Integrand size = 28, antiderivative size = 259 \[ \int \sec ^{10}(c+d x) (a \cos (c+d x)+b \sin (c+d x))^3 \, dx=\frac {5 a^3 \text {arctanh}(\sin (c+d x))}{16 d}-\frac {15 a b^2 \text {arctanh}(\sin (c+d x))}{128 d}+\frac {3 a^2 b \sec ^7(c+d x)}{7 d}-\frac {b^3 \sec ^7(c+d x)}{7 d}+\frac {b^3 \sec ^9(c+d x)}{9 d}+\frac {5 a^3 \sec (c+d x) \tan (c+d x)}{16 d}-\frac {15 a b^2 \sec (c+d x) \tan (c+d x)}{128 d}+\frac {5 a^3 \sec ^3(c+d x) \tan (c+d x)}{24 d}-\frac {5 a b^2 \sec ^3(c+d x) \tan (c+d x)}{64 d}+\frac {a^3 \sec ^5(c+d x) \tan (c+d x)}{6 d}-\frac {a b^2 \sec ^5(c+d x) \tan (c+d x)}{16 d}+\frac {3 a b^2 \sec ^7(c+d x) \tan (c+d x)}{8 d} \] Output:
5/16*a^3*arctanh(sin(d*x+c))/d-15/128*a*b^2*arctanh(sin(d*x+c))/d+3/7*a^2* b*sec(d*x+c)^7/d-1/7*b^3*sec(d*x+c)^7/d+1/9*b^3*sec(d*x+c)^9/d+5/16*a^3*se c(d*x+c)*tan(d*x+c)/d-15/128*a*b^2*sec(d*x+c)*tan(d*x+c)/d+5/24*a^3*sec(d* x+c)^3*tan(d*x+c)/d-5/64*a*b^2*sec(d*x+c)^3*tan(d*x+c)/d+1/6*a^3*sec(d*x+c )^5*tan(d*x+c)/d-1/16*a*b^2*sec(d*x+c)^5*tan(d*x+c)/d+3/8*a*b^2*sec(d*x+c) ^7*tan(d*x+c)/d
Leaf count is larger than twice the leaf count of optimal. \(810\) vs. \(2(259)=518\).
Time = 3.56 (sec) , antiderivative size = 810, normalized size of antiderivative = 3.13 \[ \int \sec ^{10}(c+d x) (a \cos (c+d x)+b \sin (c+d x))^3 \, dx =\text {Too large to display} \] Input:
Integrate[Sec[c + d*x]^10*(a*Cos[c + d*x] + b*Sin[c + d*x])^3,x]
Output:
(Sec[c + d*x]^9*(442368*a^2*b + 81920*b^3 + 147456*(3*a^2*b - b^3)*Cos[2*( c + d*x)] - 211680*a^3*Cos[3*(c + d*x)]*Log[Cos[(c + d*x)/2] - Sin[(c + d* x)/2]] + 79380*a*b^2*Cos[3*(c + d*x)]*Log[Cos[(c + d*x)/2] - Sin[(c + d*x) /2]] - 90720*a^3*Cos[5*(c + d*x)]*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 34020*a*b^2*Cos[5*(c + d*x)]*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - 22680*a^3*Cos[7*(c + d*x)]*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 850 5*a*b^2*Cos[7*(c + d*x)]*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - 2520*a ^3*Cos[9*(c + d*x)]*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 945*a*b^2*C os[9*(c + d*x)]*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - 39690*a*(8*a^2 - 3*b^2)*Cos[c + d*x]*(Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - Log[Cos[ (c + d*x)/2] + Sin[(c + d*x)/2]]) + 211680*a^3*Cos[3*(c + d*x)]*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] - 79380*a*b^2*Cos[3*(c + d*x)]*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + 90720*a^3*Cos[5*(c + d*x)]*Log[Cos[(c + d*x )/2] + Sin[(c + d*x)/2]] - 34020*a*b^2*Cos[5*(c + d*x)]*Log[Cos[(c + d*x)/ 2] + Sin[(c + d*x)/2]] + 22680*a^3*Cos[7*(c + d*x)]*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] - 8505*a*b^2*Cos[7*(c + d*x)]*Log[Cos[(c + d*x)/2] + Si n[(c + d*x)/2]] + 2520*a^3*Cos[9*(c + d*x)]*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] - 945*a*b^2*Cos[9*(c + d*x)]*Log[Cos[(c + d*x)/2] + Sin[(c + d* x)/2]] + 223776*a^3*Sin[2*(c + d*x)] + 303156*a*b^2*Sin[2*(c + d*x)] + 167 328*a^3*Sin[4*(c + d*x)] - 62748*a*b^2*Sin[4*(c + d*x)] + 43680*a^3*Sin...
