Integrand size = 21, antiderivative size = 264 \[ \int \frac {\sin ^2(c+d x)}{(a+b \tan (c+d x))^4} \, dx=\frac {\left (a^6-25 a^4 b^2+35 a^2 b^4-3 b^6\right ) x}{2 \left (a^2+b^2\right )^5}+\frac {4 a b \left (a^4-5 a^2 b^2+2 b^4\right ) \log (a \cos (c+d x)+b \sin (c+d x))}{\left (a^2+b^2\right )^5 d}-\frac {a^2 b}{3 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))^3}-\frac {a b \left (a^2-b^2\right )}{\left (a^2+b^2\right )^3 d (a+b \tan (c+d x))^2}-\frac {b \left (3 a^4-8 a^2 b^2+b^4\right )}{\left (a^2+b^2\right )^4 d (a+b \tan (c+d x))}-\frac {\cos ^2(c+d x) \left (4 a b \left (a^2-b^2\right )+\left (a^4-6 a^2 b^2+b^4\right ) \tan (c+d x)\right )}{2 \left (a^2+b^2\right )^4 d} \]
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Time = 0.77 (sec) , antiderivative size = 264, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {3597, 1661, 1643, 649, 209, 266} \[ \int \frac {\sin ^2(c+d x)}{(a+b \tan (c+d x))^4} \, dx=-\frac {a^2 b}{3 d \left (a^2+b^2\right )^2 (a+b \tan (c+d x))^3}-\frac {a b \left (a^2-b^2\right )}{d \left (a^2+b^2\right )^3 (a+b \tan (c+d x))^2}-\frac {b \left (3 a^4-8 a^2 b^2+b^4\right )}{d \left (a^2+b^2\right )^4 (a+b \tan (c+d x))}-\frac {\cos ^2(c+d x) \left (4 a b \left (a^2-b^2\right )+\left (a^4-6 a^2 b^2+b^4\right ) \tan (c+d x)\right )}{2 d \left (a^2+b^2\right )^4}+\frac {4 a b \left (a^4-5 a^2 b^2+2 b^4\right ) \log (a \cos (c+d x)+b \sin (c+d x))}{d \left (a^2+b^2\right )^5}+\frac {x \left (a^6-25 a^4 b^2+35 a^2 b^4-3 b^6\right )}{2 \left (a^2+b^2\right )^5} \]
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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^2}{(a+x)^4 \left (b^2+x^2\right )^2} \, dx,x,b \tan (c+d x)\right )}{d} \\ & = -\frac {\cos ^2(c+d x) \left (4 a b \left (a^2-b^2\right )+\left (a^4-6 a^2 b^2+b^4\right ) \tan (c+d x)\right )}{2 \left (a^2+b^2\right )^4 d}-\frac {\text {Subst}\left (\int \frac {-\frac {a^4 b^2 \left (a^4-6 a^2 b^2+b^4\right )}{\left (a^2+b^2\right )^4}+\frac {4 a^3 b^2 \left (a^4+4 a^2 b^2-b^4\right ) x}{\left (a^2+b^2\right )^4}+\frac {2 b^2 \left (3 a^4-6 a^2 b^2-b^4\right ) x^2}{\left (a^2+b^2\right )^3}+\frac {4 a b^2 \left (a^4-4 a^2 b^2-b^4\right ) x^3}{\left (a^2+b^2\right )^4}+\frac {b^2 \left (a^4-6 a^2 b^2+b^4\right ) x^4}{\left (a^2+b^2\right )^4}}{(a+x)^4 \left (b^2+x^2\right )} \, dx,x,b \tan (c+d x)\right )}{2 b d} \\ & = -\frac {\cos ^2(c+d x) \left (4 a b \left (a^2-b^2\right )+\left (a^4-6 a^2 b^2+b^4\right ) \tan (c+d x)\right )}{2 \left (a^2+b^2\right )^4 d}-\frac {\text {Subst}\left (\int \left (-\frac {2 a^2 b^2}{\left (a^2+b^2\right )^2 (a+x)^4}+\frac {4 a b^2 \left (-a^2+b^2\right )}{\left (a^2+b^2\right )^3 (a+x)^3}-\frac {2 \left (3 a^4 b^2-8 a^2 b^4+b^6\right )}{\left (a^2+b^2\right )^4 (a+x)^2}-\frac {8 a b^2 \left (a^4-5 a^2 b^2+2 b^4\right )}{\left (a^2+b^2\right )^5 (a+x)}+\frac {b^2 \left (-a^6+25 a^4 b^2-35 a^2 b^4+3 b^6+8 a \left (a^4-5 a^2 b^2+2 b^4\right ) x\right )}{\left (a^2+b^2\right )^5 \left (b^2+x^2\right )}\right ) \, dx,x,b \tan (c+d x)\right )}{2 b d} \\ & = \frac {4 a b \left (a^4-5 a^2 b^2+2 b^4\right ) \log (a+b \tan (c+d x))}{\left (a^2+b^2\right )^5 d}-\frac {a^2 b}{3 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))^3}-\frac {a b \left (a^2-b^2\right )}{\left (a^2+b^2\right )^3 d (a+b \tan (c+d x))^2}-\frac {b \left (3 a^4-8 a^2 b^2+b^4\right )}{\left (a^2+b^2\right )^4 d (a+b \tan (c+d x))}-\frac {\cos ^2(c+d x) \left (4 a b \left (a^2-b^2\right )+\left (a^4-6 a^2 b^2+b^4\right ) \tan (c+d x)\right )}{2 \left (a^2+b^2\right )^4 d}-\frac {b \text {Subst}\left (\int \frac {-a^6+25 a^4 b^2-35 a^2 b^4+3 b^6+8 a \left (a^4-5 a^2 b^2+2 b^4\right ) x}{b^2+x^2} \, dx,x,b \tan (c+d x)\right )}{2 \left (a^2+b^2\right )^5 d} \\ & = \frac {4 a b \left (a^4-5 a^2 b^2+2 b^4\right ) \log (a+b \tan (c+d x))}{\left (a^2+b^2\right )^5 d}-\frac {a^2 b}{3 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))^3}-\frac {a b \left (a^2-b^2\right )}{\left (a^2+b^2\right )^3 d (a+b \tan (c+d x))^2}-\frac {b \left (3 a^4-8 a^2 b^2+b^4\right )}{\left (a^2+b^2\right )^4 d (a+b \tan (c+d x))}-\frac {\cos ^2(c+d x) \left (4 a b \left (a^2-b^2\right )+\left (a^4-6 a^2 b^2+b^4\right ) \tan (c+d x)\right )}{2 \left (a^2+b^2\right )^4 d}-\frac {\left (4 a b \left (a^4-5 a^2 b^2+2 b^4\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 (b \left (a^6-25 a^4 b^2+35 a^2 b^4-3 b^6\right )\right ) \text {Subst}\left (\int \frac {1}{b^2+x^2} \, dx,x,b \tan (c+d x)\right )}{2 \left (a^2+b^2\right )^5 d} \\ & = \frac {\left (a^6-25 a^4 b^2+35 a^2 b^4-3 b^6\right ) x}{2 \left (a^2+b^2\right )^5}+\frac {4 a b \left (a^4-5 a^2 b^2+2 b^4\right ) \log (\cos (c+d x))}{\left (a^2+b^2\right )^5 d}+\frac {4 a b \left (a^4-5 a^2 b^2+2 b^4\right ) \log (a+b \tan (c+d x))}{\left (a^2+b^2\right )^5 d}-\frac {a^2 b}{3 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))^3}-\frac {a b \left (a^2-b^2\right )}{\left (a^2+b^2\right )^3 d (a+b \tan (c+d x))^2}-\frac {b \left (3 a^4-8 a^2 b^2+b^4\right )}{\left (a^2+b^2\right )^4 d (a+b \tan (c+d x))}-\frac {\cos ^2(c+d x) \left (4 a b \left (a^2-b^2\right )+\left (a^4-6 a^2 b^2+b^4\right ) \tan (c+d x)\right )}{2 \left (a^2+b^2\right )^4 d} \\ \end{align*}
