Integrand size = 19, antiderivative size = 221 \[ \int \frac {\cos (c+d x)}{(a+b \tan (c+d x))^3} \, dx=-\frac {3 b^2 \left (4 a^2-b^2\right ) \text {arctanh}\left (\frac {b-a \tan (c+d x)}{\sqrt {a^2+b^2} \sqrt {\sec ^2(c+d x)}}\right ) \cos (c+d x) \sqrt {\sec ^2(c+d x)}}{2 \left (a^2+b^2\right )^{7/2} d}+\frac {b \left (2 a^2-3 b^2\right ) \sec (c+d x)}{2 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))^2}+\frac {\cos (c+d x) (b+a \tan (c+d x))}{\left (a^2+b^2\right ) d (a+b \tan (c+d x))^2}+\frac {a b \left (2 a^2-13 b^2\right ) \sec (c+d x)}{2 \left (a^2+b^2\right )^3 d (a+b \tan (c+d x))} \]
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Time = 0.29 (sec) , antiderivative size = 221, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.316, Rules used = {3593, 755, 849, 821, 739, 212} \[ \int \frac {\cos (c+d x)}{(a+b \tan (c+d x))^3} \, dx=-\frac {3 b^2 \left (4 a^2-b^2\right ) \cos (c+d x) \sqrt {\sec ^2(c+d x)} \text {arctanh}\left (\frac {b-a \tan (c+d x)}{\sqrt {a^2+b^2} \sqrt {\sec ^2(c+d x)}}\right )}{2 d \left (a^2+b^2\right )^{7/2}}+\frac {\cos (c+d x) (a \tan (c+d x)+b)}{d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}+\frac {a b \left (2 a^2-13 b^2\right ) \sec (c+d x)}{2 d \left (a^2+b^2\right )^3 (a+b \tan (c+d x))}+\frac {b \left (2 a^2-3 b^2\right ) \sec (c+d x)}{2 d \left (a^2+b^2\right )^2 (a+b \tan (c+d x))^2} \]
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Rule 212
Rule 739
Rule 755
Rule 821
Rule 849
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)^3 \left (1+\frac {x^2}{b^2}\right )^{3/2}} \, dx,x,b \tan (c+d x)\right )}{b d} \\ & = \frac {\cos (c+d x) (b+a \tan (c+d x))}{\left (a^2+b^2\right ) d (a+b \tan (c+d x))^2}-\frac {\left (b \cos (c+d x) \sqrt {\sec ^2(c+d x)}\right ) \text {Subst}\left (\int \frac {-3-\frac {2 a x}{b^2}}{(a+x)^3 \sqrt {1+\frac {x^2}{b^2}}} \, dx,x,b \tan (c+d x)\right )}{\left (a^2+b^2\right ) d} \\ & = \frac {b \left (2 a^2-3 b^2\right ) \sec (c+d x)}{2 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))^2}+\frac {\cos (c+d x) (b+a \tan (c+d x))}{\left (a^2+b^2\right ) d (a+b \tan (c+d x))^2}+\frac {\left (b^3 \cos (c+d x) \sqrt {\sec ^2(c+d x)}\right ) \text {Subst}\left (\int \frac {\frac {10 a}{b^2}+\frac {\left (2 a^2-3 b^2\right ) x}{b^4}}{(a+x)^2 \sqrt {1+\frac {x^2}{b^2}}} \, dx,x,b \tan (c+d x)\right )}{2 \left (a^2+b^2\right )^2 d} \\ & = \frac {b \left (2 a^2-3 b^2\right ) \sec (c+d x)}{2 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))^2}+\frac {\cos (c+d x) (b+a \tan (c+d x))}{\left (a^2+b^2\right ) d (a+b \tan (c+d x))^2}+\frac {a b \left (2 a^2-13 b^2\right ) \sec (c+d x)}{2 \left (a^2+b^2\right )^3 d (a+b \tan (c+d x))}+\frac {\left (3 b \left (4 a^2-b^2\right ) \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 )}{2 \left (a^2+b^2\right )^3 d} \\ & = \frac {b \left (2 a^2-3 b^2\right ) \sec (c+d x)}{2 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))^2}+\frac {\cos (c+d x) (b+a \tan (c+d x))}{\left (a^2+b^2\right ) d (a+b \tan (c+d x))^2}+\frac {a b \left (2 a^2-13 b^2\right ) \sec (c+d x)}{2 \left (a^2+b^2\right )^3 d (a+b \tan (c+d x))}-\frac {\left (3 b \left (4 a^2-b^2\right ) \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 )}{2 \left (a^2+b^2\right )^3 d} \\ & = -\frac {3 b^2 \left (4 a^2-b^2\right ) \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)}}{2 \left (a^2+b^2\right )^{7/2} d}+\frac {b \left (2 a^2-3 b^2\right ) \sec (c+d x)}{2 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))^2}+\frac {\cos (c+d x) (b+a \tan (c+d x))}{\left (a^2+b^2\right ) d (a+b \tan (c+d x))^2}+\frac {a b \left (2 a^2-13 b^2\right ) \sec (c+d x)}{2 \left (a^2+b^2\right )^3 d (a+b \tan (c+d x))} \\ \end{align*}
Time = 2.71 (sec) , antiderivative size = 183, normalized size of antiderivative = 0.83 \[ \int \frac {\cos (c+d x)}{(a+b \tan (c+d x))^3} \, dx=\frac {-\frac {12 b^2 \left (-4 a^2+b^2\right ) \text {arctanh}\left (\frac {-b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{7/2}}+\frac {\sec ^2(c+d x) \left (b \left (11 a^4-22 a^2 b^2-3 b^4\right ) \cos (c+d x)+b \left (a^2+b^2\right )^2 \cos (3 (c+d x))+2 a \left (a^4+4 a^2 b^2-12 b^4+\left (a^2+b^2\right )^2 \cos (2 (c+d x))\right ) \sin (c+d x)\right )}{\left (a^2+b^2\right )^3 (a+b \tan (c+d x))^2}}{4 d} \]
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Time = 4.99 (sec) , antiderivative size = 283, normalized size of antiderivative = 1.28
method | result | size |
derivativedivides | \(\frac {-\frac {2 \left (\left (-a^{3}+3 a \,b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-3 a^{2} b +b^{3}\right )}{\left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right ) \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}-\frac {2 b^{2} \left (\frac {-\frac {b^{2} \left (9 a^{2}+2 b^{2}\right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 a}-\frac {b \left (8 a^{4}-15 a^{2} b^{2}-2 b^{4}\right ) \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 a^{2}}+\frac {b^{2} \left (23 a^{2}+2 b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 a}+4 a^{2} b +\frac {b^{3}}{2}}{{\left (\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a -2 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-a \right )}^{2}}-\frac {3 \left (4 a^{2}-b^{2}\right ) \operatorname {arctanh}\left (\frac {2 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-2 b}{2 \sqrt {a^{2}+b^{2}}}\right )}{2 \sqrt {a^{2}+b^{2}}}\right )}{\left (a^{2}+b^{2}\right )^{3}}}{d}\) | \(283\) |