Time = 0.50 (sec) , antiderivative size = 259, 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.
| \(\displaystyle \int \sec ^{10}(c+d x) (a \cos (c+d x)+b \sin (c+d x))^3 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {(a \cos (c+d x)+b \sin (c+d x))^3}{\cos (c+d x)^{10}}dx\) |
| \(\Big \downarrow \) 3569 |
| \(\displaystyle \int \left (a^3 \sec ^7(c+d x)+3 a^2 b \tan (c+d x) \sec ^7(c+d x)+3 a b^2 \tan ^2(c+d x) \sec ^7(c+d x)+b^3 \tan ^3(c+d x) \sec ^7(c+d x)\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {5 a^3 \text {arctanh}(\sin (c+d x))}{16 d}+\frac {a^3 \tan (c+d x) \sec ^5(c+d x)}{6 d}+\frac {5 a^3 \tan (c+d x) \sec ^3(c+d x)}{24 d}+\frac {5 a^3 \tan (c+d x) \sec (c+d x)}{16 d}+\frac {3 a^2 b \sec ^7(c+d x)}{7 d}-\frac {15 a b^2 \text {arctanh}(\sin (c+d x))}{128 d}+\frac {3 a b^2 \tan (c+d x) \sec ^7(c+d x)}{8 d}-\frac {a b^2 \tan (c+d x) \sec ^5(c+d x)}{16 d}-\frac {5 a b^2 \tan (c+d x) \sec ^3(c+d x)}{64 d}-\frac {15 a b^2 \tan (c+d x) \sec (c+d x)}{128 d}+\frac {b^3 \sec ^9(c+d x)}{9 d}-\frac {b^3 \sec ^7(c+d x)}{7 d}\) |
Input:
Int[Sec[c + d*x]^10*(a*Cos[c + d*x] + b*Sin[c + d*x])^3,x]
Output:
(5*a^3*ArcTanh[Sin[c + d*x]])/(16*d) - (15*a*b^2*ArcTanh[Sin[c + d*x]])/(1 28*d) + (3*a^2*b*Sec[c + d*x]^7)/(7*d) - (b^3*Sec[c + d*x]^7)/(7*d) + (b^3 *Sec[c + d*x]^9)/(9*d) + (5*a^3*Sec[c + d*x]*Tan[c + d*x])/(16*d) - (15*a* b^2*Sec[c + d*x]*Tan[c + d*x])/(128*d) + (5*a^3*Sec[c + d*x]^3*Tan[c + d*x ])/(24*d) - (5*a*b^2*Sec[c + d*x]^3*Tan[c + d*x])/(64*d) + (a^3*Sec[c + d* x]^5*Tan[c + d*x])/(6*d) - (a*b^2*Sec[c + d*x]^5*Tan[c + d*x])/(16*d) + (3 *a*b^2*Sec[c + d*x]^7*Tan[c + d*x])/(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 = 1.73 (sec) , antiderivative size = 214, normalized size of antiderivative = 0.83
| method | result | size |
| parts | \(\frac {a^{3} \left (-\left (-\frac {\sec \left (d x +c \right )^{5}}{6}-\frac {5 \sec \left (d x +c \right )^{3}}{24}-\frac {5 \sec \left (d x +c \right )}{16}\right ) \tan \left (d x +c \right )+\frac {5 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{16}\right )}{d}+\frac {b^{3} \left (\frac {\sec \left (d x +c \right )^{9}}{9}-\frac {\sec \left (d x +c \right )^{7}}{7}\right )}{d}+\frac {3 a^{2} b \sec \left (d x +c \right )^{7}}{7 d}+\frac {3 a \,b^{2} \left (\frac {\sin \left (d x +c \right )^{3}}{8 \cos \left (d x +c \right )^{8}}+\frac {5 \sin \left (d x +c \right )^{3}}{48 \cos \left (d x +c \right )^{6}}+\frac {5 \sin \left (d x +c \right )^{3}}{64 \cos \left (d x +c \right )^{4}}+\frac {5 \sin \left (d x +c \right )^{3}}{128 \cos \left (d x +c \right )^{2}}+\frac {5 \sin \left (d x +c \right )}{128}-\frac {5 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{128}\right )}{d}\) | \(214\) |
