Integrand size = 21, antiderivative size = 223 \[ \int \frac {\sin ^7(c+d x)}{a+b \sec (c+d x)} \, dx=-\frac {\left (a^2-b^2\right )^3 \cos (c+d x)}{a^7 d}-\frac {b \left (3 a^4-3 a^2 b^2+b^4\right ) \cos ^2(c+d x)}{2 a^6 d}+\frac {\left (3 a^4-3 a^2 b^2+b^4\right ) \cos ^3(c+d x)}{3 a^5 d}+\frac {b \left (3 a^2-b^2\right ) \cos ^4(c+d x)}{4 a^4 d}-\frac {\left (3 a^2-b^2\right ) \cos ^5(c+d x)}{5 a^3 d}-\frac {b \cos ^6(c+d x)}{6 a^2 d}+\frac {\cos ^7(c+d x)}{7 a d}+\frac {b \left (a^2-b^2\right )^3 \log (b+a \cos (c+d x))}{a^8 d} \]
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Time = 0.33 (sec) , antiderivative size = 223, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {3957, 2916, 12, 786} \[ \int \frac {\sin ^7(c+d x)}{a+b \sec (c+d x)} \, dx=-\frac {b \cos ^6(c+d x)}{6 a^2 d}+\frac {b \left (a^2-b^2\right )^3 \log (a \cos (c+d x)+b)}{a^8 d}-\frac {\left (a^2-b^2\right )^3 \cos (c+d x)}{a^7 d}+\frac {b \left (3 a^2-b^2\right ) \cos ^4(c+d x)}{4 a^4 d}-\frac {\left (3 a^2-b^2\right ) \cos ^5(c+d x)}{5 a^3 d}-\frac {b \left (3 a^4-3 a^2 b^2+b^4\right ) \cos ^2(c+d x)}{2 a^6 d}+\frac {\left (3 a^4-3 a^2 b^2+b^4\right ) \cos ^3(c+d x)}{3 a^5 d}+\frac {\cos ^7(c+d x)}{7 a d} \]
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Rule 12
Rule 786
Rule 2916
Rule 3957
Rubi steps \begin{align*} \text {integral}& = -\int \frac {\cos (c+d x) \sin ^7(c+d x)}{-b-a \cos (c+d x)} \, dx \\ & = \frac {\text {Subst}\left (\int \frac {x \left (a^2-x^2\right )^3}{a (-b+x)} \, dx,x,-a \cos (c+d x)\right )}{a^7 d} \\ & = \frac {\text {Subst}\left (\int \frac {x \left (a^2-x^2\right )^3}{-b+x} \, dx,x,-a \cos (c+d x)\right )}{a^8 d} \\ & = \frac {\text {Subst}\left (\int \left (\left (a^2-b^2\right )^3+\frac {b \left (-a^2+b^2\right )^3}{b-x}-b \left (3 a^4-3 a^2 b^2+b^4\right ) x-\left (3 a^4-3 a^2 b^2+b^4\right ) x^2-b \left (-3 a^2+b^2\right ) x^3+\left (3 a^2-b^2\right ) x^4-b x^5-x^6\right ) \, dx,x,-a \cos (c+d x)\right )}{a^8 d} \\ & = -\frac {\left (a^2-b^2\right )^3 \cos (c+d x)}{a^7 d}-\frac {b \left (3 a^4-3 a^2 b^2+b^4\right ) \cos ^2(c+d x)}{2 a^6 d}+\frac {\left (3 a^4-3 a^2 b^2+b^4\right ) \cos ^3(c+d x)}{3 a^5 d}+\frac {b \left (3 a^2-b^2\right ) \cos ^4(c+d x)}{4 a^4 d}-\frac {\left (3 a^2-b^2\right ) \cos ^5(c+d x)}{5 a^3 d}-\frac {b \cos ^6(c+d x)}{6 a^2 d}+\frac {\cos ^7(c+d x)}{7 a d}+\frac {b \left (a^2-b^2\right )^3 \log (b+a \cos (c+d x))}{a^8 d} \\ \end{align*}
