Optimal. Leaf size=123 \[ \frac {3 \tan ^{-1}\left (\frac {\sqrt {b \sec (e+f x)}}{\sqrt {b}}\right )}{32 b^{5/2} f}+\frac {3 \tanh ^{-1}\left (\frac {\sqrt {b \sec (e+f x)}}{\sqrt {b}}\right )}{32 b^{5/2} f}-\frac {\cot ^4(e+f x) \sqrt {b \sec (e+f x)}}{4 b^3 f}-\frac {\cot ^2(e+f x) \sqrt {b \sec (e+f x)}}{16 b^3 f} \]
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Rubi [A] time = 0.08, antiderivative size = 123, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {2622, 288, 290, 329, 212, 206, 203} \[ -\frac {\cot ^4(e+f x) \sqrt {b \sec (e+f x)}}{4 b^3 f}-\frac {\cot ^2(e+f x) \sqrt {b \sec (e+f x)}}{16 b^3 f}+\frac {3 \tan ^{-1}\left (\frac {\sqrt {b \sec (e+f x)}}{\sqrt {b}}\right )}{32 b^{5/2} f}+\frac {3 \tanh ^{-1}\left (\frac {\sqrt {b \sec (e+f x)}}{\sqrt {b}}\right )}{32 b^{5/2} f} \]
Antiderivative was successfully verified.
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Rule 203
Rule 206
Rule 212
Rule 288
Rule 290
Rule 329
Rule 2622
Rubi steps
\begin {align*} \int \frac {\csc ^5(e+f x)}{(b \sec (e+f x))^{5/2}} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {x^{3/2}}{\left (-1+\frac {x^2}{b^2}\right )^3} \, dx,x,b \sec (e+f x)\right )}{b^5 f}\\ &=-\frac {\cot ^4(e+f x) \sqrt {b \sec (e+f x)}}{4 b^3 f}+\frac {\operatorname {Subst}\left (\int \frac {1}{\sqrt {x} \left (-1+\frac {x^2}{b^2}\right )^2} \, dx,x,b \sec (e+f x)\right )}{8 b^3 f}\\ &=-\frac {\cot ^2(e+f x) \sqrt {b \sec (e+f x)}}{16 b^3 f}-\frac {\cot ^4(e+f x) \sqrt {b \sec (e+f x)}}{4 b^3 f}-\frac {3 \operatorname {Subst}\left (\int \frac {1}{\sqrt {x} \left (-1+\frac {x^2}{b^2}\right )} \, dx,x,b \sec (e+f x)\right )}{32 b^3 f}\\ &=-\frac {\cot ^2(e+f x) \sqrt {b \sec (e+f x)}}{16 b^3 f}-\frac {\cot ^4(e+f x) \sqrt {b \sec (e+f x)}}{4 b^3 f}-\frac {3 \operatorname {Subst}\left (\int \frac {1}{-1+\frac {x^4}{b^2}} \, dx,x,\sqrt {b \sec (e+f x)}\right )}{16 b^3 f}\\ &=-\frac {\cot ^2(e+f x) \sqrt {b \sec (e+f x)}}{16 b^3 f}-\frac {\cot ^4(e+f x) \sqrt {b \sec (e+f x)}}{4 b^3 f}+\frac {3 \operatorname {Subst}\left (\int \frac {1}{b-x^2} \, dx,x,\sqrt {b \sec (e+f x)}\right )}{32 b^2 f}+\frac {3 \operatorname {Subst}\left (\int \frac {1}{b+x^2} \, dx,x,\sqrt {b \sec (e+f x)}\right )}{32 b^2 f}\\ &=\frac {3 \tan ^{-1}\left (\frac {\sqrt {b \sec (e+f x)}}{\sqrt {b}}\right )}{32 b^{5/2} f}+\frac {3 \tanh ^{-1}\left (\frac {\sqrt {b \sec (e+f x)}}{\sqrt {b}}\right )}{32 b^{5/2} f}-\frac {\cot ^2(e+f x) \sqrt {b \sec (e+f x)}}{16 b^3 f}-\frac {\cot ^4(e+f x) \sqrt {b \sec (e+f x)}}{4 b^3 f}\\ \end {align*}
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Mathematica [A] time = 2.40, size = 110, normalized size = 0.89 \[ \frac {\sqrt {\sec (e+f x)} \left (-3 \log \left (1-\sqrt {\sec (e+f x)}\right )+3 \log \left (\sqrt {\sec (e+f x)}+1\right )+6 \tan ^{-1}\left (\sqrt {\sec (e+f x)}\right )-\frac {2 (3 \cos (2 (e+f x))+5) \csc ^4(e+f x)}{\sec ^{\frac {3}{2}}(e+f x)}\right )}{64 b^2 f \sqrt {b \sec (e+f x)}} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.85, size = 458, normalized size = 3.72 \[ \left [-\frac {6 \, {\left (\cos \left (f x + e\right )^{4} - 2 \, \cos \left (f x + e\right )^{2} + 1\right )} \sqrt {-b} \arctan \left (\frac {\sqrt {-b} \sqrt {\frac {b}{\cos \left (f x + e\right )}} {\left (\cos \left (f x + e\right ) + 1\right )}}{2 \, b}\right ) + 3 \, {\left (\cos \left (f x + e\right )^{4} - 2 \, \cos \left (f x + e\right )^{2} + 1\right )} \sqrt {-b} \log \left (\frac {b \cos \left (f x + e\right )^{2} + 4 \, {\left (\cos \left (f x + e\right )^{2} - \cos \left (f x + e\right )\right )} \sqrt {-b} \sqrt {\frac {b}{\cos \left (f x + e\right )}} - 6 \, b \cos \left (f x + e\right ) + b}{\cos \left (f x + e\right )^{2} + 2 \, \cos \left (f x + e\right ) + 1}\right ) + 8 \, {\left (3 \, \cos \left (f x + e\right )^{4} + \cos \left (f x + e\right )^{2}\right )} \sqrt {\frac {b}{\cos \left (f x + e\right )}}}{128 \, {\left (b^{3} f \cos \left (f x + e\right )^{4} - 2 \, b^{3} f \cos \left (f x + e\right )^{2} + b^{3} f\right )}}, -\frac {6 \, {\left (\cos \left (f x + e\right )^{4} - 2 \, \cos \left (f x + e\right )^{2} + 1\right )} \sqrt {b} \arctan \left (\frac {\sqrt {\frac {b}{\cos \left (f x + e\right )}} {\left (\cos \left (f x + e\right ) - 1\right )}}{2 \, \sqrt {b}}\right ) - 3 \, {\left (\cos \left (f x + e\right )^{4} - 2 \, \cos \left (f x + e\right )^{2} + 1\right )} \sqrt {b} \log \left (\frac {b \cos \left (f x + e\right )^{2} + 4 \, {\left (\cos \left (f x + e\right )^{2} + \cos \left (f x + e\right )\right )} \sqrt {b} \sqrt {\frac {b}{\cos \left (f x + e\right )}} + 6 \, b \cos \left (f x + e\right ) + b}{\cos \left (f x + e\right )^{2} - 2 \, \cos \left (f x + e\right ) + 1}\right ) + 8 \, {\left (3 \, \cos \left (f x + e\right )^{4} + \cos \left (f x + e\right )^{2}\right )} \sqrt {\frac {b}{\cos \left (f x + e\right )}}}{128 \, {\left (b^{3} f \cos \left (f x + e\right )^{4} - 2 \, b^{3} f \cos \left (f x + e\right )^{2} + b^{3} f\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: NotImplementedError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.20, size = 737, normalized size = 5.99 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.90, size = 136, normalized size = 1.11 \[ -\frac {b {\left (\frac {4 \, {\left (3 \, b^{2} \sqrt {\frac {b}{\cos \left (f x + e\right )}} + \left (\frac {b}{\cos \left (f x + e\right )}\right )^{\frac {5}{2}}\right )}}{b^{6} - \frac {2 \, b^{6}}{\cos \left (f x + e\right )^{2}} + \frac {b^{6}}{\cos \left (f x + e\right )^{4}}} - \frac {6 \, \arctan \left (\frac {\sqrt {\frac {b}{\cos \left (f x + e\right )}}}{\sqrt {b}}\right )}{b^{\frac {7}{2}}} + \frac {3 \, \log \left (-\frac {\sqrt {b} - \sqrt {\frac {b}{\cos \left (f x + e\right )}}}{\sqrt {b} + \sqrt {\frac {b}{\cos \left (f x + e\right )}}}\right )}{b^{\frac {7}{2}}}\right )}}{64 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {1}{{\sin \left (e+f\,x\right )}^5\,{\left (\frac {b}{\cos \left (e+f\,x\right )}\right )}^{5/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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