Integrand size = 21, antiderivative size = 202 \[ \int \csc ^9(c+d x) (a+a \sec (c+d x))^3 \, dx=-\frac {a^7}{16 d (a-a \cos (c+d x))^4}-\frac {a^6}{3 d (a-a \cos (c+d x))^3}-\frac {39 a^5}{32 d (a-a \cos (c+d x))^2}-\frac {75 a^4}{16 d (a-a \cos (c+d x))}-\frac {a^4}{32 d (a+a \cos (c+d x))}+\frac {501 a^3 \log (1-\cos (c+d x))}{64 d}-\frac {8 a^3 \log (\cos (c+d x))}{d}+\frac {11 a^3 \log (1+\cos (c+d x))}{64 d}+\frac {3 a^3 \sec (c+d x)}{d}+\frac {a^3 \sec ^2(c+d x)}{2 d} \]
[Out]
Time = 0.28 (sec) , antiderivative size = 202, 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, 2915, 12, 90} \[ \int \csc ^9(c+d x) (a+a \sec (c+d x))^3 \, dx=-\frac {a^7}{16 d (a-a \cos (c+d x))^4}-\frac {a^6}{3 d (a-a \cos (c+d x))^3}-\frac {39 a^5}{32 d (a-a \cos (c+d x))^2}-\frac {75 a^4}{16 d (a-a \cos (c+d x))}-\frac {a^4}{32 d (a \cos (c+d x)+a)}+\frac {a^3 \sec ^2(c+d x)}{2 d}+\frac {3 a^3 \sec (c+d x)}{d}+\frac {501 a^3 \log (1-\cos (c+d x))}{64 d}-\frac {8 a^3 \log (\cos (c+d x))}{d}+\frac {11 a^3 \log (\cos (c+d x)+1)}{64 d} \]
[In]
[Out]
Rule 12
Rule 90
Rule 2915
Rule 3957
Rubi steps \begin{align*} \text {integral}& = -\int (-a-a \cos (c+d x))^3 \csc ^9(c+d x) \sec ^3(c+d x) \, dx \\ & = \frac {a^9 \text {Subst}\left (\int \frac {a^3}{(-a-x)^5 x^3 (-a+x)^2} \, dx,x,-a \cos (c+d x)\right )}{d} \\ & = \frac {a^{12} \text {Subst}\left (\int \frac {1}{(-a-x)^5 x^3 (-a+x)^2} \, dx,x,-a \cos (c+d x)\right )}{d} \\ & = \frac {a^{12} \text {Subst}\left (\int \left (-\frac {1}{32 a^8 (a-x)^2}-\frac {11}{64 a^9 (a-x)}-\frac {1}{a^7 x^3}+\frac {3}{a^8 x^2}-\frac {8}{a^9 x}+\frac {1}{4 a^5 (a+x)^5}+\frac {1}{a^6 (a+x)^4}+\frac {39}{16 a^7 (a+x)^3}+\frac {75}{16 a^8 (a+x)^2}+\frac {501}{64 a^9 (a+x)}\right ) \, dx,x,-a \cos (c+d x)\right )}{d} \\ & = -\frac {a^7}{16 d (a-a \cos (c+d x))^4}-\frac {a^6}{3 d (a-a \cos (c+d x))^3}-\frac {39 a^5}{32 d (a-a \cos (c+d x))^2}-\frac {75 a^4}{16 d (a-a \cos (c+d x))}-\frac {a^4}{32 d (a+a \cos (c+d x))}+\frac {501 a^3 \log (1-\cos (c+d x))}{64 d}-\frac {8 a^3 \log (\cos (c+d x))}{d}+\frac {11 a^3 \log (1+\cos (c+d x))}{64 d}+\frac {3 a^3 \sec (c+d x)}{d}+\frac {a^3 \sec ^2(c+d x)}{2 d} \\ \end{align*}
Time = 0.89 (sec) , antiderivative size = 159, normalized size of antiderivative = 0.79 \[ \int \csc ^9(c+d x) (a+a \sec (c+d x))^3 \, dx=-\frac {a^3 (1+\cos (c+d x))^3 \sec ^6\left (\frac {1}{2} (c+d x)\right ) \left (1800 \csc ^2\left (\frac {1}{2} (c+d x)\right )+234 \csc ^4\left (\frac {1}{2} (c+d x)\right )+32 \csc ^6\left (\frac {1}{2} (c+d x)\right )+3 \csc ^8\left (\frac {1}{2} (c+d x)\right )-12 \left (22 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-512 \log (\cos (c+d x))+1002 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )-\sec ^2\left (\frac {1}{2} (c+d x)\right )+192 \sec (c+d x)+32 \sec ^2(c+d x)\right )\right )}{6144 d} \]
[In]
[Out]
Time = 1.25 (sec) , antiderivative size = 210, normalized size of antiderivative = 1.04
method | result | size |
norman | \(\frac {-\frac {a^{3}}{256 d}-\frac {19 a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{384 d}-\frac {263 a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{768 d}-\frac {431 a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{192 d}-\frac {a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{14}}{64 d}-\frac {451 a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}}{64 d}+\frac {749 a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}}{64 d}}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8} \left (-1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{2}}+\frac {501 a^{3} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{32 d}-\frac {8 a^{3} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{d}-\frac {8 a^{3} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{d}\) | \(210\) |