Time = 3.95 (sec) , antiderivative size = 395, normalized size of antiderivative = 1.50 \[ \int \frac {\sin ^2(c+d x)}{(a+b \tan (c+d x))^4} \, dx=-\frac {b \left (\frac {3 \left (a^2+b^2\right ) \left (a^4-6 a^2 b^2+b^4\right ) \arctan (\tan (c+d x))}{b}+12 a (a-b) (a+b) \left (a^2+b^2\right ) \cos ^2(c+d x)+3 \left (4 a^5-20 a^3 b^2+8 a b^4+\frac {-a^6+15 a^4 b^2-15 a^2 b^4+b^6}{\sqrt {-b^2}}\right ) \log \left (\sqrt {-b^2}-b \tan (c+d x)\right )-24 a \left (a^4-5 a^2 b^2+2 b^4\right ) \log (a+b \tan (c+d x))+3 \left (4 a^5-20 a^3 b^2+8 a b^4+\frac {a^6-15 a^4 b^2+15 a^2 b^4-b^6}{\sqrt {-b^2}}\right ) \log \left (\sqrt {-b^2}+b \tan (c+d x)\right )+\frac {3 \left (a^2+b^2\right ) \left (a^4-6 a^2 b^2+b^4\right ) \sin (2 (c+d x))}{2 b}+\frac {2 a^2 \left (a^2+b^2\right )^3}{(a+b \tan (c+d x))^3}+\frac {6 a (a-b) (a+b) \left (a^2+b^2\right )^2}{(a+b \tan (c+d x))^2}+\frac {6 \left (a^2+b^2\right ) \left (3 a^4-8 a^2 b^2+b^4\right )}{a+b \tan (c+d x)}\right )}{6 \left (a^2+b^2\right )^5 d} \]
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Time = 17.84 (sec) , antiderivative size = 288, normalized size of antiderivative = 1.09
method | result | size |
derivativedivides | \(\frac {\frac {\frac {\left (-\frac {1}{2} a^{6}+\frac {5}{2} a^{4} b^{2}+\frac {5}{2} a^{2} b^{4}-\frac {1}{2} b^{6}\right ) \tan \left (d x +c \right )-2 a^{5} b +2 a \,b^{5}}{1+\tan ^{2}\left (d x +c \right )}+\frac {\left (-8 a^{5} b +40 a^{3} b^{3}-16 a \,b^{5}\right ) \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{4}+\frac {\left (a^{6}-25 a^{4} b^{2}+35 a^{2} b^{4}-3 b^{6}\right ) \arctan \left (\tan \left (d x +c \right )\right )}{2}}{\left (a^{2}+b^{2}\right )^{5}}-\frac {a^{2} b}{3 \left (a^{2}+b^{2}\right )^{2} \left (a +b \tan \left (d x +c \right )\right )^{3}}-\frac {b \left (3 a^{4}-8 a^{2} b^{2}+b^{4}\right )}{\left (a^{2}+b^{2}\right )^{4} \left (a +b \tan \left (d x +c \right )\right )}-\frac {a b \left (a^{2}-b^{2}\right )}{\left (a^{2}+b^{2}\right )^{3} \left (a +b \tan \left (d x +c \right )\right )^{2}}+\frac {4 a b \left (a^{4}-5 a^{2} b^{2}+2 b^{4}\right ) \ln \left (a +b \tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{5}}}{d}\) | \(288\) |
default | \(\frac {\frac {\frac {\left (-\frac {1}{2} a^{6}+\frac {5}{2} a^{4} b^{2}+\frac {5}{2} a^{2} b^{4}-\frac {1}{2} b^{6}\right ) \tan \left (d x +c \right )-2 a^{5} b +2 a \,b^{5}}{1+\tan ^{2}\left (d x +c \right )}+\frac {\left (-8 a^{5} b +40 a^{3} b^{3}-16 a \,b^{5}\right ) \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{4}+\frac {\left (a^{6}-25 a^{4} b^{2}+35 a^{2} b^{4}-3 b^{6}\right ) \arctan \left (\tan \left (d x +c \right )\right )}{2}}{\left (a^{2}+b^{2}\right )^{5}}-\frac {a^{2} b}{3 \left (a^{2}+b^{2}\right )^{2} \left (a +b \tan \left (d x +c \right )\right )^{3}}-\frac {b \left (3 a^{4}-8 a^{2} b^{2}+b^{4}\right )}{\left (a^{2}+b^{2}\right )^{4} \left (a +b \tan \left (d x +c \right )\right )}-\frac {a b \left (a^{2}-b^{2}\right )}{\left (a^{2}+b^{2}\right )^{3} \left (a +b \tan \left (d x +c \right )\right )^{2}}+\frac {4 a b \left (a^{4}-5 a^{2} b^{2}+2 b^{4}\right ) \ln \left (a +b \tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{5}}}{d}\) | \(288\) |
risch | \(-\frac {3 i x b}{2 \left (5 i a^{4} b -10 i a^{2} b^{3}+i b^{5}-a^{5}+10 a^{3} b^{2}-5 a \,b^{4}\right )}-\frac {x a}{2 \left (5 i a^{4} b -10 i a^{2} b^{3}+i b^{5}-a^{5}+10 a^{3} b^{2}-5 a \,b^{4}\right )}+\frac {i {\mathrm e}^{2 i \left (d x +c \right )}}{8 \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 )}}{8 \left (4 i a^{3} b -4 i a \,b^{3}+a^{4}-6 a^{2} b^{2}+b^{4}\right ) d}-\frac {8 i a^{5} b 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 {40 i a^{3} b^{3} 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 {16 i a \,b^{5} 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 {8 i a^{5} b 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 {40 i a^{3} b^{3} 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 {16 i a \,b^{5} 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 {2 i b^{2} \left (-12 i a^{5} b \,{\mathrm e}^{2 i \left (d x +c \right )}-6 i a^{3} b^{3} {\mathrm e}^{2 i \left (d x +c \right )}-18 a^{6} {\mathrm e}^{4 i \left (d x +c \right )}+27 a^{4} b^{2} {\mathrm e}^{4 i \left (d x +c \right )}-24 a^{2} b^{4} {\mathrm e}^{4 i \left (d x +c \right )}+3 b^{6} {\mathrm e}^{4 i \left (d x +c \right )}+6 i a \,b^{5} {\mathrm e}^{2 i \left (d x +c \right )}-48 i a^{3} b^{3} {\mathrm e}^{4 i \left (d x +c \right )}+24 i a^{5} b \,{\mathrm e}^{4 i \left (d x +c \right )}-36 a^{6} {\mathrm e}^{2 i \left (d x +c \right )}+24 a^{4} b^{2} {\mathrm e}^{2 i \left (d x +c \right )}+54 a^{2} b^{4} {\mathrm e}^{2 i \left (d x +c \right )}-6 b^{6} {\mathrm e}^{2 i \left (d x +c \right )}-6 i a \,b^{5}-36 i a^{5} b +62 i a^{3} b^{3}-18 a^{6}+49 a^{4} b^{2}-34 a^{2} b^{4}+3 b^{6}\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 )^{3} \left (-i a +b \right )^{4} d \left (i a +b \right )^{5}}+\frac {4 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^{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 {20 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^{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 {8 a \,b^{5} \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 )}\) | \(1054\) |
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Leaf count of result is larger than twice the leaf count of optimal. 802 vs. \(2 (258) = 516\).