Time = 1.81 (sec) , antiderivative size = 282, normalized size of antiderivative = 1.26 \[ \int \frac {\sin ^7(c+d x)}{a+b \sec (c+d x)} \, dx=\frac {-105 a \left (35 a^6-152 a^4 b^2+176 a^2 b^4-64 b^6\right ) \cos (c+d x)-105 \left (29 a^6 b-40 a^4 b^3+16 a^2 b^5\right ) \cos (2 (c+d x))+735 a^7 \cos (3 (c+d x))-1260 a^5 b^2 \cos (3 (c+d x))+560 a^3 b^4 \cos (3 (c+d x))+420 a^6 b \cos (4 (c+d x))-210 a^4 b^3 \cos (4 (c+d x))-147 a^7 \cos (5 (c+d x))+84 a^5 b^2 \cos (5 (c+d x))-35 a^6 b \cos (6 (c+d x))+15 a^7 \cos (7 (c+d x))+6720 a^6 b \log (b+a \cos (c+d x))-20160 a^4 b^3 \log (b+a \cos (c+d x))+20160 a^2 b^5 \log (b+a \cos (c+d x))-6720 b^7 \log (b+a \cos (c+d x))}{6720 a^8 d} \]
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Time = 1.36 (sec) , antiderivative size = 275, normalized size of antiderivative = 1.23
method | result | size |
derivativedivides | \(\frac {\frac {\frac {\cos \left (d x +c \right )^{7} a^{6}}{7}-\frac {b \cos \left (d x +c \right )^{6} a^{5}}{6}-\frac {3 a^{6} \cos \left (d x +c \right )^{5}}{5}+\frac {a^{4} b^{2} \cos \left (d x +c \right )^{5}}{5}+\frac {3 \cos \left (d x +c \right )^{4} a^{5} b}{4}-\frac {\cos \left (d x +c \right )^{4} a^{3} b^{3}}{4}+\cos \left (d x +c \right )^{3} a^{6}-\cos \left (d x +c \right )^{3} a^{4} b^{2}+\frac {\cos \left (d x +c \right )^{3} a^{2} b^{4}}{3}-\frac {3 \cos \left (d x +c \right )^{2} a^{5} b}{2}+\frac {3 \cos \left (d x +c \right )^{2} a^{3} b^{3}}{2}-\frac {\cos \left (d x +c \right )^{2} a \,b^{5}}{2}-\cos \left (d x +c \right ) a^{6}+3 \cos \left (d x +c \right ) a^{4} b^{2}-3 \cos \left (d x +c \right ) a^{2} b^{4}+b^{6} \cos \left (d x +c \right )}{a^{7}}+\frac {b \left (a^{6}-3 a^{4} b^{2}+3 a^{2} b^{4}-b^{6}\right ) \ln \left (b +a \cos \left (d x +c \right )\right )}{a^{8}}}{d}\) | \(275\) |
default | \(\frac {\frac {\frac {\cos \left (d x +c \right )^{7} a^{6}}{7}-\frac {b \cos \left (d x +c \right )^{6} a^{5}}{6}-\frac {3 a^{6} \cos \left (d x +c \right )^{5}}{5}+\frac {a^{4} b^{2} \cos \left (d x +c \right )^{5}}{5}+\frac {3 \cos \left (d x +c \right )^{4} a^{5} b}{4}-\frac {\cos \left (d x +c \right )^{4} a^{3} b^{3}}{4}+\cos \left (d x +c \right )^{3} a^{6}-\cos \left (d x +c \right )^{3} a^{4} b^{2}+\frac {\cos \left (d x +c \right )^{3} a^{2} b^{4}}{3}-\frac {3 \cos \left (d x +c \right )^{2} a^{5} b}{2}+\frac {3 \cos \left (d x +c \right )^{2} a^{3} b^{3}}{2}-\frac {\cos \left (d x +c \right )^{2} a \,b^{5}}{2}-\cos \left (d x +c \right ) a^{6}+3 \cos \left (d x +c \right ) a^{4} b^{2}-3 \cos \left (d x +c \right ) a^{2} b^{4}+b^{6} \cos \left (d x +c \right )}{a^{7}}+\frac {b \left (a^{6}-3 a^{4} b^{2}+3 a^{2} b^{4}-b^{6}\right ) \ln \left (b +a \cos \left (d x +c \right )\right )}{a^{8}}}{d}\) | \(275\) |