default | \(\frac {-\frac {2 \left (\left (-a^{3}+3 a \,b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-3 a^{2} b +b^{3}\right )}{\left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right ) \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}-\frac {2 b^{2} \left (\frac {-\frac {b^{2} \left (9 a^{2}+2 b^{2}\right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 a}-\frac {b \left (8 a^{4}-15 a^{2} b^{2}-2 b^{4}\right ) \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 a^{2}}+\frac {b^{2} \left (23 a^{2}+2 b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 a}+4 a^{2} b +\frac {b^{3}}{2}}{{\left (\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a -2 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-a \right )}^{2}}-\frac {3 \left (4 a^{2}-b^{2}\right ) \operatorname {arctanh}\left (\frac {2 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-2 b}{2 \sqrt {a^{2}+b^{2}}}\right )}{2 \sqrt {a^{2}+b^{2}}}\right )}{\left (a^{2}+b^{2}\right )^{3}}}{d}\) | \(283\) |
risch | \(-\frac {i {\mathrm e}^{i \left (d x +c \right )}}{2 \left (-3 i b \,a^{2}+i b^{3}+a^{3}-3 a \,b^{2}\right ) d}+\frac {i {\mathrm e}^{-i \left (d x +c \right )}}{2 \left (3 i b \,a^{2}-i b^{3}+a^{3}-3 a \,b^{2}\right ) d}+\frac {b^{3} {\mathrm e}^{i \left (d x +c \right )} \left (-7 i a b \,{\mathrm e}^{2 i \left (d x +c \right )}+8 a^{2} {\mathrm e}^{2 i \left (d x +c \right )}+b^{2} {\mathrm e}^{2 i \left (d x +c \right )}+7 i a b +8 a^{2}+b^{2}\right )}{\left (-i a +b \right )^{3} \left (b \,{\mathrm e}^{2 i \left (d x +c \right )}+i a \,{\mathrm e}^{2 i \left (d x +c \right )}-b +i a \right )^{2} d \left (i a +b \right )^{3}}+\frac {6 b^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i a^{7}+3 i a^{5} b^{2}+3 i a^{3} b^{4}+i a \,b^{6}-a^{6} b -3 a^{4} b^{3}-3 b^{5} a^{2}-b^{7}}{\left (a^{2}+b^{2}\right )^{\frac {7}{2}}}\right ) a^{2}}{\left (a^{2}+b^{2}\right )^{\frac {7}{2}} d}-\frac {3 b^{4} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i a^{7}+3 i a^{5} b^{2}+3 i a^{3} b^{4}+i a \,b^{6}-a^{6} b -3 a^{4} b^{3}-3 b^{5} a^{2}-b^{7}}{\left (a^{2}+b^{2}\right )^{\frac {7}{2}}}\right )}{2 \left (a^{2}+b^{2}\right )^{\frac {7}{2}} d}-\frac {6 b^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-\frac {i a^{7}+3 i a^{5} b^{2}+3 i a^{3} b^{4}+i a \,b^{6}-a^{6} b -3 a^{4} b^{3}-3 b^{5} a^{2}-b^{7}}{\left (a^{2}+b^{2}\right )^{\frac {7}{2}}}\right ) a^{2}}{\left (a^{2}+b^{2}\right )^{\frac {7}{2}} d}+\frac {3 b^{4} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-\frac {i a^{7}+3 i a^{5} b^{2}+3 i a^{3} b^{4}+i a \,b^{6}-a^{6} b -3 a^{4} b^{3}-3 b^{5} a^{2}-b^{7}}{\left (a^{2}+b^{2}\right )^{\frac {7}{2}}}\right )}{2 \left (a^{2}+b^{2}\right )^{\frac {7}{2}} d}\) | \(611\) |
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Leaf count of result is larger than twice the leaf count of optimal. 480 vs. \(2 (209) = 418\).
