Integrand size = 14, antiderivative size = 256 \[ \int \frac {1}{a+b \tan ^3(c+d x)} \, dx=\frac {a x}{a^2+b^2}+\frac {\sqrt [3]{b} \left (a^{4/3}-b^{4/3}\right ) \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} \tan (c+d x)}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} a^{2/3} \left (a^2+b^2\right ) d}-\frac {b \log \left (a \cos ^3(c+d x)+b \sin ^3(c+d x)\right )}{3 \left (a^2+b^2\right ) d}+\frac {\sqrt [3]{b} \left (a^{4/3}+b^{4/3}\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} \tan (c+d x)\right )}{3 a^{2/3} \left (a^2+b^2\right ) d}-\frac {\sqrt [3]{b} \left (a^{4/3}+b^{4/3}\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \tan (c+d x)+b^{2/3} \tan ^2(c+d x)\right )}{6 a^{2/3} \left (a^2+b^2\right ) d} \]
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Time = 0.45 (sec) , antiderivative size = 256, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.857, Rules used = {3742, 6857, 649, 209, 266, 1885, 1874, 31, 648, 631, 210, 642} \[ \int \frac {1}{a+b \tan ^3(c+d x)} \, dx=-\frac {b \log \left (a \cos ^3(c+d x)+b \sin ^3(c+d x)\right )}{3 d \left (a^2+b^2\right )}+\frac {a x}{a^2+b^2}+\frac {\sqrt [3]{b} \left (a^{4/3}-b^{4/3}\right ) \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} \tan (c+d x)}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} a^{2/3} d \left (a^2+b^2\right )}-\frac {\sqrt [3]{b} \left (a^{4/3}+b^{4/3}\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \tan (c+d x)+b^{2/3} \tan ^2(c+d x)\right )}{6 a^{2/3} d \left (a^2+b^2\right )}+\frac {\sqrt [3]{b} \left (a^{4/3}+b^{4/3}\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} \tan (c+d x)\right )}{3 a^{2/3} d \left (a^2+b^2\right )} \]
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Rule 31
Rule 209
Rule 210
Rule 266
Rule 631
Rule 642
Rule 648
Rule 649
Rule 1874
Rule 1885
Rule 3742
Rule 6857
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {1}{\left (1+x^2\right ) \left (a+b x^3\right )} \, dx,x,\tan (c+d x)\right )}{d} \\ & = \frac {\text {Subst}\left (\int \left (\frac {a+b x}{\left (a^2+b^2\right ) \left (1+x^2\right )}-\frac {b \left (-b+a x+b x^2\right )}{\left (a^2+b^2\right ) \left (a+b x^3\right )}\right ) \, dx,x,\tan (c+d x)\right )}{d} \\ & = \frac {\text {Subst}\left (\int \frac {a+b x}{1+x^2} \, dx,x,\tan (c+d x)\right )}{\left (a^2+b^2\right ) d}-\frac {b \text {Subst}\left (\int \frac {-b+a x+b x^2}{a+b x^3} \, dx,x,\tan (c+d x)\right )}{\left (a^2+b^2\right ) d} \\ & = \frac {a \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\tan (c+d x)\right )}{\left (a^2+b^2\right ) d}+\frac {b \text {Subst}\left (\int \frac {x}{1+x^2} \, dx,x,\tan (c+d x)\right )}{\left (a^2+b^2\right ) d}-\frac {b \text {Subst}\left (\int \frac {-b+a x}{a+b x^3} \, dx,x,\tan (c+d x)\right )}{\left (a^2+b^2\right ) d}-\frac {b^2 \text {Subst}\left (\int \frac {x^2}{a+b x^3} \, dx,x,\tan (c+d x)\right )}{\left (a^2+b^2\right ) d} \\ & = \frac {a x}{a^2+b^2}-\frac {b \log (\cos (c+d x))}{\left (a^2+b^2\right ) d}-\frac {b \log \left (a+b \tan ^3(c+d x)\right )}{3 \left (a^2+b^2\right ) d}-\frac {b^{2/3} \text {Subst}\left (\int \frac {\sqrt [3]{a} \left (a^{4/3}-2 b^{4/3}\right )+\sqrt [3]{b} \left (a^{4/3}+b^{4/3}\right ) x}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx,x,\tan (c+d x)\right )}{3 a^{2/3} \left (a^2+b^2\right ) d}+\frac {\left (b^{2/3} \left (a^{4/3}+b^{4/3}\right )\right ) \text {Subst}\left (\int \frac {1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx,x,\tan (c+d x)\right )}{3 a^{2/3} \left (a^2+b^2\right ) d} \\ & = \frac {a x}{a^2+b^2}-\frac {b \log (\cos (c+d x))}{\left (a^2+b^2\right ) d}+\frac {\sqrt [3]{b} \left (a^{4/3}+b^{4/3}\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} \tan (c+d x)\right )}{3 a^{2/3} \left (a^2+b^2\right ) d}-\frac {b \log \left (a+b \tan ^3(c+d x)\right )}{3 \left (a^2+b^2\right ) d}-\frac {\left (b^{2/3} \left (a^{4/3}-b^{4/3}\right )\right ) \text {Subst}\left (\int \frac {1}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx,x,\tan (c+d x)\right )}{2 \sqrt [3]{a} \left (a^2+b^2\right ) d}-\frac {\left (\sqrt [3]{b} \left (a^{4/3}+b^{4/3}\right )\right ) \text {Subst}\left (\int \frac {-\sqrt [3]{a} \sqrt [3]{b}+2 b^{2/3} x}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx,x,\tan (c+d x)\right )}{6 a^{2/3} \left (a^2+b^2\right ) d} \\ & = \frac {a x}{a^2+b^2}-\frac {b \log (\cos (c+d x))}{\left (a^2+b^2\right ) d}+\frac {\sqrt [3]{b} \left (a^{4/3}+b^{4/3}\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} \tan (c+d x)\right )}{3 a^{2/3} \left (a^2+b^2\right ) d}-\frac {\sqrt [3]{b} \left (a^{4/3}+b^{4/3}\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \tan (c+d x)+b^{2/3} \tan ^2(c+d x)\right )}{6 a^{2/3} \left (a^2+b^2\right ) d}-\frac {b \log \left (a+b \tan ^3(c+d x)\right )}{3 \left (a^2+b^2\right ) d}-\frac {\left (\sqrt [3]{b} \left (a^{4/3}-b^{4/3}\right )\right ) \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\frac {2 \sqrt [3]{b} \tan (c+d x)}{\sqrt [3]{a}}\right )}{a^{2/3} \left (a^2+b^2\right ) d} \\ & = \frac {a x}{a^2+b^2}+\frac {\sqrt [3]{b} \left (a^{4/3}-b^{4/3}\right ) \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} \tan (c+d x)}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{\sqrt {3} a^{2/3} \left (a^2+b^2\right ) d}-\frac {b \log (\cos (c+d x))}{\left (a^2+b^2\right ) d}+\frac {\sqrt [3]{b} \left (a^{4/3}+b^{4/3}\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} \tan (c+d x)\right )}{3 a^{2/3} \left (a^2+b^2\right ) d}-\frac {\sqrt [3]{b} \left (a^{4/3}+b^{4/3}\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \tan (c+d x)+b^{2/3} \tan ^2(c+d x)\right )}{6 a^{2/3} \left (a^2+b^2\right ) d}-\frac {b \log \left (a+b \tan ^3(c+d x)\right )}{3 \left (a^2+b^2\right ) d} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.70 (sec) , antiderivative size = 278, normalized size of antiderivative = 1.09 \[ \int \frac {1}{a+b \tan ^3(c+d x)} \, dx=\frac {-2 \sqrt {3} b^{5/3} \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} \tan (c+d x)}{\sqrt {3} \sqrt [3]{a}}\right )-3 i a^{5/3} \log (i-\tan (c+d x))+3 a^{2/3} b \log (i-\tan (c+d x))+3 i a^{5/3} \log (i+\tan (c+d x))+3 a^{2/3} b \log (i+\tan (c+d x))+2 b^{5/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} \tan (c+d x)\right )-b^{5/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \tan (c+d x)+b^{2/3} \tan ^2(c+d x)\right )-2 a^{2/3} b \log \left (a+b \tan ^3(c+d x)\right )-3 a^{2/3} b \operatorname {Hypergeometric2F1}\left (\frac {2}{3},1,\frac {5}{3},-\frac {b \tan ^3(c+d x)}{a}\right ) \tan ^2(c+d x)}{6 a^{2/3} \left (a^2+b^2\right ) d} \]
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Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.17 (sec) , antiderivative size = 265, normalized size of antiderivative = 1.04
method | result | size |
risch | \(\frac {x}{i b +a}+\frac {2 i b \,a^{2} d^{3} x}{a^{4} d^{3}+a^{2} b^{2} d^{3}}+\frac {2 i b \,a^{2} d^{2} c}{a^{4} d^{3}+a^{2} b^{2} d^{3}}+\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\left (27 a^{4} d^{3}+27 a^{2} b^{2} d^{3}\right ) \textit {\_Z}^{3}+27 \textit {\_Z}^{2} a^{2} b \,d^{2}-b \right )}{\sum }\textit {\_R} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+\left (-\frac {18 d^{2} a^{4}}{a^{2}-b^{2}}-\frac {18 d^{2} b^{2} a^{2}}{a^{2}-b^{2}}\right ) \textit {\_R}^{2}+\left (\frac {6 i d \,a^{3}}{a^{2}-b^{2}}-\frac {6 i d \,b^{2} a}{a^{2}-b^{2}}-\frac {6 d b \,a^{2}}{a^{2}-b^{2}}\right ) \textit {\_R} +\frac {a^{2}}{a^{2}-b^{2}}+\frac {b^{2}}{a^{2}-b^{2}}\right )\right )\) | \(265\) |
