Integrand size = 21, antiderivative size = 285 \[ \int \frac {\sin ^4(c+d x)}{(a+b \tan (c+d x))^3} \, dx=\frac {3 a \left (a^6-25 a^4 b^2+35 a^2 b^4-3 b^6\right ) x}{8 \left (a^2+b^2\right )^5}+\frac {3 a^2 b \left (a^4-5 a^2 b^2+2 b^4\right ) \log (a \cos (c+d x)+b \sin (c+d x))}{\left (a^2+b^2\right )^5 d}-\frac {a^4 b}{2 \left (a^2+b^2\right )^3 d (a+b \tan (c+d x))^2}-\frac {2 a^3 b \left (a^2-2 b^2\right )}{\left (a^2+b^2\right )^4 d (a+b \tan (c+d x))}+\frac {\cos ^4(c+d x) \left (b \left (3 a^2-b^2\right )+a \left (a^2-3 b^2\right ) \tan (c+d x)\right )}{4 \left (a^2+b^2\right )^3 d}-\frac {a \cos ^2(c+d x) \left (24 a b \left (a^2-b^2\right )+\left (5 a^4-34 a^2 b^2+9 b^4\right ) \tan (c+d x)\right )}{8 \left (a^2+b^2\right )^4 d} \]
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Time = 1.24 (sec) , antiderivative size = 285, 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} \[ \int \frac {\sin ^4(c+d x)}{(a+b \tan (c+d x))^3} \, dx=\frac {\cos ^4(c+d x) \left (a \left (a^2-3 b^2\right ) \tan (c+d x)+b \left (3 a^2-b^2\right )\right )}{4 d \left (a^2+b^2\right )^3}-\frac {a^4 b}{2 d \left (a^2+b^2\right )^3 (a+b \tan (c+d x))^2}-\frac {a \cos ^2(c+d x) \left (24 a b \left (a^2-b^2\right )+\left (5 a^4-34 a^2 b^2+9 b^4\right ) \tan (c+d x)\right )}{8 d \left (a^2+b^2\right )^4}+\frac {3 a^2 b \left (a^4-5 a^2 b^2+2 b^4\right ) \log (a \cos (c+d x)+b \sin (c+d x))}{d \left (a^2+b^2\right )^5}-\frac {2 a^3 b \left (a^2-2 b^2\right )}{d \left (a^2+b^2\right )^4 (a+b \tan (c+d x))}+\frac {3 a x \left (a^6-25 a^4 b^2+35 a^2 b^4-3 b^6\right )}{8 \left (a^2+b^2\right )^5} \]
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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^4}{(a+x)^3 \left (b^2+x^2\right )^3} \, dx,x,b \tan (c+d x)\right )}{d} \\ & = \frac {\cos ^4(c+d x) \left (b \left (3 a^2-b^2\right )+a \left (a^2-3 b^2\right ) \tan (c+d x)\right )}{4 \left (a^2+b^2\right )^3 d}-\frac {\text {Subst}\left (\int \frac {\frac {a^4 b^4 \left (a^2-3 b^2\right )}{\left (a^2+b^2\right )^3}-\frac {a^3 b^4 \left (9 a^2+5 b^2\right ) x}{\left (a^2+b^2\right )^3}-\frac {a^2 b^2 \left (4 a^4+21 a^2 b^2-3 b^4\right ) x^2}{\left (a^2+b^2\right )^3}-\frac {3 a b^4 \left (a^2-3 b^2\right ) x^3}{\left (a^2+b^2\right )^3}}{(a+x)^3 \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 (b \left (3 a^2-b^2\right )+a \left (a^2-3 b^2\right ) \tan (c+d x)\right )}{4 \left (a^2+b^2\right )^3 d}-\frac {a \cos ^2(c+d x) \left (24 a b \left (a^2-b^2\right )+\left (5 a^4-34 a^2 b^2+9 b^4\right ) \tan (c+d x)\right )}{8 \left (a^2+b^2\right )^4 d}+\frac {\text {Subst}\left (\int \frac {\frac {3 a^4 b^4 \left (a^4-10 a^2 b^2+5 b^4\right )}{\left (a^2+b^2\right )^4}-\frac {a^3 b^4 \left (15 a^4+26 a^2 b^2-37 b^4\right ) x}{\left (a^2+b^2\right )^4}-\frac {3 a^2 