parallelrisch | \(-\frac {a^{3} \left (2048 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )+2048 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )-4008 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\cot \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}+\frac {38 \cot \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{3}+4 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}+\frac {263 \cot \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{3}+\frac {1724 \cot \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{3}+1804 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-2996\right )}{256 d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}\) | \(231\) |
risch | \(\frac {a^{3} \left (735 \,{\mathrm e}^{13 i \left (d x +c \right )}-3642 \,{\mathrm e}^{12 i \left (d x +c \right )}+6662 \,{\mathrm e}^{11 i \left (d x +c \right )}-4650 \,{\mathrm e}^{10 i \left (d x +c \right )}-1983 \,{\mathrm e}^{9 i \left (d x +c \right )}+8868 \,{\mathrm e}^{8 i \left (d x +c \right )}-12748 \,{\mathrm e}^{7 i \left (d x +c \right )}+8868 \,{\mathrm e}^{6 i \left (d x +c \right )}-1983 \,{\mathrm e}^{5 i \left (d x +c \right )}-4650 \,{\mathrm e}^{4 i \left (d x +c \right )}+6662 \,{\mathrm e}^{3 i \left (d x +c \right )}-3642 \,{\mathrm e}^{2 i \left (d x +c \right )}+735 \,{\mathrm e}^{i \left (d x +c \right )}\right )}{48 d \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )^{8} \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )^{2} \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{2}}+\frac {501 a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{32 d}+\frac {11 a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{32 d}-\frac {8 a^{3} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}{d}\) | \(253\) |
derivativedivides | \(\frac {a^{3} \left (-\frac {1}{8 \sin \left (d x +c \right )^{8} \cos \left (d x +c \right )^{2}}-\frac {5}{24 \sin \left (d x +c \right )^{6} \cos \left (d x +c \right )^{2}}-\frac {5}{12 \sin \left (d x +c \right )^{4} \cos \left (d x +c \right )^{2}}+\frac {5}{4 \sin \left (d x +c \right )^{2} \cos \left (d x +c \right )^{2}}-\frac {5}{2 \sin \left (d x +c \right )^{2}}+5 \ln \left (\tan \left (d x +c \right )\right )\right )+3 a^{3} \left (-\frac {1}{8 \sin \left (d x +c \right )^{8} \cos \left (d x +c \right )}-\frac {3}{16 \sin \left (d x +c \right )^{6} \cos \left (d x +c \right )}-\frac {21}{64 \sin \left (d x +c \right )^{4} \cos \left (d x +c \right )}-\frac {105}{128 \sin \left (d x +c \right )^{2} \cos \left (d x +c \right )}+\frac {315}{128 \cos \left (d x +c \right )}+\frac {315 \ln \left (-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )}{128}\right )+3 a^{3} \left (-\frac {1}{8 \sin \left (d x +c \right )^{8}}-\frac {1}{6 \sin \left (d x +c \right )^{6}}-\frac {1}{4 \sin \left (d x +c \right )^{4}}-\frac {1}{2 \sin \left (d x +c \right )^{2}}+\ln \left (\tan \left (d x +c \right )\right )\right )+a^{3} \left (\left (-\frac {\csc \left (d x +c \right )^{7}}{8}-\frac {7 \csc \left (d x +c \right )^{5}}{48}-\frac {35 \csc \left (d x +c \right )^{3}}{192}-\frac {35 \csc \left (d x +c \right )}{128}\right ) \cot \left (d x +c \right )+\frac {35 \ln \left (-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )}{128}\right )}{d}\) | \(330\) |
