Integrand size = 21, antiderivative size = 118 \[ \int \frac {\cos ^5(c+d x)}{a+b \sin (c+d x)} \, dx=\frac {\left (a^2-b^2\right )^2 \log (a+b \sin (c+d x))}{b^5 d}-\frac {a \left (a^2-2 b^2\right ) \sin (c+d x)}{b^4 d}+\frac {\left (a^2-2 b^2\right ) \sin ^2(c+d x)}{2 b^3 d}-\frac {a \sin ^3(c+d x)}{3 b^2 d}+\frac {\sin ^4(c+d x)}{4 b d} \]
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Time = 0.07 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {2747, 711} \[ \int \frac {\cos ^5(c+d x)}{a+b \sin (c+d x)} \, dx=\frac {\left (a^2-b^2\right )^2 \log (a+b \sin (c+d x))}{b^5 d}-\frac {a \left (a^2-2 b^2\right ) \sin (c+d x)}{b^4 d}+\frac {\left (a^2-2 b^2\right ) \sin ^2(c+d x)}{2 b^3 d}-\frac {a \sin ^3(c+d x)}{3 b^2 d}+\frac {\sin ^4(c+d x)}{4 b d} \]
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Rule 711
Rule 2747
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {\left (b^2-x^2\right )^2}{a+x} \, dx,x,b \sin (c+d x)\right )}{b^5 d} \\ & = \frac {\text {Subst}\left (\int \left (-a^3 \left (1-\frac {2 b^2}{a^2}\right )+\left (a^2-2 b^2\right ) x-a x^2+x^3+\frac {\left (a^2-b^2\right )^2}{a+x}\right ) \, dx,x,b \sin (c+d x)\right )}{b^5 d} \\ & = \frac {\left (a^2-b^2\right )^2 \log (a+b \sin (c+d x))}{b^5 d}-\frac {a \left (a^2-2 b^2\right ) \sin (c+d x)}{b^4 d}+\frac {\left (a^2-2 b^2\right ) \sin ^2(c+d x)}{2 b^3 d}-\frac {a \sin ^3(c+d x)}{3 b^2 d}+\frac {\sin ^4(c+d x)}{4 b d} \\ \end{align*}
Time = 0.14 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.87 \[ \int \frac {\cos ^5(c+d x)}{a+b \sin (c+d x)} \, dx=\frac {3 b^4 \cos ^4(c+d x)+12 \left (a^2-b^2\right )^2 \log (a+b \sin (c+d x))-12 a b \left (a^2-2 b^2\right ) \sin (c+d x)+6 b^2 \left (a^2-b^2\right ) \sin ^2(c+d x)-4 a b^3 \sin ^3(c+d x)}{12 b^5 d} \]
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Time = 0.87 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.90
method | result | size |
derivativedivides | \(\frac {-\frac {-\frac {\left (\sin ^{4}\left (d x +c \right )\right ) b^{3}}{4}+\frac {a \left (\sin ^{3}\left (d x +c \right )\right ) b^{2}}{3}-\frac {\left (a^{2}-2 b^{2}\right ) \left (\sin ^{2}\left (d x +c \right )\right ) b}{2}+a \left (a^{2}-2 b^{2}\right ) \sin \left (d x +c \right )}{b^{4}}+\frac {\left (a^{4}-2 a^{2} b^{2}+b^{4}\right ) \ln \left (a +b \sin \left (d x +c \right )\right )}{b^{5}}}{d}\) | \(106\) |
default | \(\frac {-\frac {-\frac {\left (\sin ^{4}\left (d x +c \right )\right ) b^{3}}{4}+\frac {a \left (\sin ^{3}\left (d x +c \right )\right ) b^{2}}{3}-\frac {\left (a^{2}-2 b^{2}\right ) \left (\sin ^{2}\left (d x +c \right )\right ) b}{2}+a \left (a^{2}-2 b^{2}\right ) \sin \left (d x +c \right )}{b^{4}}+\frac {\left (a^{4}-2 a^{2} b^{2}+b^{4}\right ) \ln \left (a +b \sin \left (d x +c \right )\right )}{b^{5}}}{d}\) | \(106\) |
parallelrisch | \(\frac {96 \left (a -b \right )^{2} \left (a +b \right )^{2} \ln \left (2 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+a \left (\sec ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )\right )-96 \left (a -b \right )^{2} \left (a +b \right )^{2} \ln \left (\sec ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-24 a^{2} b^{2}+36 b^{4}\right ) \cos \left (2 d x +2 c \right )+3 \cos \left (4 d x +4 c \right ) b^{4}+8 a \sin \left (3 d x +3 c \right ) b^{3}+\left (-96 a^{3} b +168 a \,b^{3}\right ) \sin \left (d x +c \right )+24 a^{2} b^{2}-39 b^{4}}{96 b^{5} d}\) | \(163\) |
risch | \(-\frac {i x}{b}-\frac {2 i c}{b d}+\frac {i a^{3} {\mathrm e}^{i \left (d x +c \right )}}{2 b^{4} d}-\frac {{\mathrm e}^{2 i \left (d x +c \right )} a^{2}}{8 b^{3} d}+\frac {3 \,{\mathrm e}^{2 i \left (d x +c \right )}}{16 b d}+\frac {2 i x \,a^{2}}{b^{3}}-\frac {i x \,a^{4}}{b^{5}}+\frac {7 i a \,{\mathrm e}^{-i \left (d x +c \right )}}{8 b^{2} d}-\frac {2 i a^{4} c}{b^{5} d}-\frac {{\mathrm e}^{-2 i \left (d x +c \right )} a^{2}}{8 b^{3} d}+\frac {3 \,{\mathrm e}^{-2 i \left (d x +c \right )}}{16 b d}-\frac {7 i a \,{\mathrm e}^{i \left (d x +c \right )}}{8 b^{2} d}-\frac {i a^{3} {\mathrm e}^{-i \left (d x +c \right )}}{2 b^{4} d}+\frac {4 i a^{2} c}{b^{3} d}+\frac {\ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+\frac {2 i a \,{\mathrm e}^{i \left (d x +c \right )}}{b}-1\right ) a^{4}}{b^{5} d}-\frac {2 \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+\frac {2 i a \,{\mathrm e}^{i \left (d x +c \right )}}{b}-1\right ) a^{2}}{b^{3} d}+\frac {\ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+\frac {2 i a \,{\mathrm e}^{i \left (d x +c \right )}}{b}-1\right )}{b d}+\frac {\cos \left (4 d x +4 c \right )}{32 b d}+\frac {a \sin \left (3 d x +3 c \right )}{12 b^{2} d}\) | \(366\) |
