Integrand size = 28, antiderivative size = 391 \[ \int \sec ^{10}(c+d x) (a \cos (c+d x)+b \sin (c+d x))^5 \, dx=\frac {3 a^5 \text {arctanh}(\sin (c+d x))}{8 d}-\frac {5 a^3 b^2 \text {arctanh}(\sin (c+d x))}{8 d}+\frac {15 a b^4 \text {arctanh}(\sin (c+d x))}{128 d}+\frac {a^4 b \sec ^5(c+d x)}{d}-\frac {2 a^2 b^3 \sec ^5(c+d x)}{d}+\frac {b^5 \sec ^5(c+d x)}{5 d}+\frac {10 a^2 b^3 \sec ^7(c+d x)}{7 d}-\frac {2 b^5 \sec ^7(c+d x)}{7 d}+\frac {b^5 \sec ^9(c+d x)}{9 d}+\frac {3 a^5 \sec (c+d x) \tan (c+d x)}{8 d}-\frac {5 a^3 b^2 \sec (c+d x) \tan (c+d x)}{8 d}+\frac {15 a b^4 \sec (c+d x) \tan (c+d x)}{128 d}+\frac {a^5 \sec ^3(c+d x) \tan (c+d x)}{4 d}-\frac {5 a^3 b^2 \sec ^3(c+d x) \tan (c+d x)}{12 d}+\frac {5 a b^4 \sec ^3(c+d x) \tan (c+d x)}{64 d}+\frac {5 a^3 b^2 \sec ^5(c+d x) \tan (c+d x)}{3 d}-\frac {5 a b^4 \sec ^5(c+d x) \tan (c+d x)}{16 d}+\frac {5 a b^4 \sec ^5(c+d x) \tan ^3(c+d x)}{8 d} \] Output:
3/8*a^5*arctanh(sin(d*x+c))/d-5/8*a^3*b^2*arctanh(sin(d*x+c))/d+15/128*a*b ^4*arctanh(sin(d*x+c))/d+a^4*b*sec(d*x+c)^5/d-2*a^2*b^3*sec(d*x+c)^5/d+1/5 *b^5*sec(d*x+c)^5/d+10/7*a^2*b^3*sec(d*x+c)^7/d-2/7*b^5*sec(d*x+c)^7/d+1/9 *b^5*sec(d*x+c)^9/d+3/8*a^5*sec(d*x+c)*tan(d*x+c)/d-5/8*a^3*b^2*sec(d*x+c) *tan(d*x+c)/d+15/128*a*b^4*sec(d*x+c)*tan(d*x+c)/d+1/4*a^5*sec(d*x+c)^3*ta n(d*x+c)/d-5/12*a^3*b^2*sec(d*x+c)^3*tan(d*x+c)/d+5/64*a*b^4*sec(d*x+c)^3* tan(d*x+c)/d+5/3*a^3*b^2*sec(d*x+c)^5*tan(d*x+c)/d-5/16*a*b^4*sec(d*x+c)^5 *tan(d*x+c)/d+5/8*a*b^4*sec(d*x+c)^5*tan(d*x+c)^3/d
Time = 3.57 (sec) , antiderivative size = 331, normalized size of antiderivative = 0.85 \[ \int \sec ^{10}(c+d x) (a \cos (c+d x)+b \sin (c+d x))^5 \, dx=\frac {-40320 a \left (48 a^4-80 a^2 b^2+15 b^4\right ) \left (\log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )-\log \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )\right )+\sec ^9(c+d x) \left (1935360 a^4 b-184320 a^2 b^3+223232 b^5+73728 \left (35 a^4 b-20 a^2 b^3-3 b^5\right ) \cos (2 (c+d x))+129024 \left (5 a^4 b-10 a^2 b^3+b^5\right ) \cos (4 (c+d x))+372960 a^5 \sin (4 (c+d x))+453600 a^3 b^2 \sin (4 (c+d x))-488250 a b^4 \sin (4 (c+d x))+131040 a^5 \sin (6 (c+d x))-218400 a^3 b^2 \sin (6 (c+d x))+40950 a b^4 \sin (6 (c+d x))+15120 a^5 \sin (8 (c+d x))-25200 a^3 b^2 \sin (8 (c+d x))+4725 a b^4 \sin (8 (c+d x))\right )+1260 a \left (656 a^4+2320 a^2 b^2+845 b^4\right ) \sec ^7(c+d x) \tan (c+d x)}{5160960 d} \] Input:
Integrate[Sec[c + d*x]^10*(a*Cos[c + d*x] + b*Sin[c + d*x])^5,x]
Output:
(-40320*a*(48*a^4 - 80*a^2*b^2 + 15*b^4)*(Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) + Sec[c + d*x]^9*(193 5360*a^4*b - 184320*a^2*b^3 + 223232*b^5 + 73728*(35*a^4*b - 20*a^2*b^3 - 3*b^5)*Cos[2*(c + d*x)] + 129024*(5*a^4*b - 10*a^2*b^3 + b^5)*Cos[4*(c + d *x)] + 372960*a^5*Sin[4*(c + d*x)] + 453600*a^3*b^2*Sin[4*(c + d*x)] - 488 250*a*b^4*Sin[4*(c + d*x)] + 131040*a^5*Sin[6*(c + d*x)] - 218400*a^3*b^2* Sin[6*(c + d*x)] + 40950*a*b^4*Sin[6*(c + d*x)] + 15120*a^5*Sin[8*(c + d*x )] - 25200*a^3*b^2*Sin[8*(c + d*x)] + 4725*a*b^4*Sin[8*(c + d*x)]) + 1260* a*(656*a^4 + 2320*a^2*b^2 + 845*b^4)*Sec[c + d*x]^7*Tan[c + d*x])/(5160960 *d)
Time = 0.70 (sec) , antiderivative size = 391, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.107, Rules used = {3042, 3569, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \sec ^{10}(c+d x) (a \cos (c+d x)+b \sin (c+d x))^5 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {(a \cos (c+d x)+b \sin (c+d x))^5}{\cos (c+d x)^{10}}dx\) |
