Integrand size = 28, antiderivative size = 330 \[ \int \sec ^9(c+d x) (a \cos (c+d x)+b \sin (c+d x))^4 \, dx=\frac {3 a^4 \text {arctanh}(\sin (c+d x))}{8 d}-\frac {3 a^2 b^2 \text {arctanh}(\sin (c+d x))}{8 d}+\frac {3 b^4 \text {arctanh}(\sin (c+d x))}{128 d}+\frac {4 a^3 b \sec ^5(c+d x)}{5 d}-\frac {4 a b^3 \sec ^5(c+d x)}{5 d}+\frac {4 a b^3 \sec ^7(c+d x)}{7 d}+\frac {3 a^4 \sec (c+d x) \tan (c+d x)}{8 d}-\frac {3 a^2 b^2 \sec (c+d x) \tan (c+d x)}{8 d}+\frac {3 b^4 \sec (c+d x) \tan (c+d x)}{128 d}+\frac {a^4 \sec ^3(c+d x) \tan (c+d x)}{4 d}-\frac {a^2 b^2 \sec ^3(c+d x) \tan (c+d x)}{4 d}+\frac {b^4 \sec ^3(c+d x) \tan (c+d x)}{64 d}+\frac {a^2 b^2 \sec ^5(c+d x) \tan (c+d x)}{d}-\frac {b^4 \sec ^5(c+d x) \tan (c+d x)}{16 d}+\frac {b^4 \sec ^5(c+d x) \tan ^3(c+d x)}{8 d} \]
3/8*a^4*arctanh(sin(d*x+c))/d-3/8*a^2*b^2*arctanh(sin(d*x+c))/d+3/128*b^4* arctanh(sin(d*x+c))/d+4/5*a^3*b*sec(d*x+c)^5/d-4/5*a*b^3*sec(d*x+c)^5/d+4/ 7*a*b^3*sec(d*x+c)^7/d+3/8*a^4*sec(d*x+c)*tan(d*x+c)/d-3/8*a^2*b^2*sec(d*x +c)*tan(d*x+c)/d+3/128*b^4*sec(d*x+c)*tan(d*x+c)/d+1/4*a^4*sec(d*x+c)^3*ta n(d*x+c)/d-1/4*a^2*b^2*sec(d*x+c)^3*tan(d*x+c)/d+1/64*b^4*sec(d*x+c)^3*tan (d*x+c)/d+a^2*b^2*sec(d*x+c)^5*tan(d*x+c)/d-1/16*b^4*sec(d*x+c)^5*tan(d*x+ c)/d+1/8*b^4*sec(d*x+c)^5*tan(d*x+c)^3/d
Leaf count is larger than twice the leaf count of optimal. \(1732\) vs. \(2(330)=660\).
Time = 7.98 (sec) , antiderivative size = 1732, normalized size of antiderivative = 5.25 \[ \int \sec ^9(c+d x) (a \cos (c+d x)+b \sin (c+d x))^4 \, dx =\text {Too large to display} \]
(a*b*(42*a^2 - 17*b^2)*Cos[c + d*x]^4*(a + b*Tan[c + d*x])^4)/(140*d*(a*Co s[c + d*x] + b*Sin[c + d*x])^4) - (3*(16*a^4 - 16*a^2*b^2 + b^4)*Cos[c + d *x]^4*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]]*(a + b*Tan[c + d*x])^4)/(12 8*d*(a*Cos[c + d*x] + b*Sin[c + d*x])^4) + (3*(16*a^4 - 16*a^2*b^2 + b^4)* Cos[c + d*x]^4*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]*(a + b*Tan[c + d*x ])^4)/(128*d*(a*Cos[c + d*x] + b*Sin[c + d*x])^4) + (b^4*Cos[c + d*x]^4*(a + b*Tan[c + d*x])^4)/(128*d*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^8*(a*Co s[c + d*x] + b*Sin[c + d*x])^4) + ((56*a^2*b^2 + 16*a*b^3 - 7*b^4)*Cos[c + d*x]^4*(a + b*Tan[c + d*x])^4)/(448*d*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2 ])^6*(a*Cos[c + d*x] + b*Sin[c + d*x])^4) + ((560*a^4 + 896*a^3*b - 256*a* b^3 - 35*b^4)*Cos[c + d*x]^4*(a + b*Tan[c + d*x])^4)/(8960*d*(Cos[(c + d*x )/2] - Sin[(c + d*x)/2])^4*(a*Cos[c + d*x] + b*Sin[c + d*x])^4) + ((1680*a ^4 + 1344*a^3*b - 1680*a^2*b^2 - 544*a*b^3 + 105*b^4)*Cos[c + d*x]^4*(a + b*Tan[c + d*x])^4)/(8960*d*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^2*(a*Cos[ c + d*x] + b*Sin[c + d*x])^4) + (a*b^3*Cos[c + d*x]^4*Sin[(c + d*x)/2]*(a + b*Tan[c + d*x])^4)/(14*d*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^7*(a*Cos[ c + d*x] + b*Sin[c + d*x])^4) - (b^4*Cos[c + d*x]^4*(a + b*Tan[c + d*x])^4 )/(128*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^8*(a*Cos[c + d*x] + b*Sin[c + d*x])^4) - (a*b^3*Cos[c + d*x]^4*Sin[(c + d*x)/2]*(a + b*Tan[c + d*x])^ 4)/(14*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^7*(a*Cos[c + d*x] + b*Si...