b^4 \left (5 a^4-18 a^2 b^2-7 b^4\right ) x^2}{\left (a^2+b^2\right )^4}-\frac {a b^4 \left (5 a^4-34 a^2 b^2+9 b^4\right ) x^3}{\left (a^2+b^2\right )^4}}{(a+x)^3 \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 (b \left (3 a^2-b^2\right )+a \left (a^2-3 b^2\right ) \tan (c+d x)\right )}{4 \left (a^2+b^2\right )^3 d}-\frac {a \cos ^2(c+d x) \left (24 a b \left (a^2-b^2\right )+\left (5 a^4-34 a^2 b^2+9 b^4\right ) \tan (c+d x)\right )}{8 \left (a^2+b^2\right )^4 d}+\frac {\text {Subst}\left (\int \left (\frac {8 a^4 b^4}{\left (a^2+b^2\right )^3 (a+x)^3}+\frac {16 a^3 b^4 \left (a^2-2 b^2\right )}{\left (a^2+b^2\right )^4 (a+x)^2}+\frac {24 a^2 b^4 \left (a^4-5 a^2 b^2+2 b^4\right )}{\left (a^2+b^2\right )^5 (a+x)}+\frac {3 a b^4 \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 )}{8 b^3 d} \\ & = \frac {3 a^2 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^4 b}{2 \left (a^2+b^2\right )^3 d (a+b \tan (c+d x))^2}-\frac {2 a^3 b \left (a^2-2 b^2\right )}{\left (a^2+b^2\right )^4 d (a+b \tan (c+d x))}+\frac {\cos ^4(c+d x) \left (b \left (3 a^2-b^2\right )+a \left (a^2-3 b^2\right ) \tan (c+d x)\right )}{4 \left (a^2+b^2\right )^3 d}-\frac {a \cos ^2(c+d x) \left (24 a b \left (a^2-b^2\right )+\left (5 a^4-34 a^2 b^2+9 b^4\right ) \tan (c+d x)\right )}{8 \left (a^2+b^2\right )^4 d}+\frac {(3 a 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 )}{8 \left (a^2+b^2\right )^5 d} \\ & = \frac {3 a^2 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^4 b}{2 \left (a^2+b^2\right )^3 d (a+b \tan (c+d x))^2}-\frac {2 a^3 b \left (a^2-2 b^2\right )}{\left (a^2+b^2\right )^4 d (a+b \tan (c+d x))}+\frac {\cos ^4(c+d x) \left (b \left (3 a^2-b^2\right )+a \left (a^2-3 b^2\right ) \tan (c+d x)\right )}{4 \left (a^2+b^2\right )^3 d}-\frac {a \cos ^2(c+d x) \left (24 a b \left (a^2-b^2\right )+\left (5 a^4-34 a^2 b^2+9 b^4\right ) \tan (c+d x)\right )}{8 \left (a^2+b^2\right )^4 d}-\frac {\left (3 a^2 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 (3 a 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 )}{8 \left (a^2+b^2\right )^5 d} \\ & = \frac {3 a \left (a^6-25 a^4 b^2+35 a^2 b^4-3 b^6\right ) x}{8 \left (a^2+b^2\right )^5}+\frac {3 a^2 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 {3 a^2 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^4 b}{2 \left (a^2+b^2\right )^3 d (a+b \tan (c+d x))^2}-\frac {2 a^3 b \left (a^2-2 b^2\right )}{\left (a^2+b^2\right )^4 d (a+b \tan (c+d x))}+\frac {\cos ^4(c+d x) \left (b \left (3 a^2-b^2\right )+a \left (a^2-3 b^2\right ) \tan (c+d x)\right )}{4 \left (a^2+b^2\right )^3 d}-\frac {a \cos ^2(c+d x) \left (24 a b \left (a^2-b^2\right )+\left (5 a^4-34 a^2 b^2+9 b^4\right ) \tan (c+d x)\right )}{8 \left (a^2+b^2\right )^4 d} \\ \end{align*}
