Integrand size = 21, antiderivative size = 217 \[ \int \frac {\sin ^4(c+d x)}{(a+b \tan (c+d x))^2} \, dx=\frac {\left (3 a^6-33 a^4 b^2+13 a^2 b^4+b^6\right ) x}{8 \left (a^2+b^2\right )^4}+\frac {2 a^3 b \left (a^2-2 b^2\right ) \log (a \cos (c+d x)+b \sin (c+d x))}{\left (a^2+b^2\right )^4 d}-\frac {a^4 b}{\left (a^2+b^2\right )^3 d (a+b \tan (c+d x))}+\frac {\cos ^4(c+d x) \left (2 a b+\left (a^2-b^2\right ) \tan (c+d x)\right )}{4 \left (a^2+b^2\right )^2 d}-\frac {\cos ^2(c+d x) \left (16 a^3 b+\left (5 a^4-12 a^2 b^2-b^4\right ) \tan (c+d x)\right )}{8 \left (a^2+b^2\right )^3 d} \]
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Time = 0.73 (sec) , antiderivative size = 217, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {3597, 1661, 1643, 649, 209, 266} \[ \int \frac {\sin ^4(c+d x)}{(a+b \tan (c+d x))^2} \, dx=\frac {\cos ^4(c+d x) \left (\left (a^2-b^2\right ) \tan (c+d x)+2 a b\right )}{4 d \left (a^2+b^2\right )^2}-\frac {a^4 b}{d \left (a^2+b^2\right )^3 (a+b \tan (c+d x))}+\frac {2 a^3 b \left (a^2-2 b^2\right ) \log (a \cos (c+d x)+b \sin (c+d x))}{d \left (a^2+b^2\right )^4}+\frac {x \left (3 a^6-33 a^4 b^2+13 a^2 b^4+b^6\right )}{8 \left (a^2+b^2\right )^4}-\frac {\cos ^2(c+d x) \left (16 a^3 b+\left (5 a^4-12 a^2 b^2-b^4\right ) \tan (c+d x)\right )}{8 d \left (a^2+b^2\right )^3} \]
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Rule 209
Rule 266
Rule 649
Rule 1643
Rule 1661
Rule 3597
Rubi steps \begin{align*} \text {integral}& = \frac {b \text {Subst}\left (\int \frac {x^4}{(a+x)^2 \left (b^2+x^2\right )^3} \, dx,x,b \tan (c+d x)\right )}{d} \\ & = \frac {\cos ^4(c+d x) \left (2 a b+\left (a^2-b^2\right ) \tan (c+d x)\right )}{4 \left (a^2+b^2\right )^2 d}-\frac {\text {Subst}\left (\int \frac {\frac {a^2 b^4 \left (a^2-b^2\right )}{\left (a^2+b^2\right )^2}-\frac {2 a b^4 \left (3 a^2+b^2\right ) x}{\left (a^2+b^2\right )^2}-\frac {b^2 \left (4 a^4+11 a^2 b^2+b^4\right ) x^2}{\left (a^2+b^2\right )^2}}{(a+x)^2 \left (b^2+x^2\right )^2} \, dx,x,b \tan (c+d x)\right )}{4 b d} \\ & = \frac {\cos ^4(c+d x) \left (2 a b+\left (a^2-b^2\right ) \tan (c+d x)\right )}{4 \left (a^2+b^2\right )^2 d}-\frac {\cos ^2(c+d x) \left (16 a^3 b+\left (5 a^4-12 a^2 b^2-b^4\right ) \tan (c+d x)\right )}{8 \left (a^2+b^2\right )^3 d}+\frac {\text {Subst}\left (\int \frac {\frac {a^2 b^4 \left (3 a^4-12 a^2 b^2+b^4\right )}{\left (a^2+b^2\right )^3}-\frac {2 a b^4 \left (5 a^2-b^2\right ) x}{\left (a^2+b^2\right )^2}-\frac {b^4 \left (5 a^4-12 a^2 b^2-b^4\right ) x^2}{\left (a^2+b^2\right )^3}}{(a+x)^2 \left (b^2+x^2\right )} \, dx,x,b \tan (c+d x)\right )}{8 b^3 d} \\ & = \frac {\cos ^4(c+d x) \left (2 a b+\left (a^2-b^2\right ) \tan (c+d x)\right )}{4 \left (a^2+b^2\right )^2 d}-\frac {\cos ^2(c+d x) \left (16 a^3 b+\left (5 a^4-12 a^2 b^2-b^4\right ) \tan (c+d x)\right )}{8 \left (a^2+b^2\right )^3 d}+\frac {\text {Subst}\left (\int \left (\frac {8 a^4 b^4}{\left (a^2+b^2\right )^3 (a+x)^2}+\frac {16 a^3 