Optimal. Leaf size=51 \[ -\frac{(a+b) \cot ^5(e+f x)}{5 f}+\frac{a \cot ^3(e+f x)}{3 f}-\frac{a \cot (e+f x)}{f}-a x \]
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Rubi [A] time = 0.0628051, antiderivative size = 51, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {4141, 1802, 203} \[ -\frac{(a+b) \cot ^5(e+f x)}{5 f}+\frac{a \cot ^3(e+f x)}{3 f}-\frac{a \cot (e+f x)}{f}-a x \]
Antiderivative was successfully verified.
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Rule 4141
Rule 1802
Rule 203
Rubi steps
\begin{align*} \int \cot ^6(e+f x) \left (a+b \sec ^2(e+f x)\right ) \, dx &=\frac{\operatorname{Subst}\left (\int \frac{a+b \left (1+x^2\right )}{x^6 \left (1+x^2\right )} \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac{\operatorname{Subst}\left (\int \left (\frac{a+b}{x^6}-\frac{a}{x^4}+\frac{a}{x^2}-\frac{a}{1+x^2}\right ) \, dx,x,\tan (e+f x)\right )}{f}\\ &=-\frac{a \cot (e+f x)}{f}+\frac{a \cot ^3(e+f x)}{3 f}-\frac{(a+b) \cot ^5(e+f x)}{5 f}-\frac{a \operatorname{Subst}\left (\int \frac{1}{1+x^2} \, dx,x,\tan (e+f x)\right )}{f}\\ &=-a x-\frac{a \cot (e+f x)}{f}+\frac{a \cot ^3(e+f x)}{3 f}-\frac{(a+b) \cot ^5(e+f x)}{5 f}\\ \end{align*}
Mathematica [C] time = 0.029903, size = 51, normalized size = 1. \[ -\frac{a \cot ^5(e+f x) \text{Hypergeometric2F1}\left (-\frac{5}{2},1,-\frac{3}{2},-\tan ^2(e+f x)\right )}{5 f}-\frac{b \cot ^5(e+f x)}{5 f} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.051, size = 63, normalized size = 1.2 \begin{align*}{\frac{1}{f} \left ( a \left ( -{\frac{ \left ( \cot \left ( fx+e \right ) \right ) ^{5}}{5}}+{\frac{ \left ( \cot \left ( fx+e \right ) \right ) ^{3}}{3}}-\cot \left ( fx+e \right ) -fx-e \right ) -{\frac{b \left ( \cos \left ( fx+e \right ) \right ) ^{5}}{5\, \left ( \sin \left ( fx+e \right ) \right ) ^{5}}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.56723, size = 70, normalized size = 1.37 \begin{align*} -\frac{15 \,{\left (f x + e\right )} a + \frac{15 \, a \tan \left (f x + e\right )^{4} - 5 \, a \tan \left (f x + e\right )^{2} + 3 \, a + 3 \, b}{\tan \left (f x + e\right )^{5}}}{15 \, f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 0.491892, size = 286, normalized size = 5.61 \begin{align*} -\frac{{\left (23 \, a + 3 \, b\right )} \cos \left (f x + e\right )^{5} - 35 \, a \cos \left (f x + e\right )^{3} + 15 \, a \cos \left (f x + e\right ) + 15 \,{\left (a f x \cos \left (f x + e\right )^{4} - 2 \, a f x \cos \left (f x + e\right )^{2} + a f x\right )} \sin \left (f x + e\right )}{15 \,{\left (f \cos \left (f x + e\right )^{4} - 2 \, f \cos \left (f x + e\right )^{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(-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 = 1.46628, size = 246, normalized size = 4.82 \begin{align*} \frac{3 \, a \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{5} + 3 \, b \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{5} - 35 \, a \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{3} - 15 \, b \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{3} - 480 \,{\left (f x + e\right )} a + 330 \, a \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right ) + 30 \, b \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right ) - \frac{330 \, a \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{4} + 30 \, b \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{4} - 35 \, a \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{2} - 15 \, b \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{2} + 3 \, a + 3 \, b}{\tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{5}}}{480 \, f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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