Optimal. Leaf size=244 \[ \frac {\text {ArcTan}\left (\frac {\sqrt [3]{c \cot (a+b x)}}{\sqrt [3]{c}}\right )}{b c^{4/3}}-\frac {\text {ArcTan}\left (\sqrt {3}-\frac {2 \sqrt [3]{c \cot (a+b x)}}{\sqrt [3]{c}}\right )}{2 b c^{4/3}}+\frac {\text {ArcTan}\left (\sqrt {3}+\frac {2 \sqrt [3]{c \cot (a+b x)}}{\sqrt [3]{c}}\right )}{2 b c^{4/3}}+\frac {3}{b c \sqrt [3]{c \cot (a+b x)}}+\frac {\sqrt {3} \log \left (c^{2/3}-\sqrt {3} \sqrt [3]{c} \sqrt [3]{c \cot (a+b x)}+(c \cot (a+b x))^{2/3}\right )}{4 b c^{4/3}}-\frac {\sqrt {3} \log \left (c^{2/3}+\sqrt {3} \sqrt [3]{c} \sqrt [3]{c \cot (a+b x)}+(c \cot (a+b x))^{2/3}\right )}{4 b c^{4/3}} \]
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Rubi [A]
time = 0.29, antiderivative size = 244, normalized size of antiderivative = 1.00, number of steps
used = 13, number of rules used = 9, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.750, Rules used = {3555, 3557,
335, 301, 648, 632, 210, 642, 209} \begin {gather*} \frac {\text {ArcTan}\left (\frac {\sqrt [3]{c \cot (a+b x)}}{\sqrt [3]{c}}\right )}{b c^{4/3}}-\frac {\text {ArcTan}\left (\sqrt {3}-\frac {2 \sqrt [3]{c \cot (a+b x)}}{\sqrt [3]{c}}\right )}{2 b c^{4/3}}+\frac {\text {ArcTan}\left (\frac {2 \sqrt [3]{c \cot (a+b x)}}{\sqrt [3]{c}}+\sqrt {3}\right )}{2 b c^{4/3}}+\frac {\sqrt {3} \log \left (-\sqrt {3} \sqrt [3]{c} \sqrt [3]{c \cot (a+b x)}+(c \cot (a+b x))^{2/3}+c^{2/3}\right )}{4 b c^{4/3}}-\frac {\sqrt {3} \log \left (\sqrt {3} \sqrt [3]{c} \sqrt [3]{c \cot (a+b x)}+(c \cot (a+b x))^{2/3}+c^{2/3}\right )}{4 b c^{4/3}}+\frac {3}{b c \sqrt [3]{c \cot (a+b x)}} \end {gather*}
Antiderivative was successfully verified.
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Rule 209
Rule 210
Rule 301
Rule 335
Rule 632
Rule 642
Rule 648
Rule 3555
Rule 3557
Rubi steps
\begin {align*} \int \frac {1}{(c \cot (a+b x))^{4/3}} \, dx &=\frac {3}{b c \sqrt [3]{c \cot (a+b x)}}-\frac {\int (c \cot (a+b x))^{2/3} \, dx}{c^2}\\ &=\frac {3}{b c \sqrt [3]{c \cot (a+b x)}}+\frac {\text {Subst}\left (\int \frac {x^{2/3}}{c^2+x^2} \, dx,x,c \cot (a+b x)\right )}{b c}\\ &=\frac {3}{b c \sqrt [3]{c \cot (a+b x)}}+\frac {3 \text {Subst}\left (\int \frac {x^4}{c^2+x^6} \, dx,x,\sqrt [3]{c \cot (a+b x)}\right )}{b c}\\ &=\frac {3}{b c \sqrt [3]{c \cot (a+b x)}}+\frac {\text {Subst}\left (\int \frac {-\frac {\sqrt [3]{c}}{2}+\frac {\sqrt {3} x}{2}}{c^{2/3}-\sqrt {3} \sqrt [3]{c} x+x^2} \, dx,x,\sqrt [3]{c \cot (a+b x)}\right )}{b c^{4/3}}+\frac {\text {Subst}\left (\int \frac {-\frac {\sqrt [3]{c}}{2}-\frac {\sqrt {3} x}{2}}{c^{2/3}+\sqrt {3} \sqrt [3]{c} x+x^2} \, dx,x,\sqrt [3]{c \cot (a+b x)}\right )}{b c^{4/3}}+\frac {\text {Subst}\left (\int \frac {1}{c^{2/3}+x^2} \, dx,x,\sqrt [3]{c \cot (a+b x)}\right )}{b c}\\ &=\frac {\tan ^{-1}\left (\frac {\sqrt [3]{c \cot (a+b x)}}{\sqrt [3]{c}}\right )}{b c^{4/3}}+\frac {3}{b c \sqrt [3]{c \cot (a+b x)}}+\frac {\sqrt {3} \text {Subst}\left (\int \frac {-\sqrt {3} \sqrt [3]{c}+2 x}{c^{2/3}-\sqrt {3} \sqrt [3]{c} x+x^2} \, dx,x,\sqrt [3]{c \cot (a+b x)}\right )}{4 b c^{4/3}}-\frac {\sqrt {3} \text {Subst}\left (\int \frac {\sqrt {3} \sqrt [3]{c}+2 x}{c^{2/3}+\sqrt {3} \sqrt [3]{c} x+x^2} \, dx,x,\sqrt [3]{c \cot (a+b x)}\right )}{4 b c^{4/3}}+\frac {\text {Subst}\left (\int \frac {1}{c^{2/3}-\sqrt {3} \sqrt [3]{c} x+x^2} \, dx,x,\sqrt [3]{c \cot (a+b x)}\right )}{4 b c}+\frac {\text {Subst}\left (\int \frac {1}{c^{2/3}+\sqrt {3} \sqrt [3]{c} x+x^2} \, dx,x,\sqrt [3]{c \cot (a+b x)}\right )}{4 b c}\\ &=\frac {\tan ^{-1}\left (\frac {\sqrt [3]{c \cot (a+b x)}}{\sqrt [3]{c}}\right )}{b c^{4/3}}+\frac {3}{b c \sqrt [3]{c \cot (a+b x)}}+\frac {\sqrt {3} \log \left (c^{2/3}-\sqrt {3} \sqrt [3]{c} \sqrt [3]{c \cot (a+b x)}+(c \cot (a+b x))^{2/3}\right )}{4 b c^{4/3}}-\frac {\sqrt {3} \log \left (c^{2/3}+\sqrt {3} \sqrt [3]{c} \sqrt [3]{c \cot (a+b x)}+(c \cot (a+b x))^{2/3}\right )}{4 b c^{4/3}}+\frac {\text {Subst}\left (\int \frac {1}{-\frac {1}{3}-x^2} \, dx,x,1-\frac {2 \sqrt [3]{c \cot (a+b x)}}{\sqrt {3} \sqrt [3]{c}}\right )}{2 \sqrt {3} b c^{4/3}}-\frac {\text {Subst}\left (\int \frac {1}{-\frac {1}{3}-x^2} \, dx,x,1+\frac {2 \sqrt [3]{c \cot (a+b x)}}{\sqrt {3} \sqrt [3]{c}}\right )}{2 \sqrt {3} b c^{4/3}}\\ &=\frac {\tan ^{-1}\left (\frac {\sqrt [3]{c \cot (a+b x)}}{\sqrt [3]{c}}\right )}{b c^{4/3}}-\frac {\tan ^{-1}\left (\frac {1}{3} \left (3 \sqrt {3}-\frac {6 \sqrt [3]{c \cot (a+b x)}}{\sqrt [3]{c}}\right )\right )}{2 b c^{4/3}}+\frac {\tan ^{-1}\left (\frac {1}{3} \left (3 \sqrt {3}+\frac {6 \sqrt [3]{c \cot (a+b x)}}{\sqrt [3]{c}}\right )\right )}{2 b c^{4/3}}+\frac {3}{b c \sqrt [3]{c \cot (a+b x)}}+\frac {\sqrt {3} \log \left (c^{2/3}-\sqrt {3} \sqrt [3]{c} \sqrt [3]{c \cot (a+b x)}+(c \cot (a+b x))^{2/3}\right )}{4 b c^{4/3}}-\frac {\sqrt {3} \log \left (c^{2/3}+\sqrt {3} \sqrt [3]{c} \sqrt [3]{c \cot (a+b x)}+(c \cot (a+b x))^{2/3}\right )}{4 b c^{4/3}}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 3 in
optimal.
