Optimal. Leaf size=78 \[ -\frac{\cot ^3(e+f x) \sqrt{a+b \tan ^2(e+f x)+b}}{3 f (a+b)}-\frac{(3 a+b) \cot (e+f x) \sqrt{a+b \tan ^2(e+f x)+b}}{3 f (a+b)^2} \]
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Rubi [A] time = 0.0970371, antiderivative size = 78, 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 = {4132, 453, 264} \[ -\frac{\cot ^3(e+f x) \sqrt{a+b \tan ^2(e+f x)+b}}{3 f (a+b)}-\frac{(3 a+b) \cot (e+f x) \sqrt{a+b \tan ^2(e+f x)+b}}{3 f (a+b)^2} \]
Antiderivative was successfully verified.
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Rule 4132
Rule 453
Rule 264
Rubi steps
\begin{align*} \int \frac{\csc ^4(e+f x)}{\sqrt{a+b \sec ^2(e+f x)}} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1+x^2}{x^4 \sqrt{a+b+b x^2}} \, dx,x,\tan (e+f x)\right )}{f}\\ &=-\frac{\cot ^3(e+f x) \sqrt{a+b+b \tan ^2(e+f x)}}{3 (a+b) f}+\frac{(3 a+b) \operatorname{Subst}\left (\int \frac{1}{x^2 \sqrt{a+b+b x^2}} \, dx,x,\tan (e+f x)\right )}{3 (a+b) f}\\ &=-\frac{(3 a+b) \cot (e+f x) \sqrt{a+b+b \tan ^2(e+f x)}}{3 (a+b)^2 f}-\frac{\cot ^3(e+f x) \sqrt{a+b+b \tan ^2(e+f x)}}{3 (a+b) f}\\ \end{align*}
Mathematica [A] time = 0.209077, size = 74, normalized size = 0.95 \[ \frac{\csc ^3(e+f x) \sec (e+f x) (a \cos (2 (e+f x))-2 a-b) (a \cos (2 (e+f x))+a+2 b)}{6 f (a+b)^2 \sqrt{a+b \sec ^2(e+f x)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.343, size = 66, normalized size = 0.9 \begin{align*}{\frac{ \left ( 2\,a \left ( \cos \left ( fx+e \right ) \right ) ^{2}-3\,a-b \right ) \cos \left ( fx+e \right ) }{3\,f \left ( a+b \right ) ^{2} \left ( \sin \left ( fx+e \right ) \right ) ^{3}}\sqrt{{\frac{b+a \left ( \cos \left ( fx+e \right ) \right ) ^{2}}{ \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 = 0.729017, size = 235, normalized size = 3.01 \begin{align*} -\frac{{\left (2 \, a \cos \left (f x + e\right )^{3} -{\left (3 \, a + b\right )} \cos \left (f x + e\right )\right )} \sqrt{\frac{a \cos \left (f x + e\right )^{2} + b}{\cos \left (f x + e\right )^{2}}}}{3 \,{\left ({\left (a^{2} + 2 \, a b + b^{2}\right )} f \cos \left (f x + e\right )^{2} -{\left (a^{2} + 2 \, a b + b^{2}\right )} 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(-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 )^{4}}{\sqrt{b \sec \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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