| \(\Big \downarrow \) 3569 |
| \(\displaystyle \int \left (a^5 \sec ^5(c+d x)+5 a^4 b \tan (c+d x) \sec ^5(c+d x)+10 a^3 b^2 \tan ^2(c+d x) \sec ^5(c+d x)+10 a^2 b^3 \tan ^3(c+d x) \sec ^5(c+d x)+5 a b^4 \tan ^4(c+d x) \sec ^5(c+d x)+b^5 \tan ^5(c+d x) \sec ^5(c+d x)\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {3 a^5 \text {arctanh}(\sin (c+d x))}{8 d}+\frac {a^5 \tan (c+d x) \sec ^3(c+d x)}{4 d}+\frac {3 a^5 \tan (c+d x) \sec (c+d x)}{8 d}+\frac {a^4 b \sec ^5(c+d x)}{d}-\frac {5 a^3 b^2 \text {arctanh}(\sin (c+d x))}{8 d}+\frac {5 a^3 b^2 \tan (c+d x) \sec ^5(c+d x)}{3 d}-\frac {5 a^3 b^2 \tan (c+d x) \sec ^3(c+d x)}{12 d}-\frac {5 a^3 b^2 \tan (c+d x) \sec (c+d x)}{8 d}+\frac {10 a^2 b^3 \sec ^7(c+d x)}{7 d}-\frac {2 a^2 b^3 \sec ^5(c+d x)}{d}+\frac {15 a b^4 \text {arctanh}(\sin (c+d x))}{128 d}+\frac {5 a b^4 \tan ^3(c+d x) \sec ^5(c+d x)}{8 d}-\frac {5 a b^4 \tan (c+d x) \sec ^5(c+d x)}{16 d}+\frac {5 a b^4 \tan (c+d x) \sec ^3(c+d x)}{64 d}+\frac {15 a b^4 \tan (c+d x) \sec (c+d x)}{128 d}+\frac {b^5 \sec ^9(c+d x)}{9 d}-\frac {2 b^5 \sec ^7(c+d x)}{7 d}+\frac {b^5 \sec ^5(c+d x)}{5 d}\) |
Input:
Int[Sec[c + d*x]^10*(a*Cos[c + d*x] + b*Sin[c + d*x])^5,x]
Output:
(3*a^5*ArcTanh[Sin[c + d*x]])/(8*d) - (5*a^3*b^2*ArcTanh[Sin[c + d*x]])/(8 *d) + (15*a*b^4*ArcTanh[Sin[c + d*x]])/(128*d) + (a^4*b*Sec[c + d*x]^5)/d - (2*a^2*b^3*Sec[c + d*x]^5)/d + (b^5*Sec[c + d*x]^5)/(5*d) + (10*a^2*b^3* Sec[c + d*x]^7)/(7*d) - (2*b^5*Sec[c + d*x]^7)/(7*d) + (b^5*Sec[c + d*x]^9 )/(9*d) + (3*a^5*Sec[c + d*x]*Tan[c + d*x])/(8*d) - (5*a^3*b^2*Sec[c + d*x ]*Tan[c + d*x])/(8*d) + (15*a*b^4*Sec[c + d*x]*Tan[c + d*x])/(128*d) + (a^ 5*Sec[c + d*x]^3*Tan[c + d*x])/(4*d) - (5*a^3*b^2*Sec[c + d*x]^3*Tan[c + d *x])/(12*d) + (5*a*b^4*Sec[c + d*x]^3*Tan[c + d*x])/(64*d) + (5*a^3*b^2*Se c[c + d*x]^5*Tan[c + d*x])/(3*d) - (5*a*b^4*Sec[c + d*x]^5*Tan[c + d*x])/( 16*d) + (5*a*b^4*Sec[c + d*x]^5*Tan[c + d*x]^3)/(8*d)
Int[cos[(c_.) + (d_.)*(x_)]^(m_.)*(cos[(c_.) + (d_.)*(x_)]*(a_.) + (b_.)*si n[(c_.) + (d_.)*(x_)])^(n_.), x_Symbol] :> Int[ExpandTrig[cos[c + d*x]^m*(a *cos[c + d*x] + b*sin[c + d*x])^n, x], x] /; FreeQ[{a, b, c, d}, x] && Inte gerQ[m] && IGtQ[n, 0]
Time = 1.99 (sec) , antiderivative size = 345, normalized size of antiderivative = 0.88
| method | result | size |
| parts | \(\frac {a^{5} \left (-\left (-\frac {\sec \left (d x +c \right )^{3}}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )}{d}+\frac {b^{5} \left (\frac {\sec \left (d x +c \right )^{9}}{9}-\frac {2 \sec \left (d x +c \right )^{7}}{7}+\frac {\sec \left (d x +c \right )^{5}}{5}\right )}{d}+\frac {a^{4} b \sec \left (d x +c \right )^{5}}{d}+\frac {10 a^{3} b^{2} \left (\frac {\sin \left (d x +c \right )^{3}}{6 \cos \left (d x +c \right )^{6}}+\frac {\sin \left (d x +c \right )^{3}}{8 \cos \left (d x +c \right )^{4}}+\frac {\sin \left (d x +c \right )^{3}}{16 \cos \left (d x +c \right )^{2}}+\frac {\sin \left (d x +c \right )}{16}-\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{16}\right )}{d}+\frac {10 a^{2} b^{3} \left (\frac {\sec \left (d x +c \right )^{7}}{7}-\frac {\sec \left (d x +c \right )^{5}}{5}\right )}{d}+\frac {5 b^{4} a \left (\frac {\sin \left (d x +c \right )^{5}}{8 \cos \left (d x +c \right )^{8}}+\frac {\sin \left (d x +c \right )^{5}}{16 \cos \left (d x +c \right )^{6}}+\frac {\sin \left (d x +c \right )^{5}}{64 \cos \left (d x +c \right )^{4}}-\frac {\sin \left (d x +c \right )^{5}}{128 \cos \left (d x +c \right )^{2}}-\frac {\sin \left (d x +c \right )^{3}}{128}-\frac {3 \sin \left (d x +c \right )}{128}+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{128}\right )}{d}\) | \(345\) |
