Optimal. Leaf size=123 \[ -\frac{\cot ^4(e+f x) (b \sec (e+f x))^{3/2}}{4 b^3 f}-\frac{3 \cot ^2(e+f x) (b \sec (e+f x))^{3/2}}{16 b^3 f}-\frac{3 \tan ^{-1}\left (\frac{\sqrt{b \sec (e+f x)}}{\sqrt{b}}\right )}{32 b^{3/2} f}+\frac{3 \tanh ^{-1}\left (\frac{\sqrt{b \sec (e+f x)}}{\sqrt{b}}\right )}{32 b^{3/2} f} \]
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Rubi [A] time = 0.0858205, antiderivative size = 123, normalized size of antiderivative = 1., 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, 298, 203, 206} \[ -\frac{\cot ^4(e+f x) (b \sec (e+f x))^{3/2}}{4 b^3 f}-\frac{3 \cot ^2(e+f x) (b \sec (e+f x))^{3/2}}{16 b^3 f}-\frac{3 \tan ^{-1}\left (\frac{\sqrt{b \sec (e+f x)}}{\sqrt{b}}\right )}{32 b^{3/2} f}+\frac{3 \tanh ^{-1}\left (\frac{\sqrt{b \sec (e+f x)}}{\sqrt{b}}\right )}{32 b^{3/2} f} \]
Antiderivative was successfully verified.
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Rule 2622
Rule 288
Rule 290
Rule 329
Rule 298
Rule 203
Rule 206
Rubi steps
\begin{align*} \int \frac{\csc ^5(e+f x)}{(b \sec (e+f x))^{3/2}} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{x^{5/2}}{\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))^{3/2}}{4 b^3 f}+\frac{3 \operatorname{Subst}\left (\int \frac{\sqrt{x}}{\left (-1+\frac{x^2}{b^2}\right )^2} \, dx,x,b \sec (e+f x)\right )}{8 b^3 f}\\ &=-\frac{3 \cot ^2(e+f x) (b \sec (e+f x))^{3/2}}{16 b^3 f}-\frac{\cot ^4(e+f x) (b \sec (e+f x))^{3/2}}{4 b^3 f}-\frac{3 \operatorname{Subst}\left (\int \frac{\sqrt{x}}{-1+\frac{x^2}{b^2}} \, dx,x,b \sec (e+f x)\right )}{32 b^3 f}\\ &=-\frac{3 \cot ^2(e+f x) (b \sec (e+f x))^{3/2}}{16 b^3 f}-\frac{\cot ^4(e+f x) (b \sec (e+f x))^{3/2}}{4 b^3 f}-\frac{3 \operatorname{Subst}\left (\int \frac{x^2}{-1+\frac{x^4}{b^2}} \, dx,x,\sqrt{b \sec (e+f x)}\right )}{16 b^3 f}\\ &=-\frac{3 \cot ^2(e+f x) (b \sec (e+f x))^{3/2}}{16 b^3 f}-\frac{\cot ^4(e+f x) (b \sec (e+f x))^{3/2}}{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 f}-\frac{3 \operatorname{Subst}\left (\int \frac{1}{b+x^2} \, dx,x,\sqrt{b \sec (e+f x)}\right )}{32 b f}\\ &=-\frac{3 \tan ^{-1}\left (\frac{\sqrt{b \sec (e+f x)}}{\sqrt{b}}\right )}{32 b^{3/2} f}+\frac{3 \tanh ^{-1}\left (\frac{\sqrt{b \sec (e+f x)}}{\sqrt{b}}\right )}{32 b^{3/2} f}-\frac{3 \cot ^2(e+f x) (b \sec (e+f x))^{3/2}}{16 b^3 f}-\frac{\cot ^4(e+f x) (b \sec (e+f x))^{3/2}}{4 b^3 f}\\ \end{align*}
Mathematica [A] time = 0.654316, size = 109, normalized size = 0.89 \[ \frac{-16 \csc ^4(e+f x)+4 \csc ^2(e+f x)+3 \sqrt{\sec (e+f x)} \left (\log \left (\sqrt{\sec (e+f x)}+1\right )-\log \left (1-\sqrt{\sec (e+f x)}\right )\right )-6 \sqrt{\sec (e+f x)} \tan ^{-1}\left (\sqrt{\sec (e+f x)}\right )}{64 b f \sqrt{b \sec (e+f x)}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.133, size = 729, normalized size = 5.9 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 3.05884, size = 1192, normalized size = 9.69 \begin{align*} \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 (\cos \left (f x + e\right )^{3} + 3 \, \cos \left (f x + e\right )\right )} \sqrt{\frac{b}{\cos \left (f x + e\right )}}}{128 \,{\left (b^{2} f \cos \left (f x + e\right )^{4} - 2 \, b^{2} f \cos \left (f x + e\right )^{2} + b^{2} 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 (\cos \left (f x + e\right )^{3} + 3 \, \cos \left (f x + e\right )\right )} \sqrt{\frac{b}{\cos \left (f x + e\right )}}}{128 \,{\left (b^{2} f \cos \left (f x + e\right )^{4} - 2 \, b^{2} f \cos \left (f x + e\right )^{2} + b^{2} f\right )}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\csc \left (f x + e\right )^{5}}{\left (b \sec \left (f x + e\right )\right )^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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