Time = 0.31 (sec) , antiderivative size = 480, normalized size of antiderivative = 2.17 \[ \int \frac {\cos (c+d x)}{(a+b \tan (c+d x))^3} \, dx=\frac {4 \, {\left (a^{6} b + 3 \, a^{4} b^{3} + 3 \, a^{2} b^{5} + b^{7}\right )} \cos \left (d x + c\right )^{3} - 3 \, {\left (4 \, a^{2} b^{4} - b^{6} + {\left (4 \, a^{4} b^{2} - 5 \, a^{2} b^{4} + b^{6}\right )} \cos \left (d x + c\right )^{2} + 2 \, {\left (4 \, a^{3} b^{3} - a b^{5}\right )} \cos \left (d x + c\right ) \sin \left (d x + c\right )\right )} \sqrt {a^{2} + b^{2}} \log \left (\frac {2 \, a b \cos \left (d x + c\right ) \sin \left (d x + c\right ) + {\left (a^{2} - b^{2}\right )} \cos \left (d x + c\right )^{2} - 2 \, a^{2} - b^{2} - 2 \, \sqrt {a^{2} + b^{2}} {\left (b \cos \left (d x + c\right ) - a \sin \left (d x + c\right )\right )}}{2 \, a b \cos \left (d x + c\right ) \sin \left (d x + c\right ) + {\left (a^{2} - b^{2}\right )} \cos \left (d x + c\right )^{2} + b^{2}}\right ) + 2 \, {\left (4 \, a^{6} b - 10 \, a^{4} b^{3} - 17 \, a^{2} b^{5} - 3 \, b^{7}\right )} \cos \left (d x + c\right ) + 2 \, {\left (2 \, a^{5} b^{2} - 11 \, a^{3} b^{4} - 13 \, a b^{6} + 2 \, {\left (a^{7} + 3 \, a^{5} b^{2} + 3 \, a^{3} b^{4} + a b^{6}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )}{4 \, {\left ({\left (a^{10} + 3 \, a^{8} b^{2} + 2 \, a^{6} b^{4} - 2 \, a^{4} b^{6} - 3 \, a^{2} b^{8} - b^{10}\right )} d \cos \left (d x + c\right )^{2} + 2 \, {\left (a^{9} b + 4 \, a^{7} b^{3} + 6 \, a^{5} b^{5} + 4 \, a^{3} b^{7} + a b^{9}\right )} d \cos \left (d x + c\right ) \sin \left (d x + c\right ) + {\left (a^{8} b^{2} + 4 \, a^{6} b^{4} + 6 \, a^{4} b^{6} + 4 \, a^{2} b^{8} + b^{10}\right )} d\right )}} \]
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\[ \int \frac {\cos (c+d x)}{(a+b \tan (c+d x))^3} \, dx=\int \frac {\cos {\left (c + d x \right )}}{\left (a + b \tan {\left (c + d x \right )}\right )^{3}}\, dx \]
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Leaf count of result is larger than twice the leaf count of optimal. 658 vs. \(2 (209) = 418\).
Time = 0.32 (sec) , antiderivative size = 658, normalized size of antiderivative = 2.98 \[ \int \frac {\cos (c+d x)}{(a+b \tan (c+d x))^3} \, dx=-\frac {\frac {3 \, {\left (4 \, a^{2} b^{2} - b^{4}\right )} \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 (6 \, a^{6} b - 10 \, a^{4} b^{3} - a^{2} b^{5} + \frac {{\left (2 \, a^{7} + 18 \, a^{5} b^{2} - 31 \, a^{3} b^{4} - 2 \, a b^{6}\right )} \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {2 \, {\left (2 \, a^{6} b - 2 \, a^{4} b^{3} + 12 \, a^{2} b^{5} + b^{7}\right )} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {2 \, {\left (2 \, a^{7} + 2 \, a^{5} b^{2} + 15 \, a^{3} b^{4}\right )} \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac {{\left (2 \, a^{6} b - 30 \, a^{4} b^{3} + 15 \, a^{2} b^{5} + 2 \, b^{7}\right )} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {{\left (2 \, a^{7} - 6 \, a^{5} b^{2} + 9 \, a^{3} b^{4} + 2 \, a b^{6}\right )} \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}}\right )}}{a^{10} + 3 \, a^{8} b^{2} + 3 \, a^{6} b^{4} + a^{4} b^{6} + \frac {4 \, {\left (a^{9} b + 3 \, a^{7} b^{3} + 3 \, a^{5} b^{5} + a^{3} b^{7}\right )} \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {{\left (a^{10} - a^{8} b^{2} - 9 \, a^{6} b^{4} - 11 \, a^{4} b^{6} - 4 \, a^{2} b^{8}\right )} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {{\left (a^{10} - a^{8} b^{2} - 9 \, a^{6} b^{4} - 11 \, a^{4} b^{6} - 4 \, a^{2} b^{8}\right )} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac {4 \, {\left (a^{9} b + 3 \, a^{7} b^{3} + 3 \, a^{5} b^{5} + a^{3} b^{7}\right )} \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {{\left (a^{10} + 3 \, a^{8} b^{2} + 3 \, a^{6} b^{4} + a^{4} b^{6}\right )} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}}}}{2 \, d} \]