| derivativedivides | \(\frac {a^{3} \left (-\left (-\frac {\sec \left (d x +c \right )^{5}}{6}-\frac {5 \sec \left (d x +c \right )^{3}}{24}-\frac {5 \sec \left (d x +c \right )}{16}\right ) \tan \left (d x +c \right )+\frac {5 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{16}\right )+\frac {3 a^{2} b}{7 \cos \left (d x +c \right )^{7}}+3 a \,b^{2} \left (\frac {\sin \left (d x +c \right )^{3}}{8 \cos \left (d x +c \right )^{8}}+\frac {5 \sin \left (d x +c \right )^{3}}{48 \cos \left (d x +c \right )^{6}}+\frac {5 \sin \left (d x +c \right )^{3}}{64 \cos \left (d x +c \right )^{4}}+\frac {5 \sin \left (d x +c \right )^{3}}{128 \cos \left (d x +c \right )^{2}}+\frac {5 \sin \left (d x +c \right )}{128}-\frac {5 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{128}\right )+b^{3} \left (\frac {\sin \left (d x +c \right )^{4}}{9 \cos \left (d x +c \right )^{9}}+\frac {5 \sin \left (d x +c \right )^{4}}{63 \cos \left (d x +c \right )^{7}}+\frac {\sin \left (d x +c \right )^{4}}{21 \cos \left (d x +c \right )^{5}}+\frac {\sin \left (d x +c \right )^{4}}{63 \cos \left (d x +c \right )^{3}}-\frac {\sin \left (d x +c \right )^{4}}{63 \cos \left (d x +c \right )}-\frac {\left (2+\sin \left (d x +c \right )^{2}\right ) \cos \left (d x +c \right )}{63}\right )}{d}\) | \(294\) |
| default | \(\frac {a^{3} \left (-\left (-\frac {\sec \left (d x +c \right )^{5}}{6}-\frac {5 \sec \left (d x +c \right )^{3}}{24}-\frac {5 \sec \left (d x +c \right )}{16}\right ) \tan \left (d x +c \right )+\frac {5 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{16}\right )+\frac {3 a^{2} b}{7 \cos \left (d x +c \right )^{7}}+3 a \,b^{2} \left (\frac {\sin \left (d x +c \right )^{3}}{8 \cos \left (d x +c \right )^{8}}+\frac {5 \sin \left (d x +c \right )^{3}}{48 \cos \left (d x +c \right )^{6}}+\frac {5 \sin \left (d x +c \right )^{3}}{64 \cos \left (d x +c \right )^{4}}+\frac {5 \sin \left (d x +c \right )^{3}}{128 \cos \left (d x +c \right )^{2}}+\frac {5 \sin \left (d x +c \right )}{128}-\frac {5 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{128}\right )+b^{3} \left (\frac {\sin \left (d x +c \right )^{4}}{9 \cos \left (d x +c \right )^{9}}+\frac {5 \sin \left (d x +c \right )^{4}}{63 \cos \left (d x +c \right )^{7}}+\frac {\sin \left (d x +c \right )^{4}}{21 \cos \left (d x +c \right )^{5}}+\frac {\sin \left (d x +c \right )^{4}}{63 \cos \left (d x +c \right )^{3}}-\frac {\sin \left (d x +c \right )^{4}}{63 \cos \left (d x +c \right )}-\frac {\left (2+\sin \left (d x +c \right )^{2}\right ) \cos \left (d x +c \right )}{63}\right )}{d}\) | \(294\) |
