Integrand size = 25, antiderivative size = 105 \[ \int \csc ^4(e+f x) \sqrt {a+b \sec ^2(e+f x)} \, dx=\frac {\sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} \tan (e+f x)}{\sqrt {a+b+b \tan ^2(e+f x)}}\right )}{f}-\frac {\cot (e+f x) \sqrt {a+b+b \tan ^2(e+f x)}}{f}-\frac {\cot ^3(e+f x) \left (a+b+b \tan ^2(e+f x)\right )^{3/2}}{3 (a+b) f} \]
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Time = 0.13 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {4217, 462, 283, 223, 212} \[ \int \csc ^4(e+f x) \sqrt {a+b \sec ^2(e+f x)} \, dx=\frac {\sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} \tan (e+f x)}{\sqrt {a+b \tan ^2(e+f x)+b}}\right )}{f}-\frac {\cot ^3(e+f x) \left (a+b \tan ^2(e+f x)+b\right )^{3/2}}{3 f (a+b)}-\frac {\cot (e+f x) \sqrt {a+b \tan ^2(e+f x)+b}}{f} \]
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Rule 212
Rule 223
Rule 283
Rule 462
Rule 4217
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {\left (1+x^2\right ) \sqrt {a+b+b x^2}}{x^4} \, dx,x,\tan (e+f x)\right )}{f} \\ & = -\frac {\cot ^3(e+f x) \left (a+b+b \tan ^2(e+f x)\right )^{3/2}}{3 (a+b) f}+\frac {\text {Subst}\left (\int \frac {\sqrt {a+b+b x^2}}{x^2} \, dx,x,\tan (e+f x)\right )}{f} \\ & = -\frac {\cot (e+f x) \sqrt {a+b+b \tan ^2(e+f x)}}{f}-\frac {\cot ^3(e+f x) \left (a+b+b \tan ^2(e+f x)\right )^{3/2}}{3 (a+b) f}+\frac {b \text {Subst}\left (\int \frac {1}{\sqrt {a+b+b x^2}} \, dx,x,\tan (e+f x)\right )}{f} \\ & = -\frac {\cot (e+f x) \sqrt {a+b+b \tan ^2(e+f x)}}{f}-\frac {\cot ^3(e+f x) \left (a+b+b \tan ^2(e+f x)\right )^{3/2}}{3 (a+b) f}+\frac {b \text {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {\tan (e+f x)}{\sqrt {a+b+b \tan ^2(e+f x)}}\right )}{f} \\ & = \frac {\sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} \tan (e+f x)}{\sqrt {a+b+b \tan ^2(e+f x)}}\right )}{f}-\frac {\cot (e+f x) \sqrt {a+b+b \tan ^2(e+f x)}}{f}-\frac {\cot ^3(e+f x) \left (a+b+b \tan ^2(e+f x)\right )^{3/2}}{3 (a+b) f} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 2.76 (sec) , antiderivative size = 285, normalized size of antiderivative = 2.71 \[ \int \csc ^4(e+f x) \sqrt {a+b \sec ^2(e+f x)} \, dx=-\frac {\sqrt {2} \cot (e+f x) \csc ^2(e+f x) \sqrt {a+b \sec ^2(e+f x)} \left (1-\frac {a \sin ^2(e+f x)}{a+b}\right ) \left (\frac {4 b \operatorname {Hypergeometric2F1}\left (2,2,\frac {3}{2},-\frac {b \tan ^2(e+f x)}{a+b}\right ) \sec ^2(e+f x) \sqrt {\frac {a+b \sec ^2(e+f x)}{a+b}} \left (a+b-a \sin ^2(e+f x)\right )^2 \tan ^2(e+f x)}{(a+b)^2}+\left (a+b+2 a \sin ^2(e+f x)\right ) \left (\sqrt {\frac {a+b \sec ^2(e+f x)}{a+b}}+\arcsin \left (\sqrt {-\frac {b \tan ^2(e+f x)}{a+b}}\right ) \sqrt {-\frac {b \tan ^2(e+f x)}{a+b}}\right )\right )}{3 f \sqrt {a+2 b+a \cos (2 e+2 f x)} \sqrt {\frac {a+b \sec ^2(e+f x)}{a+b}} \sqrt {a+b-a \sin ^2(e+f x)}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(1036\) vs. \(2(93)=186\).
