Optimal. Leaf size=42 \[ -\frac {\cot ^5(a+b x)}{5 b}-\frac {2 \cot ^3(a+b x)}{3 b}-\frac {\cot (a+b x)}{b} \]
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Rubi [A] time = 0.01, antiderivative size = 42, 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} \[ -\frac {\cot ^5(a+b x)}{5 b}-\frac {2 \cot ^3(a+b x)}{3 b}-\frac {\cot (a+b x)}{b} \]
Antiderivative was successfully verified.
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Rule 3767
Rubi steps
\begin {align*} \int \csc ^6(a+b x) \, dx &=-\frac {\operatorname {Subst}\left (\int \left (1+2 x^2+x^4\right ) \, dx,x,\cot (a+b x)\right )}{b}\\ &=-\frac {\cot (a+b x)}{b}-\frac {2 \cot ^3(a+b x)}{3 b}-\frac {\cot ^5(a+b x)}{5 b}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 56, normalized size = 1.33 \[ -\frac {8 \cot (a+b x)}{15 b}-\frac {\cot (a+b x) \csc ^4(a+b x)}{5 b}-\frac {4 \cot (a+b x) \csc ^2(a+b x)}{15 b} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.53, size = 64, normalized size = 1.52 \[ -\frac {8 \, \cos \left (b x + a\right )^{5} - 20 \, \cos \left (b x + a\right )^{3} + 15 \, \cos \left (b x + a\right )}{15 \, {\left (b \cos \left (b x + a\right )^{4} - 2 \, b \cos \left (b x + a\right )^{2} + b\right )} \sin \left (b x + a\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.19, size = 35, normalized size = 0.83 \[ -\frac {15 \, \tan \left (b x + a\right )^{4} + 10 \, \tan \left (b x + a\right )^{2} + 3}{15 \, b \tan \left (b x + a\right )^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.79, size = 33, normalized size = 0.79 \[ \frac {\left (-\frac {8}{15}-\frac {\left (\csc ^{4}\left (b x +a \right )\right )}{5}-\frac {4 \left (\csc ^{2}\left (b x +a \right )\right )}{15}\right ) \cot \left (b x +a \right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.35, size = 35, normalized size = 0.83 \[ -\frac {15 \, \tan \left (b x + a\right )^{4} + 10 \, \tan \left (b x + a\right )^{2} + 3}{15 \, b \tan \left (b x + a\right )^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.10, size = 38, normalized size = 0.90 \[ -\frac {\mathrm {cot}\left (a+b\,x\right )}{b}-\frac {2\,{\mathrm {cot}\left (a+b\,x\right )}^3}{3\,b}-\frac {{\mathrm {cot}\left (a+b\,x\right )}^5}{5\,b} \]
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 \[ \int \csc ^{6}{\left (a + b x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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