Integrand size = 23, antiderivative size = 525 \[ \int \frac {1}{\tan ^{\frac {5}{3}}(c+d x) (a+b \tan (c+d x))} \, dx=\frac {b \arctan \left (\sqrt {3}-2 \sqrt [3]{\tan (c+d x)}\right )}{2 \left (a^2+b^2\right ) d}-\frac {b \arctan \left (\sqrt {3}+2 \sqrt [3]{\tan (c+d x)}\right )}{2 \left (a^2+b^2\right ) d}+\frac {\sqrt {3} b^{8/3} \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} \sqrt [3]{\tan (c+d x)}}{\sqrt {3} \sqrt [3]{a}}\right )}{a^{5/3} \left (a^2+b^2\right ) d}+\frac {\sqrt {3} a \arctan \left (\frac {1-2 \tan ^{\frac {2}{3}}(c+d x)}{\sqrt {3}}\right )}{2 \left (a^2+b^2\right ) d}-\frac {b \arctan \left (\sqrt [3]{\tan (c+d x)}\right )}{\left (a^2+b^2\right ) d}-\frac {3 b^{8/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{\tan (c+d x)}\right )}{2 a^{5/3} \left (a^2+b^2\right ) d}+\frac {a \log \left (1+\tan ^{\frac {2}{3}}(c+d x)\right )}{2 \left (a^2+b^2\right ) d}+\frac {\sqrt {3} b \log \left (1-\sqrt {3} \sqrt [3]{\tan (c+d x)}+\tan ^{\frac {2}{3}}(c+d x)\right )}{4 \left (a^2+b^2\right ) d}-\frac {\sqrt {3} b \log \left (1+\sqrt {3} \sqrt [3]{\tan (c+d x)}+\tan ^{\frac {2}{3}}(c+d x)\right )}{4 \left (a^2+b^2\right ) d}+\frac {b^{8/3} \log (a+b \tan (c+d x))}{2 a^{5/3} \left (a^2+b^2\right ) d}-\frac {a \log \left (1-\tan ^{\frac {2}{3}}(c+d x)+\tan ^{\frac {4}{3}}(c+d x)\right )}{4 \left (a^2+b^2\right ) d}-\frac {3 a}{2 \left (a^2+b^2\right ) d \tan ^{\frac {2}{3}}(c+d x)}-\frac {3 b^2}{2 a \left (a^2+b^2\right ) d \tan ^{\frac {2}{3}}(c+d x)} \]
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Time = 0.83 (sec) , antiderivative size = 525, normalized size of antiderivative = 1.00, number of steps used = 30, number of rules used = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.783, Rules used = {3655, 3610, 3619, 3557, 335, 215, 648, 632, 210, 642, 209, 281, 298, 31, 3715, 53, 60, 631} \[ \int \frac {1}{\tan ^{\frac {5}{3}}(c+d x) (a+b \tan (c+d x))} \, dx=\frac {\sqrt {3} a \arctan \left (\frac {1-2 \tan ^{\frac {2}{3}}(c+d x)}{\sqrt {3}}\right )}{2 d \left (a^2+b^2\right )}+\frac {b \arctan \left (\sqrt {3}-2 \sqrt [3]{\tan (c+d x)}\right )}{2 d \left (a^2+b^2\right )}-\frac {b \arctan \left (2 \sqrt [3]{\tan (c+d x)}+\sqrt {3}\right )}{2 d \left (a^2+b^2\right )}-\frac {b \arctan \left (\sqrt [3]{\tan (c+d x)}\right )}{d \left (a^2+b^2\right )}-\frac {3 b^2}{2 a d \left (a^2+b^2\right ) \tan ^{\frac {2}{3}}(c+d x)}-\frac {3 a}{2 d \left (a^2+b^2\right ) \tan ^{\frac {2}{3}}(c+d x)}+\frac {\sqrt {3} b \log \left (\tan ^{\frac {2}{3}}(c+d x)-\sqrt {3} \sqrt [3]{\tan (c+d x)}+1\right )}{4 d \left (a^2+b^2\right )}-\frac {\sqrt {3} b \log \left (\tan ^{\frac {2}{3}}(c+d x)+\sqrt {3} \sqrt [3]{\tan (c+d x)}+1\right )}{4 d \left (a^2+b^2\right )}+\frac {a \log \left (\tan ^{\frac {2}{3}}(c+d x)+1\right )}{2 d \left (a^2+b^2\right )}-\frac {a \log \left (\tan ^{\frac {4}{3}}(c+d x)-\tan ^{\frac {2}{3}}(c+d x)+1\right )}{4 d \left (a^2+b^2\right )}+\frac {\sqrt {3} b^{8/3} \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} \sqrt [3]{\tan (c+d x)}}{\sqrt {3} \sqrt [3]{a}}\right )}{a^{5/3} d \left (a^2+b^2\right )}-\frac {3 b^{8/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{\tan (c+d x)}\right )}{2 a^{5/3} d \left (a^2+b^2\right )}+\frac {b^{8/3} \log (a+b \tan (c+d x))}{2 a^{5/3} d \left (a^2+b^2\right )} \]
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Rule 31
Rule 53
Rule 60
Rule 209
Rule 210
Rule 215
Rule 281
Rule 298
Rule 335
Rule 631
Rule 632
Rule 642
Rule 648
Rule 3557
Rule 3610
Rule 3619
Rule 3655
Rule 3715
Rubi steps \begin{align*} \text {integral}& = \frac {\int \frac {a-b \tan (c+d x)}{\tan ^{\frac {5}{3}}(c+d x)} \, dx}{a^2+b^2}+\frac {b^2 \int \frac {1+\tan ^2(c+d x)}{\tan ^{\frac {5}{3}}(c+d x) (a+b \tan (c+d x))} \, dx}{a^2+b^2} \\ & = -\frac {3 a}{2 \left (a^2+b^2\right ) d \tan ^{\frac {2}{3}}(c+d x)}+\frac {\int \frac {-b-a \tan (c+d x)}{\tan ^{\frac {2}{3}}(c+d x)} \, dx}{a^2+b^2}+\frac {b^2 \text {Subst}\left (\int \frac {1}{x^{5/3} (a+b x)} \, dx,x,\tan (c+d x)\right )}{\left (a^2+b^2\right ) d} \\ & = -\frac {3 a}{2 \left (a^2+b^2\right ) d \tan ^{\frac {2}{3}}(c+d x)}-\frac {3 b^2}{2 a \left (a^2+b^2\right ) d \tan ^{\frac {2}{3}}(c+d x)}-\frac {a \int \sqrt [3]{\tan (c+d x)} \, dx}{a^2+b^2}-\frac {b \int \frac {1}{\tan ^{\frac {2}{3}}(c+d x)} \, dx}{a^2+b^2}-\frac {b^3 \text {Subst}\left (\int \frac {1}{x^{2/3} (a+b x)} \, dx,x,\tan (c+d x)\right )}{a \left (a^2+b^2\right ) d} \\ & = \frac {b^{8/3} \log (a+b \tan (c+d x))}{2 a^{5/3} \left (a^2+b^2\right ) d}-\frac {3 a}{2 \left (a^2+b^2\right ) d \tan ^{\frac {2}{3}}(c+d x)}-\frac {3 b^2}{2 a \left (a^2+b^2\right ) d \tan ^{\frac {2}{3}}(c+d x)}-\frac {a \text {Subst}\left (\int \frac {\sqrt [3]{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 {1}{x^{2/3} \left (1+x^2\right )} \, dx,x,\tan (c+d x)\right )}{\left (a^2+b^2\right ) d}-\frac {\left (3 b^{7/3}\right ) \text {Subst}\left (\int \frac {1}{\frac {a^{2/3}}{b^{2/3}}-\frac {\sqrt [3]{a} x}{\sqrt [3]{b}}+x^2} \, dx,x,\sqrt [3]{\tan (c+d x)}\right )}{2 a^{4/3} \left (a^2+b^2\right ) d}-\frac {\left (3 b^{8/3}\right ) \text {Subst}\left (\int \frac {1}{\frac {\sqrt [3]{a}}{\sqrt [3]{b}}+x} \, dx,x,\sqrt [3]{\tan (c+d x)}\right )}{2 a^{5/3} \left (a^2+b^2\right ) d} \\ & = -\frac {3 b^{8/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{\tan (c+d x)}\right )}{2 a^{5/3} \left (a^2+b^2\right ) d}+\frac {b^{8/3} \log (a+b \tan (c+d x))}{2 a^{5/3} \left (a^2+b^2\right ) d}-\frac {3 a}{2 \left (a^2+b^2\right ) d \tan ^{\frac {2}{3}}(c+d x)}-\frac {3 b^2}{2 a \left (a^2+b^2\right ) d \tan ^{\frac {2}{3}}(c+d x)}-\frac {(3 