parallelrisch | \(\frac {320 b \left (a -b \right )^{3} \left (a +b \right )^{3} \ln \left (-2 a +\sec \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} \left (a -b \right )\right )-320 b \left (a -b \right )^{3} \left (a +b \right )^{3} \ln \left (\sec \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )-7 \left (\frac {5 \left (29 a^{5} b -40 a^{3} b^{3}+16 a \,b^{5}\right ) \cos \left (2 d x +2 c \right )}{7}+5 \left (-a^{6}+\frac {12}{7} a^{4} b^{2}-\frac {16}{21} a^{2} b^{4}\right ) \cos \left (3 d x +3 c \right )+\frac {10 \left (-2 a^{5} b +a^{3} b^{3}\right ) \cos \left (4 d x +4 c \right )}{7}+\left (a^{6}-\frac {4}{7} a^{4} b^{2}\right ) \cos \left (5 d x +5 c \right )+\frac {5 a^{5} b \cos \left (6 d x +6 c \right )}{21}-\frac {5 a^{6} \cos \left (7 d x +7 c \right )}{49}+5 \left (5 a^{6}-\frac {152}{7} a^{4} b^{2}+\frac {176}{7} a^{2} b^{4}-\frac {64}{7} b^{6}\right ) \cos \left (d x +c \right )+\frac {1024 a^{6}}{49}-\frac {380 a^{5} b}{21}-\frac {704 a^{4} b^{2}}{7}+\frac {190 a^{3} b^{3}}{7}+\frac {2560 a^{2} b^{4}}{21}-\frac {80 a \,b^{5}}{7}-\frac {320 b^{6}}{7}\right ) a}{320 a^{8} d}\) | \(300\) |
norman | \(\frac {\frac {\left (2 a^{5} b +2 a^{4} b^{2}-4 a^{3} b^{3}-4 a^{2} b^{4}+2 a \,b^{5}+2 b^{6}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{12}}{a^{7} d}+\frac {\left (14 a^{5} b +16 a^{4} b^{2}-24 a^{3} b^{3}-28 a^{2} b^{4}+10 a \,b^{5}+12 b^{6}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}}{a^{7} d}+\frac {-96 a^{6}+462 a^{4} b^{2}-560 a^{2} b^{4}+210 b^{6}}{105 a^{7} d}+\frac {\left (128 a^{5} b +174 a^{4} b^{2}-156 a^{3} b^{3}-232 a^{2} b^{4}+60 a \,b^{5}+90 b^{6}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}}{3 a^{7} d}+\frac {\left (-96 a^{6}+30 a^{5} b +432 a^{4} b^{2}-60 a^{3} b^{3}-500 a^{2} b^{4}+30 a \,b^{5}+180 b^{6}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{15 a^{7} d}+\frac {\left (-96 a^{6}+70 a^{5} b +382 a^{4} b^{2}-120 a^{3} b^{3}-420 a^{2} b^{4}+50 a \,b^{5}+150 b^{6}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{5 a^{7} d}+\frac {\left (-96 a^{6}+128 a^{5} b +288 a^{4} b^{2}-156 a^{3} b^{3}-328 a^{2} b^{4}+60 a \,b^{5}+120 b^{6}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{3 a^{7} d}}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{7}}+\frac {\left (a +b \right ) b \left (a^{5}-a^{4} b -2 a^{3} b^{2}+2 a^{2} b^{3}+a \,b^{4}-b^{5}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} b -a -b \right )}{a^{8} d}-\frac {\left (a +b \right ) b \left (a^{5}-a^{4} b -2 a^{3} b^{2}+2 a^{2} b^{3}+a \,b^{4}-b^{5}\right ) \ln \left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )}{a^{8} d}\) | \(577\) |
