Integrand size = 21, antiderivative size = 158 \[ \int \frac {\sin ^4(c+d x)}{a+b \tan (c+d x)} \, dx=\frac {a \left (3 a^4-6 a^2 b^2-b^4\right ) x}{8 \left (a^2+b^2\right )^3}+\frac {a^4 b \log (a \cos (c+d x)+b \sin (c+d x))}{\left (a^2+b^2\right )^3 d}+\frac {\cos ^4(c+d x) (b+a \tan (c+d x))}{4 \left (a^2+b^2\right ) d}-\frac {\cos ^2(c+d x) \left (4 b \left (2 a^2+b^2\right )+a \left (5 a^2+b^2\right ) \tan (c+d x)\right )}{8 \left (a^2+b^2\right )^2 d} \]
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Time = 0.48 (sec) , antiderivative size = 158, 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, 815, 649, 209, 266} \[ \int \frac {\sin ^4(c+d x)}{a+b \tan (c+d x)} \, dx=\frac {\cos ^4(c+d x) (a \tan (c+d x)+b)}{4 d \left (a^2+b^2\right )}-\frac {\cos ^2(c+d x) \left (a \left (5 a^2+b^2\right ) \tan (c+d x)+4 b \left (2 a^2+b^2\right )\right )}{8 d \left (a^2+b^2\right )^2}+\frac {a^4 b \log (a \cos (c+d x)+b \sin (c+d x))}{d \left (a^2+b^2\right )^3}+\frac {a x \left (3 a^4-6 a^2 b^2-b^4\right )}{8 \left (a^2+b^2\right )^3} \]
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Rule 209
Rule 266
Rule 649
Rule 815
Rule 1661
Rule 3597
Rubi steps \begin{align*} \text {integral}& = \frac {b \text {Subst}\left (\int \frac {x^4}{(a+x) \left (b^2+x^2\right )^3} \, dx,x,b \tan (c+d x)\right )}{d} \\ & = \frac {\cos ^4(c+d x) (b+a \tan (c+d x))}{4 \left (a^2+b^2\right ) d}-\frac {\text {Subst}\left (\int \frac {\frac {a^2 b^4}{a^2+b^2}-\frac {3 a b^4 x}{a^2+b^2}-4 b^2 x^2}{(a+x) \left (b^2+x^2\right )^2} \, dx,x,b \tan (c+d x)\right )}{4 b d} \\ & = \frac {\cos ^4(c+d x) (b+a \tan (c+d x))}{4 \left (a^2+b^2\right ) d}-\frac {\cos ^2(c+d x) \left (4 b \left (2 a^2+b^2\right )+a \left (5 a^2+b^2\right ) \tan (c+d x)\right )}{8 \left (a^2+b^2\right )^2 d}+\frac {\text {Subst}\left (\int \frac {\frac {a^2 b^4 \left (3 a^2-b^2\right )}{\left (a^2+b^2\right )^2}-\frac {a b^4 \left (5 a^2+b^2\right ) x}{\left (a^2+b^2\right )^2}}{(a+x) \left (b^2+x^2\right )} \, dx,x,b \tan (c+d x)\right )}{8 b^3 d} \\ & = \frac {\cos ^4(c+d x) (b+a \tan (c+d x))}{4 \left (a^2+b^2\right ) d}-\frac {\cos ^2(c+d x) \left (4 b \left (2 a^2+b^2\right )+a \left (5 a^2+b^2\right ) \tan (c+d x)\right )}{8 \left (a^2+b^2\right )^2 d}+\frac {\text {Subst}\left (\int \left (\frac {8 a^4 b^4}{\left (a^2+b^2\right )^3 (a+x)}+\frac {a b^4 \left (3 a^4-6 a^2 b^2-b^4-8 a^3 x\right )}{\left (a^2+b^2\right )^3 \left (b^2+x^2\right )}\right ) \, dx,x,b \tan (c+d x)\right )}{8 b^3 d} \\ & = \frac {a^4 b \log (a+b \tan (c+d x))}{\left (a^2+b^2\right )^3 d}+\frac {\cos ^4(c+d x) (b+a \tan (c+d x))}{4 \left (a^2+b^2\right ) d}-\frac {\cos ^2(c+d x) \left (4 b \left (2 a^2+b^2\right )+a \left (5 a^2+b^2\right ) \tan (c+d x)\right )}{8 \left (a^2+b^2\right )^2 d}+\frac {(a b) \text {Subst}\left (\int \frac {3 a^4-6 a^2 b^2-b^4-8 a^3 x}{b^2+x^2} \, dx,x,b \tan (c+d x)\right )}{8 \left (a^2+b^2\right )^3 d} \\ & = \frac {a^4 b \log (a+b \tan (c+d x))}{\left (a^2+b^2\right )^3 d}+\frac {\cos ^4(c+d x) (b+a \tan (c+d x))}{4 \left (a^2+b^2\right ) d}-\frac {\cos ^2(c+d x) \left (4 b \left (2 a^2+b^2\right )+a \left (5 a^2+b^2\right ) \tan (c+d x)\right )}{8 \left (a^2+b^2\right )^2 d}-\frac {\left (a^4 b\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 )^3 d}+\frac {\left (a b \left (3 a^4-6 a^2 b^2-b^4\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 )^3 d} \\ & = \frac {a \left (3 a^4-6 a^2 b^2-b^4\right ) x}{8 \left (a^2+b^2\right )^3}+\frac {a^4 b \log (\cos (c+d x))}{\left (a^2+b^2\right )^3 d}+\frac {a^4 b \log (a+b \tan (c+d x))}{\left (a^2+b^2\right )^3 d}+\frac {\cos ^4(c+d x) (b+a \tan (c+d x))}{4 \left (a^2+b^2\right ) d}-\frac {\cos ^2(c+d x) \left (4 b \left (2 a^2+b^2\right )+a \left (5 a^2+b^2\right ) \tan (c+d x)\right )}{8 \left (a^2+b^2\right )^2 d} \\ \end{align*}
Time = 4.17 (sec) , antiderivative size = 249, normalized size of antiderivative = 1.58 \[ \int \frac {\sin ^4(c+d x)}{a+b \tan (c+d x)} \, dx=-\frac {2 a b \left (5 a^4+6 a^2 b^2+b^4\right ) \arctan (\tan (c+d x))+8 b^2 \left (2 a^4+3 a^2 b^2+b^4\right ) \cos ^2(c+d x)-4 b^2 \left (a^2+b^2\right )^2 \cos ^4(c+d x)+8 a^4 \left (\left (b^2+a \sqrt {-b^2}\right ) \log \left (\sqrt {-b^2}-b \tan (c+d x)\right )-2 b^2 \log (a+b \tan (c+d x))+\left (b^2-a \sqrt {-b^2}\right ) \log \left (\sqrt {-b^2}+b \tan (c+d x)\right )\right )-4 a b \left (a^2+b^2\right )^2 \cos ^3(c+d x) \sin (c+d x)+a \left (5 a^4 b+6 a^2 b^3+b^5\right ) \sin (2 (c+d x))}{16 b \left (a^2+b^2\right )^3 d} \]
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Time = 4.69 (sec) , antiderivative size = 208, normalized size of antiderivative = 1.32
method | result | size |
derivativedivides | \(\frac {\frac {\frac {\left (-\frac {5}{8} a^{5}-\frac {3}{4} a^{3} b^{2}-\frac {1}{8} a \,b^{4}\right ) \left (\tan ^{3}\left (d x +c \right )\right )+\left (-a^{4} b -\frac {3}{2} a^{2} b^{3}-\frac {1}{2} b^{5}\right ) \left (\tan ^{2}\left (d x +c \right )\right )+\left (-\frac {3}{8} a^{5}-\frac {1}{4} a^{3} b^{2}+\frac {1}{8} a \,b^{4}\right ) \tan \left (d x +c \right )-\frac {3 a^{4} b}{4}-a^{2} b^{3}-\frac {b^{5}}{4}}{\left (1+\tan ^{2}\left (d x +c \right )\right )^{2}}+\frac {a \left (-4 a^{3} b \ln \left (1+\tan ^{2}\left (d x +c \right )\right )+\left (3 a^{4}-6 a^{2} b^{2}-b^{4}\right ) \arctan \left (\tan \left (d x +c \right )\right )\right )}{8}}{\left (a^{2}+b^{2}\right )^{3}}+\frac {a^{4} b \ln \left (a +b \tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{3}}}{d}\) | \(208\) |
default | \(\frac {\frac {\frac {\left (-\frac {5}{8} a^{5}-\frac {3}{4} a^{3} b^{2}-\frac {1}{8} a \,b^{4}\right ) \left (\tan ^{3}\left (d x +c \right )\right )+\left (-a^{4} b -\frac {3}{2} a^{2} b^{3}-\frac {1}{2} b^{5}\right ) \left (\tan ^{2}\left (d x +c \right )\right )+\left (-\frac {3}{8} a^{5}-\frac {1}{4} a^{3} b^{2}+\frac {1}{8} a \,b^{4}\right ) \tan \left (d x +c \right )-\frac {3 a^{4} b}{4}-a^{2} b^{3}-\frac {b^{5}}{4}}{\left (1+\tan ^{2}\left (d x +c \right )\right )^{2}}+\frac {a \left (-4 a^{3} b \ln \left (1+\tan ^{2}\left (d x +c \right )\right )+\left (3 a^{4}-6 a^{2} b^{2}-b^{4}\right ) \arctan \left (\tan \left (d x +c \right )\right )\right )}{8}}{\left (a^{2}+b^{2}\right )^{3}}+\frac {a^{4} b \ln \left (a +b \tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{3}}}{d}\) | \(208\) |