Time = 0.33 (sec) , antiderivative size = 802, normalized size of antiderivative = 3.04 \[ \int \frac {\sin ^2(c+d x)}{(a+b \tan (c+d x))^4} \, dx=-\frac {3 \, {\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 )^{5} + {\left (3 \, a^{8} b + 111 \, a^{6} b^{3} - 231 \, a^{4} b^{5} + 65 \, a^{2} b^{7} - 12 \, b^{9} - 3 \, {\left (a^{9} - 28 \, a^{7} b^{2} + 110 \, a^{5} b^{4} - 108 \, a^{3} b^{6} + 9 \, a b^{8}\right )} d x\right )} \cos \left (d x + c\right )^{3} - 3 \, {\left (25 \, a^{6} b^{3} - 51 \, a^{4} b^{5} + 25 \, a^{2} b^{7} - 3 \, b^{9} + 3 \, {\left (a^{7} b^{2} - 25 \, a^{5} b^{4} + 35 \, a^{3} b^{6} - 3 \, a b^{8}\right )} d x\right )} \cos \left (d x + c\right ) - 12 \, {\left ({\left (a^{8} b - 8 \, a^{6} b^{3} + 17 \, a^{4} b^{5} - 6 \, a^{2} b^{7}\right )} \cos \left (d x + c\right )^{3} + 3 \, {\left (a^{6} b^{3} - 5 \, a^{4} b^{5} + 2 \, a^{2} b^{7}\right )} \cos \left (d x + c\right ) + {\left (a^{5} b^{4} - 5 \, a^{3} b^{6} + 2 \, a b^{8} + {\left (3 \, a^{7} b^{2} - 16 \, a^{5} b^{4} + 11 \, a^{3} b^{6} - 2 \, a b^{8}\right )} \cos \left (d x + c\right )^{2}\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 (32 \, a^{5} b^{4} - 66 \, a^{3} b^{6} + 6 \, a b^{8} - 3 \, {\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 )^{4} + 3 \, {\left (a^{6} b^{3} - 25 \, a^{4} b^{5} + 35 \, a^{2} b^{7} - 3 \, b^{9}\right )} d x + {\left (45 \, a^{7} b^{2} - 143 \, a^{5} b^{4} + 219 \, a^{3} b^{6} - 9 \, a b^{8} + 3 \, {\left (3 \, a^{8} b - 76 \, a^{6} b^{3} + 130 \, a^{4} b^{5} - 44 \, a^{2} b^{7} + 3 \, b^{9}\right )} d x\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )}{6 \, {\left ({\left (a^{13} + 2 \, a^{11} b^{2} - 5 \, a^{9} b^{4} - 20 \, a^{7} b^{6} - 25 \, a^{5} b^{8} - 14 \, a^{3} b^{10} - 3 \, a b^{12}\right )} d \cos \left (d x + c\right )^{3} + 3 \, {\left (a^{11} b^{2} + 5 \, a^{9} b^{4} + 10 \, a^{7} b^{6} + 10 \, a^{5} b^{8} + 5 \, a^{3} b^{10} + a b^{12}\right )} d \cos \left (d x + c\right ) + {\left ({\left (3 \, a^{12} b + 14 \, a^{10} b^{3} + 25 \, a^{8} b^{5} + 20 \, a^{6} b^{7} + 5 \, a^{4} b^{9} - 2 \, a^{2} b^{11} - b^{13}\right )} d \cos \left (d x + c\right )^{2} + {\left (a^{10} b^{3} + 5 \, a^{8} b^{5} + 10 \, a^{6} b^{7} + 10 \, a^{4} b^{9} + 5 \, a^{2} b^{11} + b^{13}\right )} d\right )} \sin \left (d x + c\right )\right )}} \]
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Exception generated. \[ \int \frac {\sin ^2(c+d x)}{(a+b \tan (c+d x))^4} \, dx=\text {Exception raised: AttributeError} \]
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Leaf count of result is larger than twice the leaf count of optimal. 662 vs. \(2 (258) = 516\).
