Integrand size = 12, antiderivative size = 245 \[ \int \frac {1}{(b \tan (c+d x))^{4/3}} \, dx=-\frac {\arctan \left (\frac {\sqrt [3]{b \tan (c+d x)}}{\sqrt [3]{b}}\right )}{b^{4/3} d}+\frac {\arctan \left (\sqrt {3}-\frac {2 \sqrt [3]{b \tan (c+d x)}}{\sqrt [3]{b}}\right )}{2 b^{4/3} d}-\frac {\arctan \left (\sqrt {3}+\frac {2 \sqrt [3]{b \tan (c+d x)}}{\sqrt [3]{b}}\right )}{2 b^{4/3} d}-\frac {\sqrt {3} \log \left (b^{2/3}-\sqrt {3} \sqrt [3]{b} \sqrt [3]{b \tan (c+d x)}+(b \tan (c+d x))^{2/3}\right )}{4 b^{4/3} d}+\frac {\sqrt {3} \log \left (b^{2/3}+\sqrt {3} \sqrt [3]{b} \sqrt [3]{b \tan (c+d x)}+(b \tan (c+d x))^{2/3}\right )}{4 b^{4/3} d}-\frac {3}{b d \sqrt [3]{b \tan (c+d x)}} \]
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Time = 0.56 (sec) , antiderivative size = 245, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.750, Rules used = {3555, 3557, 335, 301, 648, 632, 210, 642, 209} \[ \int \frac {1}{(b \tan (c+d x))^{4/3}} \, dx=-\frac {\arctan \left (\frac {\sqrt [3]{b \tan (c+d x)}}{\sqrt [3]{b}}\right )}{b^{4/3} d}+\frac {\arctan \left (\sqrt {3}-\frac {2 \sqrt [3]{b \tan (c+d x)}}{\sqrt [3]{b}}\right )}{2 b^{4/3} d}-\frac {\arctan \left (\frac {2 \sqrt [3]{b \tan (c+d x)}}{\sqrt [3]{b}}+\sqrt {3}\right )}{2 b^{4/3} d}-\frac {\sqrt {3} \log \left (b^{2/3}-\sqrt {3} \sqrt [3]{b} \sqrt [3]{b \tan (c+d x)}+(b \tan (c+d x))^{2/3}\right )}{4 b^{4/3} d}+\frac {\sqrt {3} \log \left (b^{2/3}+\sqrt {3} \sqrt [3]{b} \sqrt [3]{b \tan (c+d x)}+(b \tan (c+d x))^{2/3}\right )}{4 b^{4/3} d}-\frac {3}{b d \sqrt [3]{b \tan (c+d x)}} \]
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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*} \text {integral}& = -\frac {3}{b d \sqrt [3]{b \tan (c+d x)}}-\frac {\int (b \tan (c+d x))^{2/3} \, dx}{b^2} \\ & = -\frac {3}{b d \sqrt [3]{b \tan (c+d x)}}-\frac {\text {Subst}\left (\int \frac {x^{2/3}}{b^2+x^2} \, dx,x,b \tan (c+d x)\right )}{b d} \\ & = -\frac {3}{b d \sqrt [3]{b \tan (c+d x)}}-\frac {3 \text {Subst}\left (\int \frac {x^4}{b^2+x^6} \, dx,x,\sqrt [3]{b \tan (c+d x)}\right )}{b d} \\ & = -\frac {3}{b d \sqrt [3]{b \tan (c+d x)}}-\frac {\text {Subst}\left (\int \frac {-\frac {\sqrt [3]{b}}{2}+\frac {\sqrt {3} x}{2}}{b^{2/3}-\sqrt {3} \sqrt [3]{b} x+x^2} \, dx,x,\sqrt [3]{b \tan (c+d x)}\right )}{b^{4/3} d}-\frac {\text {Subst}\left (\int \frac {-\frac {\sqrt [3]{b}}{2}-\frac {\sqrt {3} x}{2}}{b^{2/3}+\sqrt {3} \sqrt [3]{b} x+x^2} \, dx,x,\sqrt [3]{b \tan (c+d x)}\right )}{b^{4/3} d}-\frac {\text {Subst}\left (\int \frac {1}{b^{2/3}+x^2} \, dx,x,\sqrt [3]{b \tan (c+d x)}\right )}{b d} \\ & = -\frac {\arctan \left (\frac {\sqrt [3]{b \tan (c+d x)}}{\sqrt [3]{b}}\right )}{b^{4/3} d}-\frac {3}{b d \sqrt [3]{b \tan (c+d x)}}-\frac {\sqrt {3} \text {Subst}\left (\int \frac {-\sqrt {3} \sqrt [3]{b}+2 x}{b^{2/3}-\sqrt {3} \sqrt [3]{b} x+x^2} \, dx,x,\sqrt [3]{b \tan (c+d x)}\right )}{4 b^{4/3} d}+\frac {\sqrt {3} \text {Subst}\left (\int \frac {\sqrt {3} \sqrt [3]{b}+2 x}{b^{2/3}+\sqrt {3} \sqrt [3]{b} x+x^2} \, dx,x,\sqrt [3]{b \tan (c+d x)}\right )}{4 b^{4/3} d}-\frac {\text {Subst}\left (\int \frac {1}{b^{2/3}-\sqrt {3} \sqrt [3]{b} x+x^2} \, dx,x,\sqrt [3]{b \tan (c+d x)}\right )}{4 b d}-\frac {\text {Subst}\left (\int \frac {1}{b^{2/3}+\sqrt {3} \sqrt [3]{b} x+x^2} \, dx,x,\sqrt [3]{b \tan (c+d x)}\right )}{4 b d} \\ & = -\frac {\arctan \left (\frac {\sqrt [3]{b \tan (c+d x)}}{\sqrt [3]{b}}\right )}{b^{4/3} d}-\frac {\sqrt {3} \log \left (b^{2/3}-\sqrt {3} \sqrt [3]{b} \sqrt [3]{b \tan (c+d x)}+(b \tan (c+d x))^{2/3}\right )}{4 b^{4/3} d}+\frac {\sqrt {3} \log \left (b^{2/3}+\sqrt {3} \sqrt [3]{b} \sqrt [3]{b \tan (c+d x)}+(b \tan (c+d x))^{2/3}\right )}{4 b^{4/3} d}-\frac {3}{b d \sqrt [3]{b \tan (c+d x)}}-\frac {\text {Subst}\left (\int \frac {1}{-\frac {1}{3}-x^2} \, dx,x,1-\frac {2 \sqrt [3]{b \tan (c+d x)}}{\sqrt {3} \sqrt [3]{b}}\right )}{2 \sqrt {3} b^{4/3} d}+\frac {\text {Subst}\left (\int \frac {1}{-\frac {1}{3}-x^2} \, dx,x,1+\frac {2 \sqrt [3]{b \tan (c+d x)}}{\sqrt {3} \sqrt [3]{b}}\right )}{2 \sqrt {3} b^{4/3} d} \\ & = -\frac {\arctan \left (\frac {\sqrt [3]{b \tan (c+d x)}}{\sqrt [3]{b}}\right )}{b^{4/3} d}+\frac {\arctan \left (\frac {1}{3} \left (3 \sqrt {3}-\frac {6 \sqrt [3]{b \tan (c+d x)}}{\sqrt [3]{b}}\right )\right )}{2 b^{4/3} d}-\frac {\arctan \left (\frac {1}{3} \left (3 \sqrt {3}+\frac {6 \sqrt [3]{b \tan (c+d x)}}{\sqrt [3]{b}}\right )\right )}{2 b^{4/3} d}-\frac {\sqrt {3} \log \left (b^{2/3}-\sqrt {3} \sqrt [3]{b} \sqrt [3]{b \tan (c+d x)}+(b \tan (c+d x))^{2/3}\right )}{4 b^{4/3} d}+\frac {\sqrt {3} \log \left (b^{2/3}+\sqrt {3} \sqrt [3]{b} \sqrt [3]{b \tan (c+d x)}+(b \tan (c+d x))^{2/3}\right )}{4 b^{4/3} d}-\frac {3}{b d \sqrt [3]{b \tan (c+d x)}} \\ \end{align*}
Result contains complex when optimal does not.
