Optimal. Leaf size=22 \[ \frac{\tan (a+b x)}{b}-\frac{\cot (a+b x)}{b} \]
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Rubi [A] time = 0.0303198, antiderivative size = 22, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {2620, 14} \[ \frac{\tan (a+b x)}{b}-\frac{\cot (a+b x)}{b} \]
Antiderivative was successfully verified.
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Rule 2620
Rule 14
Rubi steps
\begin{align*} \int \csc ^2(a+b x) \sec ^2(a+b x) \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1+x^2}{x^2} \, dx,x,\tan (a+b x)\right )}{b}\\ &=\frac{\operatorname{Subst}\left (\int \left (1+\frac{1}{x^2}\right ) \, dx,x,\tan (a+b x)\right )}{b}\\ &=-\frac{\cot (a+b x)}{b}+\frac{\tan (a+b x)}{b}\\ \end{align*}
Mathematica [A] time = 0.0166702, size = 13, normalized size = 0.59 \[ -\frac{2 \cot (2 (a+b x))}{b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.044, size = 31, normalized size = 1.4 \begin{align*}{\frac{1}{b} \left ({\frac{1}{\cos \left ( bx+a \right ) \sin \left ( bx+a \right ) }}-2\,\cot \left ( bx+a \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.986127, size = 30, normalized size = 1.36 \begin{align*} -\frac{\frac{1}{\tan \left (b x + a\right )} - \tan \left (b x + a\right )}{b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.79507, size = 74, normalized size = 3.36 \begin{align*} -\frac{2 \, \cos \left (b x + a\right )^{2} - 1}{b \cos \left (b x + a\right ) \sin \left (b x + a\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 \frac{\sec ^{2}{\left (a + b x \right )}}{\sin ^{2}{\left (a + b x \right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.14322, size = 22, normalized size = 1. \begin{align*} -\frac{2}{b \tan \left (2 \, b x + 2 \, a\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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