Time = 7.32 (sec) , antiderivative size = 1037, normalized size of antiderivative = 9.88
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Leaf count of result is larger than twice the leaf count of optimal. 202 vs. \(2 (93) = 186\).
Time = 0.47 (sec) , antiderivative size = 436, normalized size of antiderivative = 4.15 \[ \int \csc ^4(e+f x) \sqrt {a+b \sec ^2(e+f x)} \, dx=\left [\frac {3 \, {\left ({\left (a + b\right )} \cos \left (f x + e\right )^{2} - a - b\right )} \sqrt {b} \log \left (\frac {{\left (a^{2} - 6 \, a b + b^{2}\right )} \cos \left (f x + e\right )^{4} + 8 \, {\left (a b - b^{2}\right )} \cos \left (f x + e\right )^{2} + 4 \, {\left ({\left (a - b\right )} \cos \left (f x + e\right )^{3} + 2 \, b \cos \left (f x + e\right )\right )} \sqrt {b} \sqrt {\frac {a \cos \left (f x + e\right )^{2} + b}{\cos \left (f x + e\right )^{2}}} \sin \left (f x + e\right ) + 8 \, b^{2}}{\cos \left (f x + e\right )^{4}}\right ) \sin \left (f x + e\right ) - 4 \, {\left ({\left (2 \, a + 3 \, b\right )} \cos \left (f x + e\right )^{3} - {\left (3 \, a + 4 \, b\right )} \cos \left (f x + e\right )\right )} \sqrt {\frac {a \cos \left (f x + e\right )^{2} + b}{\cos \left (f x + e\right )^{2}}}}{12 \, {\left ({\left (a + b\right )} f \cos \left (f x + e\right )^{2} - {\left (a + b\right )} f\right )} \sin \left (f x + e\right )}, \frac {3 \, {\left ({\left (a + b\right )} \cos \left (f x + e\right )^{2} - a - b\right )} \sqrt {-b} \arctan \left (-\frac {{\left ({\left (a - b\right )} \cos \left (f x + e\right )^{3} + 2 \, b \cos \left (f x + e\right )\right )} \sqrt {-b} \sqrt {\frac {a \cos \left (f x + e\right )^{2} + b}{\cos \left (f x + e\right )^{2}}}}{2 \, {\left (a b \cos \left (f x + e\right )^{2} + b^{2}\right )} \sin \left (f x + e\right )}\right ) \sin \left (f x + e\right ) - 2 \, {\left ({\left (2 \, a + 3 \, b\right )} \cos \left (f x + e\right )^{3} - {\left (3 \, a + 4 \, b\right )} \cos \left (f x + e\right )\right )} \sqrt {\frac {a \cos \left (f x + e\right )^{2} + b}{\cos \left (f x + e\right )^{2}}}}{6 \, {\left ({\left (a + b\right )} f \cos \left (f x + e\right )^{2} - {\left (a + b\right )} f\right )} \sin \left (f x + e\right )}\right ] \]
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\[ \int \csc ^4(e+f x) \sqrt {a+b \sec ^2(e+f x)} \, dx=\int \sqrt {a + b \sec ^{2}{\left (e + f x \right )}} \csc ^{4}{\left (e + f x \right )}\, dx \]
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Time = 0.18 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.78 \[ \int \csc ^4(e+f x) \sqrt {a+b \sec ^2(e+f x)} \, dx=\frac {3 \, \sqrt {b} \operatorname {arsinh}\left (\frac {b \tan \left (f x + e\right )}{\sqrt {{\left (a + b\right )} b}}\right ) - \frac {3 \, \sqrt {b \tan \left (f x + e\right )^{2} + a + b}}{\tan \left (f x + e\right )} - \frac {{\left (b \tan \left (f x + e\right )^{2} + a + b\right )}^{\frac {3}{2}}}{{\left (a + b\right )} \tan \left (f x + e\right )^{3}}}{3 \, f} \]
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\[ \int \csc ^4(e+f x) \sqrt {a+b \sec ^2(e+f x)} \, dx=\int { \sqrt {b \sec \left (f x + e\right )^{2} + a} \csc \left (f x + e\right )^{4} \,d x } \]
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Timed out. \[ \int \csc ^4(e+f x) \sqrt {a+b \sec ^2(e+f x)} \, dx=\int \frac {\sqrt {a+\frac {b}{{\cos \left (e+f\,x\right )}^2}}}{{\sin \left (e+f\,x\right )}^4} \,d x \]
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