Optimal. Leaf size=38 \[ \frac{\tan ^3(a+b x)}{3 b}+\frac{2 \tan (a+b x)}{b}-\frac{\cot (a+b x)}{b} \]
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Rubi [A] time = 0.0367211, antiderivative size = 38, 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, 270} \[ \frac{\tan ^3(a+b x)}{3 b}+\frac{2 \tan (a+b x)}{b}-\frac{\cot (a+b x)}{b} \]
Antiderivative was successfully verified.
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Rule 2620
Rule 270
Rubi steps
\begin{align*} \int \csc ^2(a+b x) \sec ^4(a+b x) \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\left (1+x^2\right )^2}{x^2} \, dx,x,\tan (a+b x)\right )}{b}\\ &=\frac{\operatorname{Subst}\left (\int \left (2+\frac{1}{x^2}+x^2\right ) \, dx,x,\tan (a+b x)\right )}{b}\\ &=-\frac{\cot (a+b x)}{b}+\frac{2 \tan (a+b x)}{b}+\frac{\tan ^3(a+b x)}{3 b}\\ \end{align*}
Mathematica [A] time = 0.0330181, size = 46, normalized size = 1.21 \[ \frac{5 \tan (a+b x)}{3 b}-\frac{\cot (a+b x)}{b}+\frac{\tan (a+b x) \sec ^2(a+b x)}{3 b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.022, size = 50, normalized size = 1.3 \begin{align*}{\frac{1}{b} \left ({\frac{1}{3\, \left ( \cos \left ( bx+a \right ) \right ) ^{3}\sin \left ( bx+a \right ) }}+{\frac{4}{3\,\cos \left ( bx+a \right ) \sin \left ( bx+a \right ) }}-{\frac{8\,\cot \left ( bx+a \right ) }{3}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.985613, size = 43, normalized size = 1.13 \begin{align*} \frac{\tan \left (b x + a\right )^{3} - \frac{3}{\tan \left (b x + a\right )} + 6 \, \tan \left (b x + a\right )}{3 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.78534, size = 108, normalized size = 2.84 \begin{align*} -\frac{8 \, \cos \left (b x + a\right )^{4} - 4 \, \cos \left (b x + a\right )^{2} - 1}{3 \, b \cos \left (b x + a\right )^{3} \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 ^{4}{\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 = 1.19946, size = 43, normalized size = 1.13 \begin{align*} \frac{\tan \left (b x + a\right )^{3} - \frac{3}{\tan \left (b x + a\right )} + 6 \, \tan \left (b x + a\right )}{3 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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