Integrand size = 21, antiderivative size = 295 \[ \int \frac {\cos ^4(c+d x)}{(a+b \tan (c+d x))^3} \, dx=\frac {3 a \left (a^6+7 a^4 b^2+35 a^2 b^4-35 b^6\right ) x}{8 \left (a^2+b^2\right )^5}+\frac {3 b^5 \left (7 a^2-b^2\right ) \log (a \cos (c+d x)+b \sin (c+d x))}{\left (a^2+b^2\right )^5 d}+\frac {3 b \left (a^4+5 a^2 b^2-4 b^4\right )}{8 \left (a^2+b^2\right )^3 d (a+b \tan (c+d x))^2}+\frac {\cos ^4(c+d x) (b+a \tan (c+d x))}{4 \left (a^2+b^2\right ) d (a+b \tan (c+d x))^2}+\frac {3 a b \left (a^4+6 a^2 b^2-27 b^4\right )}{8 \left (a^2+b^2\right )^4 d (a+b \tan (c+d x))}-\frac {\cos ^2(c+d x) \left (2 b \left (a^2-3 b^2\right )-a \left (3 a^2+11 b^2\right ) \tan (c+d x)\right )}{8 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))^2} \]
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Time = 0.47 (sec) , antiderivative size = 295, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {3587, 755, 837, 815, 649, 209, 266} \[ \int \frac {\cos ^4(c+d x)}{(a+b \tan (c+d x))^3} \, dx=\frac {\cos ^4(c+d x) (a \tan (c+d x)+b)}{4 d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}-\frac {\cos ^2(c+d x) \left (2 b \left (a^2-3 b^2\right )-a \left (3 a^2+11 b^2\right ) \tan (c+d x)\right )}{8 d \left (a^2+b^2\right )^2 (a+b \tan (c+d x))^2}+\frac {3 b^5 \left (7 a^2-b^2\right ) \log (a \cos (c+d x)+b \sin (c+d x))}{d \left (a^2+b^2\right )^5}+\frac {3 a b \left (a^4+6 a^2 b^2-27 b^4\right )}{8 d \left (a^2+b^2\right )^4 (a+b \tan (c+d x))}+\frac {3 b \left (a^4+5 a^2 b^2-4 b^4\right )}{8 d \left (a^2+b^2\right )^3 (a+b \tan (c+d x))^2}+\frac {3 a x \left (a^6+7 a^4 b^2+35 a^2 b^4-35 b^6\right )}{8 \left (a^2+b^2\right )^5} \]
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Rule 209
Rule 266
Rule 649
Rule 755
Rule 815
Rule 837
Rule 3587
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {1}{(a+x)^3 \left (1+\frac {x^2}{b^2}\right )^3} \, dx,x,b \tan (c+d x)\right )}{b d} \\ & = \frac {\cos ^4(c+d x) (b+a \tan (c+d x))}{4 \left (a^2+b^2\right ) d (a+b \tan (c+d x))^2}-\frac {b \text {Subst}\left (\int \frac {-3 \left (2+\frac {a^2}{b^2}\right )-\frac {5 a x}{b^2}}{(a+x)^3 \left (1+\frac {x^2}{b^2}\right )^2} \, dx,x,b \tan (c+d x)\right )}{4 \left (a^2+b^2\right ) d} \\ & = \frac {\cos ^4(c+d x) (b+a \tan (c+d x))}{4 \left (a^2+b^2\right ) d (a+b \tan (c+d x))^2}-\frac {\cos ^2(c+d x) \left (2 b \left (a^2-3 b^2\right )-a \left (3 a^2+11 b^2\right ) \tan (c+d x)\right )}{8 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))^2}+\frac {b^5 \text {Subst}\left (\int \frac {\frac {3 \left (a^4+a^2 b^2+8 b^4\right )}{b^6}+\frac {3 a \left (3 a^2+11 b^2\right ) x}{b^6}}{(a+x)^3 \left (1+\frac {x^2}{b^2}\right )} \, dx,x,b \tan (c+d x)\right )}{8 \left (a^2+b^2\right )^2 d} \\ & = \frac {\cos ^4(c+d x) (b+a \tan (c+d x))}{4 \left (a^2+b^2\right ) d (a+b \tan (c+d x))^2}-\frac {\cos ^2(c+d x) \left (2 b \left (a^2-3 b^2\right )-a \left (3 a^2+11 b^2\right ) \tan (c+d x)\right )}{8 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))^2}+\frac {b^5 \text {Subst}\left (\int \left (\frac {6 \left (-a^4-5 a^2 b^2+4 b^4\right )}{b^4 \left (a^2+b^2\right ) (a+x)^3}+\frac {3 a \left (-a^4-6 a^2 b^2+27 b^4\right )}{b^4 \left (a^2+b^2\right )^2 (a+x)^2}+\frac {24 \left (7 a^2-b^2\right )}{\left (a^2+b^2\right )^3 (a+x)}+\frac {3 \left (a \left (a^6+7 a^4 b^2+35 a^2 b^4-35 b^6\right )-8 b^4 \left (7 a^2-b^2\right ) x\right )}{b^4 \left (a^2+b^2\right )^3 \left (b^2+x^2\right )}\right ) \, dx,x,b \tan (c+d x)\right )}{8 \left (a^2+b^2\right )^2 d} \\ & = \frac {3 b^5 \left (7 a^2-b^2\right ) \log (a+b \tan (c+d x))}{\left (a^2+b^2\right )^5 d}+\frac {3 b \left (a^4+5 a^2 b^2-4 b^4\right )}{8 \left (a^2+b^2\right )^3 d (a+b \tan (c+d x))^2}+\frac {\cos ^4(c+d x) (b+a \tan (c+d x))}{4 \left (a^2+b^2\right ) d (a+b \tan (c+d x))^2}+\frac {3 a b \left (a^4+6 a^2 b^2-27 b^4\right )}{8 \left (a^2+b^2\right )^4 d (a+b \tan (c+d x))}-\frac {\cos ^2(c+d x) \left (2 b \left (a^2-3 b^2\right )-a \left (3 a^2+11 b^2\right ) \tan (c+d x)\right )}{8 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))^2}+\frac {(3 b) \text {Subst}\left (\int \frac {a \left (a^6+7 a^4 b^2+35 a^2 b^4-35 b^6\right )-8 b^4 \left (7 a^2-b^2\right ) x}{b^2+x^2} \, dx,x,b \tan (c+d x)\right )}{8 \left (a^2+b^2\right )^5 d} \\ & = \frac {3 b^5 \left (7 a^2-b^2\right ) \log (a+b \tan (c+d x))}{\left (a^2+b^2\right )^5 d}+\frac {3 b \left (a^4+5 a^2 b^2-4 b^4\right )}{8 \left (a^2+b^2\right )^3 d (a+b \tan (c+d x))^2}+\frac {\cos ^4(c+d x) (b+a \tan (c+d x))}{4 \left (a^2+b^2\right ) d (a+b \tan (c+d x))^2}+\frac {3 a b \left (a^4+6 a^2 b^2-27 b^4\right )}{8 \left (a^2+b^2\right )^4 d (a+b \tan (c+d x))}-\frac {\cos ^2(c+d x) \left (2 b \left (a^2-3 b^2\right )-a \left (3 a^2+11 b^2\right ) \tan (c+d x)\right )}{8 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))^2}-\frac {\left (3 b^5 \left (7 a^2-b^2\right )\right ) \text {Subst}\left (\int \frac {x}{b^2+x^2} \, dx,x,b \tan (c+d x)\right )}{\left (a^2+b^2\right )^5 d}+\frac {\left (3 a b \left (a^6+7 a^4 b^2+35 a^2 b^4-35 b^6\right )\right ) \text {Subst}\left (\int \frac {1}{b^2+x^2} \, dx,x,b \tan (c+d x)\right )}{8 \left (a^2+b^2\right )^5 d} \\ & = \frac {3 a \left (a^6+7 a^4 b^2+35 a^2 b^4-35 b^6\right ) x}{8 \left (a^2+b^2\right )^5}+\frac {3 b^5 \left (7 a^2-b^2\right ) \log (\cos (c+d x))}{\left (a^2+b^2\right )^5 d}+\frac {3 b^5 \left (7 a^2-b^2\right ) \log (a+b \tan (c+d x))}{\left (a^2+b^2\right )^5 d}+\frac {3 b \left (a^4+5 a^2 b^2-4 b^4\right )}{8 \left (a^2+b^2\right )^3 d (a+b \tan (c+d x))^2}+\frac {\cos ^4(c+d x) (b+a \tan (c+d x))}{4 \left (a^2+b^2\right ) d (a+b \tan (c+d x))^2}+\frac {3 a b \left (a^4+6 a^2 b^2-27 b^4\right )}{8 \left (a^2+b^2\right )^4 d (a+b \tan (c+d x))}-\frac {\cos ^2(c+d x) \left (2 b \left (a^2-3 b^2\right )-a \left (3 a^2+11 b^2\right ) \tan (c+d x)\right )}{8 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))^2} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(596\) vs. \(2(295)=590\).
