Integrand size = 25, antiderivative size = 237 \[ \int \frac {\csc ^5(e+f x)}{\left (a+b \tan ^2(e+f x)\right )^{5/2}} \, dx=-\frac {\left (3 a^2-30 a b+35 b^2\right ) \text {arctanh}\left (\frac {\sqrt {a} \sec (e+f x)}{\sqrt {a-b+b \sec ^2(e+f x)}}\right )}{8 a^{9/2} f}-\frac {(5 a-7 b) \cot (e+f x) \csc (e+f x)}{8 a^2 f \left (a-b+b \sec ^2(e+f x)\right )^{3/2}}-\frac {\cot ^3(e+f x) \csc (e+f x)}{4 a f \left (a-b+b \sec ^2(e+f x)\right )^{3/2}}-\frac {(23 a-35 b) b \sec (e+f x)}{24 a^3 f \left (a-b+b \sec ^2(e+f x)\right )^{3/2}}-\frac {5 (11 a-21 b) b \sec (e+f x)}{24 a^4 f \sqrt {a-b+b \sec ^2(e+f x)}} \]
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Time = 0.37 (sec) , antiderivative size = 237, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {3745, 481, 541, 12, 385, 213} \[ \int \frac {\csc ^5(e+f x)}{\left (a+b \tan ^2(e+f x)\right )^{5/2}} \, dx=-\frac {5 b (11 a-21 b) \sec (e+f x)}{24 a^4 f \sqrt {a+b \sec ^2(e+f x)-b}}-\frac {b (23 a-35 b) \sec (e+f x)}{24 a^3 f \left (a+b \sec ^2(e+f x)-b\right )^{3/2}}-\frac {(5 a-7 b) \cot (e+f x) \csc (e+f x)}{8 a^2 f \left (a+b \sec ^2(e+f x)-b\right )^{3/2}}-\frac {\left (3 a^2-30 a b+35 b^2\right ) \text {arctanh}\left (\frac {\sqrt {a} \sec (e+f x)}{\sqrt {a+b \sec ^2(e+f x)-b}}\right )}{8 a^{9/2} f}-\frac {\cot ^3(e+f x) \csc (e+f x)}{4 a f \left (a+b \sec ^2(e+f x)-b\right )^{3/2}} \]
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Rule 12
Rule 213
Rule 385
Rule 481
Rule 541
Rule 3745
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {x^4}{\left (-1+x^2\right )^3 \left (a-b+b x^2\right )^{5/2}} \, dx,x,\sec (e+f x)\right )}{f} \\ & = -\frac {\cot ^3(e+f x) \csc (e+f x)}{4 a f \left (a-b+b \sec ^2(e+f x)\right )^{3/2}}-\frac {\text {Subst}\left (\int \frac {-a+b-2 (2 a-3 b) x^2}{\left (-1+x^2\right )^2 \left (a-b+b x^2\right )^{5/2}} \, dx,x,\sec (e+f x)\right )}{4 a f} \\ & = -\frac {(5 a-7 b) \cot (e+f x) \csc (e+f x)}{8 a^2 f \left (a-b+b \sec ^2(e+f x)\right )^{3/2}}-\frac {\cot ^3(e+f x) \csc (e+f x)}{4 a f \left (a-b+b \sec ^2(e+f x)\right )^{3/2}}-\frac {\text {Subst}\left (\int \frac {-((3 a-7 b) (a-b))+4 (5 a-7 b) b x^2}{\left (-1+x^2\right ) \left (a-b+b x^2\right )^{5/2}} \, dx,x,\sec (e+f x)\right )}{8 a^2 f} \\ & = -\frac {(5 a-7 b) \cot (e+f x) \csc (e+f x)}{8 a^2 f \left (a-b+b \sec ^2(e+f x)\right )^{3/2}}-\frac {\cot ^3(e+f x) \csc (e+f x)}{4 a f \left (a-b+b \sec ^2(e+f x)\right )^{3/2}}-\frac {(23 a-35 b) b \sec (e+f x)}{24 a^3 f \left (a-b+b \sec ^2(e+f x)\right )^{3/2}}-\frac {\text {Subst}\left (\int \frac {-\left ((9 a-35 b) (a-b)^2\right )+2 (23 a-35 b) (a-b) b x^2}{\left (-1+x^2\right ) \left (a-b+b x^2\right )^{3/2}} \, dx,x,\sec (e+f x)\right )}{24 a^3 (a-b) f} \\ & = -\frac {(5 a-7 b) \cot (e+f x) \csc (e+f x)}{8 a^2 f \left (a-b+b \sec ^2(e+f x)\right )^{3/2}}-\frac {\cot ^3(e+f x) \csc (e+f x)}{4 a f \left (a-b+b \sec ^2(e+f x)\right )^{3/2}}-\frac {(23 a-35 b) b \sec (e+f x)}{24 a^3 f \left (a-b+b \sec ^2(e+f x)\right )^{3/2}}-\frac {5 (11 a-21 b) b \sec (e+f x)}{24 a^4 f \sqrt {a-b+b \sec ^2(e+f x)}}-\frac {\text {Subst}\left (\int -\frac {3 (a-b)^2 \left (3 a^2-30 a b+35 b^2\right )}{\left (-1+x^2\right ) \sqrt {a-b+b x^2}} \, dx,x,\sec (e+f x)\right )}{24 a^4 (a-b)^2 f} \\ & = -\frac {(5 a-7 b) \cot (e+f x) \csc (e+f x)}{8 a^2 f \left (a-b+b \sec ^2(e+f x)\right )^{3/2}}-\frac {\cot ^3(e+f x) \csc (e+f x)}{4 a f \left (a-b+b \sec ^2(e+f x)\right )^{3/2}}-\frac {(23 a-35 b) b \sec (e+f x)}{24 a^3 f \left (a-b+b \sec ^2(e+f x)\right )^{3/2}}-\frac {5 (11 a-21 b) b \sec (e+f x)}{24 a^4 f \sqrt {a-b+b \sec ^2(e+f x)}}+\frac {\left (3 a^2-30 a b+35 b^2\right ) \text {Subst}\left (\int \frac {1}{\left (-1+x^2\right ) \sqrt {a-b+b x^2}} \, dx,x,\sec (e+f x)\right )}{8 a^4 f} \\ & = -\frac {(5 a-7 b) \cot (e+f x) \csc (e+f x)}{8 a^2 f \left (a-b+b \sec ^2(e+f x)\right )^{3/2}}-\frac {\cot ^3(e+f x) \csc (e+f x)}{4 a f \left (a-b+b \sec ^2(e+f x)\right )^{3/2}}-\frac {(23 a-35 b) b \sec (e+f x)}{24 a^3 f \left (a-b+b \sec ^2(e+f x)\right )^{3/2}}-\frac {5 (11 a-21 b) b \sec (e+f x)}{24 a^4 f \sqrt {a-b+b \sec ^2(e+f x)}}+\frac {\left (3 a^2-30 a b+35 b^2\right ) \text {Subst}\left (\int \frac {1}{-1+a x^2} \, dx,x,\frac {\sec (e+f x)}{\sqrt {a-b+b \sec ^2(e+f x)}}\right )}{8 a^4 f} \\ & = -\frac {\left (3 a^2-30 a b+35 b^2\right ) \text {arctanh}\left (\frac {\sqrt {a} \sec (e+f x)}{\sqrt {a-b+b \sec ^2(e+f x)}}\right )}{8 a^{9/2} f}-\frac {(5 a-7 b) \cot (e+f x) \csc (e+f x)}{8 a^2 f \left (a-b+b \sec ^2(e+f x)\right )^{3/2}}-\frac {\cot ^3(e+f x) \csc (e+f x)}{4 a f \left (a-b+b \sec ^2(e+f x)\right )^{3/2}}-\frac {(23 a-35 b) b \sec (e+f x)}{24 a^3 f \left (a-b+b \sec ^2(e+f x)\right )^{3/2}}-\frac {5 (11 a-21 b) b \sec (e+f x)}{24 a^4 f \sqrt {a-b+b \sec ^2(e+f x)}} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(1132\) vs. \(2(237)=474\).
