Integrand size = 27, antiderivative size = 159 \[ \int \frac {\cos ^4(c+d x) \sin (c+d x)}{a+b \sin (c+d x)} \, dx=\frac {\left (8 a^4-12 a^2 b^2+3 b^4\right ) x}{8 b^5}-\frac {2 a \left (a^2-b^2\right )^{3/2} \arctan \left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{b^5 d}-\frac {\cos ^3(c+d x) (4 a-3 b \sin (c+d x))}{12 b^2 d}+\frac {\cos (c+d x) \left (8 a \left (a^2-b^2\right )-b \left (4 a^2-3 b^2\right ) \sin (c+d x)\right )}{8 b^4 d} \]
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Time = 0.23 (sec) , antiderivative size = 159, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {2944, 2814, 2739, 632, 210} \[ \int \frac {\cos ^4(c+d x) \sin (c+d x)}{a+b \sin (c+d x)} \, dx=-\frac {2 a \left (a^2-b^2\right )^{3/2} \arctan \left (\frac {a \tan \left (\frac {1}{2} (c+d x)\right )+b}{\sqrt {a^2-b^2}}\right )}{b^5 d}+\frac {\cos (c+d x) \left (8 a \left (a^2-b^2\right )-b \left (4 a^2-3 b^2\right ) \sin (c+d x)\right )}{8 b^4 d}+\frac {x \left (8 a^4-12 a^2 b^2+3 b^4\right )}{8 b^5}-\frac {\cos ^3(c+d x) (4 a-3 b \sin (c+d x))}{12 b^2 d} \]
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Rule 210
Rule 632
Rule 2739
Rule 2814
Rule 2944
Rubi steps \begin{align*} \text {integral}& = -\frac {\cos ^3(c+d x) (4 a-3 b \sin (c+d x))}{12 b^2 d}+\frac {\int \frac {\cos ^2(c+d x) \left (-a b-\left (4 a^2-3 b^2\right ) \sin (c+d x)\right )}{a+b \sin (c+d x)} \, dx}{4 b^2} \\ & = -\frac {\cos ^3(c+d x) (4 a-3 b \sin (c+d x))}{12 b^2 d}+\frac {\cos (c+d x) \left (8 a \left (a^2-b^2\right )-b \left (4 a^2-3 b^2\right ) \sin (c+d x)\right )}{8 b^4 d}+\frac {\int \frac {a b \left (4 a^2-5 b^2\right )+\left (8 a^4-12 a^2 b^2+3 b^4\right ) \sin (c+d x)}{a+b \sin (c+d x)} \, dx}{8 b^4} \\ & = \frac {\left (8 a^4-12 a^2 b^2+3 b^4\right ) x}{8 b^5}-\frac {\cos ^3(c+d x) (4 a-3 b \sin (c+d x))}{12 b^2 d}+\frac {\cos (c+d x) \left (8 a \left (a^2-b^2\right )-b \left (4 a^2-3 b^2\right ) \sin (c+d x)\right )}{8 b^4 d}-\frac {\left (a \left (a^2-b^2\right )^2\right ) \int \frac {1}{a+b \sin (c+d x)} \, dx}{b^5} \\ & = \frac {\left (8 a^4-12 a^2 b^2+3 b^4\right ) x}{8 b^5}-\frac {\cos ^3(c+d x) (4 a-3 b \sin (c+d x))}{12 b^2 d}+\frac {\cos (c+d x) \left (8 a \left (a^2-b^2\right )-b \left (4 a^2-3 b^2\right ) \sin (c+d x)\right )}{8 b^4 d}-\frac {\left (2 a \left (a^2-b^2\right )^2\right ) \text {Subst}\left (\int \frac {1}{a+2 b x+a x^2} \, dx,x,\tan \left (\frac {1}{2} (c+d x)\right )\right )}{b^5 d} \\ & = \frac {\left (8 a^4-12 a^2 b^2+3 b^4\right ) x}{8 b^5}-\frac {\cos ^3(c+d x) (4 a-3 b \sin (c+d x))}{12 b^2 d}+\frac {\cos (c+d x) \left (8 a \left (a^2-b^2\right )-b \left (4 a^2-3 b^2\right ) \sin (c+d x)\right )}{8 b^4 d}+\frac {\left (4 a \left (a^2-b^2\right )^2\right ) \text {Subst}\left (\int \frac {1}{-4 \left (a^2-b^2\right )-x^2} \, dx,x,2 b+2 a \tan \left (\frac {1}{2} (c+d x)\right )\right )}{b^5 d} \\ & = \frac {\left (8 a^4-12 a^2 b^2+3 b^4\right ) x}{8 b^5}-\frac {2 a \left (a^2-b^2\right )^{3/2} \arctan \left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{b^5 d}-\frac {\cos ^3(c+d x) (4 a-3 b \sin (c+d x))}{12 b^2 d}+\frac {\cos (c+d x) \left (8 a \left (a^2-b^2\right )-b \left (4 a^2-3 b^2\right ) \sin (c+d x)\right )}{8 b^4 d} \\ \end{align*}