a) \text {Subst}\left (\int \frac {x^3}{1+x^6} \, dx,x,\sqrt [3]{\tan (c+d x)}\right )}{\left (a^2+b^2\right ) d}-\frac {(3 b) \text {Subst}\left (\int \frac {1}{1+x^6} \, dx,x,\sqrt [3]{\tan (c+d x)}\right )}{\left (a^2+b^2\right ) d}-\frac {\left (3 b^{8/3}\right ) \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\frac {2 \sqrt [3]{b} \sqrt [3]{\tan (c+d x)}}{\sqrt [3]{a}}\right )}{a^{5/3} \left (a^2+b^2\right ) d} \\ & = \frac {\sqrt {3} b^{8/3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} \sqrt [3]{\tan (c+d x)}}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{a^{5/3} \left (a^2+b^2\right ) d}-\frac {3 b^{8/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{\tan (c+d x)}\right )}{2 a^{5/3} \left (a^2+b^2\right ) d}+\frac {b^{8/3} \log (a+b \tan (c+d x))}{2 a^{5/3} \left (a^2+b^2\right ) d}-\frac {3 a}{2 \left (a^2+b^2\right ) d \tan ^{\frac {2}{3}}(c+d x)}-\frac {3 b^2}{2 a \left (a^2+b^2\right ) d \tan ^{\frac {2}{3}}(c+d x)}-\frac {(3 a) \text {Subst}\left (\int \frac {x}{1+x^3} \, dx,x,\tan ^{\frac {2}{3}}(c+d x)\right )}{2 \left (a^2+b^2\right ) d}-\frac {b \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\sqrt [3]{\tan (c+d x)}\right )}{\left (a^2+b^2\right ) d}-\frac {b \text {Subst}\left (\int \frac {1-\frac {\sqrt {3} x}{2}}{1-\sqrt {3} x+x^2} \, dx,x,\sqrt [3]{\tan (c+d x)}\right )}{\left (a^2+b^2\right ) d}-\frac {b \text {Subst}\left (\int \frac {1+\frac {\sqrt {3} x}{2}}{1+\sqrt {3} x+x^2} \, dx,x,\sqrt [3]{\tan (c+d x)}\right )}{\left (a^2+b^2\right ) d} \\ & = \frac {\sqrt {3} b^{8/3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} \sqrt [3]{\tan (c+d x)}}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{a^{5/3} \left (a^2+b^2\right ) d}-\frac {b \arctan \left (\sqrt [3]{\tan (c+d x)}\right )}{\left (a^2+b^2\right ) d}-\frac {3 b^{8/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{\tan (c+d x)}\right )}{2 a^{5/3} \left (a^2+b^2\right ) d}+\frac {b^{8/3} \log (a+b \tan (c+d x))}{2 a^{5/3} \left (a^2+b^2\right ) d}-\frac {3 a}{2 \left (a^2+b^2\right ) d \tan ^{\frac {2}{3}}(c+d x)}-\frac {3 b^2}{2 a \left (a^2+b^2\right ) d \tan ^{\frac {2}{3}}(c+d x)}+\frac {a \text {Subst}\left (\int \frac {1}{1+x} \, dx,x,\tan ^{\frac {2}{3}}(c+d x)\right )}{2 \left (a^2+b^2\right ) d}-\frac {a \text {Subst}\left (\int \frac {1+x}{1-x+x^2} \, dx,x,\tan ^{\frac {2}{3}}(c+d x)\right )}{2 \left (a^2+b^2\right ) d}-\frac {b \text {Subst}\left (\int \frac {1}{1-\sqrt {3} x+x^2} \, dx,x,\sqrt [3]{\tan (c+d x)}\right )}{4 \left (a^2+b^2\right ) d}-\frac {b \text {Subst}\left (\int \frac {1}{1+\sqrt {3} x+x^2} \, dx,x,\sqrt [3]{\tan (c+d x)}\right )}{4 \left (a^2+b^2\right ) d}+\frac {\left (\sqrt {3} b\right ) \text {Subst}\left (\int \frac {-\sqrt {3}+2 x}{1-\sqrt {3} x+x^2} \, dx,x,\sqrt [3]{\tan (c+d x)}\right )}{4 \left (a^2+b^2\right ) d}-\frac {\left (\sqrt {3} b\right ) \text {Subst}\left (\int \frac {\sqrt {3}+2 x}{1+\sqrt {3} x+x^2} \, dx,x,\sqrt [3]{\tan (c+d x)}\right )}{4 \left (a^2+b^2\right ) d} \\ & = \frac {\sqrt {3} b^{8/3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} \sqrt [3]{\tan (c+d x)}}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{a^{5/3} \left (a^2+b^2\right ) d}-\frac {b \arctan \left (\sqrt [3]{\tan (c+d x)}\right )}{\left (a^2+b^2\right ) d}-\frac {3 b^{8/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{\tan (c+d x)}\right )}{2 a^{5/3} \left (a^2+b^2\right ) d}+\frac {a \log \left (1+\tan ^{\frac {2}{3}}(c+d x)\right )}{2 \left (a^2+b^2\right ) d}+\frac {\sqrt {3} b \log \left (1-\sqrt {3} \sqrt [3]{\tan (c+d x)}+\tan ^{\frac {2}{3}}(c+d x)\right )}{4 \left (a^2+b^2\right ) d}-\frac {\sqrt {3} b \log \left (1+\sqrt {3} \sqrt [3]{\tan (c+d x)}+\tan ^{\frac {2}{3}}(c+d x)\right )}{4 \left (a^2+b^2\right ) d}+\frac {b^{8/3} \log (a+b \tan (c+d x))}{2 a^{5/3} \left (a^2+b^2\right ) d}-\frac {3 a}{2 \left (a^2+b^2\right ) d \tan ^{\frac {2}{3}}(c+d x)}-\frac {3 b^2}{2 a \left (a^2+b^2\right ) d \tan ^{\frac {2}{3}}(c+d x)}-\frac {a \text {Subst}\left (\int \frac {-1+2 x}{1-x+x^2} \, dx,x,\tan ^{\frac {2}{3}}(c+d x)\right )}{4 \left (a^2+b^2\right ) d}-\frac {(3 a) \text {Subst}\left (\int \frac {1}{1-x+x^2} \, dx,x,\tan ^{\frac {2}{3}}(c+d x)\right )}{4 \left (a^2+b^2\right ) d}+\frac {b \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,-\sqrt {3}+2 \sqrt [3]{\tan (c+d x)}\right )}{2 \left (a^2+b^2\right ) d}+\frac {b \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,\sqrt {3}+2 \sqrt [3]{\tan (c+d x)}\right )}{2 \left (a^2+b^2\right ) d} \\ & = \frac {b \arctan \left (\sqrt {3}-2 \sqrt [3]{\tan (c+d x)}\right )}{2 \left (a^2+b^2\right ) d}-\frac {b \arctan \left (\sqrt {3}+2 \sqrt [3]{\tan (c+d x)}\right )}{2 \left (a^2+b^2\right ) d}+\frac {\sqrt {3} b^{8/3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} \sqrt [3]{\tan (c+d x)}}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{a^{5/3} \left (a^2+b^2\right ) d}-\frac {b \arctan \left (\sqrt [3]{\tan (c+d x)}\right )}{\left (a^2+b^2\right ) d}-\frac {3 b^{8/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{\tan (c+d x)}\right )}{2 a^{5/3} \left (a^2+b^2\right ) d}+\frac {a \log \left (1+\tan ^{\frac {2}{3}}(c+d x)\right )}{2 \left (a^2+b^2\right ) d}+\frac {\sqrt {3} b \log \left (1-\sqrt {3} \sqrt [3]{\tan (c+d x)}+\tan ^{\frac {2}{3}}(c+d x)\right )}{4 \left (a^2+b^2\right ) d}-\frac {\sqrt {3} b \log \left (1+\sqrt {3} \sqrt [3]{\tan (c+d x)}+\tan ^{\frac {2}{3}}(c+d x)\right )}{4 \left (a^2+b^2\right ) d}+\frac {b^{8/3} \log (a+b \tan (c+d x))}{2 a^{5/3} \left (a^2+b^2\right ) d}-\frac {a \log \left (1-\tan ^{\frac {2}{3}}(c+d x)+\tan ^{\frac {4}{3}}(c+d