Optimal. Leaf size=26 \[ \frac{b \tan (e+f x)}{f}-\frac{(a+b) \cot (e+f x)}{f} \]
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Rubi [A] time = 0.0335316, antiderivative size = 26, 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, 14} \[ \frac{b \tan (e+f x)}{f}-\frac{(a+b) \cot (e+f x)}{f} \]
Antiderivative was successfully verified.
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Rule 4132
Rule 14
Rubi steps
\begin{align*} \int \csc ^2(e+f x) \left (a+b \sec ^2(e+f x)\right ) \, dx &=\frac{\operatorname{Subst}\left (\int \frac{a+b+b x^2}{x^2} \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac{\operatorname{Subst}\left (\int \left (b+\frac{a+b}{x^2}\right ) \, dx,x,\tan (e+f x)\right )}{f}\\ &=-\frac{(a+b) \cot (e+f x)}{f}+\frac{b \tan (e+f x)}{f}\\ \end{align*}
Mathematica [A] time = 0.0644044, size = 36, normalized size = 1.38 \[ -\frac{a \cot (e+f x)}{f}+\frac{b \tan (e+f x)}{f}-\frac{b \cot (e+f x)}{f} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.039, size = 43, normalized size = 1.7 \begin{align*}{\frac{1}{f} \left ( -\cot \left ( fx+e \right ) a+b \left ({\frac{1}{\sin \left ( fx+e \right ) \cos \left ( fx+e \right ) }}-2\,\cot \left ( fx+e \right ) \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.966755, size = 35, normalized size = 1.35 \begin{align*} \frac{b \tan \left (f x + e\right ) - \frac{a + b}{\tan \left (f x + e\right )}}{f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.454164, size = 85, normalized size = 3.27 \begin{align*} -\frac{{\left (a + 2 \, b\right )} \cos \left (f x + e\right )^{2} - b}{f \cos \left (f x + e\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] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (a + b \sec ^{2}{\left (e + f x \right )}\right ) \csc ^{2}{\left (e + f x \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.31167, size = 38, normalized size = 1.46 \begin{align*} \frac{b \tan \left (f x + e\right ) - \frac{a + b}{\tan \left (f x + e\right )}}{f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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