Time = 0.76 (sec) , antiderivative size = 155, normalized size of antiderivative = 0.97 \[ \int \frac {\cos ^4(c+d x) \sin (c+d x)}{a+b \sin (c+d x)} \, dx=\frac {-192 a \left (a^2-b^2\right )^{3/2} \arctan \left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )+24 a b \left (4 a^2-5 b^2\right ) \cos (c+d x)-8 a b^3 \cos (3 (c+d x))+3 \left (4 \left (8 a^4-12 a^2 b^2+3 b^4\right ) (c+d x)+\left (-8 a^2 b^2+8 b^4\right ) \sin (2 (c+d x))+b^4 \sin (4 (c+d x))\right )}{96 b^5 d} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(308\) vs. \(2(148)=296\).
Time = 0.65 (sec) , antiderivative size = 309, normalized size of antiderivative = 1.94
method | result | size |
derivativedivides | \(\frac {-\frac {2 a \left (a^{4}-2 a^{2} b^{2}+b^{4}\right ) \arctan \left (\frac {2 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{b^{5} \sqrt {a^{2}-b^{2}}}+\frac {\frac {2 \left (\left (\frac {1}{2} a^{2} b^{2}-\frac {5}{8} b^{4}\right ) \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (a^{3} b -2 a \,b^{3}\right ) \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (\frac {1}{2} a^{2} b^{2}+\frac {3}{8} b^{4}\right ) \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (3 a^{3} b -4 a \,b^{3}\right ) \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-\frac {1}{2} a^{2} b^{2}-\frac {3}{8} b^{4}\right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (3 a^{3} b -\frac {10}{3} a \,b^{3}\right ) \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-\frac {1}{2} a^{2} b^{2}+\frac {5}{8} b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+a^{3} b -\frac {4 a \,b^{3}}{3}\right )}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{4}}+\frac {\left (8 a^{4}-12 a^{2} b^{2}+3 b^{4}\right ) \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4}}{b^{5}}}{d}\) | \(309\) |
default | \(\frac {-\frac {2 a \left (a^{4}-2 a^{2} b^{2}+b^{4}\right ) \arctan \left (\frac {2 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{b^{5} \sqrt {a^{2}-b^{2}}}+\frac {\frac {2 \left (\left (\frac {1}{2} a^{2} b^{2}-\frac {5}{8} b^{4}\right ) \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (a^{3} b -2 a \,b^{3}\right ) \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (\frac {1}{2} a^{2} b^{2}+\frac {3}{8} b^{4}\right ) \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (3 a^{3} b -4 a \,b^{3}\right ) \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-\frac {1}{2} a^{2} b^{2}-\frac {3}{8} b^{4}\right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (3 a^{3} b -\frac {10}{3} a \,b^{3}\right ) \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-\frac {1}{2} a^{2} b^{2}+\frac {5}{8} b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+a^{3} b -\frac {4 a \,b^{3}}{3}\right )}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{4}}+\frac {\left (8 a^{4}-12 a^{2} b^{2}+3 b^{4}\right ) \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4}}{b^{5}}}{d}\) | \(309\) |