Time = 6.29 (sec) , antiderivative size = 596, normalized size of antiderivative = 2.02 \[ \int \frac {\cos ^4(c+d x)}{(a+b \tan (c+d x))^3} \, dx=\frac {b^5 \left (\frac {\cos ^4(c+d x) \left (b^2+a b \tan (c+d x)\right )}{4 b^6 \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}-\frac {\frac {\cos ^2(c+d x) \left (5 a^2 b^2-3 b^2 \left (a^2+2 b^2\right )+b \left (-5 a b^2-3 a \left (a^2+2 b^2\right )\right ) \tan (c+d x)\right )}{2 b^4 \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}-\frac {\left (-3 a^2 \left (3 a^2+11 b^2\right )+3 \left (a^4+a^2 b^2+8 b^4\right )\right ) \left (-\frac {\left (3 a^2-b^2-\frac {a^3-3 a b^2}{\sqrt {-b^2}}\right ) \log \left (\sqrt {-b^2}-b \tan (c+d x)\right )}{2 \left (a^2+b^2\right )^3}+\frac {\left (3 a^2-b^2\right ) \log (a+b \tan (c+d x))}{\left (a^2+b^2\right )^3}-\frac {\left (3 a^2-b^2+\frac {a^3-3 a b^2}{\sqrt {-b^2}}\right ) \log \left (\sqrt {-b^2}+b \tan (c+d x)\right )}{2 \left (a^2+b^2\right )^3}-\frac {1}{2 \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}-\frac {2 a}{\left (a^2+b^2\right )^2 (a+b \tan (c+d x))}\right )+3 a \left (3 a^2+11 b^2\right ) \left (-\frac {\left (2 a-\frac {a^2-b^2}{\sqrt {-b^2}}\right ) \log \left (\sqrt {-b^2}-b \tan (c+d x)\right )}{2 \left (a^2+b^2\right )^2}+\frac {2 a \log (a+b \tan (c+d x))}{\left (a^2+b^2\right )^2}-\frac {\left (2 a+\frac {a^2-b^2}{\sqrt {-b^2}}\right ) \log \left (\sqrt {-b^2}+b \tan (c+d x)\right )}{2 \left (a^2+b^2\right )^2}-\frac {1}{\left (a^2+b^2\right ) (a+b \tan (c+d x))}\right )}{2 b^2 \left (a^2+b^2\right )}}{4 b^2 \left (a^2+b^2\right )}\right )}{d} \]
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Time = 43.90 (sec) , antiderivative size = 312, normalized size of antiderivative = 1.06
method | result | size |
derivativedivides | \(\frac {-\frac {b^{5}}{2 \left (a^{2}+b^{2}\right )^{3} \left (a +b \tan \left (d x +c \right )\right )^{2}}-\frac {6 b^{5} a}{\left (a^{2}+b^{2}\right )^{4} \left (a +b \tan \left (d x +c \right )\right )}+\frac {3 b^{5} \left (7 a^{2}-b^{2}\right ) \ln \left (a +b \tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{5}}+\frac {\frac {\left (\frac {3}{8} a^{7}+\frac {21}{8} a^{5} b^{2}-\frac {15}{8} a^{3} b^{4}-\frac {33}{8} a \,b^{6}\right ) \left (\tan ^{3}\left (d x +c \right )\right )+\left (5 a^{4} b^{3}+4 b^{5} a^{2}-b^{7}\right ) \left (\tan ^{2}\left (d x +c \right )\right )+\left (\frac {19}{8} a^{5} b^{2}-\frac {39}{8} a \,b^{6}+\frac {5}{8} a^{7}-\frac {25}{8} a^{3} b^{4}\right ) \tan \left (d x +c \right )+\frac {3 a^{6} b}{4}+\frac {25 a^{4} b^{3}}{4}+\frac {17 b^{5} a^{2}}{4}-\frac {5 b^{7}}{4}}{\left (1+\tan ^{2}\left (d x +c \right )\right )^{2}}+\frac {3 \left (-56 b^{5} a^{2}+8 b^{7}\right ) \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{16}+\frac {3 \left (a^{7}+7 a^{5} b^{2}+35 a^{3} b^{4}-35 a \,b^{6}\right ) \arctan \left (\tan \left (d x +c \right )\right )}{8}}{\left (a^{2}+b^{2}\right )^{5}}}{d}\) | \(312\) |