Time = 6.50 (sec) , antiderivative size = 521, normalized size of antiderivative = 1.83 \[ \int \frac {\sin ^4(c+d x)}{(a+b \tan (c+d x))^3} \, dx=\frac {b \left (-\frac {a^3 \left (a^2-5 b^2\right ) \arctan (\tan (c+d x))}{b \left (a^2+b^2\right )^4}+\frac {3 a \left (a^2-3 b^2\right ) \arctan (\tan (c+d x))}{8 b \left (a^2+b^2\right )^3}-\frac {3 a^2 (a-b) (a+b) \cos ^2(c+d x)}{\left (a^2+b^2\right )^4}+\frac {\left (3 a^2-b^2\right ) \cos ^4(c+d x)}{4 \left (a^2+b^2\right )^3}-\frac {a^2 \left (3 a^4-15 a^2 b^2+6 b^4-\frac {a^5-13 a^3 b^2+10 a b^4}{\sqrt {-b^2}}\right ) \log \left (\sqrt {-b^2}-b \tan (c+d x)\right )}{2 \left (a^2+b^2\right )^5}+\frac {3 a^2 \left (a^4-5 a^2 b^2+2 b^4\right ) \log (a+b \tan (c+d x))}{\left (a^2+b^2\right )^5}-\frac {a^2 \left (3 a^4-15 a^2 b^2+6 b^4+\frac {a^5-13 a^3 b^2+10 a b^4}{\sqrt {-b^2}}\right ) \log \left (\sqrt {-b^2}+b \tan (c+d x)\right )}{2 \left (a^2+b^2\right )^5}-\frac {a^3 \left (a^2-5 b^2\right ) \cos (c+d x) \sin (c+d x)}{b \left (a^2+b^2\right )^4}+\frac {3 a \left (a^2-3 b^2\right ) \cos (c+d x) \sin (c+d x)}{8 b \left (a^2+b^2\right )^3}+\frac {a \left (a^2-3 b^2\right ) \cos ^3(c+d x) \sin (c+d x)}{4 b \left (a^2+b^2\right )^3}-\frac {a^4}{2 \left (a^2+b^2\right )^3 (a+b \tan (c+d x))^2}-\frac {2 a^3 \left (a^2-2 b^2\right )}{\left (a^2+b^2\right )^4 (a+b \tan (c+d x))}\right )}{d} \]
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Time = 20.58 (sec) , antiderivative size = 331, normalized size of antiderivative = 1.16
method | result | size |
derivativedivides | \(\frac {\frac {\frac {\left (-\frac {5}{8} a^{7}+\frac {29}{8} a^{5} b^{2}+\frac {25}{8} a^{3} b^{4}-\frac {9}{8} a \,b^{6}\right ) \left (\tan ^{3}\left (d x +c \right )\right )+\left (-3 a^{6} b +3 b^{5} a^{2}\right ) \left (\tan ^{2}\left (d x +c \right )\right )+\left (-\frac {3}{8} a^{7}+\frac {27}{8} a^{5} b^{2}+\frac {15}{8} a^{3} b^{4}-\frac {15}{8} a \,b^{6}\right ) \tan \left (d x +c \right )-\frac {9 a^{6} b}{4}+\frac {5 a^{4} b^{3}}{4}+\frac {13 b^{5} a^{2}}{4}-\frac {b^{7}}{4}}{\left (1+\tan ^{2}\left (d x +c \right )\right )^{2}}+\frac {3 a \left (\frac {\left (-8 a^{5} b +40 a^{3} b^{3}-16 a \,b^{5}\right ) \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2}+\left (a^{6}-25 a^{4} b^{2}+35 a^{2} b^{4}-3 b^{6}\right ) \arctan \left (\tan \left (d x +c \right )\right )\right )}{8}}{\left (a^{2}+b^{2}\right )^{5}}-\frac {b \,a^{4}}{2 \left (a^{2}+b^{2}\right )^{3} \left (a +b \tan \left (d x +c \right )\right )^{2}}+\frac {3 a^{2} b \left (a^{4}-5 a^{2} b^{2}+2 b^{4}\right ) \ln \left (a +b \tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{5}}-\frac {2 a^{3} b \left (a^{2}-2 b^{2}\right )}{\left (a^{2}+b^{2}\right )^{4} \left (a +b \tan \left (d x +c \right )\right )}}{d}\) | \(331\) |