Time = 0.58 (sec) , antiderivative size = 330, 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.
| \(\displaystyle \int \sec ^9(c+d x) (a \cos (c+d x)+b \sin (c+d x))^4 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {(a \cos (c+d x)+b \sin (c+d x))^4}{\cos (c+d x)^9}dx\) |
| \(\Big \downarrow \) 3569 |
| \(\displaystyle \int \left (a^4 \sec ^5(c+d x)+4 a^3 b \tan (c+d x) \sec ^5(c+d x)+6 a^2 b^2 \tan ^2(c+d x) \sec ^5(c+d x)+4 a b^3 \tan ^3(c+d x) \sec ^5(c+d x)+b^4 \tan ^4(c+d x) \sec ^5(c+d x)\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {3 a^4 \text {arctanh}(\sin (c+d x))}{8 d}+\frac {a^4 \tan (c+d x) \sec ^3(c+d x)}{4 d}+\frac {3 a^4 \tan (c+d x) \sec (c+d x)}{8 d}+\frac {4 a^3 b \sec ^5(c+d x)}{5 d}-\frac {3 a^2 b^2 \text {arctanh}(\sin (c+d x))}{8 d}+\frac {a^2 b^2 \tan (c+d x) \sec ^5(c+d x)}{d}-\frac {a^2 b^2 \tan (c+d x) \sec ^3(c+d x)}{4 d}-\frac {3 a^2 b^2 \tan (c+d x) \sec (c+d x)}{8 d}+\frac {4 a b^3 \sec ^7(c+d x)}{7 d}-\frac {4 a b^3 \sec ^5(c+d x)}{5 d}+\frac {3 b^4 \text {arctanh}(\sin (c+d x))}{128 d}+\frac {b^4 \tan ^3(c+d x) \sec ^5(c+d x)}{8 d}-\frac {b^4 \tan (c+d x) \sec ^5(c+d x)}{16 d}+\frac {b^4 \tan (c+d x) \sec ^3(c+d x)}{64 d}+\frac {3 b^4 \tan (c+d x) \sec (c+d x)}{128 d}\) |
(3*a^4*ArcTanh[Sin[c + d*x]])/(8*d) - (3*a^2*b^2*ArcTanh[Sin[c + d*x]])/(8 *d) + (3*b^4*ArcTanh[Sin[c + d*x]])/(128*d) + (4*a^3*b*Sec[c + d*x]^5)/(5* d) - (4*a*b^3*Sec[c + d*x]^5)/(5*d) + (4*a*b^3*Sec[c + d*x]^7)/(7*d) + (3* a^4*Sec[c + d*x]*Tan[c + d*x])/(8*d) - (3*a^2*b^2*Sec[c + d*x]*Tan[c + d*x ])/(8*d) + (3*b^4*Sec[c + d*x]*Tan[c + d*x])/(128*d) + (a^4*Sec[c + d*x]^3 *Tan[c + d*x])/(4*d) - (a^2*b^2*Sec[c + d*x]^3*Tan[c + d*x])/(4*d) + (b^4* Sec[c + d*x]^3*Tan[c + d*x])/(64*d) + (a^2*b^2*Sec[c + d*x]^5*Tan[c + d*x] )/d - (b^4*Sec[c + d*x]^5*Tan[c + d*x])/(16*d) + (b^4*Sec[c + d*x]^5*Tan[c + d*x]^3)/(8*d)
3.1.88.3.1 Defintions of rubi rules used
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 = 1.95 (sec) , antiderivative size = 304, normalized size of antiderivative = 0.92
| method | result | size |
