Integrand size = 21, antiderivative size = 241 \[ \int \frac {\cos ^3(c+d x)}{(a+b \tan (c+d x))^2} \, dx=-\frac {5 a b^4 \text {arctanh}\left (\frac {b-a \tan (c+d x)}{\sqrt {a^2+b^2} \sqrt {\sec ^2(c+d x)}}\right ) \cos (c+d x) \sqrt {\sec ^2(c+d x)}}{\left (a^2+b^2\right )^{7/2} d}+\frac {b \left (2 a^4+9 a^2 b^2-8 b^4\right ) \sec (c+d x)}{3 \left (a^2+b^2\right )^3 d (a+b \tan (c+d x))}+\frac {\cos ^3(c+d x) (b+a \tan (c+d x))}{3 \left (a^2+b^2\right ) d (a+b \tan (c+d x))}-\frac {\cos (c+d x) \left (b \left (a^2-4 b^2\right )-a \left (2 a^2+7 b^2\right ) \tan (c+d x)\right )}{3 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))} \]
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Time = 0.32 (sec) , antiderivative size = 241, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {3593, 755, 837, 821, 739, 212} \[ \int \frac {\cos ^3(c+d x)}{(a+b \tan (c+d x))^2} \, dx=-\frac {5 a b^4 \cos (c+d x) \sqrt {\sec ^2(c+d x)} \text {arctanh}\left (\frac {b-a \tan (c+d x)}{\sqrt {a^2+b^2} \sqrt {\sec ^2(c+d x)}}\right )}{d \left (a^2+b^2\right )^{7/2}}+\frac {\cos ^3(c+d x) (a \tan (c+d x)+b)}{3 d \left (a^2+b^2\right ) (a+b \tan (c+d x))}-\frac {\cos (c+d x) \left (b \left (a^2-4 b^2\right )-a \left (2 a^2+7 b^2\right ) \tan (c+d x)\right )}{3 d \left (a^2+b^2\right )^2 (a+b \tan (c+d x))}+\frac {b \left (2 a^4+9 a^2 b^2-8 b^4\right ) \sec (c+d x)}{3 d \left (a^2+b^2\right )^3 (a+b \tan (c+d x))} \]
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Rule 212
Rule 739
Rule 755
Rule 821
Rule 837
Rule 3593
Rubi steps \begin{align*} \text {integral}& = \frac {\left (\cos (c+d x) \sqrt {\sec ^2(c+d x)}\right ) \text {Subst}\left (\int \frac {1}{(a+x)^2 \left (1+\frac {x^2}{b^2}\right )^{5/2}} \, dx,x,b \tan (c+d x)\right )}{b d} \\ & = \frac {\cos ^3(c+d x) (b+a \tan (c+d x))}{3 \left (a^2+b^2\right ) d (a+b \tan (c+d x))}-\frac {\left (b \cos (c+d x) \sqrt {\sec ^2(c+d x)}\right ) \text {Subst}\left (\int \frac {-2 \left (2+\frac {a^2}{b^2}\right )-\frac {3 a x}{b^2}}{(a+x)^2 \left (1+\frac {x^2}{b^2}\right )^{3/2}} \, dx,x,b \tan (c+d x)\right )}{3 \left (a^2+b^2\right ) d} \\ & = \frac {\cos ^3(c+d x) (b+a \tan (c+d x))}{3 \left (a^2+b^2\right ) d (a+b \tan (c+d x))}-\frac {\cos (c+d x) \left (b \left (a^2-4 b^2\right )-a \left (2 a^2+7 b^2\right ) \tan (c+d x)\right )}{3 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))}+\frac {\left (b^5 \cos (c+d x) \sqrt {\sec ^2(c+d x)}\right ) \text {Subst}\left (\int \frac {-\frac {2 \left (a^2-4 b^2\right )}{b^4}+\frac {a \left (2 a^2+7 b^2\right ) x}{b^6}}{(a+x)^2 \sqrt {1+\frac {x^2}{b^2}}} \, dx,x,b \tan (c+d x)\right )}{3 \left (a^2+b^2\right )^2 d} \\ & = \frac {b \left (2 a^4+9 a^2 b^2-8 b^4\right ) \sec (c+d x)}{3 \left (a^2+b^2\right )^3 d (a+b \tan (c+d x))}+\frac {\cos ^3(c+d x) (b+a \tan (c+d x))}{3 \left (a^2+b^2\right ) d (a+b \tan (c+d x))}-\frac {\cos (c+d x) \left (b \left (a^2-4 b^2\right )-a \left (2 a^2+7 b^2\right ) \tan (c+d x)\right )}{3 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))}+\frac {\left (5 a b^3 \cos (c+d x) \sqrt {\sec ^2(c+d x)}\right ) \text {Subst}\left (\int \frac {1}{(a+x) \sqrt {1+\frac {x^2}{b^2}}} \, dx,x,b \tan (c+d x)\right )}{\left (a^2+b^2\right )^3 d} \\ & = \frac {b \left (2 a^4+9 a^2 b^2-8 b^4\right ) \sec (c+d x)}{3 \left (a^2+b^2\right )^3 d (a+b \tan (c+d x))}+\frac {\cos ^3(c+d x) (b+a \tan (c+d x))}{3 \left (a^2+b^2\right ) d (a+b \tan (c+d x))}-\frac {\cos (c+d x) \left (b \left (a^2-4 b^2\right )-a \left (2 a^2+7 b^2\right ) \tan (c+d x)\right )}{3 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))}-\frac {\left (5 a b^3 \cos (c+d x) \sqrt {\sec ^2(c+d x)}\right ) \text {Subst}\left (\int \frac {1}{1+\frac {a^2}{b^2}-x^2} \, dx,x,\frac {1-\frac {a \tan (c+d x)}{b}}{\sqrt {\sec ^2(c+d x)}}\right )}{\left (a^2+b^2\right )^3 d} \\ & = -\frac {5 a b^4 \text {arctanh}\left (\frac {b \left (1-\frac {a \tan (c+d x)}{b}\right )}{\sqrt {a^2+b^2} \sqrt {\sec ^2(c+d x)}}\right ) \cos (c+d x) \sqrt {\sec ^2(c+d x)}}{\left (a^2+b^2\right )^{7/2} d}+\frac {b \left (2 a^4+9 a^2 b^2-8 b^4\right ) \sec (c+d x)}{3 \left (a^2+b^2\right )^3 d (a+b \tan (c+d x))}+\frac {\cos ^3(c+d x) (b+a \tan (c+d x))}{3 \left (a^2+b^2\right ) d (a+b \tan (c+d x))}-\frac {\cos (c+d x) \left (b \left (a^2-4 b^2\right )-a \left (2 a^2+7 b^2\right ) \tan (c+d x)\right )}{3 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))} \\ \end{align*}
Time = 1.80 (sec) , antiderivative size = 249, normalized size of antiderivative = 1.03 \[ \int \frac {\cos ^3(c+d x)}{(a+b \tan (c+d x))^2} \, dx=\frac {\sec (c+d x) \left (240 a b^4 \sqrt {a^2+b^2} \text {arctanh}\left (\frac {-b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2+b^2}}\right ) (a \cos (c+d x)+b \sin (c+d x))+\left (a^2+b^2\right ) \left (15 a^4 b+90 a^2 b^3-45 b^5+20 b^3 \left (a^2+b^2\right ) \cos (2 (c+d x))+b \left (a^2+b^2\right )^2 \cos (4 (c+d x))+10 a^5 \sin (2 (c+d x))+40 a^3 b^2 \sin (2 (c+d x))+30 a b^4 \sin (2 (c+d x))+a^5 \sin (4 (c+d x))+2 a^3 b^2 \sin (4 (c+d x))+a b^4 \sin (4 (c+d x))\right )\right )}{24 \left (a^2+b^2\right )^4 d (a+b \tan (c+d x))} \]
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Time = 8.25 (sec) , antiderivative size = 320, normalized size of antiderivative = 1.33
method | result | size |
derivativedivides | \(\frac {-\frac {2 \left (\left (-a^{4}-3 a^{2} b^{2}+2 b^{4}\right ) \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-2 a^{3} b -6 a \,b^{3}\right ) \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-\frac {2}{3} a^{4}-6 a^{2} b^{2}+\frac {8}{3} b^{4}\right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-8 a \,b^{3} \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-a^{4}-3 a^{2} b^{2}+2 b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-\frac {2 a^{3} b}{3}-\frac {14 a \,b^{3}}{3}\right )}{\left (a^{2}+b^{2}\right ) \left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3}}-\frac {2 b^{4} \left (\frac {-\frac {b^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{a}-b}{\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a -2 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-a}-\frac {5 a \,\operatorname {arctanh}\left (\frac {2 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-2 b}{2 \sqrt {a^{2}+b^{2}}}\right )}{\sqrt {a^{2}+b^{2}}}\right )}{\left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) \left (a^{2}+b^{2}\right )}}{d}\) | \(320\) |