default | \(\frac {-\frac {b^{5}}{2 \left (a^{2}+b^{2}\right )^{3} \left (a +b \tan \left (d x +c \right )\right )^{2}}-\frac {6 b^{5} a}{\left (a^{2}+b^{2}\right )^{4} \left (a +b \tan \left (d x +c \right )\right )}+\frac {3 b^{5} \left (7 a^{2}-b^{2}\right ) \ln \left (a +b \tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{5}}+\frac {\frac {\left (\frac {3}{8} a^{7}+\frac {21}{8} a^{5} b^{2}-\frac {15}{8} a^{3} b^{4}-\frac {33}{8} a \,b^{6}\right ) \left (\tan ^{3}\left (d x +c \right )\right )+\left (5 a^{4} b^{3}+4 b^{5} a^{2}-b^{7}\right ) \left (\tan ^{2}\left (d x +c \right )\right )+\left (\frac {19}{8} a^{5} b^{2}-\frac {39}{8} a \,b^{6}+\frac {5}{8} a^{7}-\frac {25}{8} a^{3} b^{4}\right ) \tan \left (d x +c \right )+\frac {3 a^{6} b}{4}+\frac {25 a^{4} b^{3}}{4}+\frac {17 b^{5} a^{2}}{4}-\frac {5 b^{7}}{4}}{\left (1+\tan ^{2}\left (d x +c \right )\right )^{2}}+\frac {3 \left (-56 b^{5} a^{2}+8 b^{7}\right ) \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{16}+\frac {3 \left (a^{7}+7 a^{5} b^{2}+35 a^{3} b^{4}-35 a \,b^{6}\right ) \arctan \left (\tan \left (d x +c \right )\right )}{8}}{\left (a^{2}+b^{2}\right )^{5}}}{d}\) | \(312\) |
risch | \(\frac {15 x a b}{8 i a^{5}-80 i a^{3} b^{2}+40 i a \,b^{4}+40 a^{4} b -80 a^{2} b^{3}+8 b^{5}}-\frac {24 i x \,b^{2}}{8 i a^{5}-80 i a^{3} b^{2}+40 i a \,b^{4}+40 a^{4} b -80 a^{2} b^{3}+8 b^{5}}-\frac {i {\mathrm e}^{2 i \left (d x +c \right )} a}{8 \left (-4 i a^{3} b +4 i a \,b^{3}+a^{4}-6 a^{2} b^{2}+b^{4}\right ) d}+\frac {6 i b^{7} x}{a^{10}+5 a^{8} b^{2}+10 a^{6} b^{4}+10 a^{4} b^{6}+5 a^{2} b^{8}+b^{10}}-\frac {5 \,{\mathrm e}^{2 i \left (d x +c \right )} b}{16 \left (-4 i a^{3} b +4 i a \,b^{3}+a^{4}-6 a^{2} b^{2}+b^{4}\right ) d}+\frac {i {\mathrm e}^{-2 i \left (d x +c \right )} a}{8 \left (i b +a \right )^{2} \left (2 i a b +a^{2}-b^{2}\right ) d}-\frac {5 \,{\mathrm e}^{-2 i \left (d x +c \right )} b}{16 \left (i b +a \right )^{2} \left (2 i a b +a^{2}-b^{2}\right ) d}-\frac {42 i b^{5} a^{2} c}{d \left (a^{10}+5 a^{8} b^{2}+10 a^{6} b^{4}+10 a^{4} b^{6}+5 a^{2} b^{8}+b^{10}\right )}+\frac {6 i b^{7} c}{d \left (a^{10}+5 a^{8} b^{2}+10 a^{6} b^{4}+10 a^{4} b^{6}+5 a^{2} b^{8}+b^{10}\right )}-\frac {42 i b^{5} a^{2} x}{a^{10}+5 a^{8} b^{2}+10 a^{6} b^{4}+10 a^{4} b^{6}+5 a^{2} b^{8}+b^{10}}+\frac {3 i