| parts | \(\frac {a^{4} \left (-\left (-\frac {\sec \left (d x +c \right )^{3}}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )}{d}+\frac {b^{4} \left (\frac {\sin \left (d x +c \right )^{5}}{8 \cos \left (d x +c \right )^{8}}+\frac {\sin \left (d x +c \right )^{5}}{16 \cos \left (d x +c \right )^{6}}+\frac {\sin \left (d x +c \right )^{5}}{64 \cos \left (d x +c \right )^{4}}-\frac {\sin \left (d x +c \right )^{5}}{128 \cos \left (d x +c \right )^{2}}-\frac {\sin \left (d x +c \right )^{3}}{128}-\frac {3 \sin \left (d x +c \right )}{128}+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{128}\right )}{d}+\frac {4 a \,b^{3} \left (\frac {\sec \left (d x +c \right )^{7}}{7}-\frac {\sec \left (d x +c \right )^{5}}{5}\right )}{d}+\frac {4 a^{3} b \sec \left (d x +c \right )^{5}}{5 d}+\frac {6 a^{2} b^{2} \left (\frac {\sin \left (d x +c \right )^{3}}{6 \cos \left (d x +c \right )^{6}}+\frac {\sin \left (d x +c \right )^{3}}{8 \cos \left (d x +c \right )^{4}}+\frac {\sin \left (d x +c \right )^{3}}{16 \cos \left (d x +c \right )^{2}}+\frac {\sin \left (d x +c \right )}{16}-\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{16}\right )}{d}\) | \(304\) |
| derivativedivides | \(\frac {a^{4} \left (-\left (-\frac {\sec \left (d x +c \right )^{3}}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )+\frac {4 a^{3} b}{5 \cos \left (d x +c \right )^{5}}+6 a^{2} b^{2} \left (\frac {\sin \left (d x +c \right )^{3}}{6 \cos \left (d x +c \right )^{6}}+\frac {\sin \left (d x +c \right )^{3}}{8 \cos \left (d x +c \right )^{4}}+\frac {\sin \left (d x +c \right )^{3}}{16 \cos \left (d x +c \right )^{2}}+\frac {\sin \left (d x +c \right )}{16}-\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{16}\right )+4 a \,b^{3} \left (\frac {\sin \left (d x +c \right )^{4}}{7 \cos \left (d x +c \right )^{7}}+\frac {3 \sin \left (d x +c \right )^{4}}{35 \cos \left (d x +c \right )^{5}}+\frac {\sin \left (d x +c \right )^{4}}{35 \cos \left (d x +c \right )^{3}}-\frac {\sin \left (d x +c \right )^{4}}{35 \cos \left (d x +c \right )}-\frac {\left (2+\sin \left (d x +c \right )^{2}\right ) \cos \left (d x +c \right )}{35}\right )+b^{4} \left (\frac {\sin \left (d x +c \right )^{5}}{8 \cos \left (d x +c \right )^{8}}+\frac {\sin \left (d x +c \right )^{5}}{16 \cos \left (d x +c \right )^{6}}+\frac {\sin \left (d x +c \right )^{5}}{64 \cos \left (d x +c \right )^{4}}-\frac {\sin \left (d x +c \right )^{5}}{128 \cos \left (d x +c \right )^{2}}-\frac {\sin \left (d x +c \right )^{3}}{128}-\frac {3 \sin \left (d x +c \right )}{128}+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{128}\right )}{d}\) | \(363\) |
| default | \(\frac {a^{4} \left (-\left (-\frac {\sec \left (d x +c \right )^{3}}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )+\frac {4 a^{3} b}{5 \cos \left (d x +c \right )^{5}}+6 a^{2} b^{2} \left (\frac {\sin \left (d x +c \right )^{3}}{6 \cos \left (d x +c \right )^{6}}+\frac {\sin \left (d x +c \right )^{3}}{8 \cos \left (d x +c \right )^{4}}+\frac {\sin \left (d x +c \right )^{3}}{16 \cos \left (d x +c \right )^{2}}+\frac {\sin \left (d x +c \right )}{16}-\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{16}\right )+4 a \,b^{3} \left (\frac {\sin \left (d x +c \right )^{4}}{7 \cos \left (d x +c \right )^{7}}+\frac {3 \sin \left (d x +c \right )^{4}}{35 \cos \left (d x +c \right )^{5}}+\frac {\sin \left (d x +c \right )^{4}}{35 \cos \left (d x +c \right )^{3}}-\frac {\sin \left (d x +c \right )^{4}}{35 \cos \left (d x +c \right )}-\frac {\left (2+\sin \left (d x +c \right )^{2}\right ) \cos \left (d x +c \right )}{35}\right )+b^{4} \left (\frac {\sin \left (d x +c \right )^{5}}{8 \cos \left (d x +c \right )^{8}}+\frac {\sin \left (d x +c \right )^{5}}{16 \cos \left (d x +c \right )^{6}}+\frac {\sin \left (d x +c \right )^{5}}{64 \cos \left (d x +c \right )^{4}}-\frac {\sin \left (d x +c \right )^{5}}{128 \cos \left (d x +c \right )^{2}}-\frac {\sin \left (d x +c \right )^{3}}{128}-\frac {3 \sin \left (d x +c \right )}{128}+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{128}\right )}{d}\) | \(363\) |
| parallelrisch | \(\frac {-13440 \left (a^{4}-a^{2} b^{2}+\frac {1}{16} b^{4}\right ) \left (\frac {35}{8}+\frac {\cos \left (8 d x +8 c \right )}{8}+\cos \left (6 d x +6 c \right )+\frac {7 \cos \left (4 d x +4 c \right )}{2}+7 \cos \left (2 d x +2 c \right )\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )+13440 \left (a^{4}-a^{2} b^{2}+\frac {1}{16} b^{4}\right ) \left (\frac {35}{8}+\frac {\cos \left (8 d x +8 c \right )}{8}+\cos \left (6 d x +6 c \right )+\frac {7 \cos \left (4 d x +4 c \right )}{2}+7 \cos \left (2 d x +2 c \right )\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )+\left (200704 a^{3} b -57344 a \,b^{3}\right ) \cos \left (2 d x +2 c \right )+\left (100352 a^{3} b -28672 a \,b^{3}\right ) \cos \left (4 d x +4 c \right )+\left (28672 a^{3} b -8192 a \,b^{3}\right ) \cos \left (6 d x +6 c \right )+\left (3584 a^{3} b -1024 a \,b^{3}\right ) \cos \left (8 d x +8 c \right )+\left (57120 a^{4}+86240 a^{2} b^{2}-23310 b^{4}\right ) \sin \left (3 d x +3 c \right )+\left (25760 a^{4}-25760 a^{2} b^{2}+1610 b^{4}\right ) \sin \left (5 d x +5 c \right )+\left (3360 a^{4}-3360 a^{2} b^{2}+210 b^{4}\right ) \sin \left (7 d x +7 c \right )+\left (114688 a^{3} b -114688 a \,b^{3}\right ) \cos \left (3 d x +3 c \right )+\left (34720 a^{4}+108640 a^{2} b^{2}+46970 b^{4}\right ) \sin \left (d x +c \right )+344064 b a \left (\left (a^{2}-\frac {b^{2}}{21}\right ) \cos \left (d x +c \right )+\frac {35 a^{2}}{96}-\frac {5 b^{2}}{48}\right )}{4480 d \left (\cos \left (8 d x +8 c \right )+8 \cos \left (6 d x +6 c \right )+28 \cos \left (4 d x +4 c \right )+56 \cos \left (2 d x +2 c \right )+35\right )}\) | \(462\) |