default | \(\frac {\frac {\frac {\left (-\frac {5}{8} a^{7}+\frac {29}{8} a^{5} b^{2}+\frac {25}{8} a^{3} b^{4}-\frac {9}{8} a \,b^{6}\right ) \left (\tan ^{3}\left (d x +c \right )\right )+\left (-3 a^{6} b +3 b^{5} a^{2}\right ) \left (\tan ^{2}\left (d x +c \right )\right )+\left (-\frac {3}{8} a^{7}+\frac {27}{8} a^{5} b^{2}+\frac {15}{8} a^{3} b^{4}-\frac {15}{8} a \,b^{6}\right ) \tan \left (d x +c \right )-\frac {9 a^{6} b}{4}+\frac {5 a^{4} b^{3}}{4}+\frac {13 b^{5} a^{2}}{4}-\frac {b^{7}}{4}}{\left (1+\tan ^{2}\left (d x +c \right )\right )^{2}}+\frac {3 a \left (\frac {\left (-8 a^{5} b +40 a^{3} b^{3}-16 a \,b^{5}\right ) \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2}+\left (a^{6}-25 a^{4} b^{2}+35 a^{2} b^{4}-3 b^{6}\right ) \arctan \left (\tan \left (d x +c \right )\right )\right )}{8}}{\left (a^{2}+b^{2}\right )^{5}}-\frac {b \,a^{4}}{2 \left (a^{2}+b^{2}\right )^{3} \left (a +b \tan \left (d x +c \right )\right )^{2}}+\frac {3 a^{2} b \left (a^{4}-5 a^{2} b^{2}+2 b^{4}\right ) \ln \left (a +b \tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{5}}-\frac {2 a^{3} b \left (a^{2}-2 b^{2}\right )}{\left (a^{2}+b^{2}\right )^{4} \left (a +b \tan \left (d x +c \right )\right )}}{d}\) | \(331\) |
risch | \(-\frac {9 i a x b}{8 \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 {3 a^{2} x}{8 \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 {12 i a^{2} 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 {{\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 {6 i a^{6} 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 {{\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 {30 i a^{4} 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 {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 {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 {12 i a^{2} 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 {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 {6 i a^{6} 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 {30 i a^{4} 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 {2 i a^{3} b^{2} \left (-3 i a^{3} {\mathrm e}^{2 i \left (d x +c \right )}+3 i a \,b^{2} {\mathrm e}^{2 i \left (d x +c \right )}-2 a^{2} b \,{\mathrm e}^{2 i \left (d x +c \right )}+4 b^{3} {\mathrm e}^{2 i \left (d x +c \right )}-3 i a^{3}+4 i a \,b^{2}+3 a^{2} b -4 b^{3}\right )}{\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} \left (-i a +b \right )^{4} d \left (i a +b \right )^{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 {3 a^{6} 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 {15 a^{4} 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 {6 a^{2} 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 )}\) | \(1052\) |
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Leaf count of result is larger than twice the leaf count of optimal. 705 vs. \(2 (277) = 554\).