x \,a^{2}}{8 i a^{5}-80 i a^{3} b^{2}+40 i a \,b^{4}+40 a^{4} b -80 a^{2} b^{3}+8 b^{5}}-\frac {i {\mathrm e}^{4 i \left (d x +c \right )}}{64 \left (-3 i b \,a^{2}+i b^{3}+a^{3}-3 a \,b^{2}\right ) d}+\frac {i {\mathrm e}^{-4 i \left (d x +c \right )}}{64 \left (2 i a b +a^{2}-b^{2}\right ) \left (i b +a \right ) d}+\frac {2 b^{6} \left (-6 i a b \,{\mathrm e}^{2 i \left (d x +c \right )}+7 a^{2} {\mathrm e}^{2 i \left (d x +c \right )}+b^{2} {\mathrm e}^{2 i \left (d x +c \right )}+7 i a b +7 a^{2}\right )}{\left (-i a +b \right )^{4} \left (b \,{\mathrm e}^{2 i \left (d x +c \right )}+i a \,{\mathrm e}^{2 i \left (d x +c \right )}-b +i a \right )^{2} d \left (i a +b \right )^{5}}+\frac {21 b^{5} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-\frac {i b +a}{i b -a}\right ) a^{2}}{d \left (a^{10}+5 a^{8} b^{2}+10 a^{6} b^{4}+10 a^{4} b^{6}+5 a^{2} b^{8}+b^{10}\right )}-\frac {3 b^{7} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-\frac {i b +a}{i b -a}\right )}{d \left (a^{10}+5 a^{8} b^{2}+10 a^{6} b^{4}+10 a^{4} b^{6}+5 a^{2} b^{8}+b^{10}\right )}\) | \(877\) |
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Leaf count of result is larger than twice the leaf count of optimal. 671 vs. \(2 (284) = 568\).
Time = 0.33 (sec) , antiderivative size = 671, normalized size of antiderivative = 2.27 \[ \int \frac {\cos ^4(c+d x)}{(a+b \tan (c+d x))^3} \, dx=-\frac {9 \, a^{6} b^{3} + 95 \, a^{4} b^{5} - 141 \, a^{2} b^{7} - 3 \, b^{9} - 8 \, {\left (a^{8} b + 4 \, a^{6} b^{3} + 6 \, a^{4} b^{5} + 4 \, a^{2} b^{7} + b^{9}\right )} \cos \left (d x + c\right )^{6} + 8 \, {\left (a^{8} b - 6 \, a^{4} b^{5} - 8 \, a^{2} b^{7} - 3 \, b^{9}\right )} \cos \left (d x + c\right )^{4} - 12 \, {\left (a^{7} b^{2} + 7 \, a^{5} b^{4} + 35 \, a^{3} b^{6} - 35 \, a b^{8}\right )} d x - {\left (15 \, a^{8} b + 82 \, a^{6} b^{3} + 68 \, a^{4} b^{5} - 498 \, a^{2} b^{7} - 51 \, b^{9} + 12 \, {\left (a^{9} + 6 \, a^{7} b^{2} + 28 \, a^{5} b^{4} - 70 \, a^{3} b^{6} + 35 \, a b^{8}\right )} d x\right )} \cos \left (d x + c\right )^{2} - 48 \, {\left (7 \, a^{2} b^{7} - b^{9} + {\left (7 \, a^{4} b^{5} - 8 \, a^{2} b^{7} + b^{9}\right )} \cos \left (d x + c\right )^{2} + 2 \, {\left (7 \, a^{3} b^{6} - a b^{8}\right )} \cos \left (d x + c\right ) \sin \left (d x + c\right )\right )} \log \left (2 \, a b \cos \left (d x + c\right ) \sin \left (d x + c\right ) + {\left (a^{2} - b^{2}\right )} \cos \left (d x + c\right )^{2} + b^{2}\right ) - 2 \, {\left (4 \, {\left (a^{9} + 4 \, a^{7} b^{2} + 6 \, a^{5} b^{4} + 4 \, a^{3} b^{6} + a b^{8}\right )} \cos \left (d x + c\right )^{5} + 2 \, {\left (3 \, a^{9} + 