Time = 8.49 (sec) , antiderivative size = 1132, normalized size of antiderivative = 4.78 \[ \int \frac {\csc ^5(e+f x)}{\left (a+b \tan ^2(e+f x)\right )^{5/2}} \, dx=\frac {\sqrt {\frac {a+b+a \cos (2 (e+f x))-b \cos (2 (e+f x))}{1+\cos (2 (e+f x))}} \left (\frac {4 b^2 \cos (e+f x)}{3 a^3 (a+b+a \cos (2 (e+f x))-b \cos (2 (e+f x)))^2}-\frac {2 \left (2 a b \cos (e+f x)-3 b^2 \cos (e+f x)\right )}{a^4 (a+b+a \cos (2 (e+f x))-b \cos (2 (e+f x)))}+\frac {(-3 a \cos (e+f x)+11 b \cos (e+f x)) \csc ^2(e+f x)}{8 a^4}-\frac {\cot (e+f x) \csc ^3(e+f x)}{4 a^3}\right )}{f}+\frac {\left (3 a^2-30 a b+35 b^2\right ) \left (\frac {(1+\cos (e+f x)) \sqrt {\frac {1+\cos (2 (e+f x))}{(1+\cos (e+f x))^2}} \sqrt {\frac {a+b+(a-b) \cos (2 (e+f x))}{1+\cos (2 (e+f x))}} \left (4 \sqrt {a} \text {arctanh}\left (\frac {-\sqrt {a} \left (-1+\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )+\sqrt {4 b \tan ^2\left (\frac {1}{2} (e+f x)\right )+a \left (-1+\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )^2}}{2 \sqrt {b}}\right )-\sqrt {b} \left (2 \text {arctanh}\left (\tan ^2\left (\frac {1}{2} (e+f x)\right )-\frac {\sqrt {4 b \tan ^2\left (\frac {1}{2} (e+f x)\right )+a \left (-1+\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )^2}}{\sqrt {a}}\right )+\log \left (a-2 b-a \tan ^2\left (\frac {1}{2} (e+f x)\right )+\sqrt {a} \sqrt {4 b \tan ^2\left (\frac {1}{2} (e+f x)\right )+a \left (-1+\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )^2}\right )\right )\right ) \left (-1+\tan ^2\left (\frac {1}{2} (e+f x)\right )\right ) \left (1+\tan ^2\left (\frac {1}{2} (e+f x)\right )\right ) \sqrt {\frac {4 b \tan ^2\left (\frac {1}{2} (e+f x)\right )+a \left (-1+\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )^2}{\left (1+\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )^2}}}{4 \sqrt {a} \sqrt {b} \sqrt {a+b+(a-b) \cos (2 (e+f x))} \sqrt {\left (-1+\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )^2} \sqrt {4 b \tan ^2\left (\frac {1}{2} (e+f x)\right )+a \left (-1+\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )^2}}-\frac {(1+\cos (e+f x)) \sqrt {\frac {1+\cos (2 (e+f x))}{(1+\cos (e+f x))^2}} \sqrt {\frac {a+b+(a-b) \cos (2 (e+f x))}{1+\cos (2 (e+f x))}} \left (4 \sqrt {a} \text {arctanh}\left (\frac {-\sqrt {a} \left (-1+\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )+\sqrt {4 b \tan ^2\left (\frac {1}{2} (e+f x)\right )+a \left (-1+\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )^2}}{2 \sqrt {b}}\right )+\sqrt {b} \left (2 \text {arctanh}\left (\tan ^2\left (\frac {1}{2} (e+f x)\right )-\frac {\sqrt {4 b \tan ^2\left (\frac {1}{2} (e+f x)\right )+a \left (-1+\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )^2}}{\sqrt {a}}\right )+\log \left (a-2 b-a \tan ^2\left (\frac {1}{2} (e+f x)\right )+\sqrt {a} \sqrt {4 b \tan ^2\left (\frac {1}{2} (e+f x)\right )+a \left (-1+\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )^2}\right )\right )\right ) \left (-1+\tan ^2\left (\frac {1}{2} (e+f x)\right )\right ) \left (1+\tan ^2\left (\frac {1}{2} (e+f x)\right )\right ) \sqrt {\frac {4 b \tan ^2\left (\frac {1}{2} (e+f x)\right )+a \left (-1+\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )^2}{\left (1+\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )^2}}}{4 \sqrt {a} \sqrt {b} \sqrt {a+b+(a-b) \cos (2 (e+f x))} \sqrt {\left (-1+\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )^2} \sqrt {4 b \tan ^2\left (\frac {1}{2} (e+f x)\right )+a \left (-1+\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )^2}}\right )}{8 a^4 f} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(18431\) vs. \(2(213)=426\).
Time = 4.46 (sec) , antiderivative size = 18432, normalized size of antiderivative = 77.77
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Leaf count of result is larger than twice the leaf count of optimal. 505 vs. \(2 (213) = 426\).
Time = 0.51 (sec) , antiderivative size = 1037, normalized size of antiderivative = 4.38 \[ \int \frac {\csc ^5(e+f x)}{\left (a+b \tan ^2(e+f x)\right )^{5/2}} \, dx=\text {Too large to display} \]
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\[ \int \frac {\csc ^5(e+f x)}{\left (a+b \tan ^2(e+f x)\right )^{5/2}} \, dx=\int \frac {\csc ^{5}{\left (e + f x \right )}}{\left (a + b \tan ^{2}{\left (e + f x \right )}\right )^{\frac {5}{2}}}\, dx \]
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Timed out. \[ \int \frac {\csc ^5(e+f x)}{\left (a+b \tan ^2(e+f x)\right )^{5/2}} \, dx=\text {Timed out} \]
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\[ \int \frac {\csc ^5(e+f x)}{\left (a+b \tan ^2(e+f x)\right )^{5/2}} \, dx=\int { \frac {\csc \left (f x + e\right )^{5}}{{\left (b \tan \left (f x + e\right )^{2} + a\right )}^{\frac {5}{2}}} \,d x } \]
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Timed out. \[ \int \frac {\csc ^5(e+f x)}{\left (a+b \tan ^2(e+f x)\right )^{5/2}} \, dx=\text {Hanged} \]
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