risch | \(\frac {x \,a^{4}}{b^{5}}-\frac {3 x \,a^{2}}{2 b^{3}}+\frac {3 x}{8 b}+\frac {a^{3} {\mathrm e}^{i \left (d x +c \right )}}{2 b^{4} d}-\frac {5 a \,{\mathrm e}^{i \left (d x +c \right )}}{8 d \,b^{2}}+\frac {a^{3} {\mathrm e}^{-i \left (d x +c \right )}}{2 b^{4} d}-\frac {5 a \,{\mathrm e}^{-i \left (d x +c \right )}}{8 b^{2} d}-\frac {i \sqrt {a^{2}-b^{2}}\, a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i \left (\sqrt {a^{2}-b^{2}}\, a +a^{2}-b^{2}\right )}{b \sqrt {a^{2}-b^{2}}}\right )}{d \,b^{5}}+\frac {i \sqrt {a^{2}-b^{2}}\, a \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i \left (\sqrt {a^{2}-b^{2}}\, a +a^{2}-b^{2}\right )}{b \sqrt {a^{2}-b^{2}}}\right )}{d \,b^{3}}+\frac {i \sqrt {a^{2}-b^{2}}\, a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i \left (\sqrt {a^{2}-b^{2}}\, a -a^{2}+b^{2}\right )}{b \sqrt {a^{2}-b^{2}}}\right )}{d \,b^{5}}-\frac {i \sqrt {a^{2}-b^{2}}\, a \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i \left (\sqrt {a^{2}-b^{2}}\, a -a^{2}+b^{2}\right )}{b \sqrt {a^{2}-b^{2}}}\right )}{d \,b^{3}}+\frac {\sin \left (4 d x +4 c \right )}{32 b d}-\frac {a \cos \left (3 d x +3 c \right )}{12 b^{2} d}-\frac {\sin \left (2 d x +2 c \right ) a^{2}}{4 b^{3} d}+\frac {\sin \left (2 d x +2 c \right )}{4 b d}\) | \(461\) |
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Time = 0.43 (sec) , antiderivative size = 414, normalized size of antiderivative = 2.60 \[ \int \frac {\cos ^4(c+d x) \sin (c+d x)}{a+b \sin (c+d x)} \, dx=\left [-\frac {8 \, a b^{3} \cos \left (d x + c\right )^{3} - 3 \, {\left (8 \, a^{4} - 12 \, a^{2} b^{2} + 3 \, b^{4}\right )} d x + 12 \, {\left (a^{3} - a b^{2}\right )} \sqrt {-a^{2} + b^{2}} \log \left (-\frac {{\left (2 \, a^{2} - b^{2}\right )} \cos \left (d x + c\right )^{2} - 2 \, a b \sin \left (d x + c\right ) - a^{2} - b^{2} - 2 \, {\left (a \cos \left (d x + c\right ) \sin \left (d x + c\right ) + b \cos \left (d x + c\right )\right )} \sqrt {-a^{2} + b^{2}}}{b^{2} \cos \left (d x + c\right )^{2} - 2 \, a b \sin \left (d x + c\right ) - a^{2} - b^{2}}\right ) - 24 \, {\left (a^{3} b - a b^{3}\right )} \cos \left (d x + c\right ) - 3 \, {\left (2 \, b^{4} \cos \left (d x + c\right )^{3} - {\left (4 \, a^{2} b^{2} - 3 \, b^{4}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{24 \, b^{5} d}, -\frac {8 \, a b^{3} \cos \left (d x + c\right )^{3} - 3 \, {\left (8 \, a^{4} - 12 \, a^{2} b^{2} + 3 \, b^{4}\right )} d x - 24 \, {\left (a^{3} - a b^{2}\right )} \sqrt {a^{2} - b^{2}} \arctan \left (-\frac {a \sin \left (d x + c\right ) + b}{\sqrt {a^{2} - b^{2}} \cos \left (d x + c\right )}\right ) - 24 \, {\left (a^{3} b - a b^{3}\right )} \cos \left (d x + c\right ) - 3 \, {\left (2 \, b^{4} \cos \left (d x + c\right )^{3} - {\left (4 \, a^{2} b^{2} - 3 \, b^{4}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{24 \, b^{5} d}\right ] \]
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Timed out. \[ \int \frac {\cos ^4(c+d x) \sin (c+d x)}{a+b \sin (c+d x)} \, dx=\text {Timed out} \]
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Exception generated. \[ \int \frac {\cos ^4(c+d x) \sin (c+d x)}{a+b \sin (c+d x)} \, dx=\text {Exception raised: ValueError} \]
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Leaf count of result is larger than twice the leaf count of optimal. 371 vs. \(2 (147) = 294\).