x)\right )}{4 \left (a^2+b^2\right ) d}-\frac {3 a}{2 \left (a^2+b^2\right ) d \tan ^{\frac {2}{3}}(c+d x)}-\frac {3 b^2}{2 a \left (a^2+b^2\right ) d \tan ^{\frac {2}{3}}(c+d x)}+\frac {(3 a) \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,-1+2 \tan ^{\frac {2}{3}}(c+d x)\right )}{2 \left (a^2+b^2\right ) d} \\ & = \frac {b \arctan \left (\sqrt {3}-2 \sqrt [3]{\tan (c+d x)}\right )}{2 \left (a^2+b^2\right ) d}-\frac {b \arctan \left (\sqrt {3}+2 \sqrt [3]{\tan (c+d x)}\right )}{2 \left (a^2+b^2\right ) d}+\frac {\sqrt {3} b^{8/3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} \sqrt [3]{\tan (c+d x)}}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{a^{5/3} \left (a^2+b^2\right ) d}+\frac {\sqrt {3} a \arctan \left (\frac {1-2 \tan ^{\frac {2}{3}}(c+d x)}{\sqrt {3}}\right )}{2 \left (a^2+b^2\right ) d}-\frac {b \arctan \left (\sqrt [3]{\tan (c+d x)}\right )}{\left (a^2+b^2\right ) d}-\frac {3 b^{8/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{\tan (c+d x)}\right )}{2 a^{5/3} \left (a^2+b^2\right ) d}+\frac {a \log \left (1+\tan ^{\frac {2}{3}}(c+d x)\right )}{2 \left (a^2+b^2\right ) d}+\frac {\sqrt {3} b \log \left (1-\sqrt {3} \sqrt [3]{\tan (c+d x)}+\tan ^{\frac {2}{3}}(c+d x)\right )}{4 \left (a^2+b^2\right ) d}-\frac {\sqrt {3} b \log \left (1+\sqrt {3} \sqrt [3]{\tan (c+d x)}+\tan ^{\frac {2}{3}}(c+d x)\right )}{4 \left (a^2+b^2\right ) d}+\frac {b^{8/3} \log (a+b \tan (c+d x))}{2 a^{5/3} \left (a^2+b^2\right ) d}-\frac {a \log \left (1-\tan ^{\frac {2}{3}}(c+d x)+\tan ^{\frac {4}{3}}(c+d x)\right )}{4 \left (a^2+b^2\right ) d}-\frac {3 a}{2 \left (a^2+b^2\right ) d \tan ^{\frac {2}{3}}(c+d x)}-\frac {3 b^2}{2 a \left (a^2+b^2\right ) d \tan ^{\frac {2}{3}}(c+d x)} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.24 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.20 \[ \int \frac {1}{\tan ^{\frac {5}{3}}(c+d x) (a+b \tan (c+d x))} \, dx=-\frac {3 \left (b^2 \operatorname {Hypergeometric2F1}\left (-\frac {2}{3},1,\frac {1}{3},-\frac {b \tan (c+d x)}{a}\right )+a \left (a \operatorname {Hypergeometric2F1}\left (-\frac {1}{3},1,\frac {2}{3},-\tan ^2(c+d x)\right )+2 b \operatorname {Hypergeometric2F1}\left (\frac {1}{6},1,\frac {7}{6},-\tan ^2(c+d x)\right ) \tan (c+d x)\right )\right )}{2 a \left (a^2+b^2\right ) d \tan ^{\frac {2}{3}}(c+d x)} \]
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Time = 0.17 (sec) , antiderivative size = 350, normalized size of antiderivative = 0.67
method | result | size |
