Optimal. Leaf size=123 \[ -\frac {\cot ^4(e+f x) (b \sec (e+f x))^{7/2}}{4 b^3 f}-\frac {7 \cot ^2(e+f x) (b \sec (e+f x))^{3/2}}{16 b f}+\frac {21 \sqrt {b} \tan ^{-1}\left (\frac {\sqrt {b \sec (e+f x)}}{\sqrt {b}}\right )}{32 f}-\frac {21 \sqrt {b} \tanh ^{-1}\left (\frac {\sqrt {b \sec (e+f x)}}{\sqrt {b}}\right )}{32 f} \]
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Rubi [A] time = 0.09, antiderivative size = 123, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {2622, 288, 329, 298, 203, 206} \[ -\frac {\cot ^4(e+f x) (b \sec (e+f x))^{7/2}}{4 b^3 f}-\frac {7 \cot ^2(e+f x) (b \sec (e+f x))^{3/2}}{16 b f}+\frac {21 \sqrt {b} \tan ^{-1}\left (\frac {\sqrt {b \sec (e+f x)}}{\sqrt {b}}\right )}{32 f}-\frac {21 \sqrt {b} \tanh ^{-1}\left (\frac {\sqrt {b \sec (e+f x)}}{\sqrt {b}}\right )}{32 f} \]
Antiderivative was successfully verified.
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Rule 203
Rule 206
Rule 288
Rule 298
Rule 329
Rule 2622
Rubi steps
\begin {align*} \int \csc ^5(e+f x) \sqrt {b \sec (e+f x)} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {x^{9/2}}{\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) (b \sec (e+f x))^{7/2}}{4 b^3 f}+\frac {7 \operatorname {Subst}\left (\int \frac {x^{5/2}}{\left (-1+\frac {x^2}{b^2}\right )^2} \, dx,x,b \sec (e+f x)\right )}{8 b^3 f}\\ &=-\frac {7 \cot ^2(e+f x) (b \sec (e+f x))^{3/2}}{16 b f}-\frac {\cot ^4(e+f x) (b \sec (e+f x))^{7/2}}{4 b^3 f}+\frac {21 \operatorname {Subst}\left (\int \frac {\sqrt {x}}{-1+\frac {x^2}{b^2}} \, dx,x,b \sec (e+f x)\right )}{32 b f}\\ &=-\frac {7 \cot ^2(e+f x) (b \sec (e+f x))^{3/2}}{16 b f}-\frac {\cot ^4(e+f x) (b \sec (e+f x))^{7/2}}{4 b^3 f}+\frac {21 \operatorname {Subst}\left (\int \frac {x^2}{-1+\frac {x^4}{b^2}} \, dx,x,\sqrt {b \sec (e+f x)}\right )}{16 b f}\\ &=-\frac {7 \cot ^2(e+f x) (b \sec (e+f x))^{3/2}}{16 b f}-\frac {\cot ^4(e+f x) (b \sec (e+f x))^{7/2}}{4 b^3 f}-\frac {(21 b) \operatorname {Subst}\left (\int \frac {1}{b-x^2} \, dx,x,\sqrt {b \sec (e+f x)}\right )}{32 f}+\frac {(21 b) \operatorname {Subst}\left (\int \frac {1}{b+x^2} \, dx,x,\sqrt {b \sec (e+f x)}\right )}{32 f}\\ &=\frac {21 \sqrt {b} \tan ^{-1}\left (\frac {\sqrt {b \sec (e+f x)}}{\sqrt {b}}\right )}{32 f}-\frac {21 \sqrt {b} \tanh ^{-1}\left (\frac {\sqrt {b \sec (e+f x)}}{\sqrt {b}}\right )}{32 f}-\frac {7 \cot ^2(e+f x) (b \sec (e+f x))^{3/2}}{16 b f}-\frac {\cot ^4(e+f x) (b \sec (e+f x))^{7/2}}{4 b^3 f}\\ \end {align*}
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Mathematica [A] time = 1.01, size = 107, normalized size = 0.87 \[ \frac {b \left (-16 \csc ^4(e+f x)-28 \csc ^2(e+f x)+21 \sqrt {\sec (e+f x)} \left (\log \left (1-\sqrt {\sec (e+f x)}\right )-\log \left (\sqrt {\sec (e+f x)}+1\right )\right )+42 \sqrt {\sec (e+f x)} \tan ^{-1}\left (\sqrt {\sec (e+f x)}\right )\right )}{64 f \sqrt {b \sec (e+f x)}} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.93, size = 438, normalized size = 3.56 \[ \left [\frac {42 \, {\left (\cos \left (f x + e\right )^{4} - 2 \, \cos \left (f x + e\right )^{2} + 1\right )} \sqrt {-b} \arctan \left (\frac {\sqrt {-b} \sqrt {\frac {b}{\cos \left (f x + e\right )}} {\left (\cos \left (f x + e\right ) + 1\right )}}{2 \, b}\right ) + 21 \, {\left (\cos \left (f x + e\right )^{4} - 2 \, \cos \left (f x + e\right )^{2} + 1\right )} \sqrt {-b} \log \left (\frac {b \cos \left (f x + e\right )^{2} - 4 \, {\left (\cos \left (f x + e\right )^{2} - \cos \left (f x + e\right )\right )} \sqrt {-b} \sqrt {\frac {b}{\cos \left (f x + e\right )}} - 6 \, b \cos \left (f x + e\right ) + b}{\cos \left (f x + e\right )^{2} + 2 \, \cos \left (f x + e\right ) + 1}\right ) + 8 \, {\left (7 \, \cos \left (f x + e\right )^{3} - 11 \, \cos \left (f x + e\right )\right )} \sqrt {\frac {b}{\cos \left (f x + e\right )}}}{128 \, {\left (f \cos \left (f x + e\right )^{4} - 2 \, f \cos \left (f x + e\right )^{2} + f\right )}}, -\frac {42 \, {\left (\cos \left (f x + e\right )^{4} - 2 \, \cos \left (f x + e\right )^{2} + 1\right )} \sqrt {b} \arctan \left (\frac {\sqrt {\frac {b}{\cos \left (f x + e\right )}} {\left (\cos \left (f x + e\right ) - 1\right )}}{2 \, \sqrt {b}}\right ) - 21 \, {\left (\cos \left (f x + e\right )^{4} - 2 \, \cos \left (f x + e\right )^{2} + 1\right )} \sqrt {b} \log \left (\frac {b \cos \left (f x + e\right )^{2} - 4 \, {\left (\cos \left (f x + e\right )^{2} + \cos \left (f x + e\right )\right )} \sqrt {b} \sqrt {\frac {b}{\cos \left (f x + e\right )}} + 6 \, b \cos \left (f x + e\right ) + b}{\cos \left (f x + e\right )^{2} - 2 \, \cos \left (f x + e\right ) + 1}\right ) - 8 \, {\left (7 \, \cos \left (f x + e\right )^{3} - 11 \, \cos \left (f x + e\right )\right )} \sqrt {\frac {b}{\cos \left (f x + e\right )}}}{128 \, {\left (f \cos \left (f x + e\right )^{4} - 2 \, f \cos \left (f x + e\right )^{2} + f\right )}}\right ] \]
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 = 0.23, size = 1089, normalized size = 8.85 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.42, size = 138, normalized size = 1.12 \[ \frac {b {\left (\frac {42 \, \arctan \left (\frac {\sqrt {\frac {b}{\cos \left (f x + e\right )}}}{\sqrt {b}}\right )}{\sqrt {b}} + \frac {21 \, \log \left (-\frac {\sqrt {b} - \sqrt {\frac {b}{\cos \left (f x + e\right )}}}{\sqrt {b} + \sqrt {\frac {b}{\cos \left (f x + e\right )}}}\right )}{\sqrt {b}} + \frac {4 \, {\left (7 \, b^{2} \left (\frac {b}{\cos \left (f x + e\right )}\right )^{\frac {3}{2}} - 11 \, \left (\frac {b}{\cos \left (f x + e\right )}\right )^{\frac {7}{2}}\right )}}{b^{4} - \frac {2 \, b^{4}}{\cos \left (f x + e\right )^{2}} + \frac {b^{4}}{\cos \left (f x + e\right )^{4}}}\right )}}{64 \, f} \]
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 \[ \int \frac {\sqrt {\frac {b}{\cos \left (e+f\,x\right )}}}{{\sin \left (e+f\,x\right )}^5} \,d x \]
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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