norman | \(\frac {\frac {\left (2 a^{2}-4 b^{2}\right ) \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{b^{3} d}+\frac {\left (2 a^{2}-4 b^{2}\right ) \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{b^{3} d}+\frac {2 \left (3 a^{2}-4 b^{2}\right ) \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{b^{3} d}+\frac {2 \left (3 a^{2}-4 b^{2}\right ) \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{b^{3} d}-\frac {2 a \left (a^{2}-2 b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{b^{4} d}-\frac {2 a \left (a^{2}-2 b^{2}\right ) \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{b^{4} d}-\frac {8 a \left (3 a^{2}-5 b^{2}\right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 b^{4} d}-\frac {8 a \left (3 a^{2}-5 b^{2}\right ) \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 b^{4} d}-\frac {4 a \left (9 a^{2}-14 b^{2}\right ) \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 b^{4} d}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{5}}+\frac {\left (a^{4}-2 a^{2} b^{2}+b^{4}\right ) \ln \left (a \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+2 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+a \right )}{b^{5} d}-\frac {\left (a^{4}-2 a^{2} b^{2}+b^{4}\right ) \ln \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{b^{5} d}\) | \(373\) |
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Time = 0.29 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.91 \[ \int \frac {\cos ^5(c+d x)}{a+b \sin (c+d x)} \, dx=\frac {3 \, b^{4} \cos \left (d x + c\right )^{4} - 6 \, {\left (a^{2} b^{2} - b^{4}\right )} \cos \left (d x + c\right )^{2} + 12 \, {\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} \log \left (b \sin \left (d x + c\right ) + a\right ) + 4 \, {\left (a b^{3} \cos \left (d x + c\right )^{2} - 3 \, a^{3} b + 5 \, a b^{3}\right )} \sin \left (d x + c\right )}{12 \, b^{5} d} \]
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Timed out. \[ \int \frac {\cos ^5(c+d x)}{a+b \sin (c+d x)} \, dx=\text {Timed out} \]
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Time = 0.18 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.92 \[ \int \frac {\cos ^5(c+d x)}{a+b \sin (c+d x)} \, dx=\frac {\frac {3 \, b^{3} \sin \left (d x + c\right )^{4} - 4 \, a b^{2} \sin \left (d x + c\right )^{3} + 6 \, {\left (a^{2} b - 2 \, b^{3}\right )} \sin \left (d x + c\right )^{2} - 12 \, {\left (a^{3} - 2 \, a b^{2}\right )} \sin \left (d x + c\right )}{b^{4}} + \frac {12 \, {\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} \log \left (b \sin \left (d x + c\right ) + a\right )}{b^{5}}}{12 \, d} \]
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Time = 0.31 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.02 \[ \int \frac {\cos ^5(c+d x)}{a+b \sin (c+d x)} \, dx=\frac {\frac {3 \, b^{3} \sin \left (d x + c\right )^{4} - 4 \, a b^{2} \sin \left (d x + c\right )^{3} + 6 \, a^{2} b \sin \left (d x + c\right )^{2} - 12 \, b^{3} \sin \left (d x + c\right )^{2} - 12 \, a^{3} \sin \left (d x + c\right ) + 24 \, a b^{2} \sin \left (d x + c\right )}{b^{4}} + \frac {12 \, {\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} \log \left ({\left | b \sin \left (d x + c\right ) + a \right |}\right )}{b^{5}}}{12 \, d} \]
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Time = 4.44 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.92 \[ \int \frac {\cos ^5(c+d x)}{a+b \sin (c+d x)} \, dx=\frac {\frac {{\sin \left (c+d\,x\right )}^4}{4\,b}-{\sin \left (c+d\,x\right )}^2\,\left (\frac {1}{b}-\frac {a^2}{2\,b^3}\right )+\frac {\ln \left (a+b\,\sin \left (c+d\,x\right )\right )\,\left (a^4-2\,a^2\,b^2+b^4\right )}{b^5}-\frac {a\,{\sin \left (c+d\,x\right )}^3}{3\,b^2}+\frac {a\,\sin \left (c+d\,x\right )\,\left (\frac {2}{b}-\frac {a^2}{b^3}\right )}{b}}{d} \]
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