b^4 \left (a^2-2 b^2\right )}{\left (a^2+b^2\right )^4 (a+x)}+\frac {b^4 \left (3 a^6-33 a^4 b^2+13 a^2 b^4+b^6-16 a^3 \left (a^2-2 b^2\right ) x\right )}{\left (a^2+b^2\right )^4 \left (b^2+x^2\right )}\right ) \, dx,x,b \tan (c+d x)\right )}{8 b^3 d} \\ & = \frac {2 a^3 b \left (a^2-2 b^2\right ) \log (a+b \tan (c+d x))}{\left (a^2+b^2\right )^4 d}-\frac {a^4 b}{\left (a^2+b^2\right )^3 d (a+b \tan (c+d x))}+\frac {\cos ^4(c+d x) \left (2 a b+\left (a^2-b^2\right ) \tan (c+d x)\right )}{4 \left (a^2+b^2\right )^2 d}-\frac {\cos ^2(c+d x) \left (16 a^3 b+\left (5 a^4-12 a^2 b^2-b^4\right ) \tan (c+d x)\right )}{8 \left (a^2+b^2\right )^3 d}+\frac {b \text {Subst}\left (\int \frac {3 a^6-33 a^4 b^2+13 a^2 b^4+b^6-16 a^3 \left (a^2-2 b^2\right ) x}{b^2+x^2} \, dx,x,b \tan (c+d x)\right )}{8 \left (a^2+b^2\right )^4 d} \\ & = \frac {2 a^3 b \left (a^2-2 b^2\right ) \log (a+b \tan (c+d x))}{\left (a^2+b^2\right )^4 d}-\frac {a^4 b}{\left (a^2+b^2\right )^3 d (a+b \tan (c+d x))}+\frac {\cos ^4(c+d x) \left (2 a b+\left (a^2-b^2\right ) \tan (c+d x)\right )}{4 \left (a^2+b^2\right )^2 d}-\frac {\cos ^2(c+d x) \left (16 a^3 b+\left (5 a^4-12 a^2 b^2-b^4\right ) \tan (c+d x)\right )}{8 \left (a^2+b^2\right )^3 d}-\frac {\left (2 a^3 b \left (a^2-2 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 )^4 d}+\frac {\left (b \left (3 a^6-33 a^4 b^2+13 a^2 b^4+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 )^4 d} \\ & = \frac {\left (3 a^6-33 a^4 b^2+13 a^2 b^4+b^6\right ) x}{8 \left (a^2+b^2\right )^4}+\frac {2 a^3 b \left (a^2-2 b^2\right ) \log (\cos (c+d x))}{\left (a^2+b^2\right )^4 d}+\frac {2 a^3 b \left (a^2-2 b^2\right ) \log (a+b \tan (c+d x))}{\left (a^2+b^2\right )^4 d}-\frac {a^4 b}{\left (a^2+b^2\right )^3 d (a+b \tan (c+d x))}+\frac {\cos ^4(c+d x) \left (2 a b+\left (a^2-b^2\right ) \tan (c+d x)\right )}{4 \left (a^2+b^2\right )^2 d}-\frac {\cos ^2(c+d x) \left (16 a^3 b+\left (5 a^4-12 a^2 b^2-b^4\right ) \tan (c+d x)\right )}{8 \left (a^2+b^2\right )^3 d} \\ \end{align*}
Time = 4.47 (sec) , antiderivative size = 392, normalized size of antiderivative = 1.81 \[ \int \frac {\sin ^4(c+d x)}{(a+b \tan (c+d x))^2} \, dx=\frac {b \left (\frac {3 \left (a^2-b^2\right ) \left (a^2+b^2\right )^2 \arctan (\tan (c+d x))}{b}+\frac {4 \left (a^2+b^2\right ) \left (-2 a^4+3 a^2 b^2+b^4\right ) \arctan (\tan (c+d x))}{b}-16 a^3 \left (a^2+b^2\right ) \cos ^2(c+d x)+4 a \left (a^2+b^2\right )^2 \cos ^4(c+d x)-4 a^3 \left (2 a^2-4 b^2+\frac {-a^3+5 a b^2}{\sqrt {-b^2}}\right ) \log \left (\sqrt {-b^2}-b \tan (c+d x)\right )+16 a^3 \left (a^2-2 b^2\right ) \log (a+b \tan (c+d x))-4 a^3 \left (2 a^2-4 b^2+\frac {a^3-5 a b^2}{\sqrt {-b^2}}\right ) \log \left (\sqrt {-b^2}+b \tan (c+d x)\right )+\frac {2 \left (a^2-b^2\right ) \left (a^2+b^2\right )^2 \cos ^3(c+d x) \sin (c+d x)}{b}+\frac {3 (a-b) (a+b) \left (a^2+b^2\right )^2 \sin (2 (c+d x))}{2 b}+\frac {2 \left (a^2+b^2\right ) \left (-2 a^4+3 a^2 b^2+b^4\right ) \sin (2 (c+d x))}{b}-\frac {8 a^4 \left (a^2+b^2\right )}{a+b \tan (c+d x)}\right )}{8 \left (a^2+b^2\right )^4 d} \]