risch | \(\frac {i a x b}{24 i b \,a^{2}-8 i b^{3}-8 a^{3}+24 a \,b^{2}}-\frac {3 a^{2} x}{8 \left (3 i b \,a^{2}-i b^{3}-a^{3}+3 a \,b^{2}\right )}+\frac {{\mathrm e}^{2 i \left (d x +c \right )} b}{16 \left (-2 i a b +a^{2}-b^{2}\right ) d}+\frac {i {\mathrm e}^{2 i \left (d x +c \right )} a}{8 \left (-2 i a b +a^{2}-b^{2}\right ) d}+\frac {{\mathrm e}^{-2 i \left (d x +c \right )} b}{16 \left (i b +a \right )^{2} d}-\frac {i {\mathrm e}^{-2 i \left (d x +c \right )} a}{8 \left (i b +a \right )^{2} d}-\frac {2 i a^{4} b x}{a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}}-\frac {2 i a^{4} b c}{d \left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right )}+\frac {a^{4} b \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-\frac {i b +a}{i b -a}\right )}{d \left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right )}-\frac {b \cos \left (4 d x +4 c \right )}{32 d \left (-a^{2}-b^{2}\right )}-\frac {a \sin \left (4 d x +4 c \right )}{32 d \left (-a^{2}-b^{2}\right )}\) | \(365\) |
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Time = 0.28 (sec) , antiderivative size = 216, normalized size of antiderivative = 1.37 \[ \int \frac {\sin ^4(c+d x)}{a+b \tan (c+d x)} \, dx=\frac {4 \, a^{4} b \log \left (2 \, a b \cos \left (d x + c\right ) \sin \left (d x + c\right ) + {\left (a^{2} - b^{2}\right )} \cos \left (d x + c\right )^{2} + b^{2}\right ) + 2 \, {\left (a^{4} b + 2 \, a^{2} b^{3} + b^{5}\right )} \cos \left (d x + c\right )^{4} + {\left (3 \, a^{5} - 6 \, a^{3} b^{2} - a b^{4}\right )} d x - 4 \, {\left (2 \, a^{4} b + 3 \, a^{2} b^{3} + b^{5}\right )} \cos \left (d x + c\right )^{2} + {\left (2 \, {\left (a^{5} + 2 \, a^{3} b^{2} + a b^{4}\right )} \cos \left (d x + c\right )^{3} - {\left (5 \, a^{5} + 6 \, a^{3} b^{2} + a b^{4}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{8 \, {\left (a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}\right )} d} \]
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Timed out. \[ \int \frac {\sin ^4(c+d x)}{a+b \tan (c+d x)} \, dx=\text {Timed out} \]
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Time = 0.30 (sec) , antiderivative size = 280, normalized size of antiderivative = 1.77 \[ \int \frac {\sin ^4(c+d x)}{a+b \tan (c+d x)} \, dx=\frac {\frac {8 \, a^{4} b \log \left (b \tan \left (d x + c\right ) + a\right )}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} - \frac {4 \, a^{4} b \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} + \frac {{\left (3 \, a^{5} - 6 \, a^{3} b^{2} - a b^{4}\right )} {\left (d x + c\right )}}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} - \frac {{\left (5 \, a^{3} + a b^{2}\right )} \tan \left (d x + c\right )^{3} + 6 \, a^{2} b + 2 \, b^{3} + 4 \, {\left (2 \, a^{2} b + b^{3}\right )} \tan \left (d x + c\right )^{2} + {\left (3 \, a^{3} - a b^{2}\right )} \tan \left (d x + c\right )}{{\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} \tan \left (d x + c\right )^{4} + a^{4} + 2 \, a^{2} b^{2} + b^{4} + 2 \, {\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} \tan \left (d x + c\right )^{2}}}{8 \, d} \]
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Leaf count of result is larger than twice the leaf count of optimal. 334 vs. \(2 (152) = 304\).