| risch | \(-\frac {i {\mathrm e}^{i \left (d x +c \right )} \left (-1680 a^{4}-105 b^{4}+1680 a^{2} b^{2}+28560 a^{4} {\mathrm e}^{10 i \left (d x +c \right )}-805 b^{4} {\mathrm e}^{2 i \left (d x +c \right )}-23485 b^{4} {\mathrm e}^{6 i \left (d x +c \right )}-11655 b^{4} {\mathrm e}^{10 i \left (d x +c \right )}-12880 a^{2} b^{2} {\mathrm e}^{12 i \left (d x +c \right )}-1680 a^{2} b^{2} {\mathrm e}^{14 i \left (d x +c \right )}-57344 i a \,b^{3} {\mathrm e}^{10 i \left (d x +c \right )}+57344 i a^{3} b \,{\mathrm e}^{10 i \left (d x +c \right )}+43120 a^{2} b^{2} {\mathrm e}^{10 i \left (d x +c \right )}+172032 i a^{3} b \,{\mathrm e}^{8 i \left (d x +c \right )}+57344 i a^{3} b \,{\mathrm e}^{4 i \left (d x +c \right )}+172032 i a^{3} b \,{\mathrm e}^{6 i \left (d x +c \right )}-8192 i a \,b^{3} {\mathrm e}^{8 i \left (d x +c \right )}-8192 i a \,b^{3} {\mathrm e}^{6 i \left (d x +c \right )}-57344 i a \,b^{3} {\mathrm e}^{4 i \left (d x +c \right )}+17360 a^{4} {\mathrm e}^{8 i \left (d x +c \right )}+23485 b^{4} {\mathrm e}^{8 i \left (d x +c \right )}-17360 a^{4} {\mathrm e}^{6 i \left (d x +c \right )}-28560 a^{4} {\mathrm e}^{4 i \left (d x +c \right )}+11655 b^{4} {\mathrm e}^{4 i \left (d x +c \right )}-12880 a^{4} {\mathrm e}^{2 i \left (d x +c \right )}+54320 a^{2} b^{2} {\mathrm e}^{8 i \left (d x +c \right )}-54320 a^{2} b^{2} {\mathrm e}^{6 i \left (d x +c \right )}-43120 a^{2} b^{2} {\mathrm e}^{4 i \left (d x +c \right )}+12880 a^{2} b^{2} {\mathrm e}^{2 i \left (d x +c \right )}+12880 a^{4} {\mathrm e}^{12 i \left (d x +c \right )}+1680 a^{4} {\mathrm e}^{14 i \left (d x +c \right )}+105 b^{4} {\mathrm e}^{14 i \left (d x +c \right )}+805 b^{4} {\mathrm e}^{12 i \left (d x +c \right )}\right )}{2240 d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{8}}-\frac {3 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) a^{4}}{8 d}+\frac {3 b^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) a^{2}}{8 d}-\frac {3 b^{4} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{128 d}+\frac {3 \ln \left (i+{\mathrm e}^{i \left (d x +c \right )}\right ) a^{4}}{8 d}-\frac {3 b^{2} \ln \left (i+{\mathrm e}^{i \left (d x +c \right )}\right ) a^{2}}{8 d}+\frac {3 b^{4} \ln \left (i+{\mathrm e}^{i \left (d x +c \right )}\right )}{128 d}\) | \(684\) |
a^4/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^4/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)))+4*a*b^3/d*( 1/7*sec(d*x+c)^7-1/5*sec(d*x+c)^5)+4/5*a^3*b*sec(d*x+c)^5/d+6*a^2*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(sec(d*x+c)+tan(d*x+c)))