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Time = 9.21 (sec) , antiderivative size = 269, normalized size of antiderivative = 1.24
method | result | size |
derivativedivides | \(\frac {-\frac {a^{4} b}{\left (a^{2}+b^{2}\right )^{3} \left (a +b \tan \left (d x +c \right )\right )}+\frac {2 a^{3} b \left (a^{2}-2 b^{2}\right ) \ln \left (a +b \tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{4}}+\frac {\frac {\left (-\frac {5}{8} a^{6}+\frac {7}{8} a^{4} b^{2}+\frac {13}{8} a^{2} b^{4}+\frac {1}{8} b^{6}\right ) \left (\tan ^{3}\left (d x +c \right )\right )+\left (-2 a^{5} b -2 a^{3} b^{3}\right ) \left (\tan ^{2}\left (d x +c \right )\right )+\left (-\frac {3}{8} a^{6}+\frac {9}{8} a^{4} b^{2}+\frac {11}{8} a^{2} b^{4}-\frac {1}{8} b^{6}\right ) \tan \left (d x +c \right )-\frac {3 a^{5} b}{2}-a^{3} b^{3}+\frac {a \,b^{5}}{2}}{\left (1+\tan ^{2}\left (d x +c \right )\right )^{2}}+\frac {\left (-16 a^{5} b +32 a^{3} b^{3}\right ) \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{16}+\frac {\left (3 a^{6}-33 a^{4} b^{2}+13 a^{2} b^{4}+b^{6}\right ) \arctan \left (\tan \left (d x +c \right )\right )}{8}}{\left (a^{2}+b^{2}\right )^{4}}}{d}\) | \(269\) |
default | \(\frac {-\frac {a^{4} b}{\left (a^{2}+b^{2}\right )^{3} \left (a +b \tan \left (d x +c \right )\right )}+\frac {2 a^{3} b \left (a^{2}-2 b^{2}\right ) \ln \left (a +b \tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{4}}+\frac {\frac {\left (-\frac {5}{8} a^{6}+\frac {7}{8} a^{4} b^{2}+\frac {13}{8} a^{2} b^{4}+\frac {1}{8} b^{6}\right ) \left (\tan ^{3}\left (d x +c \right )\right )+\left (-2 a^{5} b -2 a^{3} b^{3}\right ) \left (\tan ^{2}\left (d x +c \right )\right )+\left (-\frac {3}{8} a^{6}+\frac {9}{8} a^{4} b^{2}+\frac {11}{8} a^{2} b^{4}-\frac {1}{8} b^{6}\right ) \tan \left (d x +c \right )-\frac {3 a^{5} b}{2}-a^{3} b^{3}+\frac {a \,b^{5}}{2}}{\left (1+\tan ^{2}\left (d x +c \right )\right )^{2}}+\frac {\left (-16 a^{5} b +32 a^{3} b^{3}\right ) \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{16}+\frac {\left (3 a^{6}-33 a^{4} b^{2}+13 a^{2} b^{4}+b^{6}\right ) \arctan \left (\tan \left (d x +c \right )\right )}{8}}{\left (a^{2}+b^{2}\right )^{4}}}{d}\) | \(269\) |
risch | \(-\frac {i x a b}{2 \left (4 i a^{3} b -4 i a \,b^{3}-a^{4}+6 a^{2} b^{2}-b^{4}\right )}-\frac {3 x \,a^{2}}{8 \left (4 i a^{3} b -4 i a \,b^{3}-a^{4}+6 a^{2} b^{2}-b^{4}\right )}-\frac {x \,b^{2}}{8 \left (4 i a^{3} b -4 i a \,b^{3}-a^{4}+6 a^{2} b^{2}-b^{4}\right )}-\frac {i {\mathrm e}^{4 i \left (d x +c \right )}}{64 \left (-2 i a b +a^{2}-b^{2}\right ) d}+\frac {i a \,{\mathrm e}^{2 i \left (d x +c \right )}}{8 \left (-3 i b \,a^{2}+i b^{3}+a^{3}-3 a \,b^{2}\right ) d}-\frac {i a \,{\mathrm e}^{-2 i \left (d x +c \right )}}{8 \left (2 i a b +a^{2}-b^{2}\right ) \left (i b +a \right ) d}+\frac {i {\mathrm e}^{-4 i \left (d x +c \right )}}{64 \left (2 i a b +a^{2}-b^{2}\right ) d}-\frac {4 i a^{5} b x}{a^{8}+4 a^{6} b^{2}+6 a^{4} b^{4}+4 b^{6} a^{2}+b^{8}}+\frac {8 i a^{3} b^{3} x}{a^{8}+4 a^{6} b^{2}+6 a^{4} b^{4}+4 b^{6} a^{2}+b^{8}}-\frac {4 i a^{5} b c}{d \left (a^{8}+4 a^{6} b^{2}+6 a^{4} b^{4}+4 b^{6} a^{2}+b^{8}\right )}+\frac {8 i a^{3} b^{3} c}{d \left (a^{8}+4 a^{6} b^{2}+6 a^{4} b^{4}+4 b^{6} a^{2}+b^{8}\right )}+\frac {2 i a^{4} b^{2}}{\left (i b +a \right )^{3} d \left (-i b +a \right )^{4} \left (-i b \,{\mathrm e}^{2 i \left (d x +c \right )}+a \,{\mathrm e}^{2 i \left (d x +c \right )}+i b +a \right )}+\frac {2 a^{5} b \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-\frac {i b +a}{i b -a}\right )}{d \left (a^{8}+4 a^{6} b^{2}+6 a^{4} b^{4}+4 b^{6} a^{2}+b^{8}\right )}-\frac {4 a^{3} b^{3} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-\frac {i b +a}{i b -a}\right )}{d \left (a^{8}+4 a^{6} b^{2}+6 a^{4} b^{4}+4 b^{6} a^{2}+b^{8}\right )}\) | \(646\) |
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Leaf count of result is larger than twice the leaf count of optimal. 444 vs. \(2 (211) = 422\).
Time = 0.31 (sec) , antiderivative size = 444, normalized size of antiderivative = 2.05 \[ \int \frac {\sin ^4(c+d x)}{(a+b \tan (c+d x))^2} \, dx=\frac {4 \, {\left (a^{6} b + 3 \, a^{4} b^{3} + 3 \, a^{2} b^{5} + b^{7}\right )} \cos \left (d x + c\right )^{5} - 6 \, {\left (3 \, a^{6} b + 7 \, a^{4} b^{3} + 5 \, a^{2} b^{5} + b^{7}\right )} \cos \left (d x + c\right )^{3} + {\left (3 \, a^{6} b + 8 \, a^{4} b^{3} + 23 \, a^{2} b^{5} + 2 \, b^{7} + 2 \, {\left (3 \, a^{7} - 33 \, a^{5} b^{2} + 13 \, a^{3} b^{4} + a b^{6}\right )} d x\right )} \cos \left (d x + c\right ) + 16 \, {\left ({\left (a^{6} b - 2 \, a^{4} b^{3}\right )} \cos \left (d x + c\right ) + {\left (a^{5} b^{2} - 2 \, a^{3} b^{4}\right )} \sin \left (d x + c\right )\right )} \log \left (2 \, a b \cos \left (d x + c\right ) \sin \left (d x + c\right ) + {\left (a^{2} - b^{2}\right )} \cos \left (d x + c\right )^{2} + b^{2}\right ) + {\left (29 \, a^{5} b^{2} + 10 \, a^{3} b^{4} - 3 \, a b^{6} + 4 \, {\left (a^{7} + 3 \, a^{5} b^{2} + 3 \, a^{3} b^{4} + a b^{6}\right )} \cos \left (d x + c\right )^{4} + 2 \, {\left (3 \, a^{6} b - 33 \, a^{4} b^{3} + 13 \, a^{2} b^{5} + b^{7}\right )} d x - 2 \, {\left (5 \, a^{7} + 9 \, a^{5} b^{2} + 3 \, a^{3} b^{4} - a b^{6}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )}{16 \, {\left ({\left (a^{9} + 4 \, a^{7} b^{2} + 6 \, a^{5} b^{4} + 4 \, a^{3} b^{6} + a b^{8}\right )} d \cos \left (d x + c\right ) + {\left (a^{8} b + 4 \, a^{6} b^{3} + 6 \, a^{4} b^{5} + 4 \, a^{2} b^{7} + b^{9}\right )} d \sin \left (d x + c\right )\right )}} \]
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Timed out. \[ \int \frac {\sin ^4(c+d x)}{(a+b \tan (c+d x))^2} \, dx=\text {Timed out} \]
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Leaf count of result is larger than twice the leaf count of optimal. 507 vs. \(2 (211) = 422\).