risch | \(-\frac {7 \cos \left (5 d x +5 c \right )}{320 d a}-\frac {b \cos \left (6 d x +6 c \right )}{192 d \,a^{2}}+\frac {7 \cos \left (3 d x +3 c \right )}{64 a d}+\frac {\cos \left (7 d x +7 c \right )}{448 a d}+\frac {\cos \left (5 d x +5 c \right ) b^{2}}{80 d \,a^{3}}+\frac {b \cos \left (4 d x +4 c \right )}{16 d \,a^{2}}-\frac {b^{3} \cos \left (4 d x +4 c \right )}{32 d \,a^{4}}-\frac {3 \cos \left (3 d x +3 c \right ) b^{2}}{16 a^{3} d}+\frac {\cos \left (3 d x +3 c \right ) b^{4}}{12 a^{5} d}+\frac {b \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+\frac {2 b \,{\mathrm e}^{i \left (d x +c \right )}}{a}+1\right )}{a^{2} d}-\frac {3 b^{3} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+\frac {2 b \,{\mathrm e}^{i \left (d x +c \right )}}{a}+1\right )}{a^{4} d}+\frac {3 b^{5} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+\frac {2 b \,{\mathrm e}^{i \left (d x +c \right )}}{a}+1\right )}{a^{6} d}-\frac {b^{7} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+\frac {2 b \,{\mathrm e}^{i \left (d x +c \right )}}{a}+1\right )}{a^{8} d}-\frac {29 b \,{\mathrm e}^{-2 i \left (d x +c \right )}}{128 a^{2} d}+\frac {5 b^{3} {\mathrm e}^{-2 i \left (d x +c \right )}}{16 a^{4} d}-\frac {b^{5} {\mathrm e}^{-2 i \left (d x +c \right )}}{8 a^{6} d}+\frac {19 \,{\mathrm e}^{i \left (d x +c \right )} b^{2}}{16 a^{3} d}-\frac {11 \,{\mathrm e}^{i \left (d x +c \right )} b^{4}}{8 a^{5} d}+\frac {{\mathrm e}^{i \left (d x +c \right )} b^{6}}{2 a^{7} d}+\frac {19 \,{\mathrm e}^{-i \left (d x +c \right )} b^{2}}{16 a^{3} d}-\frac {11 \,{\mathrm e}^{-i \left (d x +c \right )} b^{4}}{8 a^{5} d}+\frac {{\mathrm e}^{-i \left (d x +c \right )} b^{6}}{2 a^{7} d}-\frac {i x b}{a^{2}}+\frac {3 i x \,b^{3}}{a^{4}}-\frac {3 i x \,b^{5}}{a^{6}}+\frac {i x \,b^{7}}{a^{8}}-\frac {29 b \,{\mathrm e}^{2 i \left (d x +c \right )}}{128 a^{2} d}+\frac {5 b^{3} {\mathrm e}^{2 i \left (d x +c \right )}}{16 a^{4} d}-\frac {b^{5} {\mathrm e}^{2 i \left (d x +c \right )}}{8 a^{6} d}-\frac {35 \,{\mathrm e}^{-i \left (d x +c \right )}}{128 a d}-\frac {35 \,{\mathrm e}^{i \left (d x +c \right )}}{128 d a}-\frac {2 i b c}{a^{2} d}+\frac {6 i b^{3} c}{a^{4} d}-\frac {6 i b^{5} c}{a^{6} d}+\frac {2 i b^{7} c}{a^{8} d}\) | \(676\) |
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Time = 0.28 (sec) , antiderivative size = 222, normalized size of antiderivative = 1.00 \[ \int \frac {\sin ^7(c+d x)}{a+b \sec (c+d x)} \, dx=\frac {60 \, a^{7} \cos \left (d x + c\right )^{7} - 70 \, a^{6} b \cos \left (d x + c\right )^{6} - 84 \, {\left (3 \, a^{7} - a^{5} b^{2}\right )} \cos \left (d x + c\right )^{5} + 105 \, {\left (3 \, a^{6} b - a^{4} b^{3}\right )} \cos \left (d x + c\right )^{4} + 140 \, {\left (3 \, a^{7} - 3 \, a^{5} b^{2} + a^{3} b^{4}\right )} \cos \left (d x + c\right )^{3} - 210 \, {\left (3 \, a^{6} b - 3 \, a^{4} b^{3} + a^{2} b^{5}\right )} \cos \left (d x + c\right )^{2} - 420 \, {\left (a^{7} - 3 \, a^{5} b^{2} + 3 \, a^{3} b^{4} - a b^{6}\right )} \cos \left (d x + c\right ) + 420 \, {\left (a^{6} b - 3 \, a^{4} b^{3} + 3 \, a^{2} b^{5} - b^{7}\right )} \log \left (a \cos \left (d x + c\right ) + b\right )}{420 \, a^{8} d} \]