Time = 0.34 (sec) , antiderivative size = 705, normalized size of antiderivative = 2.47 \[ \int \frac {\sin ^4(c+d x)}{(a+b \tan (c+d x))^3} \, dx=\frac {119 \, a^{6} b^{3} - 159 \, a^{4} b^{5} - 51 \, 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 (5 \, a^{8} b + 16 \, a^{6} b^{3} + 18 \, a^{4} b^{5} + 8 \, a^{2} b^{7} + b^{9}\right )} \cos \left (d x + c\right )^{4} + 12 \, {\left (a^{7} b^{2} - 25 \, a^{5} b^{4} + 35 \, a^{3} b^{6} - 3 \, a b^{8}\right )} d x - {\left (a^{8} b + 110 \, a^{6} b^{3} - 420 \, a^{4} b^{5} - 78 \, a^{2} b^{7} + 3 \, b^{9} - 12 \, {\left (a^{9} - 26 \, a^{7} b^{2} + 60 \, a^{5} b^{4} - 38 \, a^{3} b^{6} + 3 \, a b^{8}\right )} d x\right )} \cos \left (d x + c\right )^{2} + 48 \, {\left (a^{6} b^{3} - 5 \, a^{4} b^{5} + 2 \, a^{2} b^{7} + {\left (a^{8} b - 6 \, a^{6} b^{3} + 7 \, a^{4} b^{5} - 2 \, a^{2} b^{7}\right )} \cos \left (d x + c\right )^{2} + 2 \, {\left (a^{7} b^{2} - 5 \, a^{5} b^{4} + 2 \, a^{3} b^{6}\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 (5 \, a^{9} + 12 \, a^{7} b^{2} + 6 \, a^{5} b^{4} - 4 \, a^{3} b^{6} - 3 \, a b^{8}\right )} \cos \left (d x + c\right )^{3} + {\left (77 \, a^{7} b^{2} - 69 \, a^{5} b^{4} + 63 \, a^{3} b^{6} - 15 \, a b^{8} + 12 \, {\left (a^{8} b - 25 \, a^{6} b^{3} + 35 \, a^{4} b^{5} - 3 \, 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. \[ \int \frac {\sin ^4(c+d x)}{(a+b \tan (c+d x))^3} \, dx=\text {Exception raised: AttributeError} \]
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Leaf count of result is larger than twice the leaf count of optimal. 744 vs. \(2 (277) = 554\).
Time = 0.55 (sec) , antiderivative size = 744, normalized size of antiderivative = 2.61 \[ \int \frac {\sin ^4(c+d x)}{(a+b \tan (c+d x))^3} \, dx=\frac {\frac {3 \, {\left (a^{7} - 25 \, a^{5} b^{2} + 35 \, a^{3} b^{4} - 3 \, 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 (a^{6} b - 5 \, a^{4} b^{3} + 2 \, a^{2} 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^{6} b - 5 \, a^{4} b^{3} + 2 \, a^{2} 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^{5} b^{2} - 22 \, a^{3} b^{4} + 3 \, a b^{6}\right )} \tan \left (d x + c\right )^{5} + 6 \, {\left (5 \, a^{6} b - 12 \, a^{4} b^{3} - a^{2} b^{5}\right )} \tan \left (d x + c\right )^{4} + {\left (5 \, a^{7} + 49 \, a^{5} b^{2} - 133 \, a^{3} b^{4} + 15 \, a b^{6}\right )} \tan \left (d x + c\right )^{3} + 2 \, {\left (35 \, a^{6} b - 61 \, a^{4} b^{3} + a^{2} b^{5} + b^{7}\right )} \tan \left (d x + c\right )^{2} + {\left (3 \, a^{7} + 22 \, a^{5} b^{2} - 73 \, a^{3} b^{4} + 4 \, 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. 588 vs. \(2 (277) = 554\).