| derivativedivides | \(\frac {a^{5} \left (-\left (-\frac {\sec \left (d x +c \right )^{3}}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )+\frac {a^{4} b}{\cos \left (d x +c \right )^{5}}+10 a^{3} b^{2} \left (\frac {\sin \left (d x +c \right )^{3}}{6 \cos \left (d x +c \right )^{6}}+\frac {\sin \left (d x +c \right )^{3}}{8 \cos \left (d x +c \right )^{4}}+\frac {\sin \left (d x +c \right )^{3}}{16 \cos \left (d x +c \right )^{2}}+\frac {\sin \left (d x +c \right )}{16}-\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{16}\right )+10 a^{2} b^{3} \left (\frac {\sin \left (d x +c \right )^{4}}{7 \cos \left (d x +c \right )^{7}}+\frac {3 \sin \left (d x +c \right )^{4}}{35 \cos \left (d x +c \right )^{5}}+\frac {\sin \left (d x +c \right )^{4}}{35 \cos \left (d x +c \right )^{3}}-\frac {\sin \left (d x +c \right )^{4}}{35 \cos \left (d x +c \right )}-\frac {\left (2+\sin \left (d x +c \right )^{2}\right ) \cos \left (d x +c \right )}{35}\right )+5 b^{4} a \left (\frac {\sin \left (d x +c \right )^{5}}{8 \cos \left (d x +c \right )^{8}}+\frac {\sin \left (d x +c \right )^{5}}{16 \cos \left (d x +c \right )^{6}}+\frac {\sin \left (d x +c \right )^{5}}{64 \cos \left (d x +c \right )^{4}}-\frac {\sin \left (d x +c \right )^{5}}{128 \cos \left (d x +c \right )^{2}}-\frac {\sin \left (d x +c \right )^{3}}{128}-\frac {3 \sin \left (d x +c \right )}{128}+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{128}\right )+b^{5} \left (\frac {\sin \left (d x +c \right )^{6}}{9 \cos \left (d x +c \right )^{9}}+\frac {\sin \left (d x +c \right )^{6}}{21 \cos \left (d x +c \right )^{7}}+\frac {\sin \left (d x +c \right )^{6}}{105 \cos \left (d x +c \right )^{5}}-\frac {\sin \left (d x +c \right )^{6}}{315 \cos \left (d x +c \right )^{3}}+\frac {\sin \left (d x +c \right )^{6}}{105 \cos \left (d x +c \right )}+\frac {\left (\frac {8}{3}+\sin \left (d x +c \right )^{4}+\frac {4 \sin \left (d x +c \right )^{2}}{3}\right ) \cos \left (d x +c \right )}{105}\right )}{d}\) | \(489\) |
| default | \(\frac {a^{5} \left (-\left (-\frac {\sec \left (d x +c \right )^{3}}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )+\frac {a^{4} b}{\cos \left (d x +c \right )^{5}}+10 a^{3} b^{2} \left (\frac {\sin \left (d x +c \right )^{3}}{6 \cos \left (d x +c \right )^{6}}+\frac {\sin \left (d x +c \right )^{3}}{8 \cos \left (d x +c \right )^{4}}+\frac {\sin \left (d x +c \right )^{3}}{16 \cos \left (d x +c \right )^{2}}+\frac {\sin \left (d x +c \right )}{16}-\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{16}\right )+10 a^{2} b^{3} \left (\frac {\sin \left (d x +c \right )^{4}}{7 \cos \left (d x +c \right )^{7}}+\frac {3 \sin \left (d x +c \right )^{4}}{35 \cos \left (d x +c \right )^{5}}+\frac {\sin \left (d x +c \right )^{4}}{35 \cos \left (d x +c \right )^{3}}-\frac {\sin \left (d x +c \right )^{4}}{35 \cos \left (d x +c \right )}-\frac {\left (2+\sin \left (d x +c \right )^{2}\right ) \cos \left (d x +c \right )}{35}\right )+5 b^{4} a \left (\frac {\sin \left (d x +c \right )^{5}}{8 \cos \left (d x +c \right )^{8}}+\frac {\sin \left (d x +c \right )^{5}}{16 \cos \left (d x +c \right )^{6}}+\frac {\sin \left (d x +c \right )^{5}}{64 \cos \left (d x +c \right )^{4}}-\frac {\sin \left (d x +c \right )^{5}}{128 \cos \left (d x +c \right )^{2}}-\frac {\sin \left (d x +c \right )^{3}}{128}-\frac {3 \sin \left (d x +c \right )}{128}+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{128}\right )+b^{5} \left (\frac {\sin \left (d x +c \right )^{6}}{9 \cos \left (d x +c \right )^{9}}+\frac {\sin \left (d x +c \right )^{6}}{21 \cos \left (d x +c \right )^{7}}+\frac {\sin \left (d x +c \right )^{6}}{105 \cos \left (d x +c \right )^{5}}-\frac {\sin \left (d x +c \right )^{6}}{315 \cos \left (d x +c \right )^{3}}+\frac {\sin \left (d x +c \right )^{6}}{105 \cos \left (d x +c \right )}+\frac {\left (\frac {8}{3}+\sin \left (d x +c \right )^{4}+\frac {4 \sin \left (d x +c \right )^{2}}{3}\right ) \cos \left (d x +c \right )}{105}\right )}{d}\) | \(489\) |
| parallelrisch | \(\frac {-544320 \left (a^{4}-\frac {5}{3} a^{2} b^{2}+\frac {5}{16} b^{4}\right ) \left (\frac {\cos \left (9 d x +9 c \right )}{36}+\frac {\cos \left (7 d x +7 c \right )}{4}+\cos \left (5 d x +5 c \right )+\frac {7 \cos \left (3 d x +3 c \right )}{3}+\frac {7 \cos \left (d x +c \right )}{2}\right ) a \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )+544320 \left (a^{4}-\frac {5}{3} a^{2} b^{2}+\frac {5}{16} b^{4}\right ) \left (\frac {\cos \left (9 d x +9 c \right )}{36}+\frac {\cos \left (7 d x +7 c \right )}{4}+\cos \left (5 d x +5 c \right )+\frac {7 \cos \left (3 d x +3 c \right )}{3}+\frac {7 \cos \left (d x +c \right )}{2}\right ) a \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )+\left (3386880 a^{4} b -1935360 a^{2} b^{3}+86016 b^{5}\right ) \cos \left (3 d x +3 c \right )+\left (1451520 a^{4} b -829440 a^{2} b^{3}+36864 b^{5}\right ) \cos \left (5 d x +5 c \right )+\left (362880 a^{4} b -207360 a^{2} b^{3}+9216 b^{5}\right ) \cos \left (7 d x +7 c \right )+\left (40320 a^{4} b -23040 a^{2} b^{3}+1024 b^{5}\right ) \cos \left (9 d x +9 c \right )+\left (5160960 a^{4} b -2949120 a^{2} b^{3}-442368 b^{5}\right ) \cos \left (2 d x +2 c \right )+1290240 \left (a^{4}-2 a^{2} b^{2}+\frac {1}{5} b^{4}\right ) b \cos \left (4 d x +4 c \right )+\left (826560 a^{5}+2923200 a^{3} b^{2}+1064700 b^{4} a \right ) \sin \left (2 d x +2 c \right )+\left (745920 a^{5}+907200 a^{3} b^{2}-976500 b^{4} a \right ) \sin \left (4 d x +4 c \right )+\left (262080 a^{5}-436800 a^{3} b^{2}+81900 b^{4} a \right ) \sin \left (6 d x +6 c \right )+\left (30240 a^{5}-50400 a^{3} b^{2}+9450 b^{4} a \right ) \sin \left (8 d x +8 c \right )+5080320 \left (\left (a^{4}-\frac {4}{7} a^{2} b^{2}+\frac {8}{315} b^{4}\right ) \cos \left (d x +c \right )+\frac {16 a^{4}}{21}-\frac {32 a^{2} b^{2}}{441}+\frac {1744 b^{4}}{19845}\right ) b}{40320 d \left (\cos \left (9 d x +9 c \right )+9 \cos \left (7 d x +7 c \right )+36 \cos \left (5 d x +5 c \right )+84 \cos \left (3 d x +3 c \right )+126 \cos \left (d x +c \right )\right )}\) | \(571\) |
| risch | \(\frac {{\mathrm e}^{i \left (d x +c \right )} \left (25200 i a^{3} b^{2} {\mathrm e}^{16 i \left (d x +c \right )}-4725 i a \,b^{4} {\mathrm e}^{16 i \left (d x +c \right )}+218400 i a^{3} b^{2} {\mathrm e}^{14 i \left (d x +c \right )}-40950 i a \,b^{4} {\mathrm e}^{14 i \left (d x +c \right )}+446464 b^{5} {\mathrm e}^{8 i \left (d x +c \right )}+129024 b^{5} {\mathrm e}^{4 i \left (d x +c \right )}+1461600 i a^{3} b^{2} {\mathrm e}^{6 i \left (d x +c \right )}+532350 i a \,b^{4} {\mathrm e}^{6 i \left (d x +c \right )}+453600 i a^{3} b^{2} {\mathrm e}^{4 i \left (d x +c \right )}-488250 i a \,b^{4} {\mathrm e}^{4 i \left (d x +c \right )}-1461600 i a^{3} b^{2} {\mathrm e}^{10 i \left (d x +c \right )}-532350 i a \,b^{4} {\mathrm e}^{10 i \left (d x +c \right )}-218400 i a^{3} b^{2} {\mathrm e}^{2 i \left (d x +c \right )}+40950 i a \,b^{4} {\mathrm e}^{2 i \left (d x +c \right )}-372960 i a^{5} {\mathrm e}^{12 i \left (d x +c \right )}-1290240 a^{2} b^{3} {\mathrm e}^{12 i \left (d x +c \right )}+645120 a^{4} b \,{\mathrm e}^{12 i \left (d x +c \right )}+131040 i a^{5} {\mathrm e}^{2 i \left (d x +c \right )}-413280 i a^{5} {\mathrm e}^{10 i \left (d x +c \right )}+413280 i a^{5} {\mathrm e}^{6 i \left (d x +c \right )}+372960 i a^{5} {\mathrm e}^{4 i \left (d x +c \right )}-368640 a^{2} b^{3} {\mathrm e}^{8 i \left (d x +c \right )}-453600 i a^{3} b^{2} {\mathrm e}^{12 i \left (d x +c \right )}+488250 i a \,b^{4} {\mathrm e}^{12 i \left (d x +c \right )}+129024 b^{5} {\mathrm e}^{12 i \left (d x +c \right )}+2580480 a^{4} b \,{\mathrm e}^{6 i \left (d x +c \right )}-1474560 a^{2} b^{3} {\mathrm e}^{6 i \left (d x +c \right )}+645120 a^{4} b \,{\mathrm e}^{4 i \left (d x +c \right )}-1290240 a^{2} b^{3} {\mathrm e}^{4 i \left (d x +c \right )}+2580480 a^{4} b \,{\mathrm e}^{10 i \left (d x +c \right )}-1474560 a^{2} b^{3} {\mathrm e}^{10 i \left (d x +c \right )}+3870720 a^{4} b \,{\mathrm e}^{8 i \left (d x +c \right )}+4725 i a \,b^{4}-25200 i a^{3} b^{2}+15120 i a^{5}-221184 b^{5} {\mathrm e}^{6 i \left (d x +c \right )}-221184 b^{5} {\mathrm e}^{10 i \left (d x +c \right )}-15120 i a^{5} {\mathrm e}^{16 i \left (d x +c \right )}-131040 i a^{5} {\mathrm e}^{14 i \left (d x +c \right )}\right )}{20160 d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{9}}-\frac {3 a^{5} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{8 d}+\frac {5 a^{3} b^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{8 d}-\frac {15 b^{4} a \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{128 d}+\frac {3 a^{5} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{8 d}-\frac {5 a^{3} b^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{8 d}+\frac {15 b^{4} a \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{128 d}\) | \(833\) |
Input:
int(sec(d*x+c)^10*(a*cos(d*x+c)+b*sin(d*x+c))^5,x,method=_RETURNVERBOSE)
Output:
a^5/d*(-(-1/4*sec(d*x+c)^3-3/8*sec(d*x+c))*tan(d*x+c)+3/8*ln(sec(d*x+c)+ta n(d*x+c)))+b^5/d*(1/9*sec(d*x+c)^9-2/7*sec(d*x+c)^7+1/5*sec(d*x+c)^5)+a^4* b*sec(d*x+c)^5/d+10*a^3*b^2/d*(1/6*sin(d*x+c)^3/cos(d*x+c)^6+1/8*sin(d*x+c )^3/cos(d*x+c)^4+1/16*sin(d*x+c)^3/cos(d*x+c)^2+1/16*sin(d*x+c)-1/16*ln(se c(d*x+c)+tan(d*x+c)))+10*a^2*b^3/d*(1/7*sec(d*x+c)^7-1/5*sec(d*x+c)^5)+5*b ^4*a/d*(1/8*sin(d*x+c)^5/cos(d*x+c)^8+1/16*sin(d*x+c)^5/cos(d*x+c)^6+1/64* sin(d*x+c)^5/cos(d*x+c)^4-1/128*sin(d*x+c)^5/cos(d*x+c)^2-1/128*sin(d*x+c) ^3-3/128*sin(d*x+c)+3/128*ln(sec(d*x+c)+tan(d*x+c)))
Time = 0.10 (sec) , antiderivative size = 257, normalized size of antiderivative = 0.66 \[ \int \sec ^{10}(c+d x) (a \cos (c+d x)+b \sin (c+d x))^5 \, dx=\frac {315 \, {\left (48 \, a^{5} - 80 \, a^{3} b^{2} + 15 \, a b^{4}\right )} \cos \left (d x + c\right )^{9} \log \left (\sin \left (d x + c\right ) + 1\right ) - 315 \, {\left (48 \, a^{5} - 80 \, a^{3} b^{2} + 15 \, a b^{4}\right )} \cos \left (d x + c\right )^{9} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 8960 \, b^{5} + 16128 \, {\left (5 \, a^{4} b - 10 \, a^{2} b^{3} + b^{5}\right )} \cos \left (d x + c\right )^{4} + 23040 \, {\left (5 \, a^{2} b^{3} - b^{5}\right )} \cos \left (d x + c\right )^{2} + 210 \, {\left (3 \, {\left (48 \, a^{5} - 80 \, a^{3} b^{2} + 15 \, a b^{4}\right )} \cos \left (d x + c\right )^{7} + 240 \, a b^{4} \cos \left (d x + c\right ) + 2 \, {\left (48 \, a^{5} - 80 \, a^{3} b^{2} + 15 \, a b^{4}\right )} \cos \left (d x + c\right )^{5} + 40 \, {\left (16 \, a^{3} b^{2} - 9 \, a b^{4}\right )} \cos \left (d x + c\right )^{3}\right )} \sin \left (d x + c\right )}{80640 \, d \cos \left (d x + c\right )^{9}} \] Input:
integrate(sec(d*x+c)^10*(a*cos(d*x+c)+b*sin(d*x+c))^5,x, algorithm="fricas ")
Output:
1/80640*(315*(48*a^5 - 80*a^3*b^2 + 15*a*b^4)*cos(d*x + c)^9*log(sin(d*x + c) + 1) - 315*(48*a^5 - 80*a^3*b^2 + 15*a*b^4)*cos(d*x + c)^9*log(-sin(d* x + c) + 1) + 8960*b^5 + 16128*(5*a^4*b - 10*a^2*b^3 + b^5)*cos(d*x + c)^4 + 23040*(5*a^2*b^3 - b^5)*cos(d*x + c)^2 + 210*(3*(48*a^5 - 80*a^3*b^2 + 15*a*b^4)*cos(d*x + c)^7 + 240*a*b^4*cos(d*x + c) + 2*(48*a^5 - 80*a^3*b^2 + 15*a*b^4)*cos(d*x + c)^5 + 40*(16*a^3*b^2 - 9*a*b^4)*cos(d*x + c)^3)*si n(d*x + c))/(d*cos(d*x + c)^9)
Timed out. \[ \int \sec ^{10}(c+d x) (a \cos (c+d x)+b \sin (c+d x))^5 \, dx=\text {Timed out} \] Input:
integrate(sec(d*x+c)**10*(a*cos(d*x+c)+b*sin(d*x+c))**5,x)
Output:
Timed out
Time = 0.04 (sec) , antiderivative size = 360, normalized size of antiderivative = 0.92 \[ \int \sec ^{10}(c+d x) (a \cos (c+d x)+b \sin (c+d x))^5 \, dx=-\frac {1575 \, a b^{4} {\left (\frac {2 \, {\left (3 \, \sin \left (d x + c\right )^{7} - 11 \, \sin \left (d x + c\right )^{5} - 11 \, \sin \left (d x + c\right )^{3} + 3 \, \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{8} - 4 \, \sin \left (d x + c\right )^{6} + 6 \, \sin \left (d x + c\right )^{4} - 4 \, \sin \left (d x + c\right )^{2} + 1} - 3 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 8400 \, a^{3} b^{2} {\left (\frac {2 \, {\left (3 \, \sin \left (d x + c\right )^{5} - 8 \, \sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{6} - 3 \, \sin \left (d x + c\right )^{4} + 3 \, \sin \left (d x + c\right )^{2} - 1} - 3 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 5040 \, a^{5} {\left (\frac {2 \, {\left (3 \, \sin \left (d x + c\right )^{3} - 5 \, \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{4} - 2 \, \sin \left (d x + c\right )^{2} + 1} - 3 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - \frac {80640 \, a^{4} b}{\cos \left (d x + c\right )^{5}} + \frac {23040 \, {\left (7 \, \cos \left (d x + c\right )^{2} - 5\right )} a^{2} b^{3}}{\cos \left (d x + c\right )^{7}} - \frac {256 \, {\left (63 \, \cos \left (d x + c\right )^{4} - 90 \, \cos \left (d x + c\right )^{2} + 35\right )} b^{5}}{\cos \left (d x + c\right )^{9}}}{80640 \, d} \] Input:
integrate(sec(d*x+c)^10*(a*cos(d*x+c)+b*sin(d*x+c))^5,x, algorithm="maxima ")
Output:
-1/80640*(1575*a*b^4*(2*(3*sin(d*x + c)^7 - 11*sin(d*x + c)^5 - 11*sin(d*x + c)^3 + 3*sin(d*x + c))/(sin(d*x + c)^8 - 4*sin(d*x + c)^6 + 6*sin(d*x + c)^4 - 4*sin(d*x + c)^2 + 1) - 3*log(sin(d*x + c) + 1) + 3*log(sin(d*x + c) - 1)) - 8400*a^3*b^2*(2*(3*sin(d*x + c)^5 - 8*sin(d*x + c)^3 - 3*sin(d* x + c))/(sin(d*x + c)^6 - 3*sin(d*x + c)^4 + 3*sin(d*x + c)^2 - 1) - 3*log (sin(d*x + c) + 1) + 3*log(sin(d*x + c) - 1)) + 5040*a^5*(2*(3*sin(d*x + c )^3 - 5*sin(d*x + c))/(sin(d*x + c)^4 - 2*sin(d*x + c)^2 + 1) - 3*log(sin( d*x + c) + 1) + 3*log(sin(d*x + c) - 1)) - 80640*a^4*b/cos(d*x + c)^5 + 23 040*(7*cos(d*x + c)^2 - 5)*a^2*b^3/cos(d*x + c)^7 - 256*(63*cos(d*x + c)^4 - 90*cos(d*x + c)^2 + 35)*b^5/cos(d*x + c)^9)/d
Leaf count of result is larger than twice the leaf count of optimal. 888 vs. \(2 (359) = 718\).