Time = 0.27 (sec) , antiderivative size = 214, normalized size of antiderivative = 0.65 \[ \int \sec ^9(c+d x) (a \cos (c+d x)+b \sin (c+d x))^4 \, dx=\frac {105 \, {\left (16 \, a^{4} - 16 \, a^{2} b^{2} + b^{4}\right )} \cos \left (d x + c\right )^{8} \log \left (\sin \left (d x + c\right ) + 1\right ) - 105 \, {\left (16 \, a^{4} - 16 \, a^{2} b^{2} + b^{4}\right )} \cos \left (d x + c\right )^{8} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 5120 \, a b^{3} \cos \left (d x + c\right ) + 7168 \, {\left (a^{3} b - a b^{3}\right )} \cos \left (d x + c\right )^{3} + 70 \, {\left (3 \, {\left (16 \, a^{4} - 16 \, a^{2} b^{2} + b^{4}\right )} \cos \left (d x + c\right )^{6} + 2 \, {\left (16 \, a^{4} - 16 \, a^{2} b^{2} + b^{4}\right )} \cos \left (d x + c\right )^{4} + 16 \, b^{4} + 8 \, {\left (16 \, a^{2} b^{2} - 3 \, b^{4}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )}{8960 \, d \cos \left (d x + c\right )^{8}} \]
1/8960*(105*(16*a^4 - 16*a^2*b^2 + b^4)*cos(d*x + c)^8*log(sin(d*x + c) + 1) - 105*(16*a^4 - 16*a^2*b^2 + b^4)*cos(d*x + c)^8*log(-sin(d*x + c) + 1) + 5120*a*b^3*cos(d*x + c) + 7168*(a^3*b - a*b^3)*cos(d*x + c)^3 + 70*(3*( 16*a^4 - 16*a^2*b^2 + b^4)*cos(d*x + c)^6 + 2*(16*a^4 - 16*a^2*b^2 + b^4)* cos(d*x + c)^4 + 16*b^4 + 8*(16*a^2*b^2 - 3*b^4)*cos(d*x + c)^2)*sin(d*x + c))/(d*cos(d*x + c)^8)
Timed out. \[ \int \sec ^9(c+d x) (a \cos (c+d x)+b \sin (c+d x))^4 \, dx=\text {Timed out} \]
Time = 0.24 (sec) , antiderivative size = 322, normalized size of antiderivative = 0.98 \[ \int \sec ^9(c+d x) (a \cos (c+d x)+b \sin (c+d x))^4 \, dx=-\frac {35 \, b^{4} {\left (\frac {2 \, {\left (3 \, \sin \left (d x + c\right )^{7} - 11 \, \sin \left (d x + c\right )^{5} - 11 \, \sin \left (d x + c\right )^{3} + 3 \, \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{8} - 4 \, \sin \left (d x + c\right )^{6} + 6 \, \sin \left (d x + c\right )^{4} - 4 \, \sin \left (d x + c\right )^{2} + 1} - 3 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 560 \, a^{2} b^{2} {\left (\frac {2 \, {\left (3 \, \sin \left (d x + c\right )^{5} - 8 \, \sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{6} - 3 \, \sin \left (d x + c\right )^{4} + 3 \, \sin \left (d x + c\right )^{2} - 1} - 3 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 560 \, a^{4} {\left (\frac {2 \, {\left (3 \, \sin \left (d x + c\right )^{3} - 5 \, \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{4} - 2 \, \sin \left (d x + c\right )^{2} + 1} - 3 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - \frac {7168 \, a^{3} b}{\cos \left (d x + c\right )^{5}} + \frac {1024 \, {\left (7 \, \cos \left (d x + c\right )^{2} - 5\right )} a b^{3}}{\cos \left (d x + c\right )^{7}}}{8960 \, d} \]
-1/8960*(35*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)) - 560*a^2*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)) + 560*a^4*(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)) - 7168*a^3*b/cos(d*x + c)^5 + 1024*(7*co s(d*x + c)^2 - 5)*a*b^3/cos(d*x + c)^7)/d
Leaf count of result is larger than twice the leaf count of optimal. 706 vs. \(2 (302) = 604\).