Time = 0.33 (sec) , antiderivative size = 371, normalized size of antiderivative = 2.33 \[ \int \frac {\cos ^4(c+d x) \sin (c+d x)}{a+b \sin (c+d x)} \, dx=\frac {\frac {3 \, {\left (8 \, a^{4} - 12 \, a^{2} b^{2} + 3 \, b^{4}\right )} {\left (d x + c\right )}}{b^{5}} - \frac {48 \, {\left (a^{5} - 2 \, a^{3} b^{2} + a b^{4}\right )} {\left (\pi \left \lfloor \frac {d x + c}{2 \, \pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\left (a\right ) + \arctan \left (\frac {a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + b}{\sqrt {a^{2} - b^{2}}}\right )\right )}}{\sqrt {a^{2} - b^{2}} b^{5}} + \frac {2 \, {\left (12 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 15 \, b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 24 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 48 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 12 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 9 \, b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 72 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 96 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 12 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 9 \, b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 72 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 80 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 12 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 15 \, b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 24 \, a^{3} - 32 \, a b^{2}\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{4} b^{4}}}{24 \, d} \]
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Time = 13.15 (sec) , antiderivative size = 453, normalized size of antiderivative = 2.85 \[ \int \frac {\cos ^4(c+d x) \sin (c+d x)}{a+b \sin (c+d x)} \, dx=\frac {3\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{4\,b\,d}+\frac {\sin \left (2\,c+2\,d\,x\right )}{4\,b\,d}+\frac {\sin \left (4\,c+4\,d\,x\right )}{32\,b\,d}-\frac {a\,\cos \left (3\,c+3\,d\,x\right )}{12\,b^2\,d}+\frac {a^3\,\cos \left (c+d\,x\right )}{b^4\,d}-\frac {3\,a^2\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{b^3\,d}+\frac {2\,a^4\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{b^5\,d}-\frac {a^2\,\sin \left (2\,c+2\,d\,x\right )}{4\,b^3\,d}-\frac {5\,a\,\cos \left (c+d\,x\right )}{4\,b^2\,d}-\frac {2\,a\,\mathrm {atanh}\left (\frac {2\,b^2\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\sqrt {-a^6+3\,a^4\,b^2-3\,a^2\,b^4+b^6}-a^2\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\sqrt {-a^6+3\,a^4\,b^2-3\,a^2\,b^4+b^6}+a\,b\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\sqrt {-a^6+3\,a^4\,b^2-3\,a^2\,b^4+b^6}}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,a^5+2\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,a^4\,b-2\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,a^3\,b^2-4\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,a^2\,b^3+\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,a\,b^4+2\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,b^5}\right )\,\sqrt {-a^6+3\,a^4\,b^2-3\,a^2\,b^4+b^6}}{b^5\,d} \]
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