Time = 0.31 (sec) , antiderivative size = 888, normalized size of antiderivative = 2.27 \[ \int \sec ^{10}(c+d x) (a \cos (c+d x)+b \sin (c+d x))^5 \, dx=\text {Too large to display} \] Input:
integrate(sec(d*x+c)^10*(a*cos(d*x+c)+b*sin(d*x+c))^5,x, algorithm="giac")
Output:
1/40320*(315*(48*a^5 - 80*a^3*b^2 + 15*a*b^4)*log(abs(tan(1/2*d*x + 1/2*c) + 1)) - 315*(48*a^5 - 80*a^3*b^2 + 15*a*b^4)*log(abs(tan(1/2*d*x + 1/2*c) - 1)) + 2*(25200*a^5*tan(1/2*d*x + 1/2*c)^17 + 25200*a^3*b^2*tan(1/2*d*x + 1/2*c)^17 - 4725*a*b^4*tan(1/2*d*x + 1/2*c)^17 - 201600*a^4*b*tan(1/2*d* x + 1/2*c)^16 - 110880*a^5*tan(1/2*d*x + 1/2*c)^15 + 319200*a^3*b^2*tan(1/ 2*d*x + 1/2*c)^15 + 40950*a*b^4*tan(1/2*d*x + 1/2*c)^15 + 806400*a^4*b*tan (1/2*d*x + 1/2*c)^14 - 806400*a^2*b^3*tan(1/2*d*x + 1/2*c)^14 + 191520*a^5 *tan(1/2*d*x + 1/2*c)^13 - 453600*a^3*b^2*tan(1/2*d*x + 1/2*c)^13 + 488250 *a*b^4*tan(1/2*d*x + 1/2*c)^13 - 1612800*a^4*b*tan(1/2*d*x + 1/2*c)^12 + 8 06400*a^2*b^3*tan(1/2*d*x + 1/2*c)^12 - 215040*b^5*tan(1/2*d*x + 1/2*c)^12 - 151200*a^5*tan(1/2*d*x + 1/2*c)^11 - 151200*a^3*b^2*tan(1/2*d*x + 1/2*c )^11 + 532350*a*b^4*tan(1/2*d*x + 1/2*c)^11 + 2419200*a^4*b*tan(1/2*d*x + 1/2*c)^10 - 806400*a^2*b^3*tan(1/2*d*x + 1/2*c)^10 - 322560*b^5*tan(1/2*d* x + 1/2*c)^10 - 2661120*a^4*b*tan(1/2*d*x + 1/2*c)^8 + 2096640*a^2*b^3*tan (1/2*d*x + 1/2*c)^8 - 451584*b^5*tan(1/2*d*x + 1/2*c)^8 + 151200*a^5*tan(1 /2*d*x + 1/2*c)^7 + 151200*a^3*b^2*tan(1/2*d*x + 1/2*c)^7 - 532350*a*b^4*t an(1/2*d*x + 1/2*c)^7 + 1774080*a^4*b*tan(1/2*d*x + 1/2*c)^6 - 1128960*a^2 *b^3*tan(1/2*d*x + 1/2*c)^6 - 129024*b^5*tan(1/2*d*x + 1/2*c)^6 - 191520*a ^5*tan(1/2*d*x + 1/2*c)^5 + 453600*a^3*b^2*tan(1/2*d*x + 1/2*c)^5 - 488250 *a*b^4*tan(1/2*d*x + 1/2*c)^5 - 645120*a^4*b*tan(1/2*d*x + 1/2*c)^4 + 2...