derivativedivides | \(\frac {\frac {\frac {b \ln \left (1+\tan \left (d x +c \right )^{2}\right )}{2}+a \arctan \left (\tan \left (d x +c \right )\right )}{a^{2}+b^{2}}-\frac {\left (-b \left (\frac {\ln \left (\tan \left (d x +c \right )+\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}-\frac {\ln \left (\tan \left (d x +c \right )^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} \tan \left (d x +c \right )+\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \tan \left (d x +c \right )}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}\right )+a \left (-\frac {\ln \left (\tan \left (d x +c \right )+\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\ln \left (\tan \left (d x +c \right )^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} \tan \left (d x +c \right )+\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \tan \left (d x +c \right )}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )+\frac {\ln \left (a +b \tan \left (d x +c \right )^{3}\right )}{3}\right ) b}{a^{2}+b^{2}}}{d}\) | \(293\) |
default | \(\frac {\frac {\frac {b \ln \left (1+\tan \left (d x +c \right )^{2}\right )}{2}+a \arctan \left (\tan \left (d x +c \right )\right )}{a^{2}+b^{2}}-\frac {\left (-b \left (\frac {\ln \left (\tan \left (d x +c \right )+\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}-\frac {\ln \left (\tan \left (d x +c \right )^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} \tan \left (d x +c \right )+\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \tan \left (d x +c \right )}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}\right )+a \left (-\frac {\ln \left (\tan \left (d x +c \right )+\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\ln \left (\tan \left (d x +c \right )^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} \tan \left (d x +c \right )+\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \tan \left (d x +c \right )}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )+\frac {\ln \left (a +b \tan \left (d x +c \right )^{3}\right )}{3}\right ) b}{a^{2}+b^{2}}}{d}\) | \(293\) |
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Result contains complex when optimal does not.
Time = 0.97 (sec) , antiderivative size = 4817, normalized size of antiderivative = 18.82 \[ \int \frac {1}{a+b \tan ^3(c+d x)} \, dx=\text {Too large to display} \]
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Timed out. \[ \int \frac {1}{a+b \tan ^3(c+d x)} \, dx=\text {Timed out} \]
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Time = 0.29 (sec) , antiderivative size = 291, normalized size of antiderivative = 1.14 \[ \int \frac {1}{a+b \tan ^3(c+d x)} \, dx=-\frac {\frac {2 \, \sqrt {3} {\left (a {\left (3 \, \left (\frac {a}{b}\right )^{\frac {2}{3}} - 2\right )} - b {\left (3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}} - \frac {2 \, a}{b}\right )}\right )} \arctan \left (-\frac {\sqrt {3} {\left (\left (\frac {a}{b}\right )^{\frac {1}{3}} - 2 \, \tan \left (d x + c\right )\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{{\left (a^{2} \left (\frac {a}{b}\right )^{\frac {2}{3}} + b^{2} \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )} \left (\frac {a}{b}\right )^{\frac {1}{3}}} - \frac {18 \, {\left (d x + c\right )} a}{a^{2} + b^{2}} + \frac {3 \, {\left (b {\left (2 \, \left (\frac {a}{b}\right )^{\frac {2}{3}} + 1\right )} + a \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )} \log \left (\tan \left (d x + c\right )^{2} - \left (\frac {a}{b}\right )^{\frac {1}{3}} \tan \left (d x + c\right ) + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{a^{2} \left (\frac {a}{b}\right )^{\frac {2}{3}} + b^{2} \left (\frac {a}{b}\right )^{\frac {2}{3}}} - \frac {9 \, b \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{a^{2} + b^{2}} + \frac {6 \, {\left (b {\left (\left (\frac {a}{b}\right )^{\frac {2}{3}} - 1\right )} - a \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )} \log \left (\left (\frac {a}{b}\right )^{\frac {1}{3}} + \tan \left (d x + c\right )\right )}{a^{2} \left (\frac {a}{b}\right )^{\frac {2}{3}} + b^{2} \left (\frac {a}{b}\right )^{\frac {2}{3}}}}{18 \, d} \]
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Time = 0.53 (sec) , antiderivative size = 334, normalized size of antiderivative = 1.30 \[ \int \frac {1}{a+b \tan ^3(c+d x)} \, dx=\frac {\frac {2 \, {\left (a^{3} b^{2} \left (-\frac {a}{b}\right )^{\frac {1}{3}} + a b^{4} \left (-\frac {a}{b}\right )^{\frac {1}{3}} - a^{2} b^{3} - b^{5}\right )} \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left ({\left | -\left (-\frac {a}{b}\right )^{\frac {1}{3}} + \tan \left (d x + c\right ) \right |}\right )}{a^{5} b + 2 \, a^{3} b^{3} + a b^{5}} + \frac {6 \, {\left (\pi \left \lfloor \frac {d x + c}{\pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\left (\left (-\frac {a}{b}\right )^{\frac {1}{3}}\right ) + \arctan \left (\frac {\sqrt {3} {\left (\left (-\frac {a}{b}\right )^{\frac {1}{3}} + 2 \, \tan \left (d x + c\right )\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )\right )} {\left (\left (-a b^{2}\right )^{\frac {1}{3}} b^{2} + \left (-a b^{2}\right )^{\frac {2}{3}} a\right )}}{\sqrt {3} a^{3} b + \sqrt {3} a b^{3}} + \frac {6 \, {\left (d x + c\right )} a}{a^{2} + b^{2}} + \frac {{\left (\left (-a b^{2}\right )^{\frac {1}{3}} b^{2} - \left (-a b^{2}\right )^{\frac {2}{3}} a\right )} \log \left (\tan \left (d x + c\right )^{2} + \left (-\frac {a}{b}\right )^{\frac {1}{3}} \tan \left (d x + c\right ) + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{a^{3} b + a b^{3}} + \frac {3 \, b \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{a^{2} + b^{2}} - \frac {2 \, b \log \left ({\left | b \tan \left (d x + c\right )^{3} + a \right |}\right )}{a^{2} + b^{2}}}{6 \, d} \]
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Time = 13.18 (sec) , antiderivative size = 342, normalized size of antiderivative = 1.34 \[ \int \frac {1}{a+b \tan ^3(c+d x)} \, dx=\frac {\sum _{k=1}^3\ln \left (\mathrm {root}\left (27\,a^2\,b^2\,z^3+27\,a^4\,z^3+27\,a^2\,b\,z^2-b,z,k\right )\,\left (\mathrm {root}\left (27\,a^2\,b^2\,z^3+27\,a^4\,z^3+27\,a^2\,b\,z^2-b,z,k\right )\,\left (\mathrm {root}\left (27\,a^2\,b^2\,z^3+27\,a^4\,z^3+27\,a^2\,b\,z^2-b,z,k\right )\,\left (\mathrm {tan}\left (c+d\,x\right )\,\left (12\,b^6-69\,a^2\,b^4\right )+\mathrm {root}\left (27\,a^2\,b^2\,z^3+27\,a^4\,z^3+27\,a^2\,b\,z^2-b,z,k\right )\,\left (36\,a\,b^6-180\,a^3\,b^4+\mathrm {tan}\left (c+d\,x\right )\,\left (162\,a^2\,b^5-54\,a^4\,b^3\right )\right )-36\,a\,b^5+27\,a^3\,b^3\right )+13\,a\,b^4-16\,b^5\,\mathrm {tan}\left (c+d\,x\right )\right )+5\,b^4\,\mathrm {tan}\left (c+d\,x\right )\right )\right )\,\mathrm {root}\left (27\,a^2\,b^2\,z^3+27\,a^4\,z^3+27\,a^2\,b\,z^2-b,z,k\right )}{d}+\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )-\mathrm {i}\right )}{2\,d\,\left (b+a\,1{}\mathrm {i}\right )}+\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )\,1{}\mathrm {i}}{2\,d\,\left (a+b\,1{}\mathrm {i}\right )} \]
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