20 \, a^{7} b^{2} + 42 \, a^{5} b^{4} + 36 \, a^{3} b^{6} + 11 \, a b^{8}\right )} \cos \left (d x + c\right )^{3} - {\left (3 \, a^{7} b^{2} + 53 \, a^{5} b^{4} - 15 \, a^{3} b^{6} + 159 \, a b^{8} - 12 \, {\left (a^{8} b + 7 \, a^{6} b^{3} + 35 \, a^{4} b^{5} - 35 \, a^{2} b^{7}\right )} d x\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{32 \, {\left ({\left (a^{12} + 4 \, a^{10} b^{2} + 5 \, a^{8} b^{4} - 5 \, a^{4} b^{8} - 4 \, a^{2} b^{10} - b^{12}\right )} d \cos \left (d x + c\right )^{2} + 2 \, {\left (a^{11} b + 5 \, a^{9} b^{3} + 10 \, a^{7} b^{5} + 10 \, a^{5} b^{7} + 5 \, a^{3} b^{9} + a b^{11}\right )} d \cos \left (d x + c\right ) \sin \left (d x + c\right ) + {\left (a^{10} b^{2} + 5 \, a^{8} b^{4} + 10 \, a^{6} b^{6} + 10 \, a^{4} b^{8} + 5 \, a^{2} b^{10} + b^{12}\right )} d\right )}} \]
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Exception generated. \[ \int \frac {\cos ^4(c+d x)}{(a+b \tan (c+d x))^3} \, dx=\text {Exception raised: AttributeError} \]
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Leaf count of result is larger than twice the leaf count of optimal. 738 vs. \(2 (284) = 568\).
Time = 0.53 (sec) , antiderivative size = 738, normalized size of antiderivative = 2.50 \[ \int \frac {\cos ^4(c+d x)}{(a+b \tan (c+d x))^3} \, dx=\frac {\frac {3 \, {\left (a^{7} + 7 \, a^{5} b^{2} + 35 \, a^{3} b^{4} - 35 \, a b^{6}\right )} {\left (d x + c\right )}}{a^{10} + 5 \, a^{8} b^{2} + 10 \, a^{6} b^{4} + 10 \, a^{4} b^{6} + 5 \, a^{2} b^{8} + b^{10}} + \frac {24 \, {\left (7 \, a^{2} b^{5} - b^{7}\right )} \log \left (b \tan \left (d x + c\right ) + a\right )}{a^{10} + 5 \, a^{8} b^{2} + 10 \, a^{6} b^{4} + 10 \, a^{4} b^{6} + 5 \, a^{2} b^{8} + b^{10}} - \frac {12 \, {\left (7 \, a^{2} b^{5} - b^{7}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{a^{10} + 5 \, a^{8} b^{2} + 10 \, a^{6} b^{4} + 10 \, a^{4} b^{6} + 5 \, a^{2} b^{8} + b^{10}} + \frac {6 \, a^{6} b + 44 \, a^{4} b^{3} - 62 \, a^{2} b^{5} - 4 \, b^{7} + 3 \, {\left (a^{5} b^{2} + 6 \, a^{3} b^{4} - 27 \, a b^{6}\right )} \tan \left (d x + c\right )^{5} + 6 \, {\left (a^{6} b + 6 \, a^{4} b^{3} - 13 \, a^{2} b^{5} - 2 \, b^{7}\right )} \tan \left (d x + c\right )^{4} + {\left (3 \, a^{7} + 23 \, a^{5} b^{2} + 61 \, a^{3} b^{4} - 151 \, a b^{6}\right )} \tan \left (d x + c\right )^{3} + 2 \, {\left (5 \, a^{6} b + 37 \, a^{4} b^{3} - 73 \, a^{2} b^{5} - 9 \, b^{7}\right )} \tan \left (d x + c\right )^{2} + {\left (5 \, a^{7} + 26 \, a^{5} b^{2} + 49 \, a^{3} b^{4} - 68 \, a b^{6}\right )} \tan \left (d x + c\right )}{a^{10} + 4 \, a^{8} b^{2} + 6 \, a^{6} b^{4} + 4 \, a^{4} b^{6} + a^{2} b^{8} + {\left (a^{8} b^{2} + 4 \, a^{6} b^{4} + 6 \, a^{4} b^{6} + 4 \, a^{2} b^{8} + b^{10}\right )} \tan \left (d x + c\right )^{6} + 2 \, {\left (a^{9} b + 4 \, a^{7} b^{3} + 6 \, a^{5} b^{5} + 4 \, a^{3} b^{7} + a b^{9}\right )} \tan \left (d x + c\right )^{5} + {\left (a^{10} + 6 \, a^{8} b^{2} + 14 \, a^{6} b^{4} + 16 \, a^{4} b^{6} + 9 \, a^{2} b^{8} + 2 \, b^{10}\right )} \tan \left (d x + c\right )^{4} + 4 \, {\left (a^{9} b + 4 \, a^{7} b^{3} + 6 \, a^{5} b^{5} + 4 \, a^{3} b^{7} + a b^{9}\right )} \tan \left (d x + c\right )^{3} + {\left (2 \, a^{10} + 9 \, a^{8} b^{2} + 16 \, a^{6} b^{4} + 14 \, a^{4} b^{6} + 6 \, a^{2} b^{8} + b^{10}\right )} \tan \left (d x + c\right )^{2} + 2 \, {\left (a^{9} b + 4 \, a^{7} b^{3} + 6 \, a^{5} b^{5} + 4 \, a^{3} b^{7} + a b^{9}\right )} \tan \left (d x + c\right )}}{8 \, d} \]
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Leaf count of result is larger than twice the leaf count of optimal. 587 vs. \(2 (284) = 568\).
Time = 0.61 (sec) , antiderivative size = 587, normalized size of antiderivative = 1.99 \[ \int \frac {\cos ^4(c+d x)}{(a+b \tan (c+d x))^3} \, dx=\frac {\frac {3 \, {\left (a^{7} + 7 \, a^{5} b^{2} + 35 \, a^{3} b^{4} - 35 \, a b^{6}\right )} {\left (d x + c\right )}}{a^{10} + 5 \, a^{8} b^{2} + 10 \, a^{6} b^{4} + 10 \, a^{4} b^{6} + 5 \, a^{2} b^{8} + b^{10}} - \frac {12 \, {\left (7 \, a^{2} b^{5} - b^{7}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{a^{10} + 5 \, a^{8} b^{2} + 10 \, a^{6} b^{4} + 10 \, a^{4} b^{6} + 5 \, a^{2} b^{8} + b^{10}} + \frac {24 \, {\left (7 \, a^{2} b^{6} - b^{8}\right )} \log \left ({\left | b \tan \left (d x + c\right ) + a \right |}\right )}{a^{10} b + 5 \, a^{8} b^{3} + 10 \, a^{6} b^{5} + 10 \, a^{4} b^{7} + 5 \, a^{2} b^{9} + b^{11}} + \frac {3 \, a^{5} b^{2} \tan \left (d x + c\right )^{5} + 18 \, a^{3} b^{4} \tan \left (d x + c\right )^{5} - 81 \, a b^{6} \tan \left (d x + c\right )^{5} + 6 \, a^{6} b \tan \left (d x + c\right )^{4} + 36 \, a^{4} b^{3} \tan \left (d x + c\right )^{4} - 78 \, a^{2} b^{5} \tan \left (d x + c\right )^{4} - 12 \, b^{7} \tan \left (d x + c\right )^{4} + 3 \, a^{7} \tan \left (d x + c\right )^{3} + 23 \, a^{5} b^{2} \tan \left (d x + c\right )^{3} + 61 \, a^{3} b^{4} \tan \left (d x + c\right )^{3} - 151 \, a b^{6} \tan \left (d x + c\right )^{3} + 10 \, a^{6} b \tan \left (d x + c\right )^{2} + 74 \, a^{4} b^{3} \tan \left (d x + c\right )^{2} - 146 \, a^{2} b^{5} \tan \left (d x + c\right )^{2} - 18 \, b^{7} \tan \left (d x + c\right )^{2} + 5 \, a^{7} \tan \left (d x + c\right ) + 26 \, a^{5} b^{2} \tan \left (d x + c\right ) + 49 \, a^{3} b^{4} \tan \left (d x + c\right ) - 68 \, a b^{6} \tan \left (d x + c\right ) + 6 \, a^{6} b + 44 \, a^{4} b^{3} - 62 \, a^{2} b^{5} - 4 \, b^{7}}{{\left (a^{8} + 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} + 4 \, a^{2} b^{6} + b^{8}\right )} {\left (b \tan \left (d x + c\right )^{3} + a \tan \left (d x + c\right )^{2} + b \tan \left (d x + c\right ) + a\right )}^{2}}}{8 \, d} \]