Time = 20.29 (sec) , antiderivative size = 675, normalized size of antiderivative = 1.73 \[ \int \sec ^{10}(c+d x) (a \cos (c+d x)+b \sin (c+d x))^5 \, dx =\text {Too large to display} \] Input:
int((a*cos(c + d*x) + b*sin(c + d*x))^5/cos(c + d*x)^10,x)
Output:
(atanh(tan(c/2 + (d*x)/2))*((15*a*b^4)/64 + (3*a^5)/4 - (5*a^3*b^2)/4))/d - (tan(c/2 + (d*x)/2)*((5*a^5)/4 - (15*a*b^4)/64 + (5*a^3*b^2)/4) - tan(c/ 2 + (d*x)/2)^14*(40*a^4*b - 40*a^2*b^3) - tan(c/2 + (d*x)/2)^17*((5*a^5)/4 - (15*a*b^4)/64 + (5*a^3*b^2)/4) + tan(c/2 + (d*x)/2)^3*((65*a*b^4)/32 - (11*a^5)/2 + (95*a^3*b^2)/6) - tan(c/2 + (d*x)/2)^15*((65*a*b^4)/32 - (11* a^5)/2 + (95*a^3*b^2)/6) + tan(c/2 + (d*x)/2)^5*((775*a*b^4)/32 + (19*a^5) /2 - (45*a^3*b^2)/2) - tan(c/2 + (d*x)/2)^13*((775*a*b^4)/32 + (19*a^5)/2 - (45*a^3*b^2)/2) - tan(c/2 + (d*x)/2)^7*((15*a^5)/2 - (845*a*b^4)/32 + (1 5*a^3*b^2)/2) + tan(c/2 + (d*x)/2)^11*((15*a^5)/2 - (845*a*b^4)/32 + (15*a ^3*b^2)/2) - tan(c/2 + (d*x)/2)^2*(8*a^4*b + (16*b^5)/35 - (72*a^2*b^3)/7) + tan(c/2 + (d*x)/2)^4*(32*a^4*b + (64*b^5)/35 - (8*a^2*b^3)/7) + tan(c/2 + (d*x)/2)^12*(80*a^4*b + (32*b^5)/3 - 40*a^2*b^3) + tan(c/2 + (d*x)/2)^1 0*(16*b^5 - 120*a^4*b + 40*a^2*b^3) + tan(c/2 + (d*x)/2)^6*((32*b^5)/5 - 8 8*a^4*b + 56*a^2*b^3) + tan(c/2 + (d*x)/2)^8*(132*a^4*b + (112*b^5)/5 - 10 4*a^2*b^3) + 2*a^4*b + (16*b^5)/315 - (8*a^2*b^3)/7 + 10*a^4*b*tan(c/2 + ( d*x)/2)^16)/(d*(9*tan(c/2 + (d*x)/2)^2 - 36*tan(c/2 + (d*x)/2)^4 + 84*tan( c/2 + (d*x)/2)^6 - 126*tan(c/2 + (d*x)/2)^8 + 126*tan(c/2 + (d*x)/2)^10 - 84*tan(c/2 + (d*x)/2)^12 + 36*tan(c/2 + (d*x)/2)^14 - 9*tan(c/2 + (d*x)/2) ^16 + tan(c/2 + (d*x)/2)^18 - 1))
Time = 0.17 (sec) , antiderivative size = 1602, normalized size of antiderivative = 4.10 \[ \int \sec ^{10}(c+d x) (a \cos (c+d x)+b \sin (c+d x))^5 \, dx =\text {Too large to display} \] Input:
int(sec(d*x+c)^10*(a*cos(d*x+c)+b*sin(d*x+c))^5,x)
Output:
( - 15120*cos(c + d*x)*log(tan((c + d*x)/2) - 1)*sin(c + d*x)**8*a**5 + 25 200*cos(c + d*x)*log(tan((c + d*x)/2) - 1)*sin(c + d*x)**8*a**3*b**2 - 472 5*cos(c + d*x)*log(tan((c + d*x)/2) - 1)*sin(c + d*x)**8*a*b**4 + 60480*co s(c + d*x)*log(tan((c + d*x)/2) - 1)*sin(c + d*x)**6*a**5 - 100800*cos(c + d*x)*log(tan((c + d*x)/2) - 1)*sin(c + d*x)**6*a**3*b**2 + 18900*cos(c + d*x)*log(tan((c + d*x)/2) - 1)*sin(c + d*x)**6*a*b**4 - 90720*cos(c + d*x) *log(tan((c + d*x)/2) - 1)*sin(c + d*x)**4*a**5 + 151200*cos(c + d*x)*log( tan((c + d*x)/2) - 1)*sin(c + d*x)**4*a**3*b**2 - 28350*cos(c + d*x)*log(t an((c + d*x)/2) - 1)*sin(c + d*x)**4*a*b**4 + 60480*cos(c + d*x)*log(tan(( c + d*x)/2) - 1)*sin(c + d*x)**2*a**5 - 100800*cos(c + d*x)*log(tan((c + d *x)/2) - 1)*sin(c + d*x)**2*a**3*b**2 + 18900*cos(c + d*x)*log(tan((c + d* x)/2) - 1)*sin(c + d*x)**2*a*b**4 - 15120*cos(c + d*x)*log(tan((c + d*x)/2 ) - 1)*a**5 + 25200*cos(c + d*x)*log(tan((c + d*x)/2) - 1)*a**3*b**2 - 472 5*cos(c + d*x)*log(tan((c + d*x)/2) - 1)*a*b**4 + 15120*cos(c + d*x)*log(t an((c + d*x)/2) + 1)*sin(c + d*x)**8*a**5 - 25200*cos(c + d*x)*log(tan((c + d*x)/2) + 1)*sin(c + d*x)**8*a**3*b**2 + 4725*cos(c + d*x)*log(tan((c + d*x)/2) + 1)*sin(c + d*x)**8*a*b**4 - 60480*cos(c + d*x)*log(tan((c + d*x) /2) + 1)*sin(c + d*x)**6*a**5 + 100800*cos(c + d*x)*log(tan((c + d*x)/2) + 1)*sin(c + d*x)**6*a**3*b**2 - 18900*cos(c + d*x)*log(tan((c + d*x)/2) + 1)*sin(c + d*x)**6*a*b**4 + 90720*cos(c + d*x)*log(tan((c + d*x)/2) + 1...