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 = 0.01, antiderivative size = 27, normalized size of antiderivative = 1.00, 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*}
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Mathematica [A] time = 0.01, size = 35, normalized size = 1.30 \[ -\frac {2 \cot (a+b x)}{3 b}-\frac {\cot (a+b x) \csc ^2(a+b x)}{3 b} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.69, size = 45, normalized size = 1.67 \[ -\frac {2 \, \cos \left (b x + a\right )^{3} - 3 \, \cos \left (b x + a\right )}{3 \, {\left (b \cos \left (b x + a\right )^{2} - b\right )} \sin \left (b x + a\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 25, normalized size = 0.93 \[ -\frac {3 \, \tan \left (b x + a\right )^{2} + 1}{3 \, b \tan \left (b x + a\right )^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.87, size = 23, normalized size = 0.85 \[ \frac {\left (-\frac {2}{3}-\frac {\left (\csc ^{2}\left (b x +a \right )\right )}{3}\right ) \cot \left (b x +a \right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.32, size = 25, normalized size = 0.93 \[ -\frac {3 \, \tan \left (b x + a\right )^{2} + 1}{3 \, b \tan \left (b x + a\right )^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.08, size = 21, normalized size = 0.78 \[ -\frac {\mathrm {cot}\left (a+b\,x\right )\,\left ({\mathrm {cot}\left (a+b\,x\right )}^2+3\right )}{3\,b} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \csc ^{4}{\left (a + b x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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