Optimal. Leaf size=44 \[ -\frac{a^2 \cot ^3(e+f x)}{3 f}+\frac{a (a-2 b) \cot (e+f x)}{f}+x (a-b)^2 \]
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Rubi [A] time = 0.0707845, antiderivative size = 44, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.13, Rules used = {3670, 461, 203} \[ -\frac{a^2 \cot ^3(e+f x)}{3 f}+\frac{a (a-2 b) \cot (e+f x)}{f}+x (a-b)^2 \]
Antiderivative was successfully verified.
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Rule 3670
Rule 461
Rule 203
Rubi steps
\begin{align*} \int \cot ^4(e+f x) \left (a+b \tan ^2(e+f x)\right )^2 \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\left (a+b x^2\right )^2}{x^4 \left (1+x^2\right )} \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac{\operatorname{Subst}\left (\int \left (\frac{a^2}{x^4}-\frac{a (a-2 b)}{x^2}+\frac{(a-b)^2}{1+x^2}\right ) \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac{a (a-2 b) \cot (e+f x)}{f}-\frac{a^2 \cot ^3(e+f x)}{3 f}+\frac{(a-b)^2 \operatorname{Subst}\left (\int \frac{1}{1+x^2} \, dx,x,\tan (e+f x)\right )}{f}\\ &=(a-b)^2 x+\frac{a (a-2 b) \cot (e+f x)}{f}-\frac{a^2 \cot ^3(e+f x)}{3 f}\\ \end{align*}
Mathematica [A] time = 1.2306, size = 71, normalized size = 1.61 \[ -\frac{\cot (e+f x) \left (a \left (a \cot ^2(e+f x)-3 a+6 b\right )+3 (a-b)^2 \sqrt{-\tan ^2(e+f x)} \tanh ^{-1}\left (\sqrt{-\tan ^2(e+f x)}\right )\right )}{3 f} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.047, size = 60, normalized size = 1.4 \begin{align*}{\frac{1}{f} \left ({b}^{2} \left ( fx+e \right ) +2\,ab \left ( -\cot \left ( fx+e \right ) -fx-e \right ) +{a}^{2} \left ( -{\frac{ \left ( \cot \left ( fx+e \right ) \right ) ^{3}}{3}}+\cot \left ( fx+e \right ) +fx+e \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.61901, size = 77, normalized size = 1.75 \begin{align*} \frac{3 \,{\left (a^{2} - 2 \, a b + b^{2}\right )}{\left (f x + e\right )} + \frac{3 \,{\left (a^{2} - 2 \, a b\right )} \tan \left (f x + e\right )^{2} - a^{2}}{\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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Fricas [A] time = 1.07965, size = 143, normalized size = 3.25 \begin{align*} \frac{3 \,{\left (a^{2} - 2 \, a b + b^{2}\right )} f x \tan \left (f x + e\right )^{3} + 3 \,{\left (a^{2} - 2 \, a b\right )} \tan \left (f x + e\right )^{2} - a^{2}}{3 \, f \tan \left (f x + e\right )^{3}} \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 [B] time = 1.74893, size = 165, normalized size = 3.75 \begin{align*} \frac{a^{2} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{3} - 15 \, a^{2} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right ) + 24 \, a b \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right ) + 24 \,{\left (a^{2} - 2 \, a b + b^{2}\right )}{\left (f x + e\right )} + \frac{15 \, a^{2} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{2} - 24 \, a b \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{2} - a^{2}}{\tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{3}}}{24 \, f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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