Optimal. Leaf size=46 \[ -\frac{(a+b) \cot ^3(e+f x)}{3 f}-\frac{(a+2 b) \cot (e+f x)}{f}+\frac{b \tan (e+f x)}{f} \]
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Rubi [A] time = 0.0461213, antiderivative size = 46, 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 = {4132, 448} \[ -\frac{(a+b) \cot ^3(e+f x)}{3 f}-\frac{(a+2 b) \cot (e+f x)}{f}+\frac{b \tan (e+f x)}{f} \]
Antiderivative was successfully verified.
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Rule 4132
Rule 448
Rubi steps
\begin{align*} \int \csc ^4(e+f x) \left (a+b \sec ^2(e+f x)\right ) \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\left (1+x^2\right ) \left (a+b+b x^2\right )}{x^4} \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac{\operatorname{Subst}\left (\int \left (b+\frac{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{(a+b) \cot ^3(e+f x)}{3 f}+\frac{b \tan (e+f x)}{f}\\ \end{align*}
Mathematica [A] time = 0.0463745, size = 84, normalized size = 1.83 \[ -\frac{2 a \cot (e+f x)}{3 f}-\frac{a \cot (e+f x) \csc ^2(e+f x)}{3 f}+\frac{b \tan (e+f x)}{f}-\frac{5 b \cot (e+f x)}{3 f}-\frac{b \cot (e+f x) \csc ^2(e+f x)}{3 f} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.048, size = 73, normalized size = 1.6 \begin{align*}{\frac{1}{f} \left ( a \left ( -{\frac{2}{3}}-{\frac{ \left ( \csc \left ( fx+e \right ) \right ) ^{2}}{3}} \right ) \cot \left ( fx+e \right ) +b \left ( -{\frac{1}{3\, \left ( \sin \left ( fx+e \right ) \right ) ^{3}\cos \left ( fx+e \right ) }}+{\frac{4}{3\,\sin \left ( fx+e \right ) \cos \left ( fx+e \right ) }}-{\frac{8\,\cot \left ( fx+e \right ) }{3}} \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.00292, size = 58, normalized size = 1.26 \begin{align*} \frac{3 \, b \tan \left (f x + e\right ) - \frac{3 \,{\left (a + 2 \, b\right )} \tan \left (f x + e\right )^{2} + a + b}{\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 = 0.456512, size = 163, normalized size = 3.54 \begin{align*} -\frac{2 \,{\left (a + 4 \, b\right )} \cos \left (f x + e\right )^{4} - 3 \,{\left (a + 4 \, b\right )} \cos \left (f x + e\right )^{2} + 3 \, b}{3 \,{\left (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 = 1.29862, size = 73, normalized size = 1.59 \begin{align*} \frac{3 \, b \tan \left (f x + e\right ) - \frac{3 \, a \tan \left (f x + e\right )^{2} + 6 \, b \tan \left (f x + e\right )^{2} + a + b}{\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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