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Timed out. \[ \int \frac {\sin ^7(c+d x)}{a+b \sec (c+d x)} \, dx=\text {Timed out} \]
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Time = 0.20 (sec) , antiderivative size = 224, normalized size of antiderivative = 1.00 \[ \int \frac {\sin ^7(c+d x)}{a+b \sec (c+d x)} \, dx=\frac {\frac {60 \, a^{6} \cos \left (d x + c\right )^{7} - 70 \, a^{5} b \cos \left (d x + c\right )^{6} - 84 \, {\left (3 \, a^{6} - a^{4} b^{2}\right )} \cos \left (d x + c\right )^{5} + 105 \, {\left (3 \, a^{5} b - a^{3} b^{3}\right )} \cos \left (d x + c\right )^{4} + 140 \, {\left (3 \, a^{6} - 3 \, a^{4} b^{2} + a^{2} b^{4}\right )} \cos \left (d x + c\right )^{3} - 210 \, {\left (3 \, a^{5} b - 3 \, a^{3} b^{3} + a b^{5}\right )} \cos \left (d x + c\right )^{2} - 420 \, {\left (a^{6} - 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} - b^{6}\right )} \cos \left (d x + c\right )}{a^{7}} + \frac {420 \, {\left (a^{6} b - 3 \, a^{4} b^{3} + 3 \, a^{2} b^{5} - b^{7}\right )} \log \left (a \cos \left (d x + c\right ) + b\right )}{a^{8}}}{420 \, d} \]
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Leaf count of result is larger than twice the leaf count of optimal. 1559 vs. \(2 (211) = 422\).
Time = 0.36 (sec) , antiderivative size = 1559, normalized size of antiderivative = 6.99 \[ \int \frac {\sin ^7(c+d x)}{a+b \sec (c+d x)} \, dx=\text {Too large to display} \]
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Time = 0.15 (sec) , antiderivative size = 249, normalized size of antiderivative = 1.12 \[ \int \frac {\sin ^7(c+d x)}{a+b \sec (c+d x)} \, dx=\frac {{\cos \left (c+d\,x\right )}^3\,\left (\frac {1}{a}-\frac {b^2\,\left (\frac {1}{a}-\frac {b^2}{3\,a^3}\right )}{a^2}\right )-{\cos \left (c+d\,x\right )}^5\,\left (\frac {3}{5\,a}-\frac {b^2}{5\,a^3}\right )-\cos \left (c+d\,x\right )\,\left (\frac {1}{a}-\frac {b^2\,\left (\frac {3}{a}-\frac {b^2\,\left (\frac {3}{a}-\frac {b^2}{a^3}\right )}{a^2}\right )}{a^2}\right )+\frac {{\cos \left (c+d\,x\right )}^7}{7\,a}+\frac {\ln \left (b+a\,\cos \left (c+d\,x\right )\right )\,\left (a^6\,b-3\,a^4\,b^3+3\,a^2\,b^5-b^7\right )}{a^8}-\frac {b\,{\cos \left (c+d\,x\right )}^6}{6\,a^2}-\frac {b\,{\cos \left (c+d\,x\right )}^2\,\left (\frac {3}{a}-\frac {b^2\,\left (\frac {3}{a}-\frac {b^2}{a^3}\right )}{a^2}\right )}{2\,a}+\frac {b\,{\cos \left (c+d\,x\right )}^4\,\left (\frac {3}{a}-\frac {b^2}{a^3}\right )}{4\,a}}{d} \]
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