 \, {\left (\sqrt {-a + b} \cos \left (x\right ) - \sqrt {-a \cos \left (x\right )^{2} + b \cos \left (x\right )^{2} + a}\right )}^{4} {\left (2 \, a - b\right )} \sqrt {-a + b} - 6 \, {\left (\sqrt {-a + b} \cos \left (x\right ) - \sqrt {-a \cos \left (x\right )^{2} + b \cos \left (x\right )^{2} + a}\right )}^{2} a^{2} \sqrt {-a + b} + {\left (4 \, a^{3} - a^{2} b\right )} \sqrt {-a + b}\right )}}{{\left ({\left (\sqrt {-a + b} \cos \left (x\right ) - \sqrt {-a \cos \left (x\right )^{2} + b \cos \left (x\right )^{2} + a}\right )}^{2} - a\right )}^{3}}\right )} \mathrm {sgn}\left (\sin \left (x\right )\right ) + \frac {{\left (3 \, a^{2} \sqrt {-a + b} \log \left (-a - 2 \, \sqrt {-a + b} \sqrt {b} + 2 \, b\right ) - 9 \, a^{2} \sqrt {b} \log \left (-a - 2 \, \sqrt {-a + b} \sqrt {b} + 2 \, b\right ) - 15 \, a \sqrt {-a + b} b \log \left (-a - 2 \, \sqrt {-a + b} \sqrt {b} + 2 \, b\right ) + 21 \, a b^{\frac {3}{2}} \log \left (-a - 2 \, \sqrt {-a + b} \sqrt {b} + 2 \, b\right ) + 12 \, \sqrt {-a + b} b^{2} \log \left (-a - 2 \, \sqrt {-a + b} \sqrt {b} + 2 \, b\right ) - 12 \, b^{\frac {5}{2}} \log \left (-a - 2 \, \sqrt {-a + b} \sqrt {b} + 2 \, b\right ) + 8 \, a^{2} \sqrt {-a + b} - 18 \, a^{2} \sqrt {b} - 24 \, a \sqrt {-a + b} b + 30 \, a b^{\frac {3}{2}} + 12 \, \sqrt {-a + b} b^{2} - 12 \, b^{\frac {5}{2}}\right )} \mathrm {sgn}\left (\sin \left (x\right )\right )}{6 \, {\left (a^{2} + 3 \, a \sqrt {-a + b} \sqrt {b} - 5 \, a b - 4 \, \sqrt {-a + b} b^{\frac {3}{2}} + 4 \, b^{2}\right )}} \]
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Timed out. \[ \int \sqrt {a+b \cot ^2(x)} \tan ^4(x) \, dx=\int {\mathrm {tan}\left (x\right )}^4\,\sqrt {b\,{\mathrm {cot}\left (x\right )}^2+a} \,d x \]
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