Time = 0.66 (sec) , antiderivative size = 507, normalized size of antiderivative = 2.34 \[ \int \frac {\sin ^4(c+d x)}{(a+b \tan (c+d x))^2} \, dx=\frac {\frac {{\left (3 \, a^{6} - 33 \, a^{4} b^{2} + 13 \, a^{2} b^{4} + b^{6}\right )} {\left (d x + c\right )}}{a^{8} + 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} + 4 \, a^{2} b^{6} + b^{8}} + \frac {16 \, {\left (a^{5} b - 2 \, a^{3} b^{3}\right )} \log \left (b \tan \left (d x + c\right ) + a\right )}{a^{8} + 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} + 4 \, a^{2} b^{6} + b^{8}} - \frac {8 \, {\left (a^{5} b - 2 \, a^{3} b^{3}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{a^{8} + 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} + 4 \, a^{2} b^{6} + b^{8}} - \frac {20 \, a^{4} b - 4 \, a^{2} b^{3} + {\left (13 \, a^{4} b - 12 \, a^{2} b^{3} - b^{5}\right )} \tan \left (d x + c\right )^{4} + {\left (5 \, a^{5} + 4 \, a^{3} b^{2} - a b^{4}\right )} \tan \left (d x + c\right )^{3} + {\left (35 \, a^{4} b - 12 \, a^{2} b^{3} + b^{5}\right )} \tan \left (d x + c\right )^{2} + 3 \, {\left (a^{5} - a b^{4}\right )} \tan \left (d x + c\right )}{a^{7} + 3 \, a^{5} b^{2} + 3 \, a^{3} b^{4} + a b^{6} + {\left (a^{6} b + 3 \, a^{4} b^{3} + 3 \, a^{2} b^{5} + b^{7}\right )} \tan \left (d x + c\right )^{5} + {\left (a^{7} + 3 \, a^{5} b^{2} + 3 \, a^{3} b^{4} + a b^{6}\right )} \tan \left (d x + c\right )^{4} + 2 \, {\left (a^{6} b + 3 \, a^{4} b^{3} + 3 \, a^{2} b^{5} + b^{7}\right )} \tan \left (d x + c\right )^{3} + 2 \, {\left (a^{7} + 3 \, a^{5} b^{2} + 3 \, a^{3} b^{4} + a b^{6}\right )} \tan \left (d x + c\right )^{2} + {\left (a^{6} b + 3 \, a^{4} b^{3} + 3 \, a^{2} b^{5} + b^{7}\right )} \tan \left (d x + c\right )}}{8 \, d} \]
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Leaf count of result is larger than twice the leaf count of optimal. 513 vs. \(2 (211) = 422\).