default | \(\frac {a^{3} \left (-\frac {1}{8 \sin \left (d x +c \right )^{8} \cos \left (d x +c \right )^{2}}-\frac {5}{24 \sin \left (d x +c \right )^{6} \cos \left (d x +c \right )^{2}}-\frac {5}{12 \sin \left (d x +c \right )^{4} \cos \left (d x +c \right )^{2}}+\frac {5}{4 \sin \left (d x +c \right )^{2} \cos \left (d x +c \right )^{2}}-\frac {5}{2 \sin \left (d x +c \right )^{2}}+5 \ln \left (\tan \left (d x +c \right )\right )\right )+3 a^{3} \left (-\frac {1}{8 \sin \left (d x +c \right )^{8} \cos \left (d x +c \right )}-\frac {3}{16 \sin \left (d x +c \right )^{6} \cos \left (d x +c \right )}-\frac {21}{64 \sin \left (d x +c \right )^{4} \cos \left (d x +c \right )}-\frac {105}{128 \sin \left (d x +c \right )^{2} \cos \left (d x +c \right )}+\frac {315}{128 \cos \left (d x +c \right )}+\frac {315 \ln \left (-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )}{128}\right )+3 a^{3} \left (-\frac {1}{8 \sin \left (d x +c \right )^{8}}-\frac {1}{6 \sin \left (d x +c \right )^{6}}-\frac {1}{4 \sin \left (d x +c \right )^{4}}-\frac {1}{2 \sin \left (d x +c \right )^{2}}+\ln \left (\tan \left (d x +c \right )\right )\right )+a^{3} \left (\left (-\frac {\csc \left (d x +c \right )^{7}}{8}-\frac {7 \csc \left (d x +c \right )^{5}}{48}-\frac {35 \csc \left (d x +c \right )^{3}}{192}-\frac {35 \csc \left (d x +c \right )}{128}\right ) \cot \left (d x +c \right )+\frac {35 \ln \left (-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )}{128}\right )}{d}\) | \(330\) |
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 419 vs. \(2 (190) = 380\).
Time = 0.29 (sec) , antiderivative size = 419, normalized size of antiderivative = 2.07 \[ \int \csc ^9(c+d x) (a+a \sec (c+d x))^3 \, dx=\frac {1470 \, a^{3} \cos \left (d x + c\right )^{6} - 3642 \, a^{3} \cos \left (d x + c\right )^{5} + 1126 \, a^{3} \cos \left (d x + c\right )^{4} + 3390 \, a^{3} \cos \left (d x + c\right )^{3} - 2752 \, a^{3} \cos \left (d x + c\right )^{2} + 288 \, a^{3} \cos \left (d x + c\right ) + 96 \, a^{3} - 1536 \, {\left (a^{3} \cos \left (d x + c\right )^{7} - 3 \, a^{3} \cos \left (d x + c\right )^{6} + 2 \, a^{3} \cos \left (d x + c\right )^{5} + 2 \, a^{3} \cos \left (d x + c\right )^{4} - 3 \, a^{3} \cos \left (d x + c\right )^{3} + a^{3} \cos \left (d x + c\right )^{2}\right )} \log \left (-\cos \left (d x + c\right )\right ) + 33 \, {\left (a^{3} \cos \left (d x + c\right )^{7} - 3 \, a^{3} \cos \left (d x + c\right )^{6} + 2 \, a^{3} \cos \left (d x + c\right )^{5} + 2 \, a^{3} \cos \left (d x + c\right )^{4} - 3 \, a^{3} \cos \left (d x + c\right )^{3} + a^{3} \cos \left (d x + c\right )^{2}\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 1503 \, {\left (a^{3} \cos \left (d x + c\right )^{7} - 3 \, a^{3} \cos \left (d x + c\right )^{6} + 2 \, a^{3} \cos \left (d x + c\right )^{5} + 2 \, a^{3} \cos \left (d x + c\right )^{4} - 3 \, a^{3} \cos \left (d x + c\right )^{3} + a^{3} \cos \left (d x + c\right )^{2}\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right )}{192 \, {\left (d \cos \left (d x + c\right )^{7} - 3 \, d \cos \left (d x + c\right )^{6} + 2 \, d \cos \left (d x + c\right )^{5} + 2 \, d \cos \left (d x + c\right )^{4} - 3 \, d \cos \left (d x + c\right )^{3} + d \cos \left (d x + c\right )^{2}\right )}} \]