time = 0.06, size = 38, normalized size = 0.16 \begin {gather*} \frac {3 \, _2F_1\left (-\frac {1}{6},1;\frac {5}{6};-\cot ^2(a+b x)\right )}{b c \sqrt [3]{c \cot (a+b x)}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.33, size = 212, normalized size = 0.87
method | result | size |
derivativedivides | \(-\frac {3 c \left (-\frac {1}{c^{2} \left (c \cot \left (b x +a \right )\right )^{\frac {1}{3}}}-\frac {\frac {\sqrt {3}\, \left (c^{2}\right )^{\frac {5}{6}} \ln \left (\left (c \cot \left (b x +a \right )\right )^{\frac {2}{3}}-\sqrt {3}\, \left (c^{2}\right )^{\frac {1}{6}} \left (c \cot \left (b x +a \right )\right )^{\frac {1}{3}}+\left (c^{2}\right )^{\frac {1}{3}}\right )}{12 c^{2}}+\frac {\arctan \left (\frac {2 \left (c \cot \left (b x +a \right )\right )^{\frac {1}{3}}}{\left (c^{2}\right )^{\frac {1}{6}}}-\sqrt {3}\right )}{6 \left (c^{2}\right )^{\frac {1}{6}}}+\frac {\arctan \left (\frac {\left (c \cot \left (b x +a \right )\right )^{\frac {1}{3}}}{\left (c^{2}\right )^{\frac {1}{6}}}\right )}{3 \left (c^{2}\right )^{\frac {1}{6}}}-\frac {\sqrt {3}\, \left (c^{2}\right )^{\frac {5}{6}} \ln \left (\left (c \cot \left (b x +a \right )\right )^{\frac {2}{3}}+\sqrt {3}\, \left (c^{2}\right )^{\frac {1}{6}} \left (c \cot \left (b x +a \right )\right )^{\frac {1}{3}}+\left (c^{2}\right )^{\frac {1}{3}}\right )}{12 c^{2}}+\frac {\arctan \left (\frac {2 \left (c \cot \left (b x +a \right )\right )^{\frac {1}{3}}}{\left (c^{2}\right )^{\frac {1}{6}}}+\sqrt {3}\right )}{6 \left (c^{2}\right )^{\frac {1}{6}}}}{c^{2}}\right )}{b}\) | \(212\) |
default | \(-\frac {3 c \left (-\frac {1}{c^{2} \left (c \cot \left (b x +a \right )\right )^{\frac {1}{3}}}-\frac {\frac {\sqrt {3}\, \left (c^{2}\right )^{\frac {5}{6}} \ln \left (\left (c \cot \left (b x +a \right )\right )^{\frac {2}{3}}-\sqrt {3}\, \left (c^{2}\right )^{\frac {1}{6}} \left (c \cot \left (b x +a \right )\right )^{\frac {1}{3}}+\left (c^{2}\right )^{\frac {1}{3}}\right )}{12 c^{2}}+\frac {\arctan \left (\frac {2 \left (c \cot \left (b x +a \right )\right )^{\frac {1}{3}}}{\left (c^{2}\right )^{\frac {1}{6}}}-\sqrt {3}\right )}{6 \left (c^{2}\right )^{\frac {1}{6}}}+\frac {\arctan \left (\frac {\left (c \cot \left (b x +a \right )\right )^{\frac {1}{3}}}{\left (c^{2}\right )^{\frac {1}{6}}}\right )}{3 \left (c^{2}\right )^{\frac {1}{6}}}-\frac {\sqrt {3}\, \left (c^{2}\right )^{\frac {5}{6}} \ln \left (\left (c \cot \left (b x +a \right )\right )^{\frac {2}{3}}+\sqrt {3}\, \left (c^{2}\right )^{\frac {1}{6}} \left (c \cot \left (b x +a \right )\right )^{\frac {1}{3}}+\left (c^{2}\right )^{\frac {1}{3}}\right )}{12 c^{2}}+\frac {\arctan \left (\frac {2 \left (c \cot \left (b x +a \right )\right )^{\frac {1}{3}}}{\left (c^{2}\right )^{\frac {1}{6}}}+\sqrt {3}\right )}{6 \left (c^{2}\right )^{\frac {1}{6}}}}{c^{2}}\right )}{b}\) | \(212\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.49, size = 204, normalized size = 0.84 \begin {gather*} -\frac {c {\left (\frac {\frac {\sqrt {3} \log \left (\sqrt {3} c^{\frac {1}{3}} \left (\frac {c}{\tan \left (b x + a\right )}\right )^{\frac {1}{3}} + c^{\frac {2}{3}} + \left (\frac {c}{\tan \left (b x + a\right )}\right )^{\frac {2}{3}}\right )}{c^{\frac {1}{3}}} - \frac {\sqrt {3} \log \left (-\sqrt {3} c^{\frac {1}{3}} \left (\frac {c}{\tan \left (b x + a\right )}\right )^{\frac {1}{3}} + c^{\frac {2}{3}} + \left (\frac {c}{\tan \left (b x + a\right )}\right )^{\frac {2}{3}}\right )}{c^{\frac {1}{3}}} - \frac {2 \, \arctan \left (\frac {\sqrt {3} c^{\frac {1}{3}} + 2 \, \left (\frac {c}{\tan \left (b x + a\right )}\right )^{\frac {1}{3}}}{c^{\frac {1}{3}}}\right )}{c^{\frac {1}{3}}} - \frac {2 \, \arctan \left (-\frac {\sqrt {3} c^{\frac {1}{3}} - 2 \, \left (\frac {c}{\tan \left (b x + a\right )}\right )^{\frac {1}{3}}}{c^{\frac {1}{3}}}\right )}{c^{\frac {1}{3}}} - \frac {4 \, \arctan \left (\frac {\left (\frac {c}{\tan \left (b x + a\right )}\right )^{\frac {1}{3}}}{c^{\frac {1}{3}}}\right )}{c^{\frac {1}{3}}}}{c^{2}} - \frac {12}{c^{2} \left (\frac {c}{\tan \left (b x + a\right )}\right )^{\frac {1}{3}}}\right )}}{4 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \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 \frac {1}{\left (c \cot {\left (a + b x \right )}\right )^{\frac {4}{3}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.43, size = 277, normalized size = 1.14 \begin {gather*} \frac {3}{b\,c\,{\left (c\,\mathrm {cot}\left (a+b\,x\right )\right )}^{1/3}}+\frac {{\left (-1\right )}^{1/6}\,\mathrm {atan}\left (\frac {{\left (-1\right )}^{2/3}\,{\left (c\,\mathrm {cot}\left (a+b\,x\right )\right )}^{1/3}}{c^{1/3}}\right )\,1{}\mathrm {i}}{b\,c^{4/3}}-\frac {{\left (-1\right )}^{1/6}\,\ln \left (972\,b^6\,c^{12}+972\,{\left (-1\right )}^{1/6}\,b^6\,c^{35/3}\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,{\left (c\,\mathrm {cot}\left (a+b\,x\right )\right )}^{1/3}\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{2\,b\,c^{4/3}}-\frac {{\left (-1\right )}^{1/6}\,\ln \left (972\,b^6\,c^{12}+972\,{\left (-1\right )}^{1/6}\,b^6\,c^{35/3}\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,{\left (c\,\mathrm {cot}\left (a+b\,x\right )\right )}^{1/3}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{2\,b\,c^{4/3}}+\frac {{\left (-1\right )}^{1/6}\,\ln \left (972\,b^6\,c^{12}-1944\,{\left (-1\right )}^{1/6}\,b^6\,c^{35/3}\,\left (-\frac {1}{4}+\frac {\sqrt {3}\,1{}\mathrm {i}}{4}\right )\,{\left (c\,\mathrm {cot}\left (a+b\,x\right )\right )}^{1/3}\right )\,\left (-\frac {1}{4}+\frac {\sqrt {3}\,1{}\mathrm {i}}{4}\right )}{b\,c^{4/3}}+\frac {{\left (-1\right )}^{1/6}\,\ln \left (972\,b^6\,c^{12}-1944\,{\left (-1\right )}^{1/6}\,b^6\,c^{35/3}\,\left (\frac {1}{4}+\frac {\sqrt {3}\,1{}\mathrm {i}}{4}\right )\,{\left (c\,\mathrm {cot}\left (a+b\,x\right )\right )}^{1/3}\right )\,\left (\frac {1}{4}+\frac {\sqrt {3}\,1{}\mathrm {i}}{4}\right )}{b\,c^{4/3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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