Time = 0.23 (sec) , antiderivative size = 254, normalized size of antiderivative = 1.04 \[ \int \frac {1}{(b \tan (c+d x))^{4/3}} \, dx=\frac {-6-i \log \left (1-i \sqrt [6]{\tan ^2(c+d x)}\right ) \sqrt [6]{\tan ^2(c+d x)}+i \log \left (1+i \sqrt [6]{\tan ^2(c+d x)}\right ) \sqrt [6]{\tan ^2(c+d x)}-\sqrt [6]{-1} \log \left (1-\sqrt [6]{-1} \sqrt [6]{\tan ^2(c+d x)}\right ) \sqrt [6]{\tan ^2(c+d x)}+\sqrt [6]{-1} \log \left (1+\sqrt [6]{-1} \sqrt [6]{\tan ^2(c+d x)}\right ) \sqrt [6]{\tan ^2(c+d x)}-(-1)^{5/6} \log \left (1-(-1)^{5/6} \sqrt [6]{\tan ^2(c+d x)}\right ) \sqrt [6]{\tan ^2(c+d x)}+(-1)^{5/6} \log \left (1+(-1)^{5/6} \sqrt [6]{\tan ^2(c+d x)}\right ) \sqrt [6]{\tan ^2(c+d x)}}{2 b d \sqrt [3]{b \tan (c+d x)}} \]
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Time = 0.10 (sec) , antiderivative size = 215, normalized size of antiderivative = 0.88
method | result | size |
derivativedivides | \(\frac {3 b \left (-\frac {1}{b^{2} \left (b \tan \left (d x +c \right )\right )^{\frac {1}{3}}}-\frac {\frac {\sqrt {3}\, \left (b^{2}\right )^{\frac {5}{6}} \ln \left (-\left (b \tan \left (d x +c \right )\right )^{\frac {2}{3}}+\sqrt {3}\, \left (b^{2}\right )^{\frac {1}{6}} \left (b \tan \left (d x +c \right )\right )^{\frac {1}{3}}-\left (b^{2}\right )^{\frac {1}{3}}\right )}{12 b^{2}}+\frac {\arctan \left (\frac {2 \left (b \tan \left (d x +c \right )\right )^{\frac {1}{3}}}{\left (b^{2}\right )^{\frac {1}{6}}}-\sqrt {3}\right )}{6 \left (b^{2}\right )^{\frac {1}{6}}}-\frac {\sqrt {3}\, \left (b^{2}\right )^{\frac {5}{6}} \ln \left (\left (b \tan \left (d x +c \right )\right )^{\frac {2}{3}}+\sqrt {3}\, \left (b^{2}\right )^{\frac {1}{6}} \left (b \tan \left (d x +c \right )\right )^{\frac {1}{3}}+\left (b^{2}\right )^{\frac {1}{3}}\right )}{12 b^{2}}+\frac {\arctan \left (\frac {2 \left (b \tan \left (d x +c \right )\right )^{\frac {1}{3}}}{\left (b^{2}\right )^{\frac {1}{6}}}+\sqrt {3}\right )}{6 \left (b^{2}\right )^{\frac {1}{6}}}+\frac {\arctan \left (\frac {\left (b \tan \left (d x +c \right )\right )^{\frac {1}{3}}}{\left (b^{2}\right )^{\frac {1}{6}}}\right )}{3 \left (b^{2}\right )^{\frac {1}{6}}}}{b^{2}}\right )}{d}\) | \(215\) |
default | \(\frac {3 b \left (-\frac {1}{b^{2} \left (b \tan \left (d x +c \right )\right )^{\frac {1}{3}}}-\frac {\frac {\sqrt {3}\, \left (b^{2}\right )^{\frac {5}{6}} \ln \left (-\left (b \tan \left (d x +c \right )\right )^{\frac {2}{3}}+\sqrt {3}\, \left (b^{2}\right )^{\frac {1}{6}} \left (b \tan \left (d x +c \right )\right )^{\frac {1}{3}}-\left (b^{2}\right )^{\frac {1}{3}}\right )}{12 b^{2}}+\frac {\arctan \left (\frac {2 \left (b \tan \left (d x +c \right )\right )^{\frac {1}{3}}}{\left (b^{2}\right )^{\frac {1}{6}}}-\sqrt {3}\right )}{6 \left (b^{2}\right )^{\frac {1}{6}}}-\frac {\sqrt {3}\, \left (b^{2}\right )^{\frac {5}{6}} \ln \left (\left (b \tan \left (d x +c \right )\right )^{\frac {2}{3}}+\sqrt {3}\, \left (b^{2}\right )^{\frac {1}{6}} \left (b \tan \left (d x +c \right )\right )^{\frac {1}{3}}+\left (b^{2}\right )^{\frac {1}{3}}\right )}{12 b^{2}}+\frac {\arctan \left (\frac {2 \left (b \tan \left (d x +c \right )\right )^{\frac {1}{3}}}{\left (b^{2}\right )^{\frac {1}{6}}}+\sqrt {3}\right )}{6 \left (b^{2}\right )^{\frac {1}{6}}}+\frac {\arctan \left (\frac {\left (b \tan \left (d x +c \right )\right )^{\frac {1}{3}}}{\left (b^{2}\right )^{\frac {1}{6}}}\right )}{3 \left (b^{2}\right )^{\frac {1}{6}}}}{b^{2}}\right )}{d}\) | \(215\) |
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Leaf count of result is larger than twice the leaf count of optimal. 430 vs. \(2 (187) = 374\).
Time = 0.25 (sec) , antiderivative size = 430, normalized size of antiderivative = 1.76 \[ \int \frac {1}{(b \tan (c+d x))^{4/3}} \, dx=-\frac {2 \, b^{2} d \left (-\frac {1}{b^{8} d^{6}}\right )^{\frac {1}{6}} \log \left (b^{7} d^{5} \left (-\frac {1}{b^{8} d^{6}}\right )^{\frac {5}{6}} + \left (b \tan \left (d x + c\right )\right )^{\frac {1}{3}}\right ) \tan \left (d x + c\right ) - 2 \, b^{2} d \left (-\frac {1}{b^{8} d^{6}}\right )^{\frac {1}{6}} \log \left (-b^{7} d^{5} \left (-\frac {1}{b^{8} d^{6}}\right )^{\frac {5}{6}} + \left (b \tan \left (d x + c\right )\right )^{\frac {1}{3}}\right ) \tan \left (d x + c\right ) - {\left (\sqrt {-3} b^{2} d - b^{2} d\right )} \left (-\frac {1}{b^{8} d^{6}}\right )^{\frac {1}{6}} \log \left (\frac {1}{2} \, {\left (\sqrt {-3} b^{7} d^{5} + b^{7} d^{5}\right )} \left (-\frac {1}{b^{8} d^{6}}\right )^{\frac {5}{6}} + \left (b \tan \left (d x + c\right )\right )^{\frac {1}{3}}\right ) \tan \left (d x + c\right ) + {\left (\sqrt {-3} b^{2} d - b^{2} d\right )} \left (-\frac {1}{b^{8} d^{6}}\right )^{\frac {1}{6}} \log \left (-\frac {1}{2} \, {\left (\sqrt {-3} b^{7} d^{5} + b^{7} d^{5}\right )} \left (-\frac {1}{b^{8} d^{6}}\right )^{\frac {5}{6}} + \left (b \tan \left (d x + c\right )\right )^{\frac {1}{3}}\right ) \tan \left (d x + c\right ) - {\left (\sqrt {-3} b^{2} d + b^{2} d\right )} \left (-\frac {1}{b^{8} d^{6}}\right )^{\frac {1}{6}} \log \left (\frac {1}{2} \, {\left (\sqrt {-3} b^{7} d^{5} - b^{7} d^{5}\right )} \left (-\frac {1}{b^{8} d^{6}}\right )^{\frac {5}{6}} + \left (b \tan \left (d x + c\right )\right )^{\frac {1}{3}}\right ) \tan \left (d x + c\right ) + {\left (\sqrt {-3} b^{2} d + b^{2} d\right )} \left (-\frac {1}{b^{8} d^{6}}\right )^{\frac {1}{6}} \log \left (-\frac {1}{2} \, {\left (\sqrt {-3} b^{7} d^{5} - b^{7} d^{5}\right )} \left (-\frac {1}{b^{8} d^{6}}\right )^{\frac {5}{6}} + \left (b \tan \left (d x + c\right )\right )^{\frac {1}{3}}\right ) \tan \left (d x + c\right ) + 12 \, \left (b \tan \left (d x + c\right )\right )^{\frac {2}{3}}}{4 \, b^{2} d \tan \left (d x + c\right )} \]
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\[ \int \frac {1}{(b \tan (c+d x))^{4/3}} \, dx=\int \frac {1}{\left (b \tan {\left (c + d x \right )}\right )^{\frac {4}{3}}}\, dx \]
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Time = 0.31 (sec) , antiderivative size = 182, normalized size of antiderivative = 0.74 \[ \int \frac {1}{(b \tan (c+d x))^{4/3}} \, dx=\frac {\frac {\sqrt {3} \log \left (\sqrt {3} \left (b \tan \left (d x + c\right )\right )^{\frac {1}{3}} b^{\frac {1}{3}} + \left (b \tan \left (d x + c\right )\right )^{\frac {2}{3}} + b^{\frac {2}{3}}\right )}{b^{\frac {1}{3}}} - \frac {\sqrt {3} \log \left (-\sqrt {3} \left (b \tan \left (d x + c\right )\right )^{\frac {1}{3}} b^{\frac {1}{3}} + \left (b \tan \left (d x + c\right )\right )^{\frac {2}{3}} + b^{\frac {2}{3}}\right )}{b^{\frac {1}{3}}} - \frac {2 \, \arctan \left (\frac {\sqrt {3} b^{\frac {1}{3}} + 2 \, \left (b \tan \left (d x + c\right )\right )^{\frac {1}{3}}}{b^{\frac {1}{3}}}\right )}{b^{\frac {1}{3}}} - \frac {2 \, \arctan \left (-\frac {\sqrt {3} b^{\frac {1}{3}} - 2 \, \left (b \tan \left (d x + c\right )\right )^{\frac {1}{3}}}{b^{\frac {1}{3}}}\right )}{b^{\frac {1}{3}}} - \frac {4 \, \arctan \left (\frac {\left (b \tan \left (d x + c\right )\right )^{\frac {1}{3}}}{b^{\frac {1}{3}}}\right )}{b^{\frac {1}{3}}} - \frac {12}{\left (b \tan \left (d x + c\right )\right )^{\frac {1}{3}}}}{4 \, b d} \]
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Time = 0.48 (sec) , antiderivative size = 227, normalized size of antiderivative = 0.93 \[ \int \frac {1}{(b \tan (c+d x))^{4/3}} \, dx=\frac {1}{4} \, b {\left (\frac {\sqrt {3} {\left | b \right |}^{\frac {5}{3}} \log \left (\sqrt {3} \left (b \tan \left (d x + c\right )\right )^{\frac {1}{3}} {\left | b \right |}^{\frac {1}{3}} + \left (b \tan \left (d x + c\right )\right )^{\frac {2}{3}} + {\left | b \right |}^{\frac {2}{3}}\right )}{b^{4} d} - \frac {\sqrt {3} {\left | b \right |}^{\frac {5}{3}} \log \left (-\sqrt {3} \left (b \tan \left (d x + c\right )\right )^{\frac {1}{3}} {\left | b \right |}^{\frac {1}{3}} + \left (b \tan \left (d x + c\right )\right )^{\frac {2}{3}} + {\left | b \right |}^{\frac {2}{3}}\right )}{b^{4} d} - \frac {2 \, {\left | b \right |}^{\frac {5}{3}} \arctan \left (\frac {\sqrt {3} {\left | b \right |}^{\frac {1}{3}} + 2 \, \left (b \tan \left (d x + c\right )\right )^{\frac {1}{3}}}{{\left | b \right |}^{\frac {1}{3}}}\right )}{b^{4} d} - \frac {2 \, {\left | b \right |}^{\frac {5}{3}} \arctan \left (-\frac {\sqrt {3} {\left | b \right |}^{\frac {1}{3}} - 2 \, \left (b \tan \left (d x + c\right )\right )^{\frac {1}{3}}}{{\left | b \right |}^{\frac {1}{3}}}\right )}{b^{4} d} - \frac {4 \, {\left | b \right |}^{\frac {5}{3}} \arctan \left (\frac {\left (b \tan \left (d x + c\right )\right )^{\frac {1}{3}}}{{\left | b \right |}^{\frac {1}{3}}}\right )}{b^{4} d} - \frac {12}{\left (b \tan \left (d x + c\right )\right )^{\frac {1}{3}} b^{2} d}\right )} \]
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Time = 3.17 (sec) , antiderivative size = 278, normalized size of antiderivative = 1.13 \[ \int \frac {1}{(b \tan (c+d x))^{4/3}} \, dx=-\frac {3}{b\,d\,{\left (b\,\mathrm {tan}\left (c+d\,x\right )\right )}^{1/3}}-\frac {{\left (-1\right )}^{1/6}\,\mathrm {atan}\left (\frac {{\left (-1\right )}^{2/3}\,{\left (b\,\mathrm {tan}\left (c+d\,x\right )\right )}^{1/3}}{b^{1/3}}\right )\,1{}\mathrm {i}}{b^{4/3}\,d}-\frac {{\left (-1\right )}^{1/6}\,\ln \left (972\,b^{12}\,d^6-972\,{\left (-1\right )}^{1/6}\,b^{35/3}\,d^6\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,{\left (b\,\mathrm {tan}\left (c+d\,x\right )\right )}^{1/3}\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{2\,b^{4/3}\,d}-\frac {{\left (-1\right )}^{1/6}\,\ln \left (972\,b^{12}\,d^6-972\,{\left (-1\right )}^{1/6}\,b^{35/3}\,d^6\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,{\left (b\,\mathrm {tan}\left (c+d\,x\right )\right )}^{1/3}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{2\,b^{4/3}\,d}+\frac {{\left (-1\right )}^{1/6}\,\ln \left (972\,b^{12}\,d^6+1944\,{\left (-1\right )}^{1/6}\,b^{35/3}\,d^6\,\left (-\frac {1}{4}+\frac {\sqrt {3}\,1{}\mathrm {i}}{4}\right )\,{\left (b\,\mathrm {tan}\left (c+d\,x\right )\right )}^{1/3}\right )\,\left (-\frac {1}{4}+\frac {\sqrt {3}\,1{}\mathrm {i}}{4}\right )}{b^{4/3}\,d}+\frac {{\left (-1\right )}^{1/6}\,\ln \left (972\,b^{12}\,d^6+1944\,{\left (-1\right )}^{1/6}\,b^{35/3}\,d^6\,\left (\frac {1}{4}+\frac {\sqrt {3}\,1{}\mathrm {i}}{4}\right )\,{\left (b\,\mathrm {tan}\left (c+d\,x\right )\right )}^{1/3}\right )\,\left (\frac {1}{4}+\frac {\sqrt {3}\,1{}\mathrm {i}}{4}\right )}{b^{4/3}\,d} \]
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