Optimal. Leaf size=127 \[ -\frac{3 b \sec (e+f x)}{2 a^2 f \sqrt{a+b \sec ^2(e+f x)-b}}-\frac{(a-3 b) \tanh ^{-1}\left (\frac{\sqrt{a} \sec (e+f x)}{\sqrt{a+b \sec ^2(e+f x)-b}}\right )}{2 a^{5/2} f}-\frac{\cot (e+f x) \csc (e+f x)}{2 a f \sqrt{a+b \sec ^2(e+f x)-b}} \]
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Rubi [A] time = 0.157731, antiderivative size = 127, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.24, Rules used = {3664, 471, 527, 12, 377, 207} \[ -\frac{3 b \sec (e+f x)}{2 a^2 f \sqrt{a+b \sec ^2(e+f x)-b}}-\frac{(a-3 b) \tanh ^{-1}\left (\frac{\sqrt{a} \sec (e+f x)}{\sqrt{a+b \sec ^2(e+f x)-b}}\right )}{2 a^{5/2} f}-\frac{\cot (e+f x) \csc (e+f x)}{2 a f \sqrt{a+b \sec ^2(e+f x)-b}} \]
Antiderivative was successfully verified.
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Rule 3664
Rule 471
Rule 527
Rule 12
Rule 377
Rule 207
Rubi steps
\begin{align*} \int \frac{\csc ^3(e+f x)}{\left (a+b \tan ^2(e+f x)\right )^{3/2}} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{x^2}{\left (-1+x^2\right )^2 \left (a-b+b x^2\right )^{3/2}} \, dx,x,\sec (e+f x)\right )}{f}\\ &=-\frac{\cot (e+f x) \csc (e+f x)}{2 a f \sqrt{a-b+b \sec ^2(e+f x)}}+\frac{\operatorname{Subst}\left (\int \frac{a-b-2 b x^2}{\left (-1+x^2\right ) \left (a-b+b x^2\right )^{3/2}} \, dx,x,\sec (e+f x)\right )}{2 a f}\\ &=-\frac{\cot (e+f x) \csc (e+f x)}{2 a f \sqrt{a-b+b \sec ^2(e+f x)}}-\frac{3 b \sec (e+f x)}{2 a^2 f \sqrt{a-b+b \sec ^2(e+f x)}}+\frac{\operatorname{Subst}\left (\int \frac{(a-3 b) (a-b)}{\left (-1+x^2\right ) \sqrt{a-b+b x^2}} \, dx,x,\sec (e+f x)\right )}{2 a^2 (a-b) f}\\ &=-\frac{\cot (e+f x) \csc (e+f x)}{2 a f \sqrt{a-b+b \sec ^2(e+f x)}}-\frac{3 b \sec (e+f x)}{2 a^2 f \sqrt{a-b+b \sec ^2(e+f x)}}+\frac{(a-3 b) \operatorname{Subst}\left (\int \frac{1}{\left (-1+x^2\right ) \sqrt{a-b+b x^2}} \, dx,x,\sec (e+f x)\right )}{2 a^2 f}\\ &=-\frac{\cot (e+f x) \csc (e+f x)}{2 a f \sqrt{a-b+b \sec ^2(e+f x)}}-\frac{3 b \sec (e+f x)}{2 a^2 f \sqrt{a-b+b \sec ^2(e+f x)}}+\frac{(a-3 b) \operatorname{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 )}{2 a^2 f}\\ &=-\frac{(a-3 b) \tanh ^{-1}\left (\frac{\sqrt{a} \sec (e+f x)}{\sqrt{a-b+b \sec ^2(e+f x)}}\right )}{2 a^{5/2} f}-\frac{\cot (e+f x) \csc (e+f x)}{2 a f \sqrt{a-b+b \sec ^2(e+f x)}}-\frac{3 b \sec (e+f x)}{2 a^2 f \sqrt{a-b+b \sec ^2(e+f x)}}\\ \end{align*}
Mathematica [B] time = 4.09678, size = 308, normalized size = 2.43 \[ -\frac{\frac{\csc ^2(e+f x) \sec (e+f x) ((a-3 b) \cos (2 (e+f x))+a+3 b)}{\sqrt{2} a^2 \sqrt{\sec ^2(e+f x) ((a-b) \cos (2 (e+f x))+a+b)}}+\frac{(a-3 b) \cos (e+f x) \sec ^2\left (\frac{1}{2} (e+f x)\right ) \sqrt{\sec ^2(e+f x) ((a-b) \cos (2 (e+f x))+a+b)} \left (\tanh ^{-1}\left (\frac{a-(a-2 b) \tan ^2\left (\frac{1}{2} (e+f x)\right )}{\sqrt{a} \sqrt{a \left (\tan ^2\left (\frac{1}{2} (e+f x)\right )-1\right )^2+4 b \tan ^2\left (\frac{1}{2} (e+f x)\right )}}\right )+\tanh ^{-1}\left (\frac{a \left (\tan ^2\left (\frac{1}{2} (e+f x)\right )-1\right )+2 b}{\sqrt{a} \sqrt{a \left (\tan ^2\left (\frac{1}{2} (e+f x)\right )-1\right )^2+4 b \tan ^2\left (\frac{1}{2} (e+f x)\right )}}\right )\right )}{2 a^{5/2} \sqrt{\sec ^4\left (\frac{1}{2} (e+f x)\right ) ((a-b) \cos (2 (e+f x))+a+b)}}}{2 f} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.261, size = 5633, normalized size = 44.4 \begin{align*} \text{output too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [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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Fricas [A] time = 3.54322, size = 1087, normalized size = 8.56 \begin{align*} \left [-\frac{{\left ({\left (a^{2} - 4 \, a b + 3 \, b^{2}\right )} \cos \left (f x + e\right )^{4} -{\left (a^{2} - 5 \, a b + 6 \, b^{2}\right )} \cos \left (f x + e\right )^{2} - a b + 3 \, b^{2}\right )} \sqrt{a} \log \left (-\frac{2 \,{\left ({\left (a - b\right )} \cos \left (f x + e\right )^{2} + 2 \, \sqrt{a} \sqrt{\frac{{\left (a - b\right )} \cos \left (f x + e\right )^{2} + b}{\cos \left (f x + e\right )^{2}}} \cos \left (f x + e\right ) + a + b\right )}}{\cos \left (f x + e\right )^{2} - 1}\right ) - 2 \,{\left ({\left (a^{2} - 3 \, a b\right )} \cos \left (f x + e\right )^{3} + 3 \, a b \cos \left (f x + e\right )\right )} \sqrt{\frac{{\left (a - b\right )} \cos \left (f x + e\right )^{2} + b}{\cos \left (f x + e\right )^{2}}}}{4 \,{\left ({\left (a^{4} - a^{3} b\right )} f \cos \left (f x + e\right )^{4} - a^{3} b f -{\left (a^{4} - 2 \, a^{3} b\right )} f \cos \left (f x + e\right )^{2}\right )}}, \frac{{\left ({\left (a^{2} - 4 \, a b + 3 \, b^{2}\right )} \cos \left (f x + e\right )^{4} -{\left (a^{2} - 5 \, a b + 6 \, b^{2}\right )} \cos \left (f x + e\right )^{2} - a b + 3 \, b^{2}\right )} \sqrt{-a} \arctan \left (\frac{\sqrt{-a} \sqrt{\frac{{\left (a - b\right )} \cos \left (f x + e\right )^{2} + b}{\cos \left (f x + e\right )^{2}}} \cos \left (f x + e\right )}{a}\right ) +{\left ({\left (a^{2} - 3 \, a b\right )} \cos \left (f x + e\right )^{3} + 3 \, a b \cos \left (f x + e\right )\right )} \sqrt{\frac{{\left (a - b\right )} \cos \left (f x + e\right )^{2} + b}{\cos \left (f x + e\right )^{2}}}}{2 \,{\left ({\left (a^{4} - a^{3} b\right )} f \cos \left (f x + e\right )^{4} - a^{3} b f -{\left (a^{4} - 2 \, a^{3} b\right )} f \cos \left (f x + e\right )^{2}\right )}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\csc ^{3}{\left (e + f x \right )}}{\left (a + b \tan ^{2}{\left (e + f x \right )}\right )^{\frac{3}{2}}}\, dx \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 )^{3}}{{\left (b \tan \left (f x + e\right )^{2} + a\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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