derivativedivides | \(\frac {-\frac {3}{2 a \tan \left (d x +c \right )^{\frac {2}{3}}}+\frac {\frac {3 \left (-\sqrt {3}\, b -a \right ) \ln \left (1+\sqrt {3}\, \left (\tan ^{\frac {1}{3}}\left (d x +c \right )\right )+\tan ^{\frac {2}{3}}\left (d x +c \right )\right )}{4}+3 \left (-2 b -\frac {\left (-\sqrt {3}\, b -a \right ) \sqrt {3}}{2}\right ) \arctan \left (\sqrt {3}+2 \left (\tan ^{\frac {1}{3}}\left (d x +c \right )\right )\right )-\frac {3 \left (-\sqrt {3}\, b +a \right ) \ln \left (1-\sqrt {3}\, \left (\tan ^{\frac {1}{3}}\left (d x +c \right )\right )+\tan ^{\frac {2}{3}}\left (d x +c \right )\right )}{4}-3 \left (2 b +\frac {\left (-\sqrt {3}\, b +a \right ) \sqrt {3}}{2}\right ) \arctan \left (-\sqrt {3}+2 \left (\tan ^{\frac {1}{3}}\left (d x +c \right )\right )\right )}{3 a^{2}+3 b^{2}}+\frac {\frac {3 a \ln \left (1+\tan ^{\frac {2}{3}}\left (d x +c \right )\right )}{2}-3 b \arctan \left (\tan ^{\frac {1}{3}}\left (d x +c \right )\right )}{3 a^{2}+3 b^{2}}-\frac {3 \left (\frac {\ln \left (\tan ^{\frac {1}{3}}\left (d x +c \right )+\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}-\frac {\ln \left (\tan ^{\frac {2}{3}}\left (d x +c \right )-\left (\frac {a}{b}\right )^{\frac {1}{3}} \left (\tan ^{\frac {1}{3}}\left (d x +c \right )\right )+\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \left (\tan ^{\frac {1}{3}}\left (d x +c \right )\right )}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}\right ) b^{3}}{a \left (a^{2}+b^{2}\right )}}{d}\) | \(350\) |
default | \(\frac {-\frac {3}{2 a \tan \left (d x +c \right )^{\frac {2}{3}}}+\frac {\frac {3 \left (-\sqrt {3}\, b -a \right ) \ln \left (1+\sqrt {3}\, \left (\tan ^{\frac {1}{3}}\left (d x +c \right )\right )+\tan ^{\frac {2}{3}}\left (d x +c \right )\right )}{4}+3 \left (-2 b -\frac {\left (-\sqrt {3}\, b -a \right ) \sqrt {3}}{2}\right ) \arctan \left (\sqrt {3}+2 \left (\tan ^{\frac {1}{3}}\left (d x +c \right )\right )\right )-\frac {3 \left (-\sqrt {3}\, b +a \right ) \ln \left (1-\sqrt {3}\, \left (\tan ^{\frac {1}{3}}\left (d x +c \right )\right )+\tan ^{\frac {2}{3}}\left (d x +c \right )\right )}{4}-3 \left (2 b +\frac {\left (-\sqrt {3}\, b +a \right ) \sqrt {3}}{2}\right ) \arctan \left (-\sqrt {3}+2 \left (\tan ^{\frac {1}{3}}\left (d x +c \right )\right )\right )}{3 a^{2}+3 b^{2}}+\frac {\frac {3 a \ln \left (1+\tan ^{\frac {2}{3}}\left (d x +c \right )\right )}{2}-3 b \arctan \left (\tan ^{\frac {1}{3}}\left (d x +c \right )\right )}{3 a^{2}+3 b^{2}}-\frac {3 \left (\frac {\ln \left (\tan ^{\frac {1}{3}}\left (d x +c \right )+\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}-\frac {\ln \left (\tan ^{\frac {2}{3}}\left (d x +c \right )-\left (\frac {a}{b}\right )^{\frac {1}{3}} \left (\tan ^{\frac {1}{3}}\left (d x +c \right )\right )+\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \left (\tan ^{\frac {1}{3}}\left (d x +c \right )\right )}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}\right ) b^{3}}{a \left (a^{2}+b^{2}\right )}}{d}\) | \(350\) |
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Result contains complex when optimal does not.
