Optimal. Leaf size=74 \[ -\frac{(3 a-2 b) \cot (e+f x) \sqrt{a+b \tan ^2(e+f x)}}{3 a^2 f}-\frac{\cot ^3(e+f x) \sqrt{a+b \tan ^2(e+f x)}}{3 a f} \]
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Rubi [A] time = 0.0900209, antiderivative size = 74, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.12, Rules used = {3663, 453, 264} \[ -\frac{(3 a-2 b) \cot (e+f x) \sqrt{a+b \tan ^2(e+f x)}}{3 a^2 f}-\frac{\cot ^3(e+f x) \sqrt{a+b \tan ^2(e+f x)}}{3 a f} \]
Antiderivative was successfully verified.
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Rule 3663
Rule 453
Rule 264
Rubi steps
\begin{align*} \int \frac{\csc ^4(e+f x)}{\sqrt{a+b \tan ^2(e+f x)}} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1+x^2}{x^4 \sqrt{a+b x^2}} \, dx,x,\tan (e+f x)\right )}{f}\\ &=-\frac{\cot ^3(e+f x) \sqrt{a+b \tan ^2(e+f x)}}{3 a f}+\frac{(3 a-2 b) \operatorname{Subst}\left (\int \frac{1}{x^2 \sqrt{a+b x^2}} \, dx,x,\tan (e+f x)\right )}{3 a f}\\ &=-\frac{(3 a-2 b) \cot (e+f x) \sqrt{a+b \tan ^2(e+f x)}}{3 a^2 f}-\frac{\cot ^3(e+f x) \sqrt{a+b \tan ^2(e+f x)}}{3 a f}\\ \end{align*}
Mathematica [A] time = 0.413838, size = 68, normalized size = 0.92 \[ -\frac{\cot (e+f x) \left (a \csc ^2(e+f x)+2 a-2 b\right ) \sqrt{\sec ^2(e+f x) ((a-b) \cos (2 (e+f x))+a+b)}}{3 \sqrt{2} a^2 f} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.194, size = 86, normalized size = 1.2 \begin{align*}{\frac{ \left ( 2\,a \left ( \cos \left ( fx+e \right ) \right ) ^{2}-2\, \left ( \cos \left ( fx+e \right ) \right ) ^{2}b-3\,a+2\,b \right ) \cos \left ( fx+e \right ) }{3\,f{a}^{2} \left ( \sin \left ( fx+e \right ) \right ) ^{3}}\sqrt{{\frac{a \left ( \cos \left ( fx+e \right ) \right ) ^{2}- \left ( \cos \left ( fx+e \right ) \right ) ^{2}b+b}{ \left ( \cos \left ( fx+e \right ) \right ) ^{2}}}}} \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 [A] time = 3.12206, size = 211, normalized size = 2.85 \begin{align*} -\frac{{\left (2 \,{\left (a - b\right )} \cos \left (f x + e\right )^{3} -{\left (3 \, a - 2 \, b\right )} \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}}}}{3 \,{\left (a^{2} f \cos \left (f x + e\right )^{2} - a^{2} f\right )} \sin \left (f x + e\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 ^{4}{\left (e + f x \right )}}{\sqrt{a + b \tan ^{2}{\left (e + f x \right )}}}\, 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 )^{4}}{\sqrt{b \tan \left (f x + e\right )^{2} + a}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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