Optimal. Leaf size=27 \[ -\frac {\cot (a+b x)}{b}-\frac {\cot ^3(a+b x)}{3 b} \]
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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 = {3852}
\begin {gather*} -\frac {\cot ^3(a+b x)}{3 b}-\frac {\cot (a+b x)}{b} \end {gather*}
Antiderivative was successfully verified.
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Rule 3852
Rubi steps
\begin {align*} \int \csc ^4(a+b x) \, dx &=-\frac {\text {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 = 0.02, size = 35, normalized size = 1.30 \begin {gather*} -\frac {2 \cot (a+b x)}{3 b}-\frac {\cot (a+b x) \csc ^2(a+b x)}{3 b} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.04, size = 23, normalized size = 0.85
method | result | size |
derivativedivides | \(\frac {\left (-\frac {2}{3}-\frac {\left (\csc ^{2}\left (x b +a \right )\right )}{3}\right ) \cot \left (x b +a \right )}{b}\) | \(23\) |
default | \(\frac {\left (-\frac {2}{3}-\frac {\left (\csc ^{2}\left (x b +a \right )\right )}{3}\right ) \cot \left (x b +a \right )}{b}\) | \(23\) |
risch | \(\frac {4 i \left (3 \,{\mathrm e}^{2 i \left (x b +a \right )}-1\right )}{3 b \left ({\mathrm e}^{2 i \left (x b +a \right )}-1\right )^{3}}\) | \(33\) |
norman | \(\frac {-\frac {1}{24 b}-\frac {3 \left (\tan ^{2}\left (\frac {a}{2}+\frac {x b}{2}\right )\right )}{8 b}+\frac {3 \left (\tan ^{4}\left (\frac {a}{2}+\frac {x b}{2}\right )\right )}{8 b}+\frac {\tan ^{6}\left (\frac {a}{2}+\frac {x b}{2}\right )}{24 b}}{\tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{3}}\) | \(67\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 25, normalized size = 0.93 \begin {gather*} -\frac {3 \, \tan \left (b x + a\right )^{2} + 1}{3 \, b \tan \left (b x + a\right )^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.89, size = 45, normalized size = 1.67 \begin {gather*} -\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 {gather*}
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 \begin {gather*} \int \csc ^{4}{\left (a + b x \right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.43, size = 25, normalized size = 0.93 \begin {gather*} -\frac {3 \, \tan \left (b x + a\right )^{2} + 1}{3 \, b \tan \left (b x + a\right )^{3}} \end {gather*}
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 \begin {gather*} -\frac {\mathrm {cot}\left (a+b\,x\right )\,\left ({\mathrm {cot}\left (a+b\,x\right )}^2+3\right )}{3\,b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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