Time = 0.34 (sec) , antiderivative size = 662, normalized size of antiderivative = 2.51 \[ \int \frac {\sin ^2(c+d x)}{(a+b \tan (c+d x))^4} \, dx=\frac {\frac {3 \, {\left (a^{6} - 25 \, a^{4} b^{2} + 35 \, a^{2} b^{4} - 3 \, 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 (a^{5} b - 5 \, a^{3} b^{3} + 2 \, a b^{5}\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 (a^{5} b - 5 \, a^{3} b^{3} + 2 \, a b^{5}\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 {38 \, a^{6} b - 56 \, a^{4} b^{3} + 2 \, a^{2} b^{5} + 3 \, {\left (7 \, a^{4} b^{3} - 22 \, a^{2} b^{5} + 3 \, b^{7}\right )} \tan \left (d x + c\right )^{4} + 3 \, {\left (17 \, a^{5} b^{2} - 46 \, a^{3} b^{4} + a b^{6}\right )} \tan \left (d x + c\right )^{3} + {\left (35 \, a^{6} b - 44 \, a^{4} b^{3} - 73 \, a^{2} b^{5} + 6 \, b^{7}\right )} \tan \left (d x + c\right )^{2} + 3 \, {\left (a^{7} + 20 \, a^{5} b^{2} - 43 \, a^{3} b^{4} + 2 \, a b^{6}\right )} \tan \left (d x + c\right )}{a^{11} + 4 \, a^{9} b^{2} + 6 \, a^{7} b^{4} + 4 \, a^{5} b^{6} + a^{3} b^{8} + {\left (a^{8} b^{3} + 4 \, a^{6} b^{5} + 6 \, a^{4} b^{7} + 4 \, a^{2} b^{9} + b^{11}\right )} \tan \left (d x + c\right )^{5} + 3 \, {\left (a^{9} b^{2} + 4 \, a^{7} b^{4} + 6 \, a^{5} b^{6} + 4 \, a^{3} b^{8} + a b^{10}\right )} \tan \left (d x + c\right )^{4} + {\left (3 \, a^{10} b + 13 \, a^{8} b^{3} + 22 \, a^{6} b^{5} + 18 \, a^{4} b^{7} + 7 \, a^{2} b^{9} + b^{11}\right )} \tan \left (d x + c\right )^{3} + {\left (a^{11} + 7 \, a^{9} b^{2} + 18 \, a^{7} b^{4} + 22 \, a^{5} b^{6} + 13 \, a^{3} b^{8} + 3 \, a b^{10}\right )} \tan \left (d x + c\right )^{2} + 3 \, {\left (a^{10} b + 4 \, a^{8} b^{3} + 6 \, a^{6} b^{5} + 4 \, a^{4} b^{7} + a^{2} b^{9}\right )} \tan \left (d x + c\right )}}{6 \, d} \]
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Leaf count of result is larger than twice the leaf count of optimal. 642 vs. \(2 (258) = 516\).
Time = 0.83 (sec) , antiderivative size = 642, normalized size of antiderivative = 2.43 \[ \int \frac {\sin ^2(c+d x)}{(a+b \tan (c+d x))^4} \, dx=\frac {\frac {3 \, {\left (a^{6} - 25 \, a^{4} b^{2} + 35 \, a^{2} b^{4} - 3 \, 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 (a^{5} b - 5 \, a^{3} b^{3} + 2 \, a b^{5}\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 (a^{5} b^{2} - 5 \, a^{3} b^{4} + 2 \, a b^{6}\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 \, {\left (4 \, a^{5} b \tan \left (d x + c\right )^{2} - 20 \, a^{3} b^{3} \tan \left (d x + c\right )^{2} + 8 \, a b^{5} \tan \left (d x + c\right )^{2} - a^{6} \tan \left (d x + c\right ) + 5 \, a^{4} b^{2} \tan \left (d x + c\right ) + 5 \, a^{2} b^{4} \tan \left (d x + c\right ) - b^{6} \tan \left (d x + c\right ) - 20 \, a^{3} b^{3} + 12 \, a b^{5}\right )}}{{\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 )} {\left (\tan \left (d x + c\right )^{2} + 1\right )}} - \frac {2 \, {\left (22 \, a^{5} b^{4} \tan \left (d x + c\right )^{3} - 110 \, a^{3} b^{6} \tan \left (d x + c\right )^{3} + 44 \, a b^{8} \tan \left (d x + c\right )^{3} + 75 \, a^{6} b^{3} \tan \left (d x + c\right )^{2} - 345 \, a^{4} b^{5} \tan \left (d x + c\right )^{2} + 111 \, a^{2} b^{7} \tan \left (d x + c\right )^{2} + 3 \, b^{9} \tan \left (d x + c\right )^{2} + 87 \, a^{7} b^{2} \tan \left (d x + c\right ) - 357 \, a^{5} b^{4} \tan \left (d x + c\right ) + 87 \, a^{3} b^{6} \tan \left (d x + c\right ) + 3 \, a b^{8} \tan \left (d x + c\right ) + 35 \, a^{8} b - 119 \, a^{6} b^{3} + 23 \, a^{4} b^{5} + a^{2} b^{7}\right )}}{{\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 )} {\left (b \tan \left (d x + c\right ) + a\right )}^{3}}}{6 \, d} \]
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Time = 5.61 (sec) , antiderivative size = 597, normalized size of antiderivative = 2.26 \[ \int \frac {\sin ^2(c+d x)}{(a+b \tan (c+d x))^4} \, dx=\frac {\ln \left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )\,\left (\frac {4\,a\,b}{{\left (a^2+b^2\right )}^3}-\frac {28\,a\,b^3}{{\left (a^2+b^2\right )}^4}+\frac {32\,a\,b^5}{{\left (a^2+b^2\right )}^5}\right )}{d}-\frac {\frac {{\mathrm {tan}\left (c+d\,x\right )}^2\,\left (35\,a^4\,b-79\,a^2\,b^3+6\,b^5\right )}{6\,\left (a^6+3\,a^4\,b^2+3\,a^2\,b^4+b^6\right )}+\frac {{\mathrm {tan}\left (c+d\,x\right )}^4\,\left (7\,a^4\,b^3-22\,a^2\,b^5+3\,b^7\right )}{2\,\left (a^8+4\,a^6\,b^2+6\,a^4\,b^4+4\,a^2\,b^6+b^8\right )}+\frac {{\mathrm {tan}\left (c+d\,x\right )}^3\,\left (17\,a^5\,b^2-46\,a^3\,b^4+a\,b^6\right )}{2\,\left (a^8+4\,a^6\,b^2+6\,a^4\,b^4+4\,a^2\,b^6+b^8\right )}+\frac {a^2\,\left (19\,a^4\,b-28\,a^2\,b^3+b^5\right )}{3\,\left (a^2+b^2\right )\,\left (a^6+3\,a^4\,b^2+3\,a^2\,b^4+b^6\right )}+\frac {a\,\mathrm {tan}\left (c+d\,x\right )\,\left (a^6+20\,a^4\,b^2-43\,a^2\,b^4+2\,b^6\right )}{2\,\left (a^2+b^2\right )\,\left (a^6+3\,a^4\,b^2+3\,a^2\,b^4+b^6\right )}}{d\,\left (a^3+{\mathrm {tan}\left (c+d\,x\right )}^2\,\left (a^3+3\,a\,b^2\right )+{\mathrm {tan}\left (c+d\,x\right )}^3\,\left (3\,a^2\,b+b^3\right )+b^3\,{\mathrm {tan}\left (c+d\,x\right )}^5+3\,a\,b^2\,{\mathrm {tan}\left (c+d\,x\right )}^4+3\,a^2\,b\,\mathrm {tan}\left (c+d\,x\right )\right )}-\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )-\mathrm {i}\right )\,\left (3\,b+a\,1{}\mathrm {i}\right )}{4\,d\,\left (a^5+a^4\,b\,5{}\mathrm {i}-10\,a^3\,b^2-a^2\,b^3\,10{}\mathrm {i}+5\,a\,b^4+b^5\,1{}\mathrm {i}\right )}+\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )\,\left (-3\,b+a\,1{}\mathrm {i}\right )}{4\,d\,\left (a^5-a^4\,b\,5{}\mathrm {i}-10\,a^3\,b^2+a^2\,b^3\,10{}\mathrm {i}+5\,a\,b^4-b^5\,1{}\mathrm {i}\right )} \]
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