Time = 1.85 (sec) , antiderivative size = 74494, normalized size of antiderivative = 141.89 \[ \int \frac {1}{\tan ^{\frac {5}{3}}(c+d x) (a+b \tan (c+d x))} \, dx=\text {Too large to display} \]
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\[ \int \frac {1}{\tan ^{\frac {5}{3}}(c+d x) (a+b \tan (c+d x))} \, dx=\int \frac {1}{\left (a + b \tan {\left (c + d x \right )}\right ) \tan ^{\frac {5}{3}}{\left (c + d x \right )}}\, dx \]
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Timed out. \[ \int \frac {1}{\tan ^{\frac {5}{3}}(c+d x) (a+b \tan (c+d x))} \, dx=\text {Timed out} \]
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Time = 1.23 (sec) , antiderivative size = 457, normalized size of antiderivative = 0.87 \[ \int \frac {1}{\tan ^{\frac {5}{3}}(c+d x) (a+b \tan (c+d x))} \, dx=\frac {b^{3} \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 )^{\frac {1}{3}} \right |}\right )}{a^{4} d + a^{2} b^{2} d} - \frac {3 \, \left (-a b^{2}\right )^{\frac {1}{3}} b^{2} \arctan \left (\frac {\sqrt {3} {\left (\left (-\frac {a}{b}\right )^{\frac {1}{3}} + 2 \, \tan \left (d x + c\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{{\left (\sqrt {3} a^{4} + \sqrt {3} a^{2} b^{2}\right )} d} - \frac {\left (-a b^{2}\right )^{\frac {1}{3}} b^{2} \log \left (\left (-\frac {a}{b}\right )^{\frac {2}{3}} + \left (-\frac {a}{b}\right )^{\frac {1}{3}} \tan \left (d x + c\right )^{\frac {1}{3}} + \tan \left (d x + c\right )^{\frac {2}{3}}\right )}{2 \, {\left (a^{4} + a^{2} b^{2}\right )} d} + \frac {{\left (\sqrt {3} a - b\right )} \arctan \left (\sqrt {3} + 2 \, \tan \left (d x + c\right )^{\frac {1}{3}}\right )}{2 \, {\left (a^{2} d + b^{2} d\right )}} - \frac {{\left (\sqrt {3} a + b\right )} \arctan \left (-\sqrt {3} + 2 \, \tan \left (d x + c\right )^{\frac {1}{3}}\right )}{2 \, {\left (a^{2} d + b^{2} d\right )}} - \frac {b \arctan \left (\tan \left (d x + c\right )^{\frac {1}{3}}\right )}{a^{2} d + b^{2} d} - \frac {a \log \left (\tan \left (d x + c\right )^{\frac {4}{3}} - \tan \left (d x + c\right )^{\frac {2}{3}} + 1\right )}{4 \, {\left (a^{2} d + b^{2} d\right )}} - \frac {3 \, b \log \left (\sqrt {3} \tan \left (d x + c\right )^{\frac {1}{3}} + \tan \left (d x + c\right )^{\frac {2}{3}} + 1\right )}{4 \, {\left (\sqrt {3} a^{2} d + \sqrt {3} b^{2} d\right )}} + \frac {3 \, b \log \left (-\sqrt {3} \tan \left (d x + c\right )^{\frac {1}{3}} + \tan \left (d x + c\right )^{\frac {2}{3}} + 1\right )}{4 \, {\left (\sqrt {3} a^{2} d + \sqrt {3} b^{2} d\right )}} + \frac {a \log \left (\tan \left (d x + c\right )^{\frac {2}{3}} + 1\right )}{2 \, {\left (a^{2} d + b^{2} d\right )}} - \frac {3}{2 \, a d \tan \left (d x + c\right )^{\frac {2}{3}}} \]
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Time = 6.77 (sec) , antiderivative size = 3159, normalized size of antiderivative = 6.02 \[ \int \frac {1}{\tan ^{\frac {5}{3}}(c+d x) (a+b \tan (c+d x))} \, dx=\text {Too large to display} \]
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