Integrand size = 14, antiderivative size = 415 \[ \int \sqrt [3]{a+b \tan (c+d x)} \, dx=-\frac {1}{4} \sqrt [3]{a-\sqrt {-b^2}} x-\frac {1}{4} \sqrt [3]{a+\sqrt {-b^2}} x+\frac {\sqrt {3} b \sqrt [3]{a-\sqrt {-b^2}} \arctan \left (\frac {1+\frac {2 \sqrt [3]{a+b \tan (c+d x)}}{\sqrt [3]{a-\sqrt {-b^2}}}}{\sqrt {3}}\right )}{2 \sqrt {-b^2} d}-\frac {\sqrt {3} b \sqrt [3]{a+\sqrt {-b^2}} \arctan \left (\frac {1+\frac {2 \sqrt [3]{a+b \tan (c+d x)}}{\sqrt [3]{a+\sqrt {-b^2}}}}{\sqrt {3}}\right )}{2 \sqrt {-b^2} d}-\frac {b \sqrt [3]{a-\sqrt {-b^2}} \log (\cos (c+d x))}{4 \sqrt {-b^2} d}+\frac {b \sqrt [3]{a+\sqrt {-b^2}} \log (\cos (c+d x))}{4 \sqrt {-b^2} d}-\frac {3 b \sqrt [3]{a-\sqrt {-b^2}} \log \left (\sqrt [3]{a-\sqrt {-b^2}}-\sqrt [3]{a+b \tan (c+d x)}\right )}{4 \sqrt {-b^2} d}+\frac {3 b \sqrt [3]{a+\sqrt {-b^2}} \log \left (\sqrt [3]{a+\sqrt {-b^2}}-\sqrt [3]{a+b \tan (c+d x)}\right )}{4 \sqrt {-b^2} d} \]
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Time = 0.45 (sec) , antiderivative size = 415, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {3566, 726, 52, 59, 631, 210, 31} \[ \int \sqrt [3]{a+b \tan (c+d x)} \, dx=\frac {\sqrt {3} b \sqrt [3]{a-\sqrt {-b^2}} \arctan \left (\frac {\frac {2 \sqrt [3]{a+b \tan (c+d x)}}{\sqrt [3]{a-\sqrt {-b^2}}}+1}{\sqrt {3}}\right )}{2 \sqrt {-b^2} d}-\frac {\sqrt {3} b \sqrt [3]{a+\sqrt {-b^2}} \arctan \left (\frac {\frac {2 \sqrt [3]{a+b \tan (c+d x)}}{\sqrt [3]{a+\sqrt {-b^2}}}+1}{\sqrt {3}}\right )}{2 \sqrt {-b^2} d}-\frac {3 b \sqrt [3]{a-\sqrt {-b^2}} \log \left (\sqrt [3]{a-\sqrt {-b^2}}-\sqrt [3]{a+b \tan (c+d x)}\right )}{4 \sqrt {-b^2} d}+\frac {3 b \sqrt [3]{a+\sqrt {-b^2}} \log \left (\sqrt [3]{a+\sqrt {-b^2}}-\sqrt [3]{a+b \tan (c+d x)}\right )}{4 \sqrt {-b^2} d}-\frac {b \sqrt [3]{a-\sqrt {-b^2}} \log (\cos (c+d x))}{4 \sqrt {-b^2} d}+\frac {b \sqrt [3]{a+\sqrt {-b^2}} \log (\cos (c+d x))}{4 \sqrt {-b^2} d}-\frac {1}{4} x \sqrt [3]{a-\sqrt {-b^2}}-\frac {1}{4} x \sqrt [3]{a+\sqrt {-b^2}} \]
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Rule 31
Rule 52
Rule 59
Rule 210
Rule 631
Rule 726
Rule 3566
Rubi steps \begin{align*} \text {integral}& = \frac {b \text {Subst}\left (\int \frac {\sqrt [3]{a+x}}{b^2+x^2} \, dx,x,b \tan (c+d x)\right )}{d} \\ & = \frac {b \text {Subst}\left (\int \left (\frac {\sqrt {-b^2} \sqrt [3]{a+x}}{2 b^2 \left (\sqrt {-b^2}-x\right )}+\frac {\sqrt {-b^2} \sqrt [3]{a+x}}{2 b^2 \left (\sqrt {-b^2}+x\right )}\right ) \, dx,x,b \tan (c+d x)\right )}{d} \\ & = -\frac {b \text {Subst}\left (\int \frac {\sqrt [3]{a+x}}{\sqrt {-b^2}-x} \, dx,x,b \tan (c+d x)\right )}{2 \sqrt {-b^2} d}-\frac {b \text {Subst}\left (\int \frac {\sqrt [3]{a+x}}{\sqrt {-b^2}+x} \, dx,x,b \tan (c+d x)\right )}{2 \sqrt {-b^2} d} \\ & = -\frac {\left (b \left (a+\sqrt {-b^2}\right )\right ) \text {Subst}\left (\int \frac {1}{\left (\sqrt {-b^2}-x\right ) (a+x)^{2/3}} \, dx,x,b \tan (c+d x)\right )}{2 \sqrt {-b^2} d}+\frac {\left (b^2+a \sqrt {-b^2}\right ) \text {Subst}\left (\int \frac {1}{(a+x)^{2/3} \left (\sqrt {-b^2}+x\right )} \, dx,x,b \tan (c+d x)\right )}{2 b d} \\ & = -\frac {1}{4} \sqrt [3]{a-\sqrt {-b^2}} x-\frac {1}{4} \sqrt [3]{a+\sqrt {-b^2}} x+\frac {\sqrt {-b^2} \sqrt [3]{a-\sqrt {-b^2}} \log (\cos (c+d x))}{4 b d}+\frac {b \sqrt [3]{a+\sqrt {-b^2}} \log (\cos (c+d x))}{4 \sqrt {-b^2} d}-\frac {\left (3 b \sqrt [3]{a+\sqrt {-b^2}}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [3]{a+\sqrt {-b^2}}-x} \, dx,x,\sqrt [3]{a+b \tan (c+d x)}\right )}{4 \sqrt {-b^2} d}-\frac {\left (3 b \left (a+\sqrt {-b^2}\right )^{2/3}\right ) \text {Subst}\left (\int \frac {1}{\left (a+\sqrt {-b^2}\right )^{2/3}+\sqrt [3]{a+\sqrt {-b^2}} x+x^2} \, dx,x,\sqrt [3]{a+b \tan (c+d x)}\right )}{4 \sqrt {-b^2} d}-\frac {\left (3 \left (b^2+a \sqrt {-b^2}\right )\right ) \text {Subst}\left (\int \frac {1}{\sqrt [3]{a-\sqrt {-b^2}}-x} \, dx,x,\sqrt [3]{a+b \tan (c+d x)}\right )}{4 b \left (a-\sqrt {-b^2}\right )^{2/3} d}-\frac {\left (3 \left (b^2+a \sqrt {-b^2}\right )\right ) \text {Subst}\left (\int \frac {1}{\left (a-\sqrt {-b^2}\right )^{2/3}+\sqrt [3]{a-\sqrt {-b^2}} x+x^2} \, dx,x,\sqrt [3]{a+b \tan (c+d x)}\right )}{4 b \sqrt [3]{a-\sqrt {-b^2}} d} \\ & = -\frac {1}{4} \sqrt [3]{a-\sqrt {-b^2}} x-\frac {1}{4} \sqrt [3]{a+\sqrt {-b^2}} x+\frac {\sqrt {-b^2} \sqrt [3]{a-\sqrt {-b^2}} \log (\cos (c+d x))}{4 b d}+\frac {b \sqrt [3]{a+\sqrt {-b^2}} \log (\cos (c+d x))}{4 \sqrt {-b^2} d}+\frac {3 \sqrt {-b^2} \sqrt [3]{a-\sqrt {-b^2}} \log \left (\sqrt [3]{a-\sqrt {-b^2}}-\sqrt [3]{a+b \tan (c+d x)}\right )}{4 b d}+\frac {3 b \sqrt [3]{a+\sqrt {-b^2}} \log \left (\sqrt [3]{a+\sqrt {-b^2}}-\sqrt [3]{a+b \tan (c+d x)}\right )}{4 \sqrt {-b^2} d}+\frac {\left (3 b \sqrt [3]{a+\sqrt {-b^2}}\right ) \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+\frac {2 \sqrt [3]{a+b \tan (c+d x)}}{\sqrt [3]{a+\sqrt {-b^2}}}\right )}{2 \sqrt {-b^2} d}+\frac {\left (3 \left (b^2+a \sqrt {-b^2}\right )\right ) \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+\frac {2 \sqrt [3]{a+b \tan (c+d x)}}{\sqrt [3]{a-\sqrt {-b^2}}}\right )}{2 b \left (a-\sqrt {-b^2}\right )^{2/3} d} \\ & = -\frac {1}{4} \sqrt [3]{a-\sqrt {-b^2}} x-\frac {1}{4} \sqrt [3]{a+\sqrt {-b^2}} x-\frac {\sqrt {3} \sqrt {-b^2} \sqrt [3]{a-\sqrt {-b^2}} \arctan \left (\frac {1+\frac {2 \sqrt [3]{a+b \tan (c+d x)}}{\sqrt [3]{a-\sqrt {-b^2}}}}{\sqrt {3}}\right )}{2 b d}-\frac {\sqrt {3} b \sqrt [3]{a+\sqrt {-b^2}} \arctan \left (\frac {1+\frac {2 \sqrt [3]{a+b \tan (c+d x)}}{\sqrt [3]{a+\sqrt {-b^2}}}}{\sqrt {3}}\right )}{2 \sqrt {-b^2} d}+\frac {\sqrt {-b^2} \sqrt [3]{a-\sqrt {-b^2}} \log (\cos (c+d x))}{4 b d}+\frac {b \sqrt [3]{a+\sqrt {-b^2}} \log (\cos (c+d x))}{4 \sqrt {-b^2} d}+\frac {3 \sqrt {-b^2} \sqrt [3]{a-\sqrt {-b^2}} \log \left (\sqrt [3]{a-\sqrt {-b^2}}-\sqrt [3]{a+b \tan (c+d x)}\right )}{4 b d}+\frac {3 b \sqrt [3]{a+\sqrt {-b^2}} \log \left (\sqrt [3]{a+\sqrt {-b^2}}-\sqrt [3]{a+b \tan (c+d x)}\right )}{4 \sqrt {-b^2} d} \\ \end{align*}
Result contains complex when optimal does not.
Time = 0.25 (sec) , antiderivative size = 294, normalized size of antiderivative = 0.71 \[ \int \sqrt [3]{a+b \tan (c+d x)} \, dx=\frac {-i \sqrt [3]{a-i b} \left (2 \sqrt {3} \arctan \left (\frac {1+\frac {2 \sqrt [3]{a+b \tan (c+d x)}}{\sqrt [3]{a-i b}}}{\sqrt {3}}\right )-2 \log \left (\sqrt [3]{a-i b}-\sqrt [3]{a+b \tan (c+d x)}\right )+\log \left ((a-i b)^{2/3}+\sqrt [3]{a-i b} \sqrt [3]{a+b \tan (c+d x)}+(a+b \tan (c+d x))^{2/3}\right )\right )+i \sqrt [3]{a+i b} \left (2 \sqrt {3} \arctan \left (\frac {1+\frac {2 \sqrt [3]{a+b \tan (c+d x)}}{\sqrt [3]{a+i b}}}{\sqrt {3}}\right )-2 \log \left (\sqrt [3]{a+i b}-\sqrt [3]{a+b \tan (c+d x)}\right )+\log \left ((a+i b)^{2/3}+\sqrt [3]{a+i b} \sqrt [3]{a+b \tan (c+d x)}+(a+b \tan (c+d x))^{2/3}\right )\right )}{4 d} \]
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Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.76 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.14
method | result | size |
derivativedivides | \(\frac {b \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{6}-2 a \,\textit {\_Z}^{3}+a^{2}+b^{2}\right )}{\sum }\frac {\textit {\_R}^{3} \ln \left (\left (a +b \tan \left (d x +c \right )\right )^{\frac {1}{3}}-\textit {\_R} \right )}{\textit {\_R}^{5}-\textit {\_R}^{2} a}\right )}{2 d}\) | \(60\) |
default | \(\frac {b \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{6}-2 a \,\textit {\_Z}^{3}+a^{2}+b^{2}\right )}{\sum }\frac {\textit {\_R}^{3} \ln \left (\left (a +b \tan \left (d x +c \right )\right )^{\frac {1}{3}}-\textit {\_R} \right )}{\textit {\_R}^{5}-\textit {\_R}^{2} a}\right )}{2 d}\) | \(60\) |
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Time = 0.25 (sec) , antiderivative size = 536, normalized size of antiderivative = 1.29 \[ \int \sqrt [3]{a+b \tan (c+d x)} \, dx=-\frac {1}{4} \, {\left (\sqrt {-3} + 1\right )} \left (-\frac {d^{3} \sqrt {-\frac {a^{2}}{d^{6}}} + b}{d^{3}}\right )^{\frac {1}{3}} \log \left (-\frac {1}{2} \, {\left (\sqrt {-3} d^{4} + d^{4}\right )} \left (-\frac {d^{3} \sqrt {-\frac {a^{2}}{d^{6}}} + b}{d^{3}}\right )^{\frac {1}{3}} \sqrt {-\frac {a^{2}}{d^{6}}} + {\left (b \tan \left (d x + c\right ) + a\right )}^{\frac {1}{3}} a\right ) + \frac {1}{4} \, {\left (\sqrt {-3} - 1\right )} \left (-\frac {d^{3} \sqrt {-\frac {a^{2}}{d^{6}}} + b}{d^{3}}\right )^{\frac {1}{3}} \log \left (\frac {1}{2} \, {\left (\sqrt {-3} d^{4} - d^{4}\right )} \left (-\frac {d^{3} \sqrt {-\frac {a^{2}}{d^{6}}} + b}{d^{3}}\right )^{\frac {1}{3}} \sqrt {-\frac {a^{2}}{d^{6}}} + {\left (b \tan \left (d x + c\right ) + a\right )}^{\frac {1}{3}} a\right ) - \frac {1}{4} \, {\left (\sqrt {-3} + 1\right )} \left (\frac {d^{3} \sqrt {-\frac {a^{2}}{d^{6}}} - b}{d^{3}}\right )^{\frac {1}{3}} \log \left (\frac {1}{2} \, {\left (\sqrt {-3} d^{4} + d^{4}\right )} \left (\frac {d^{3} \sqrt {-\frac {a^{2}}{d^{6}}} - b}{d^{3}}\right )^{\frac {1}{3}} \sqrt {-\frac {a^{2}}{d^{6}}} + {\left (b \tan \left (d x + c\right ) + a\right )}^{\frac {1}{3}} a\right ) + \frac {1}{4} \, {\left (\sqrt {-3} - 1\right )} \left (\frac {d^{3} \sqrt {-\frac {a^{2}}{d^{6}}} - b}{d^{3}}\right )^{\frac {1}{3}} \log \left (-\frac {1}{2} \, {\left (\sqrt {-3} d^{4} - d^{4}\right )} \left (\frac {d^{3} \sqrt {-\frac {a^{2}}{d^{6}}} - b}{d^{3}}\right )^{\frac {1}{3}} \sqrt {-\frac {a^{2}}{d^{6}}} + {\left (b \tan \left (d x + c\right ) + a\right )}^{\frac {1}{3}} a\right ) + \frac {1}{2} \, \left (-\frac {d^{3} \sqrt {-\frac {a^{2}}{d^{6}}} + b}{d^{3}}\right )^{\frac {1}{3}} \log \left (d^{4} \left (-\frac {d^{3} \sqrt {-\frac {a^{2}}{d^{6}}} + b}{d^{3}}\right )^{\frac {1}{3}} \sqrt {-\frac {a^{2}}{d^{6}}} + {\left (b \tan \left (d x + c\right ) + a\right )}^{\frac {1}{3}} a\right ) + \frac {1}{2} \, \left (\frac {d^{3} \sqrt {-\frac {a^{2}}{d^{6}}} - b}{d^{3}}\right )^{\frac {1}{3}} \log \left (-d^{4} \left (\frac {d^{3} \sqrt {-\frac {a^{2}}{d^{6}}} - b}{d^{3}}\right )^{\frac {1}{3}} \sqrt {-\frac {a^{2}}{d^{6}}} + {\left (b \tan \left (d x + c\right ) + a\right )}^{\frac {1}{3}} a\right ) \]
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\[ \int \sqrt [3]{a+b \tan (c+d x)} \, dx=\int \sqrt [3]{a + b \tan {\left (c + d x \right )}}\, dx \]
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\[ \int \sqrt [3]{a+b \tan (c+d x)} \, dx=\int { {\left (b \tan \left (d x + c\right ) + a\right )}^{\frac {1}{3}} \,d x } \]
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\[ \int \sqrt [3]{a+b \tan (c+d x)} \, dx=\int { {\left (b \tan \left (d x + c\right ) + a\right )}^{\frac {1}{3}} \,d x } \]
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Time = 7.96 (sec) , antiderivative size = 863, normalized size of antiderivative = 2.08 \[ \int \sqrt [3]{a+b \tan (c+d x)} \, dx=\text {Too large to display} \]
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