Optimal. Leaf size=76 \[ -\frac{a \sqrt{b} \tan ^{-1}\left (\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b}}\right )}{f (a+b)^{5/2}}-\frac{\cot ^3(e+f x)}{3 f (a+b)}-\frac{a \cot (e+f x)}{f (a+b)^2} \]
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Rubi [A] time = 0.0944464, antiderivative size = 76, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.174, Rules used = {4132, 453, 325, 205} \[ -\frac{a \sqrt{b} \tan ^{-1}\left (\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b}}\right )}{f (a+b)^{5/2}}-\frac{\cot ^3(e+f x)}{3 f (a+b)}-\frac{a \cot (e+f x)}{f (a+b)^2} \]
Antiderivative was successfully verified.
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Rule 4132
Rule 453
Rule 325
Rule 205
Rubi steps
\begin{align*} \int \frac{\csc ^4(e+f x)}{a+b \sec ^2(e+f x)} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1+x^2}{x^4 \left (a+b+b x^2\right )} \, dx,x,\tan (e+f x)\right )}{f}\\ &=-\frac{\cot ^3(e+f x)}{3 (a+b) f}+\frac{a \operatorname{Subst}\left (\int \frac{1}{x^2 \left (a+b+b x^2\right )} \, dx,x,\tan (e+f x)\right )}{(a+b) f}\\ &=-\frac{a \cot (e+f x)}{(a+b)^2 f}-\frac{\cot ^3(e+f x)}{3 (a+b) f}-\frac{(a b) \operatorname{Subst}\left (\int \frac{1}{a+b+b x^2} \, dx,x,\tan (e+f x)\right )}{(a+b)^2 f}\\ &=-\frac{a \sqrt{b} \tan ^{-1}\left (\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b}}\right )}{(a+b)^{5/2} f}-\frac{a \cot (e+f x)}{(a+b)^2 f}-\frac{\cot ^3(e+f x)}{3 (a+b) f}\\ \end{align*}
Mathematica [C] time = 2.13727, size = 226, normalized size = 2.97 \[ \frac{\sec ^2(e+f x) (a \cos (2 (e+f x))+a+2 b) \left (\frac{1}{4} \sqrt{a+b} \csc (e) \sqrt{b (\cos (e)-i \sin (e))^4} \csc ^3(e+f x) ((b-2 a) \sin (2 e+3 f x)+6 a \sin (f x)-3 b \sin (2 e+f x))+3 a b (\cos (2 e)-i \sin (2 e)) \tan ^{-1}\left (\frac{(\cos (2 e)-i \sin (2 e)) \sec (f x) (a \sin (2 e+f x)-(a+2 b) \sin (f x))}{2 \sqrt{a+b} \sqrt{b (\cos (e)-i \sin (e))^4}}\right )\right )}{6 f (a+b)^{5/2} \sqrt{b (\cos (e)-i \sin (e))^4} \left (a+b \sec ^2(e+f x)\right )} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.097, size = 74, normalized size = 1. \begin{align*} -{\frac{1}{3\,f \left ( a+b \right ) \left ( \tan \left ( fx+e \right ) \right ) ^{3}}}-{\frac{a}{f \left ( a+b \right ) ^{2}\tan \left ( fx+e \right ) }}-{\frac{ab}{f \left ( a+b \right ) ^{2}}\arctan \left ({\tan \left ( fx+e \right ) b{\frac{1}{\sqrt{ \left ( a+b \right ) b}}}} \right ){\frac{1}{\sqrt{ \left ( a+b \right ) b}}}} \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 = 0.573484, size = 954, normalized size = 12.55 \begin{align*} \left [-\frac{4 \,{\left (2 \, a - b\right )} \cos \left (f x + e\right )^{3} - 3 \,{\left (a \cos \left (f x + e\right )^{2} - a\right )} \sqrt{-\frac{b}{a + b}} \log \left (\frac{{\left (a^{2} + 8 \, a b + 8 \, b^{2}\right )} \cos \left (f x + e\right )^{4} - 2 \,{\left (3 \, a b + 4 \, b^{2}\right )} \cos \left (f x + e\right )^{2} + 4 \,{\left ({\left (a^{2} + 3 \, a b + 2 \, b^{2}\right )} \cos \left (f x + e\right )^{3} -{\left (a b + b^{2}\right )} \cos \left (f x + e\right )\right )} \sqrt{-\frac{b}{a + b}} \sin \left (f x + e\right ) + b^{2}}{a^{2} \cos \left (f x + e\right )^{4} + 2 \, a b \cos \left (f x + e\right )^{2} + b^{2}}\right ) \sin \left (f x + e\right ) - 12 \, a \cos \left (f x + e\right )}{12 \,{\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 )}, -\frac{2 \,{\left (2 \, a - b\right )} \cos \left (f x + e\right )^{3} - 3 \,{\left (a \cos \left (f x + e\right )^{2} - a\right )} \sqrt{\frac{b}{a + b}} \arctan \left (\frac{{\left ({\left (a + 2 \, b\right )} \cos \left (f x + e\right )^{2} - b\right )} \sqrt{\frac{b}{a + b}}}{2 \, b \cos \left (f x + e\right ) \sin \left (f x + e\right )}\right ) \sin \left (f x + e\right ) - 6 \, a \cos \left (f x + e\right )}{6 \,{\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 )}\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 [A] time = 1.30426, size = 144, normalized size = 1.89 \begin{align*} -\frac{\frac{3 \,{\left (\pi \left \lfloor \frac{f x + e}{\pi } + \frac{1}{2} \right \rfloor \mathrm{sgn}\left (b\right ) + \arctan \left (\frac{b \tan \left (f x + e\right )}{\sqrt{a b + b^{2}}}\right )\right )} a b}{{\left (a^{2} + 2 \, a b + b^{2}\right )} \sqrt{a b + b^{2}}} + \frac{3 \, a \tan \left (f x + e\right )^{2} + a + b}{{\left (a^{2} + 2 \, a b + b^{2}\right )} \tan \left (f x + e\right )^{3}}}{3 \, f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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