| parallelrisch | \(\frac {-22680 \left (a^{2}-\frac {3 b^{2}}{8}\right ) \left (\frac {\cos \left (9 d x +9 c \right )}{9}+\cos \left (7 d x +7 c \right )+4 \cos \left (5 d x +5 c \right )+\frac {28 \cos \left (3 d x +3 c \right )}{3}+14 \cos \left (d x +c \right )\right ) a \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )+22680 \left (a^{2}-\frac {3 b^{2}}{8}\right ) \left (\frac {\cos \left (9 d x +9 c \right )}{9}+\cos \left (7 d x +7 c \right )+4 \cos \left (5 d x +5 c \right )+\frac {28 \cos \left (3 d x +3 c \right )}{3}+14 \cos \left (d x +c \right )\right ) a \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )+\left (290304 a^{2} b -21504 b^{3}\right ) \cos \left (3 d x +3 c \right )+\left (124416 a^{2} b -9216 b^{3}\right ) \cos \left (5 d x +5 c \right )+\left (31104 a^{2} b -2304 b^{3}\right ) \cos \left (7 d x +7 c \right )+\left (3456 a^{2} b -256 b^{3}\right ) \cos \left (9 d x +9 c \right )+\left (442368 a^{2} b -147456 b^{3}\right ) \cos \left (2 d x +2 c \right )+\left (223776 a^{3}+303156 a \,b^{2}\right ) \sin \left (2 d x +2 c \right )+\left (167328 a^{3}-62748 a \,b^{2}\right ) \sin \left (4 d x +4 c \right )+\left (43680 a^{3}-16380 a \,b^{2}\right ) \sin \left (6 d x +6 c \right )+\left (5040 a^{3}-1890 a \,b^{2}\right ) \sin \left (8 d x +8 c \right )+\left (435456 a^{2} b -32256 b^{3}\right ) \cos \left (d x +c \right )+442368 a^{2} b +81920 b^{3}}{8064 d \left (\cos \left (9 d x +9 c \right )+9 \cos \left (7 d x +7 c \right )+36 \cos \left (5 d x +5 c \right )+84 \cos \left (3 d x +3 c \right )+126 \cos \left (d x +c \right )\right )}\) | \(438\) |
| risch | \(-\frac {{\mathrm e}^{i \left (d x +c \right )} \left (945 i a \,b^{2}+83664 i a^{3} {\mathrm e}^{12 i \left (d x +c \right )}-83664 i a^{3} {\mathrm e}^{4 i \left (d x +c \right )}-8190 i a \,b^{2} {\mathrm e}^{14 i \left (d x +c \right )}-21840 i a^{3} {\mathrm e}^{2 i \left (d x +c \right )}-31374 i a \,b^{2} {\mathrm e}^{12 i \left (d x +c \right )}-2520 i a^{3}-111888 i a^{3} {\mathrm e}^{6 i \left (d x +c \right )}-221184 a^{2} b \,{\mathrm e}^{10 i \left (d x +c \right )}+73728 b^{3} {\mathrm e}^{10 i \left (d x +c \right )}-442368 a^{2} b \,{\mathrm e}^{8 i \left (d x +c \right )}-81920 b^{3} {\mathrm e}^{8 i \left (d x +c \right )}-945 i a \,b^{2} {\mathrm e}^{16 i \left (d x +c \right )}+151578 i a \,b^{2} {\mathrm e}^{10 i \left (d x +c \right )}-221184 a^{2} b \,{\mathrm e}^{6 i \left (d x +c \right )}+73728 b^{3} {\mathrm e}^{6 i \left (d x +c \right )}+8190 i a \,b^{2} {\mathrm e}^{2 i \left (d x +c \right )}+31374 i a \,b^{2} {\mathrm e}^{4 i \left (d x +c \right )}+2520 i a^{3} {\mathrm e}^{16 i \left (d x +c \right )}+111888 i a^{3} {\mathrm e}^{10 i \left (d x +c \right )}-151578 i a \,b^{2} {\mathrm e}^{6 i \left (d x +c \right )}+21840 i a^{3} {\mathrm e}^{14 i \left (d x +c \right )}\right )}{4032 d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{9}}+\frac {5 a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{16 d}-\frac {15 a \,b^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{128 d}-\frac {5 a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{16 d}+\frac {15 a \,b^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{128 d}\) | \(475\) |