Time = 0.53 (sec) , antiderivative size = 513, normalized size of antiderivative = 2.36 \[ \int \frac {\sin ^4(c+d x)}{(a+b \tan (c+d x))^2} \, dx=\frac {\frac {{\left (3 \, a^{6} - 33 \, a^{4} b^{2} + 13 \, a^{2} b^{4} + b^{6}\right )} {\left (d x + c\right )}}{a^{8} + 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} + 4 \, a^{2} b^{6} + b^{8}} - \frac {8 \, {\left (a^{5} b - 2 \, a^{3} b^{3}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{a^{8} + 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} + 4 \, a^{2} b^{6} + b^{8}} + \frac {16 \, {\left (a^{5} b^{2} - 2 \, a^{3} b^{4}\right )} \log \left ({\left | b \tan \left (d x + c\right ) + a \right |}\right )}{a^{8} b + 4 \, a^{6} b^{3} + 6 \, a^{4} b^{5} + 4 \, a^{2} b^{7} + b^{9}} - \frac {8 \, {\left (2 \, a^{5} b^{2} \tan \left (d x + c\right ) - 4 \, a^{3} b^{4} \tan \left (d x + c\right ) + 3 \, a^{6} b - 3 \, a^{4} b^{3}\right )}}{{\left (a^{8} + 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} + 4 \, a^{2} b^{6} + b^{8}\right )} {\left (b \tan \left (d x + c\right ) + a\right )}} + \frac {12 \, a^{5} b \tan \left (d x + c\right )^{4} - 24 \, a^{3} b^{3} \tan \left (d x + c\right )^{4} - 5 \, a^{6} \tan \left (d x + c\right )^{3} + 7 \, a^{4} b^{2} \tan \left (d x + c\right )^{3} + 13 \, a^{2} b^{4} \tan \left (d x + c\right )^{3} + b^{6} \tan \left (d x + c\right )^{3} + 8 \, a^{5} b \tan \left (d x + c\right )^{2} - 64 \, a^{3} b^{3} \tan \left (d x + c\right )^{2} - 3 \, a^{6} \tan \left (d x + c\right ) + 9 \, a^{4} b^{2} \tan \left (d x + c\right ) + 11 \, a^{2} b^{4} \tan \left (d x + c\right ) - b^{6} \tan \left (d x + c\right ) - 32 \, a^{3} b^{3} + 4 \, a b^{5}}{{\left (a^{8} + 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} + 4 \, a^{2} b^{6} + b^{8}\right )} {\left (\tan \left (d x + c\right )^{2} + 1\right )}^{2}}}{8 \, d} \]
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Time = 5.82 (sec) , antiderivative size = 481, normalized size of antiderivative = 2.22 \[ \int \frac {\sin ^4(c+d x)}{(a+b \tan (c+d x))^2} \, dx=\frac {\frac {{\mathrm {tan}\left (c+d\,x\right )}^3\,\left (a\,b^2-5\,a^3\right )}{8\,\left (a^4+2\,a^2\,b^2+b^4\right )}+\frac {{\mathrm {tan}\left (c+d\,x\right )}^4\,\left (-13\,a^4\,b+12\,a^2\,b^3+b^5\right )}{8\,\left (a^6+3\,a^4\,b^2+3\,a^2\,b^4+b^6\right )}+\frac {3\,\mathrm {tan}\left (c+d\,x\right )\,\left (a\,b^2-a^3\right )}{8\,\left (a^4+2\,a^2\,b^2+b^4\right )}-\frac {{\mathrm {tan}\left (c+d\,x\right )}^2\,\left (35\,a^4\,b-12\,a^2\,b^3+b^5\right )}{8\,\left (a^2+b^2\right )\,\left (a^4+2\,a^2\,b^2+b^4\right )}+\frac {a\,\left (a\,b^3-5\,a^3\,b\right )}{2\,\left (a^2+b^2\right )\,\left (a^4+2\,a^2\,b^2+b^4\right )}}{d\,\left (b\,{\mathrm {tan}\left (c+d\,x\right )}^5+a\,{\mathrm {tan}\left (c+d\,x\right )}^4+2\,b\,{\mathrm {tan}\left (c+d\,x\right )}^3+2\,a\,{\mathrm {tan}\left (c+d\,x\right )}^2+b\,\mathrm {tan}\left (c+d\,x\right )+a\right )}+\frac {\ln \left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )\,\left (\frac {2\,a\,b}{{\left (a^2+b^2\right )}^2}-\frac {8\,a\,b^3}{{\left (a^2+b^2\right )}^3}+\frac {6\,a\,b^5}{{\left (a^2+b^2\right )}^4}\right )}{d}+\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )-\mathrm {i}\right )\,\left (3\,a^2-a\,b\,4{}\mathrm {i}+b^2\right )}{16\,d\,\left (a^4\,1{}\mathrm {i}-4\,a^3\,b-a^2\,b^2\,6{}\mathrm {i}+4\,a\,b^3+b^4\,1{}\mathrm {i}\right )}-\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )\,\left (3\,a^2+a\,b\,4{}\mathrm {i}+b^2\right )}{16\,d\,\left (a^4\,1{}\mathrm {i}+4\,a^3\,b-a^2\,b^2\,6{}\mathrm {i}-4\,a\,b^3+b^4\,1{}\mathrm {i}\right )} \]
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