Integrand size = 28, antiderivative size = 77 \[ \int \cos ^3(c+d x) (a \sin (c+d x)+b \tan (c+d x))^3 \, dx=-\frac {\left (a^2-b^2\right ) (b+a \cos (c+d x))^4}{4 a^3 d}-\frac {2 b (b+a \cos (c+d x))^5}{5 a^3 d}+\frac {(b+a \cos (c+d x))^6}{6 a^3 d} \]
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Time = 0.22 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.107, Rules used = {4482, 2747, 711} \[ \int \cos ^3(c+d x) (a \sin (c+d x)+b \tan (c+d x))^3 \, dx=\frac {(a \cos (c+d x)+b)^6}{6 a^3 d}-\frac {2 b (a \cos (c+d x)+b)^5}{5 a^3 d}-\frac {\left (a^2-b^2\right ) (a \cos (c+d x)+b)^4}{4 a^3 d} \]
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Rule 711
Rule 2747
Rule 4482
Rubi steps \begin{align*} \text {integral}& = \int (b+a \cos (c+d x))^3 \sin ^3(c+d x) \, dx \\ & = -\frac {\text {Subst}\left (\int (b+x)^3 \left (a^2-x^2\right ) \, dx,x,a \cos (c+d x)\right )}{a^3 d} \\ & = -\frac {\text {Subst}\left (\int \left (\left (a^2-b^2\right ) (b+x)^3+2 b (b+x)^4-(b+x)^5\right ) \, dx,x,a \cos (c+d x)\right )}{a^3 d} \\ & = -\frac {\left (a^2-b^2\right ) (b+a \cos (c+d x))^4}{4 a^3 d}-\frac {2 b (b+a \cos (c+d x))^5}{5 a^3 d}+\frac {(b+a \cos (c+d x))^6}{6 a^3 d} \\ \end{align*}
Time = 0.97 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.48 \[ \int \cos ^3(c+d x) (a \sin (c+d x)+b \tan (c+d x))^3 \, dx=\frac {-360 b \left (a^2+2 b^2\right ) \cos (c+d x)-45 \left (a^3+8 a b^2\right ) \cos (2 (c+d x))-60 a^2 b \cos (3 (c+d x))+80 b^3 \cos (3 (c+d x))+90 a b^2 \cos (4 (c+d x))+36 a^2 b \cos (5 (c+d x))+5 a^3 \cos (6 (c+d x))}{960 d} \]
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Time = 30.63 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.30
method | result | size |
derivativedivides | \(\frac {\frac {a^{3} \cos \left (d x +c \right )^{6}}{6}+\frac {3 a^{2} b \cos \left (d x +c \right )^{5}}{5}+\frac {\left (-a^{3}+3 a \,b^{2}\right ) \cos \left (d x +c \right )^{4}}{4}+\frac {\left (-3 a^{2} b +b^{3}\right ) \cos \left (d x +c \right )^{3}}{3}-\frac {3 a \,b^{2} \cos \left (d x +c \right )^{2}}{2}-\cos \left (d x +c \right ) b^{3}}{d}\) | \(100\) |
default | \(\frac {\frac {a^{3} \cos \left (d x +c \right )^{6}}{6}+\frac {3 a^{2} b \cos \left (d x +c \right )^{5}}{5}+\frac {\left (-a^{3}+3 a \,b^{2}\right ) \cos \left (d x +c \right )^{4}}{4}+\frac {\left (-3 a^{2} b +b^{3}\right ) \cos \left (d x +c \right )^{3}}{3}-\frac {3 a \,b^{2} \cos \left (d x +c \right )^{2}}{2}-\cos \left (d x +c \right ) b^{3}}{d}\) | \(100\) |
risch | \(-\frac {3 a^{2} b \cos \left (d x +c \right )}{8 d}-\frac {3 b^{3} \cos \left (d x +c \right )}{4 d}+\frac {a^{3} \cos \left (6 d x +6 c \right )}{192 d}+\frac {3 b \,a^{2} \cos \left (5 d x +5 c \right )}{80 d}+\frac {3 a \,b^{2} \cos \left (4 d x +4 c \right )}{32 d}-\frac {b \cos \left (3 d x +3 c \right ) a^{2}}{16 d}+\frac {b^{3} \cos \left (3 d x +3 c \right )}{12 d}-\frac {3 a^{3} \cos \left (2 d x +2 c \right )}{64 d}-\frac {3 a \cos \left (2 d x +2 c \right ) b^{2}}{8 d}\) | \(154\) |
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Time = 0.25 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.30 \[ \int \cos ^3(c+d x) (a \sin (c+d x)+b \tan (c+d x))^3 \, dx=\frac {10 \, a^{3} \cos \left (d x + c\right )^{6} + 36 \, a^{2} b \cos \left (d x + c\right )^{5} - 90 \, a b^{2} \cos \left (d x + c\right )^{2} - 15 \, {\left (a^{3} - 3 \, a b^{2}\right )} \cos \left (d x + c\right )^{4} - 60 \, b^{3} \cos \left (d x + c\right ) - 20 \, {\left (3 \, a^{2} b - b^{3}\right )} \cos \left (d x + c\right )^{3}}{60 \, d} \]
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\[ \int \cos ^3(c+d x) (a \sin (c+d x)+b \tan (c+d x))^3 \, dx=\int \left (a \sin {\left (c + d x \right )} + b \tan {\left (c + d x \right )}\right )^{3} \cos ^{3}{\left (c + d x \right )}\, dx \]
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Time = 0.21 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.23 \[ \int \cos ^3(c+d x) (a \sin (c+d x)+b \tan (c+d x))^3 \, dx=\frac {45 \, a b^{2} \sin \left (d x + c\right )^{4} - 5 \, {\left (2 \, \sin \left (d x + c\right )^{6} - 3 \, \sin \left (d x + c\right )^{4}\right )} a^{3} + 12 \, {\left (3 \, \cos \left (d x + c\right )^{5} - 5 \, \cos \left (d x + c\right )^{3}\right )} a^{2} b + 20 \, {\left (\cos \left (d x + c\right )^{3} - 3 \, \cos \left (d x + c\right )\right )} b^{3}}{60 \, d} \]
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Timed out. \[ \int \cos ^3(c+d x) (a \sin (c+d x)+b \tan (c+d x))^3 \, dx=\text {Timed out} \]
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Time = 22.71 (sec) , antiderivative size = 149, normalized size of antiderivative = 1.94 \[ \int \cos ^3(c+d x) (a \sin (c+d x)+b \tan (c+d x))^3 \, dx=\frac {32\,a^3}{3\,d\,{\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}^6}+\frac {4\,{\left (a-b\right )}^3}{d\,{\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}^2}-\frac {32\,a^2\,\left (5\,a-3\,b\right )}{5\,d\,{\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}^5}-\frac {8\,{\left (a-b\right )}^2\,\left (7\,a-b\right )}{3\,d\,{\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}^3}+\frac {12\,a\,\left (3\,a^2-4\,a\,b+b^2\right )}{d\,{\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}^4} \]
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