Optimal. Leaf size=27 \[ -\frac{\cot ^3(a+b x)}{3 b}-\frac{\cot (a+b x)}{b} \]
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Rubi [A] time = 0.0113776, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {3767} \[ -\frac{\cot ^3(a+b x)}{3 b}-\frac{\cot (a+b x)}{b} \]
Antiderivative was successfully verified.
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Rule 3767
Rubi steps
\begin{align*} \int \csc ^4(a+b x) \, dx &=-\frac{\operatorname{Subst}\left (\int \left (1+x^2\right ) \, dx,x,\cot (a+b x)\right )}{b}\\ &=-\frac{\cot (a+b x)}{b}-\frac{\cot ^3(a+b x)}{3 b}\\ \end{align*}
Mathematica [A] time = 0.012798, size = 35, normalized size = 1.3 \[ -\frac{2 \cot (a+b x)}{3 b}-\frac{\cot (a+b x) \csc ^2(a+b x)}{3 b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.018, size = 23, normalized size = 0.9 \begin{align*}{\frac{\cot \left ( bx+a \right ) }{b} \left ( -{\frac{2}{3}}-{\frac{ \left ( \csc \left ( bx+a \right ) \right ) ^{2}}{3}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.05666, size = 34, normalized size = 1.26 \begin{align*} -\frac{3 \, \tan \left (b x + a\right )^{2} + 1}{3 \, b \tan \left (b x + a\right )^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.450437, size = 108, normalized size = 4. \begin{align*} -\frac{2 \, \cos \left (b x + a\right )^{3} - 3 \, \cos \left (b x + a\right )}{3 \,{\left (b \cos \left (b x + a\right )^{2} - b\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 \csc ^{4}{\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 = 1.2539, size = 34, normalized size = 1.26 \begin{align*} -\frac{3 \, \tan \left (b x + a\right )^{2} + 1}{3 \, b \tan \left (b x + a\right )^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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