Time = 0.47 (sec) , antiderivative size = 706, normalized size of antiderivative = 2.14 \[ \int \sec ^9(c+d x) (a \cos (c+d x)+b \sin (c+d x))^4 \, dx=\text {Too large to display} \]
1/4480*(105*(16*a^4 - 16*a^2*b^2 + b^4)*log(abs(tan(1/2*d*x + 1/2*c) + 1)) - 105*(16*a^4 - 16*a^2*b^2 + b^4)*log(abs(tan(1/2*d*x + 1/2*c) - 1)) + 2* (2800*a^4*tan(1/2*d*x + 1/2*c)^15 + 1680*a^2*b^2*tan(1/2*d*x + 1/2*c)^15 - 105*b^4*tan(1/2*d*x + 1/2*c)^15 - 17920*a^3*b*tan(1/2*d*x + 1/2*c)^14 - 9 520*a^4*tan(1/2*d*x + 1/2*c)^13 + 22960*a^2*b^2*tan(1/2*d*x + 1/2*c)^13 + 805*b^4*tan(1/2*d*x + 1/2*c)^13 + 53760*a^3*b*tan(1/2*d*x + 1/2*c)^12 - 35 840*a*b^3*tan(1/2*d*x + 1/2*c)^12 + 11760*a^4*tan(1/2*d*x + 1/2*c)^11 - 72 80*a^2*b^2*tan(1/2*d*x + 1/2*c)^11 + 11655*b^4*tan(1/2*d*x + 1/2*c)^11 - 8 9600*a^3*b*tan(1/2*d*x + 1/2*c)^10 - 5040*a^4*tan(1/2*d*x + 1/2*c)^9 - 173 60*a^2*b^2*tan(1/2*d*x + 1/2*c)^9 + 23485*b^4*tan(1/2*d*x + 1/2*c)^9 + 125 440*a^3*b*tan(1/2*d*x + 1/2*c)^8 - 35840*a*b^3*tan(1/2*d*x + 1/2*c)^8 - 50 40*a^4*tan(1/2*d*x + 1/2*c)^7 - 17360*a^2*b^2*tan(1/2*d*x + 1/2*c)^7 + 234 85*b^4*tan(1/2*d*x + 1/2*c)^7 - 111104*a^3*b*tan(1/2*d*x + 1/2*c)^6 + 5734 4*a*b^3*tan(1/2*d*x + 1/2*c)^6 + 11760*a^4*tan(1/2*d*x + 1/2*c)^5 - 7280*a ^2*b^2*tan(1/2*d*x + 1/2*c)^5 + 11655*b^4*tan(1/2*d*x + 1/2*c)^5 + 46592*a ^3*b*tan(1/2*d*x + 1/2*c)^4 + 7168*a*b^3*tan(1/2*d*x + 1/2*c)^4 - 9520*a^4 *tan(1/2*d*x + 1/2*c)^3 + 22960*a^2*b^2*tan(1/2*d*x + 1/2*c)^3 + 805*b^4*t an(1/2*d*x + 1/2*c)^3 - 10752*a^3*b*tan(1/2*d*x + 1/2*c)^2 + 8192*a*b^3*ta n(1/2*d*x + 1/2*c)^2 + 2800*a^4*tan(1/2*d*x + 1/2*c) + 1680*a^2*b^2*tan(1/ 2*d*x + 1/2*c) - 105*b^4*tan(1/2*d*x + 1/2*c) + 3584*a^3*b - 1024*a*b^3...