[In]
[Out]
Timed out. \[ \int \csc ^9(c+d x) (a+a \sec (c+d x))^3 \, dx=\text {Timed out} \]
[In]
[Out]
none
Time = 0.21 (sec) , antiderivative size = 189, normalized size of antiderivative = 0.94 \[ \int \csc ^9(c+d x) (a+a \sec (c+d x))^3 \, dx=\frac {33 \, a^{3} \log \left (\cos \left (d x + c\right ) + 1\right ) + 1503 \, a^{3} \log \left (\cos \left (d x + c\right ) - 1\right ) - 1536 \, a^{3} \log \left (\cos \left (d x + c\right )\right ) + \frac {2 \, {\left (735 \, a^{3} \cos \left (d x + c\right )^{6} - 1821 \, a^{3} \cos \left (d x + c\right )^{5} + 563 \, a^{3} \cos \left (d x + c\right )^{4} + 1695 \, a^{3} \cos \left (d x + c\right )^{3} - 1376 \, a^{3} \cos \left (d x + c\right )^{2} + 144 \, a^{3} \cos \left (d x + c\right ) + 48 \, a^{3}\right )}}{\cos \left (d x + c\right )^{7} - 3 \, \cos \left (d x + c\right )^{6} + 2 \, \cos \left (d x + c\right )^{5} + 2 \, \cos \left (d x + c\right )^{4} - 3 \, \cos \left (d x + c\right )^{3} + \cos \left (d x + c\right )^{2}}}{192 \, d} \]
[In]
[Out]
none
Time = 0.45 (sec) , antiderivative size = 292, normalized size of antiderivative = 1.45 \[ \int \csc ^9(c+d x) (a+a \sec (c+d x))^3 \, dx=\frac {6012 \, a^{3} \log \left (\frac {{\left | -\cos \left (d x + c\right ) + 1 \right |}}{{\left | \cos \left (d x + c\right ) + 1 \right |}}\right ) - 6144 \, a^{3} \log \left ({\left | -\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} - 1 \right |}\right ) + \frac {12 \, a^{3} {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} - \frac {{\left (3 \, a^{3} - \frac {44 \, a^{3} {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac {348 \, a^{3} {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {2376 \, a^{3} {\left (\cos \left (d x + c\right ) - 1\right )}^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {12525 \, a^{3} {\left (\cos \left (d x + c\right ) - 1\right )}^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}}\right )} {\left (\cos \left (d x + c\right ) + 1\right )}^{4}}{{\left (\cos \left (d x + c\right ) - 1\right )}^{4}} + \frac {1536 \, {\left (9 \, a^{3} + \frac {14 \, a^{3} {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac {6 \, a^{3} {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}}\right )}}{{\left (\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} + 1\right )}^{2}}}{768 \, d} \]
[In]
[Out]
Time = 13.43 (sec) , antiderivative size = 195, normalized size of antiderivative = 0.97 \[ \int \csc ^9(c+d x) (a+a \sec (c+d x))^3 \, dx=\frac {501\,a^3\,\ln \left (\cos \left (c+d\,x\right )-1\right )}{64\,d}+\frac {11\,a^3\,\ln \left (\cos \left (c+d\,x\right )+1\right )}{64\,d}+\frac {\frac {245\,a^3\,{\cos \left (c+d\,x\right )}^6}{32}-\frac {607\,a^3\,{\cos \left (c+d\,x\right )}^5}{32}+\frac {563\,a^3\,{\cos \left (c+d\,x\right )}^4}{96}+\frac {565\,a^3\,{\cos \left (c+d\,x\right )}^3}{32}-\frac {43\,a^3\,{\cos \left (c+d\,x\right )}^2}{3}+\frac {3\,a^3\,\cos \left (c+d\,x\right )}{2}+\frac {a^3}{2}}{d\,\left ({\cos \left (c+d\,x\right )}^7-3\,{\cos \left (c+d\,x\right )}^6+2\,{\cos \left (c+d\,x\right )}^5+2\,{\cos \left (c+d\,x\right )}^4-3\,{\cos \left (c+d\,x\right )}^3+{\cos \left (c+d\,x\right )}^2\right )}-\frac {8\,a^3\,\ln \left (\cos \left (c+d\,x\right )\right )}{d} \]
[In]
[Out]