Optimal. Leaf size=64 \[ -\frac{(2 a+b) \cot ^3(e+f x)}{3 f}-\frac{(a+2 b) \cot (e+f x)}{f}-\frac{a \cot ^5(e+f x)}{5 f}+\frac{b \tan (e+f x)}{f} \]
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Rubi [A] time = 0.0528497, antiderivative size = 64, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095, Rules used = {3663, 448} \[ -\frac{(2 a+b) \cot ^3(e+f x)}{3 f}-\frac{(a+2 b) \cot (e+f x)}{f}-\frac{a \cot ^5(e+f x)}{5 f}+\frac{b \tan (e+f x)}{f} \]
Antiderivative was successfully verified.
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Rule 3663
Rule 448
Rubi steps
\begin{align*} \int \csc ^6(e+f x) \left (a+b \tan ^2(e+f x)\right ) \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\left (1+x^2\right )^2 \left (a+b x^2\right )}{x^6} \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac{\operatorname{Subst}\left (\int \left (b+\frac{a}{x^6}+\frac{2 a+b}{x^4}+\frac{a+2 b}{x^2}\right ) \, dx,x,\tan (e+f x)\right )}{f}\\ &=-\frac{(a+2 b) \cot (e+f x)}{f}-\frac{(2 a+b) \cot ^3(e+f x)}{3 f}-\frac{a \cot ^5(e+f x)}{5 f}+\frac{b \tan (e+f x)}{f}\\ \end{align*}
Mathematica [A] time = 0.0491428, size = 106, normalized size = 1.66 \[ -\frac{8 a \cot (e+f x)}{15 f}-\frac{a \cot (e+f x) \csc ^4(e+f x)}{5 f}-\frac{4 a \cot (e+f x) \csc ^2(e+f x)}{15 f}+\frac{b \tan (e+f x)}{f}-\frac{5 b \cot (e+f x)}{3 f}-\frac{b \cot (e+f x) \csc ^2(e+f x)}{3 f} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.072, size = 83, normalized size = 1.3 \begin{align*}{\frac{1}{f} \left ( b \left ( -{\frac{1}{3\, \left ( \sin \left ( fx+e \right ) \right ) ^{3}\cos \left ( fx+e \right ) }}+{\frac{4}{3\,\cos \left ( fx+e \right ) \sin \left ( fx+e \right ) }}-{\frac{8\,\cot \left ( fx+e \right ) }{3}} \right ) +a \left ( -{\frac{8}{15}}-{\frac{ \left ( \csc \left ( fx+e \right ) \right ) ^{4}}{5}}-{\frac{4\, \left ( \csc \left ( fx+e \right ) \right ) ^{2}}{15}} \right ) \cot \left ( fx+e \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.11731, size = 80, normalized size = 1.25 \begin{align*} \frac{15 \, b \tan \left (f x + e\right ) - \frac{15 \,{\left (a + 2 \, b\right )} \tan \left (f x + e\right )^{4} + 5 \,{\left (2 \, a + b\right )} \tan \left (f x + e\right )^{2} + 3 \, a}{\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 [A] time = 1.87629, size = 236, normalized size = 3.69 \begin{align*} -\frac{8 \,{\left (a + 5 \, b\right )} \cos \left (f x + e\right )^{6} - 20 \,{\left (a + 5 \, b\right )} \cos \left (f x + e\right )^{4} + 15 \,{\left (a + 5 \, b\right )} \cos \left (f x + e\right )^{2} - 15 \, b}{15 \,{\left (f \cos \left (f x + e\right )^{5} - 2 \, f \cos \left (f x + e\right )^{3} + f \cos \left (f x + e\right )\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 [A] time = 1.47163, size = 107, normalized size = 1.67 \begin{align*} \frac{15 \, b \tan \left (f x + e\right ) - \frac{15 \, a \tan \left (f x + e\right )^{4} + 30 \, b \tan \left (f x + e\right )^{4} + 10 \, a \tan \left (f x + e\right )^{2} + 5 \, b \tan \left (f x + e\right )^{2} + 3 \, a}{\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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