Time = 0.40 (sec) , antiderivative size = 334, normalized size of antiderivative = 2.11 \[ \int \frac {\sin ^4(c+d x)}{a+b \tan (c+d x)} \, dx=\frac {\frac {8 \, a^{4} b^{2} \log \left ({\left | b \tan \left (d x + c\right ) + a \right |}\right )}{a^{6} b + 3 \, a^{4} b^{3} + 3 \, a^{2} b^{5} + b^{7}} - \frac {4 \, a^{4} b \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} + \frac {{\left (3 \, a^{5} - 6 \, a^{3} b^{2} - a b^{4}\right )} {\left (d x + c\right )}}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} + \frac {6 \, a^{4} b \tan \left (d x + c\right )^{4} - 5 \, a^{5} \tan \left (d x + c\right )^{3} - 6 \, a^{3} b^{2} \tan \left (d x + c\right )^{3} - a b^{4} \tan \left (d x + c\right )^{3} + 4 \, a^{4} b \tan \left (d x + c\right )^{2} - 12 \, a^{2} b^{3} \tan \left (d x + c\right )^{2} - 4 \, b^{5} \tan \left (d x + c\right )^{2} - 3 \, a^{5} \tan \left (d x + c\right ) - 2 \, a^{3} b^{2} \tan \left (d x + c\right ) + a b^{4} \tan \left (d x + c\right ) - 8 \, a^{2} b^{3} - 2 \, b^{5}}{{\left (a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}\right )} {\left (\tan \left (d x + c\right )^{2} + 1\right )}^{2}}}{8 \, d} \]
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Time = 5.20 (sec) , antiderivative size = 313, normalized size of antiderivative = 1.98 \[ \int \frac {\sin ^4(c+d x)}{a+b \tan (c+d x)} \, dx=\frac {a^4\,b\,\ln \left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )}{d\,{\left (a^2+b^2\right )}^3}-\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )-\mathrm {i}\right )\,\left (a\,b-a^2\,3{}\mathrm {i}\right )}{16\,d\,\left (-a^3-a^2\,b\,3{}\mathrm {i}+3\,a\,b^2+b^3\,1{}\mathrm {i}\right )}-\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )\,\left (-3\,a^2+a\,b\,1{}\mathrm {i}\right )}{16\,d\,\left (-a^3\,1{}\mathrm {i}-3\,a^2\,b+a\,b^2\,3{}\mathrm {i}+b^3\right )}-\frac {\frac {3\,a^2\,b+b^3}{4\,\left (a^4+2\,a^2\,b^2+b^4\right )}+\frac {{\mathrm {tan}\left (c+d\,x\right )}^3\,\left (5\,a^3+a\,b^2\right )}{8\,\left (a^4+2\,a^2\,b^2+b^4\right )}+\frac {{\mathrm {tan}\left (c+d\,x\right )}^2\,\left (2\,a^2\,b+b^3\right )}{2\,\left (a^4+2\,a^2\,b^2+b^4\right )}+\frac {a\,\mathrm {tan}\left (c+d\,x\right )\,\left (3\,a^2-b^2\right )}{8\,\left (a^4+2\,a^2\,b^2+b^4\right )}}{d\,\left ({\mathrm {tan}\left (c+d\,x\right )}^4+2\,{\mathrm {tan}\left (c+d\,x\right )}^2+1\right )} \]
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