Time = 26.24 (sec) , antiderivative size = 566, normalized size of antiderivative = 1.72 \[ \int \sec ^9(c+d x) (a \cos (c+d x)+b \sin (c+d x))^4 \, dx=\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{15}\,\left (\frac {5\,a^4}{4}+\frac {3\,a^2\,b^2}{4}-\frac {3\,b^4}{64}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (-\frac {17\,a^4}{4}+\frac {41\,a^2\,b^2}{4}+\frac {23\,b^4}{64}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}\,\left (-\frac {17\,a^4}{4}+\frac {41\,a^2\,b^2}{4}+\frac {23\,b^4}{64}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,\left (\frac {21\,a^4}{4}-\frac {13\,a^2\,b^2}{4}+\frac {333\,b^4}{64}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}\,\left (\frac {21\,a^4}{4}-\frac {13\,a^2\,b^2}{4}+\frac {333\,b^4}{64}\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7\,\left (\frac {9\,a^4}{4}+\frac {31\,a^2\,b^2}{4}-\frac {671\,b^4}{64}\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9\,\left (\frac {9\,a^4}{4}+\frac {31\,a^2\,b^2}{4}-\frac {671\,b^4}{64}\right )-\frac {16\,a\,b^3}{35}+\frac {8\,a^3\,b}{5}+\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (\frac {5\,a^4}{4}+\frac {3\,a^2\,b^2}{4}-\frac {3\,b^4}{64}\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}\,\left (16\,a\,b^3-24\,a^3\,b\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\,\left (16\,a\,b^3-56\,a^3\,b\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\left (\frac {104\,a^3\,b}{5}+\frac {16\,a\,b^3}{5}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (\frac {128\,a\,b^3}{35}-\frac {24\,a^3\,b}{5}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,\left (\frac {128\,a\,b^3}{5}-\frac {248\,a^3\,b}{5}\right )-40\,a^3\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}-8\,a^3\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{16}-8\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}+28\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}-56\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+70\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8-56\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+28\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-8\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}+\frac {\mathrm {atanh}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )\,\left (\frac {3\,a^4}{4}-\frac {3\,a^2\,b^2}{4}+\frac {3\,b^4}{64}\right )}{d} \]
(tan(c/2 + (d*x)/2)^15*((5*a^4)/4 - (3*b^4)/64 + (3*a^2*b^2)/4) + tan(c/2 + (d*x)/2)^3*((23*b^4)/64 - (17*a^4)/4 + (41*a^2*b^2)/4) + tan(c/2 + (d*x) /2)^13*((23*b^4)/64 - (17*a^4)/4 + (41*a^2*b^2)/4) + tan(c/2 + (d*x)/2)^5* ((21*a^4)/4 + (333*b^4)/64 - (13*a^2*b^2)/4) + tan(c/2 + (d*x)/2)^11*((21* a^4)/4 + (333*b^4)/64 - (13*a^2*b^2)/4) - tan(c/2 + (d*x)/2)^7*((9*a^4)/4 - (671*b^4)/64 + (31*a^2*b^2)/4) - tan(c/2 + (d*x)/2)^9*((9*a^4)/4 - (671* b^4)/64 + (31*a^2*b^2)/4) - (16*a*b^3)/35 + (8*a^3*b)/5 + tan(c/2 + (d*x)/ 2)*((5*a^4)/4 - (3*b^4)/64 + (3*a^2*b^2)/4) - tan(c/2 + (d*x)/2)^12*(16*a* b^3 - 24*a^3*b) - tan(c/2 + (d*x)/2)^8*(16*a*b^3 - 56*a^3*b) + tan(c/2 + ( d*x)/2)^4*((16*a*b^3)/5 + (104*a^3*b)/5) + tan(c/2 + (d*x)/2)^2*((128*a*b^ 3)/35 - (24*a^3*b)/5) + tan(c/2 + (d*x)/2)^6*((128*a*b^3)/5 - (248*a^3*b)/ 5) - 40*a^3*b*tan(c/2 + (d*x)/2)^10 - 8*a^3*b*tan(c/2 + (d*x)/2)^14)/(d*(2 8*tan(c/2 + (d*x)/2)^4 - 8*tan(c/2 + (d*x)/2)^2 - 56*tan(c/2 + (d*x)/2)^6 + 70*tan(c/2 + (d*x)/2)^8 - 56*tan(c/2 + (d*x)/2)^10 + 28*tan(c/2 + (d*x)/ 2)^12 - 8*tan(c/2 + (d*x)/2)^14 + tan(c/2 + (d*x)/2)^16 + 1)) + (atanh(tan (c/2 + (d*x)/2))*((3*a^4)/4 + (3*b^4)/64 - (3*a^2*b^2)/4))/d