default | \(\frac {-\frac {2 \left (\left (-a^{4}-3 a^{2} b^{2}+2 b^{4}\right ) \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-2 a^{3} b -6 a \,b^{3}\right ) \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-\frac {2}{3} a^{4}-6 a^{2} b^{2}+\frac {8}{3} b^{4}\right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-8 a \,b^{3} \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-a^{4}-3 a^{2} b^{2}+2 b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-\frac {2 a^{3} b}{3}-\frac {14 a \,b^{3}}{3}\right )}{\left (a^{2}+b^{2}\right ) \left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3}}-\frac {2 b^{4} \left (\frac {-\frac {b^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{a}-b}{\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a -2 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-a}-\frac {5 a \,\operatorname {arctanh}\left (\frac {2 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-2 b}{2 \sqrt {a^{2}+b^{2}}}\right )}{\sqrt {a^{2}+b^{2}}}\right )}{\left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) \left (a^{2}+b^{2}\right )}}{d}\) | \(320\) |
risch | \(-\frac {i {\mathrm e}^{3 i \left (d x +c \right )}}{24 \left (-2 i a b +a^{2}-b^{2}\right ) d}-\frac {7 \,{\mathrm e}^{i \left (d x +c \right )} b}{8 \left (-3 i b \,a^{2}+i b^{3}+a^{3}-3 a \,b^{2}\right ) d}-\frac {3 i {\mathrm e}^{i \left (d x +c \right )} a}{8 \left (-3 i b \,a^{2}+i b^{3}+a^{3}-3 a \,b^{2}\right ) d}-\frac {7 \,{\mathrm e}^{-i \left (d x +c \right )} b}{8 \left (i b +a \right )^{3} d}+\frac {3 i {\mathrm e}^{-i \left (d x +c \right )} a}{8 \left (i b +a \right )^{3} d}+\frac {i {\mathrm e}^{-3 i \left (d x +c \right )}}{24 \left (i b +a \right )^{2} d}-\frac {2 i b^{5} {\mathrm e}^{i \left (d x +c \right )}}{\left (-i a +b \right )^{3} d \left (i a +b \right )^{3} \left (b \,{\mathrm e}^{2 i \left (d x +c \right )}+i a \,{\mathrm e}^{2 i \left (d x +c \right )}-b +i a \right )}+\frac {5 b^{4} a \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i a^{7}+3 i a^{5} b^{2}+3 i a^{3} b^{4}+i a \,b^{6}-a^{6} b -3 a^{4} b^{3}-3 b^{5} a^{2}-b^{7}}{\left (a^{2}+b^{2}\right )^{\frac {7}{2}}}\right )}{\left (a^{2}+b^{2}\right )^{\frac {7}{2}} d}-\frac {5 b^{4} a \ln \left ({\mathrm e}^{i \left (d x +c \right )}-\frac {i a^{7}+3 i a^{5} b^{2}+3 i a^{3} b^{4}+i a \,b^{6}-a^{6} b -3 a^{4} b^{3}-3 b^{5} a^{2}-b^{7}}{\left (a^{2}+b^{2}\right )^{\frac {7}{2}}}\right )}{\left (a^{2}+b^{2}\right )^{\frac {7}{2}} d}\) | \(449\) |
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Time = 0.30 (sec) , antiderivative size = 418, normalized size of antiderivative = 1.73 \[ \int \frac {\cos ^3(c+d x)}{(a+b \tan (c+d x))^2} \, dx=\frac {4 \, a^{6} b + 22 \, a^{4} b^{3} + 2 \, a^{2} b^{5} - 16 \, b^{7} + 2 \, {\left (a^{6} b + 3 \, a^{4} b^{3} + 3 \, a^{2} b^{5} + b^{7}\right )} \cos \left (d x + c\right )^{4} - 2 \, {\left (a^{6} b - 2 \, a^{4} b^{3} - 7 \, a^{2} b^{5} - 4 \, b^{7}\right )} \cos \left (d x + c\right )^{2} + 15 \, {\left (a^{2} b^{4} \cos \left (d x + c\right ) + a b^{5} \sin \left (d x + c\right )\right )} \sqrt {a^{2} + b^{2}} \log \left (-\frac {2 \, a b \cos \left (d x + c\right ) \sin \left (d x + c\right ) + {\left (a^{2} - b^{2}\right )} \cos \left (d x + c\right )^{2} - 2 \, a^{2} - b^{2} + 2 \, \sqrt {a^{2} + b^{2}} {\left (b \cos \left (d x + c\right ) - a \sin \left (d x + c\right )\right )}}{2 \, a b \cos \left (d x + c\right ) \sin \left (d x + c\right ) + {\left (a^{2} - b^{2}\right )} \cos \left (d x + c\right )^{2} + b^{2}}\right ) + 2 \, {\left ({\left (a^{7} + 3 \, a^{5} b^{2} + 3 \, a^{3} b^{4} + a b^{6}\right )} \cos \left (d x + c\right )^{3} + {\left (2 \, a^{7} + 11 \, a^{5} b^{2} + 16 \, a^{3} b^{4} + 7 \, a b^{6}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{6 \, {\left ({\left (a^{9} + 4 \, a^{7} b^{2} + 6 \, a^{5} b^{4} + 4 \, a^{3} b^{6} + a b^{8}\right )} d \cos \left (d x + c\right ) + {\left (a^{8} b + 4 \, a^{6} b^{3} + 6 \, a^{4} b^{5} + 4 \, a^{2} b^{7} + b^{9}\right )} d \sin \left (d x + c\right )\right )}} \]