Time = 0.68 (sec) , antiderivative size = 588, normalized size of antiderivative = 2.06 \[ \int \frac {\sin ^4(c+d x)}{(a+b \tan (c+d x))^3} \, dx=\frac {\frac {3 \, {\left (a^{7} - 25 \, a^{5} b^{2} + 35 \, a^{3} b^{4} - 3 \, a b^{6}\right )} {\left (d x + c\right )}}{a^{10} + 5 \, a^{8} b^{2} + 10 \, a^{6} b^{4} + 10 \, a^{4} b^{6} + 5 \, a^{2} b^{8} + b^{10}} - \frac {12 \, {\left (a^{6} b - 5 \, a^{4} b^{3} + 2 \, a^{2} b^{5}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{a^{10} + 5 \, a^{8} b^{2} + 10 \, a^{6} b^{4} + 10 \, a^{4} b^{6} + 5 \, a^{2} b^{8} + b^{10}} + \frac {24 \, {\left (a^{6} b^{2} - 5 \, a^{4} b^{4} + 2 \, a^{2} b^{6}\right )} \log \left ({\left | b \tan \left (d x + c\right ) + a \right |}\right )}{a^{10} b + 5 \, a^{8} b^{3} + 10 \, a^{6} b^{5} + 10 \, a^{4} b^{7} + 5 \, a^{2} b^{9} + b^{11}} - \frac {21 \, a^{5} b^{2} \tan \left (d x + c\right )^{5} - 66 \, a^{3} b^{4} \tan \left (d x + c\right )^{5} + 9 \, a b^{6} \tan \left (d x + c\right )^{5} + 30 \, a^{6} b \tan \left (d x + c\right )^{4} - 72 \, a^{4} b^{3} \tan \left (d x + c\right )^{4} - 6 \, a^{2} b^{5} \tan \left (d x + c\right )^{4} + 5 \, a^{7} \tan \left (d x + c\right )^{3} + 49 \, a^{5} b^{2} \tan \left (d x + c\right )^{3} - 133 \, a^{3} b^{4} \tan \left (d x + c\right )^{3} + 15 \, a b^{6} \tan \left (d x + c\right )^{3} + 70 \, a^{6} b \tan \left (d x + c\right )^{2} - 122 \, a^{4} b^{3} \tan \left (d x + c\right )^{2} + 2 \, a^{2} b^{5} \tan \left (d x + c\right )^{2} + 2 \, b^{7} \tan \left (d x + c\right )^{2} + 3 \, a^{7} \tan \left (d x + c\right ) + 22 \, a^{5} b^{2} \tan \left (d x + c\right ) - 73 \, a^{3} b^{4} \tan \left (d x + c\right ) + 4 \, a b^{6} \tan \left (d x + c\right ) + 38 \, a^{6} b - 56 \, a^{4} b^{3} + 2 \, a^{2} b^{5}}{{\left (a^{8} + 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} + 4 \, a^{2} b^{6} + b^{8}\right )} {\left (b \tan \left (d x + c\right )^{3} + a \tan \left (d x + c\right )^{2} + b \tan \left (d x + c\right ) + a\right )}^{2}}}{8 \, d} \]
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Time = 6.09 (sec) , antiderivative size = 717, normalized size of antiderivative = 2.52 \[ \int \frac {\sin ^4(c+d x)}{(a+b \tan (c+d x))^3} \, dx=\frac {\ln \left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )\,\left (\frac {3\,b}{{\left (a^2+b^2\right )}^2}-\frac {24\,b^3}{{\left (a^2+b^2\right )}^3}+\frac {45\,b^5}{{\left (a^2+b^2\right )}^4}-\frac {24\,b^7}{{\left (a^2+b^2\right )}^5}\right )}{d}-\frac {\frac {19\,a^6\,b-28\,a^4\,b^3+a^2\,b^5}{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 (35\,a^6\,b-61\,a^4\,b^3+a^2\,b^5+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 {3\,{\mathrm {tan}\left (c+d\,x\right )}^4\,\left (-5\,a^6\,b+12\,a^4\,b^3+a^2\,b^5\right )}{4\,\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 (7\,a^5\,b^2-22\,a^3\,b^4+3\,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 (5\,a^7+49\,a^5\,b^2-133\,a^3\,b^4+15\,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 {a\,\mathrm {tan}\left (c+d\,x\right )\,\left (3\,a^6+22\,a^4\,b^2-73\,a^2\,b^4+4\,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 )}}{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 {3\,\ln \left (\mathrm {tan}\left (c+d\,x\right )-\mathrm {i}\right )\,\left (a^2\,1{}\mathrm {i}+3\,b\,a\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 (3\,a\,b-a^2\,1{}\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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