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Time = 6.49 (sec) , antiderivative size = 715, normalized size of antiderivative = 2.42 \[ \int \frac {\cos ^4(c+d x)}{(a+b \tan (c+d x))^3} \, dx=\frac {\frac {3\,a^6\,b+22\,a^4\,b^3-31\,a^2\,b^5-2\,b^7}{4\,\left (a^8+4\,a^6\,b^2+6\,a^4\,b^4+4\,a^2\,b^6+b^8\right )}+\frac {\mathrm {tan}\left (c+d\,x\right )\,\left (5\,a^7+26\,a^5\,b^2+49\,a^3\,b^4-68\,a\,b^6\right )}{8\,\left (a^8+4\,a^6\,b^2+6\,a^4\,b^4+4\,a^2\,b^6+b^8\right )}+\frac {3\,{\mathrm {tan}\left (c+d\,x\right )}^5\,\left (a^5\,b^2+6\,a^3\,b^4-27\,a\,b^6\right )}{8\,\left (a^8+4\,a^6\,b^2+6\,a^4\,b^4+4\,a^2\,b^6+b^8\right )}+\frac {{\mathrm {tan}\left (c+d\,x\right )}^3\,\left (3\,a^7+23\,a^5\,b^2+61\,a^3\,b^4-151\,a\,b^6\right )}{8\,\left (a^8+4\,a^6\,b^2+6\,a^4\,b^4+4\,a^2\,b^6+b^8\right )}+\frac {3\,{\mathrm {tan}\left (c+d\,x\right )}^4\,\left (a^6\,b+6\,a^4\,b^3-13\,a^2\,b^5-2\,b^7\right )}{4\,\left (a^8+4\,a^6\,b^2+6\,a^4\,b^4+4\,a^2\,b^6+b^8\right )}+\frac {{\mathrm {tan}\left (c+d\,x\right )}^2\,\left (5\,a^6\,b+37\,a^4\,b^3-73\,a^2\,b^5-9\,b^7\right )}{4\,\left (a^8+4\,a^6\,b^2+6\,a^4\,b^4+4\,a^2\,b^6+b^8\right )}}{d\,\left ({\mathrm {tan}\left (c+d\,x\right )}^2\,\left (2\,a^2+b^2\right )+{\mathrm {tan}\left (c+d\,x\right )}^4\,\left (a^2+2\,b^2\right )+a^2+b^2\,{\mathrm {tan}\left (c+d\,x\right )}^6+2\,a\,b\,\mathrm {tan}\left (c+d\,x\right )+4\,a\,b\,{\mathrm {tan}\left (c+d\,x\right )}^3+2\,a\,b\,{\mathrm {tan}\left (c+d\,x\right )}^5\right )}+\frac {\ln \left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )\,\left (\frac {21\,b^5}{{\left (a^2+b^2\right )}^4}-\frac {24\,b^7}{{\left (a^2+b^2\right )}^5}\right )}{d}+\frac {3\,\ln \left (\mathrm {tan}\left (c+d\,x\right )-\mathrm {i}\right )\,\left (-a^2\,1{}\mathrm {i}+5\,a\,b+b^2\,8{}\mathrm {i}\right )}{16\,d\,\left (a^5+a^4\,b\,5{}\mathrm {i}-10\,a^3\,b^2-a^2\,b^3\,10{}\mathrm {i}+5\,a\,b^4+b^5\,1{}\mathrm {i}\right )}+\frac {3\,\ln \left (\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )\,\left (a^2\,1{}\mathrm {i}+5\,a\,b-b^2\,8{}\mathrm {i}\right )}{16\,d\,\left (a^5-a^4\,b\,5{}\mathrm {i}-10\,a^3\,b^2+a^2\,b^3\,10{}\mathrm {i}+5\,a\,b^4-b^5\,1{}\mathrm {i}\right )} \]
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