Optimal. Leaf size=28 \[ -\frac {\cot ^2(a+b x)}{2 b}-\frac {\log (\sin (a+b x))}{b} \]
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Rubi [A]
time = 0.01, antiderivative size = 28, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3554, 3556}
\begin {gather*} -\frac {\cot ^2(a+b x)}{2 b}-\frac {\log (\sin (a+b x))}{b} \end {gather*}
Antiderivative was successfully verified.
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Rule 3554
Rule 3556
Rubi steps
\begin {align*} \int \cot ^3(a+b x) \, dx &=-\frac {\cot ^2(a+b x)}{2 b}-\int \cot (a+b x) \, dx\\ &=-\frac {\cot ^2(a+b x)}{2 b}-\frac {\log (\sin (a+b x))}{b}\\ \end {align*}
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Mathematica [A]
time = 0.09, size = 34, normalized size = 1.21 \begin {gather*} -\frac {\cot ^2(a+b x)+2 \log (\cos (a+b x))+2 \log (\tan (a+b x))}{2 b} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.06, size = 29, normalized size = 1.04
method | result | size |
derivativedivides | \(\frac {-\frac {\left (\cot ^{2}\left (b x +a \right )\right )}{2}+\frac {\ln \left (\cot ^{2}\left (b x +a \right )+1\right )}{2}}{b}\) | \(29\) |
default | \(\frac {-\frac {\left (\cot ^{2}\left (b x +a \right )\right )}{2}+\frac {\ln \left (\cot ^{2}\left (b x +a \right )+1\right )}{2}}{b}\) | \(29\) |
norman | \(-\frac {1}{2 b \tan \left (b x +a \right )^{2}}-\frac {\ln \left (\tan \left (b x +a \right )\right )}{b}+\frac {\ln \left (1+\tan ^{2}\left (b x +a \right )\right )}{2 b}\) | \(43\) |
risch | \(i x +\frac {2 i a}{b}+\frac {2 \,{\mathrm e}^{2 i \left (b x +a \right )}}{b \left ({\mathrm e}^{2 i \left (b x +a \right )}-1\right )^{2}}-\frac {\ln \left ({\mathrm e}^{2 i \left (b x +a \right )}-1\right )}{b}\) | \(57\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 23, normalized size = 0.82 \begin {gather*} -\frac {\frac {1}{\sin \left (b x + a\right )^{2}} + \log \left (\sin \left (b x + a\right )^{2}\right )}{2 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 3.01, size = 47, normalized size = 1.68 \begin {gather*} -\frac {{\left (\cos \left (2 \, b x + 2 \, a\right ) - 1\right )} \log \left (-\frac {1}{2} \, \cos \left (2 \, b x + 2 \, a\right ) + \frac {1}{2}\right ) - 2}{2 \, {\left (b \cos \left (2 \, b x + 2 \, a\right ) - b\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 58 vs.
\(2 (22) = 44\).
time = 0.21, size = 58, normalized size = 2.07 \begin {gather*} \begin {cases} \tilde {\infty } x & \text {for}\: \left (a = 0 \vee a = - b x\right ) \wedge \left (a = - b x \vee b = 0\right ) \\x \cot ^{3}{\left (a \right )} & \text {for}\: b = 0 \\\frac {\log {\left (\tan ^{2}{\left (a + b x \right )} + 1 \right )}}{2 b} - \frac {\log {\left (\tan {\left (a + b x \right )} \right )}}{b} - \frac {1}{2 b \tan ^{2}{\left (a + b x \right )}} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 118 vs.
\(2 (26) = 52\).
time = 0.45, size = 118, normalized size = 4.21 \begin {gather*} \frac {\frac {{\left (\frac {4 \, {\left (\cos \left (b x + a\right ) - 1\right )}}{\cos \left (b x + a\right ) + 1} + 1\right )} {\left (\cos \left (b x + a\right ) + 1\right )}}{\cos \left (b x + a\right ) - 1} + \frac {\cos \left (b x + a\right ) - 1}{\cos \left (b x + a\right ) + 1} - 4 \, \log \left (\frac {{\left | -\cos \left (b x + a\right ) + 1 \right |}}{{\left | \cos \left (b x + a\right ) + 1 \right |}}\right ) + 8 \, \log \left ({\left | -\frac {\cos \left (b x + a\right ) - 1}{\cos \left (b x + a\right ) + 1} + 1 \right |}\right )}{8 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 3.03, size = 78, normalized size = 2.79 \begin {gather*} x\,1{}\mathrm {i}-\frac {\ln \left ({\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\mathrm {e}}^{b\,x\,2{}\mathrm {i}}-1\right )}{b}+\frac {2}{b\,\left ({\mathrm {e}}^{a\,2{}\mathrm {i}+b\,x\,2{}\mathrm {i}}-1\right )}+\frac {2}{b\,\left (1+{\mathrm {e}}^{a\,4{}\mathrm {i}+b\,x\,4{}\mathrm {i}}-2\,{\mathrm {e}}^{a\,2{}\mathrm {i}+b\,x\,2{}\mathrm {i}}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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