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Time = 0.63 (sec) , antiderivative size = 399, normalized size of antiderivative = 1.81 \[ \int \frac {\cos (c+d x)}{(a+b \tan (c+d x))^3} \, dx=-\frac {\frac {3 \, {\left (4 \, a^{2} b^{2} - b^{4}\right )} \log \left (\frac {{\left | 2 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 2 \, b - 2 \, \sqrt {a^{2} + b^{2}} \right |}}{{\left | 2 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 2 \, b + 2 \, \sqrt {a^{2} + b^{2}} \right |}}\right )}{{\left (a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}\right )} \sqrt {a^{2} + b^{2}}} - \frac {4 \, {\left (a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 3 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 3 \, a^{2} b - b^{3}\right )}}{{\left (a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}\right )} {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}} - \frac {2 \, {\left (9 \, a^{3} b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 2 \, a b^{6} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 8 \, a^{4} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 15 \, a^{2} b^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 2 \, b^{7} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 23 \, a^{3} b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 2 \, a b^{6} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 8 \, a^{4} b^{3} - a^{2} b^{5}\right )}}{{\left (a^{8} + 3 \, a^{6} b^{2} + 3 \, a^{4} b^{4} + a^{2} b^{6}\right )} {\left (a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 2 \, b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - a\right )}^{2}}}{2 \, d} \]
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Time = 8.10 (sec) , antiderivative size = 610, normalized size of antiderivative = 2.76 \[ \int \frac {\cos (c+d x)}{(a+b \tan (c+d x))^3} \, dx=-\frac {\frac {-6\,a^4\,b+10\,a^2\,b^3+b^5}{a^6+3\,a^4\,b^2+3\,a^2\,b^4+b^6}+\frac {2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (2\,a^5+2\,a^3\,b^2+15\,a\,b^4\right )}{a^6+3\,a^4\,b^2+3\,a^2\,b^4+b^6}+\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\left (2\,a^6\,b-30\,a^4\,b^3+15\,a^2\,b^5+2\,b^7\right )}{a^2\,\left (a^6+3\,a^4\,b^2+3\,a^2\,b^4+b^6\right )}-\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (2\,a^6+18\,a^4\,b^2-31\,a^2\,b^4-2\,b^6\right )}{a\,\left (a^6+3\,a^4\,b^2+3\,a^2\,b^4+b^6\right )}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,\left (2\,a^6-6\,a^4\,b^2+9\,a^2\,b^4+2\,b^6\right )}{a\,\left (a^6+3\,a^4\,b^2+3\,a^2\,b^4+b^6\right )}+\frac {2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (2\,a^6\,b-2\,a^4\,b^3+12\,a^2\,b^5+b^7\right )}{a^2\,\left (a^6+3\,a^4\,b^2+3\,a^2\,b^4+b^6\right )}}{d\,\left (a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+a^2-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (a^2-4\,b^2\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\left (a^2-4\,b^2\right )-4\,a\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+4\,a\,b\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}-\frac {\mathrm {atan}\left (\frac {-1{}\mathrm {i}\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,a^7+a^6\,b\,1{}\mathrm {i}-3{}\mathrm {i}\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,a^5\,b^2+a^4\,b^3\,3{}\mathrm {i}-3{}\mathrm {i}\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,a^3\,b^4+a^2\,b^5\,3{}\mathrm {i}-1{}\mathrm {i}\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,a\,b^6+b^7\,1{}\mathrm {i}}{{\left (a^2+b^2\right )}^{7/2}}\right )\,\left (3\,b^4-12\,a^2\,b^2\right )\,1{}\mathrm {i}}{d\,{\left (a^2+b^2\right )}^{7/2}} \]
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