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Timed out. \[ \int \frac {\cos ^3(c+d x)}{(a+b \tan (c+d x))^2} \, dx=\text {Timed out} \]
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Leaf count of result is larger than twice the leaf count of optimal. 772 vs. \(2 (229) = 458\).
Time = 0.46 (sec) , antiderivative size = 772, normalized size of antiderivative = 3.20 \[ \int \frac {\cos ^3(c+d x)}{(a+b \tan (c+d x))^2} \, dx=-\frac {\frac {15 \, a b^{4} \log \left (\frac {b - \frac {a \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \sqrt {a^{2} + b^{2}}}{b - \frac {a \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \sqrt {a^{2} + b^{2}}}\right )}{{\left (a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}\right )} \sqrt {a^{2} + b^{2}}} - \frac {2 \, {\left (2 \, a^{5} b + 14 \, a^{3} b^{3} - 3 \, a b^{5} - \frac {15 \, a b^{5} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac {{\left (3 \, a^{6} + 13 \, a^{4} b^{2} + 22 \, a^{2} b^{4} - 3 \, b^{6}\right )} \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {{\left (4 \, a^{5} b + 28 \, a^{3} b^{3} - 21 \, a b^{5}\right )} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {{\left (a^{6} - 9 \, a^{4} b^{2} - 46 \, a^{2} b^{4} + 9 \, b^{6}\right )} \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {5 \, {\left (2 \, a^{5} b + 6 \, a^{3} b^{3} - 5 \, a b^{5}\right )} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {{\left (a^{6} + 3 \, a^{4} b^{2} + 38 \, a^{2} b^{4} - 9 \, b^{6}\right )} \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac {3 \, {\left (a^{6} + 3 \, a^{4} b^{2} - 2 \, a^{2} b^{4} + b^{6}\right )} \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}}\right )}}{a^{8} + 3 \, a^{6} b^{2} + 3 \, a^{4} b^{4} + a^{2} b^{6} + \frac {2 \, {\left (a^{7} b + 3 \, a^{5} b^{3} + 3 \, a^{3} b^{5} + a b^{7}\right )} \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {2 \, {\left (a^{8} + 3 \, a^{6} b^{2} + 3 \, a^{4} b^{4} + a^{2} b^{6}\right )} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {6 \, {\left (a^{7} b + 3 \, a^{5} b^{3} + 3 \, a^{3} b^{5} + a b^{7}\right )} \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {6 \, {\left (a^{7} b + 3 \, a^{5} b^{3} + 3 \, a^{3} b^{5} + a b^{7}\right )} \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac {2 \, {\left (a^{8} + 3 \, a^{6} b^{2} + 3 \, a^{4} b^{4} + a^{2} b^{6}\right )} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac {2 \, {\left (a^{7} b + 3 \, a^{5} b^{3} + 3 \, a^{3} b^{5} + a b^{7}\right )} \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} - \frac {{\left (a^{8} + 3 \, a^{6} b^{2} + 3 \, a^{4} b^{4} + a^{2} b^{6}\right )} \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}}}}{3 \, d} \]
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Time = 0.51 (sec) , antiderivative size = 438, normalized size of antiderivative = 1.82 \[ \int \frac {\cos ^3(c+d x)}{(a+b \tan (c+d x))^2} \, dx=-\frac {\frac {15 \, a b^{4} \log \left (\frac {{\left | 2 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 2 \, b - 2 \, \sqrt {a^{2} + b^{2}} \right |}}{{\left | 2 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 2 \, b + 2 \, \sqrt {a^{2} + b^{2}} \right |}}\right )}{{\left (a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}\right )} \sqrt {a^{2} + b^{2}}} - \frac {6 \, {\left (b^{6} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + a b^{5}\right )}}{{\left (a^{7} + 3 \, a^{5} b^{2} + 3 \, a^{3} b^{4} + a b^{6}\right )} {\left (a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 2 \, b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - a\right )}} - \frac {2 \, {\left (3 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 9 \, a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 6 \, b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 6 \, a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 18 \, a b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 2 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 18 \, a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 8 \, b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 24 \, a b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 3 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 9 \, a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 6 \, b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 2 \, a^{3} b + 14 \, a b^{3}\right )}}{{\left (a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}\right )} {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{3}}}{3 \, d} \]
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Time = 8.30 (sec) , antiderivative size = 674, normalized size of antiderivative = 2.80 \[ \int \frac {\cos ^3(c+d x)}{(a+b \tan (c+d x))^2} \, dx=\frac {\frac {2\,\left (2\,a^4\,b+14\,a^2\,b^3-3\,b^5\right )}{3\,\left (a^2+b^2\right )\,\left (a^4+2\,a^2\,b^2+b^4\right )}-\frac {10\,b^5\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{a^6+3\,a^4\,b^2+3\,a^2\,b^4+b^6}+\frac {10\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\left (2\,a^4+6\,a^2\,b^2-5\,b^4\right )}{3\,\left (a^6+3\,a^4\,b^2+3\,a^2\,b^4+b^6\right )}+\frac {2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (4\,a^4\,b+28\,a^2\,b^3-21\,b^5\right )}{3\,\left (a^2+b^2\right )\,\left (a^4+2\,a^2\,b^2+b^4\right )}-\frac {2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7\,\left (a^6+3\,a^4\,b^2-2\,a^2\,b^4+b^6\right )}{a\,\left (a^6+3\,a^4\,b^2+3\,a^2\,b^4+b^6\right )}+\frac {2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (3\,a^6+13\,a^4\,b^2+22\,a^2\,b^4-3\,b^6\right )}{3\,a\,\left (a^2+b^2\right )\,\left (a^4+2\,a^2\,b^2+b^4\right )}+\frac {2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,\left (a^6+3\,a^4\,b^2+38\,a^2\,b^4-9\,b^6\right )}{3\,a\,\left (a^2+b^2\right )\,\left (a^4+2\,a^2\,b^2+b^4\right )}-\frac {2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (a^6-9\,a^4\,b^2-46\,a^2\,b^4+9\,b^6\right )}{3\,a\,\left (a^2+b^2\right )\,\left (a^4+2\,a^2\,b^2+b^4\right )}}{d\,\left (-a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+2\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7-2\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+6\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+6\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+2\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+2\,b\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+a\right )}-\frac {10\,a\,b^4\,\mathrm {atanh}\left (\frac {2\,a^6\,b+2\,b^7+6\,a^2\,b^5+6\,a^4\,b^3-2\,a\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (a^6+3\,a^4\,b^2+3\,a^2\,b^4+b^6\right )}{2\,{\left (a^2+b^2\right )}^{7/2}}\right )}{d\,{\left (a^2+b^2\right )}^{7/2}} \]
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