Input:
int(sec(d*x+c)^10*(a*cos(d*x+c)+b*sin(d*x+c))^3,x,method=_RETURNVERBOSE)
Output:
a^3/d*(-(-1/6*sec(d*x+c)^5-5/24*sec(d*x+c)^3-5/16*sec(d*x+c))*tan(d*x+c)+5 /16*ln(sec(d*x+c)+tan(d*x+c)))+b^3/d*(1/9*sec(d*x+c)^9-1/7*sec(d*x+c)^7)+3 /7*a^2*b*sec(d*x+c)^7/d+3*a*b^2/d*(1/8*sin(d*x+c)^3/cos(d*x+c)^8+5/48*sin( d*x+c)^3/cos(d*x+c)^6+5/64*sin(d*x+c)^3/cos(d*x+c)^4+5/128*sin(d*x+c)^3/co s(d*x+c)^2+5/128*sin(d*x+c)-5/128*ln(sec(d*x+c)+tan(d*x+c)))
Time = 0.10 (sec) , antiderivative size = 192, normalized size of antiderivative = 0.74 \[ \int \sec ^{10}(c+d x) (a \cos (c+d x)+b \sin (c+d x))^3 \, dx=\frac {315 \, {\left (8 \, a^{3} - 3 \, a b^{2}\right )} \cos \left (d x + c\right )^{9} \log \left (\sin \left (d x + c\right ) + 1\right ) - 315 \, {\left (8 \, a^{3} - 3 \, a b^{2}\right )} \cos \left (d x + c\right )^{9} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 1792 \, b^{3} + 2304 \, {\left (3 \, a^{2} b - b^{3}\right )} \cos \left (d x + c\right )^{2} + 42 \, {\left (15 \, {\left (8 \, a^{3} - 3 \, a b^{2}\right )} \cos \left (d x + c\right )^{7} + 10 \, {\left (8 \, a^{3} - 3 \, a b^{2}\right )} \cos \left (d x + c\right )^{5} + 144 \, a b^{2} \cos \left (d x + c\right ) + 8 \, {\left (8 \, a^{3} - 3 \, a b^{2}\right )} \cos \left (d x + c\right )^{3}\right )} \sin \left (d x + c\right )}{16128 \, d \cos \left (d x + c\right )^{9}} \] Input:
integrate(sec(d*x+c)^10*(a*cos(d*x+c)+b*sin(d*x+c))^3,x, algorithm="fricas ")
Output:
1/16128*(315*(8*a^3 - 3*a*b^2)*cos(d*x + c)^9*log(sin(d*x + c) + 1) - 315* (8*a^3 - 3*a*b^2)*cos(d*x + c)^9*log(-sin(d*x + c) + 1) + 1792*b^3 + 2304* (3*a^2*b - b^3)*cos(d*x + c)^2 + 42*(15*(8*a^3 - 3*a*b^2)*cos(d*x + c)^7 + 10*(8*a^3 - 3*a*b^2)*cos(d*x + c)^5 + 144*a*b^2*cos(d*x + c) + 8*(8*a^3 - 3*a*b^2)*cos(d*x + c)^3)*sin(d*x + c))/(d*cos(d*x + c)^9)
Timed out. \[ \int \sec ^{10}(c+d x) (a \cos (c+d x)+b \sin (c+d x))^3 \, dx=\text {Timed out} \] Input:
integrate(sec(d*x+c)**10*(a*cos(d*x+c)+b*sin(d*x+c))**3,x)
Output:
Timed out
Time = 0.03 (sec) , antiderivative size = 248, normalized size of antiderivative = 0.96 \[ \int \sec ^{10}(c+d x) (a \cos (c+d x)+b \sin (c+d x))^3 \, dx=\frac {63 \, a b^{2} {\left (\frac {2 \, {\left (15 \, \sin \left (d x + c\right )^{7} - 55 \, \sin \left (d x + c\right )^{5} + 73 \, \sin \left (d x + c\right )^{3} + 15 \, \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{8} - 4 \, \sin \left (d x + c\right )^{6} + 6 \, \sin \left (d x + c\right )^{4} - 4 \, \sin \left (d x + c\right )^{2} + 1} - 15 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 15 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 168 \, a^{3} {\left (\frac {2 \, {\left (15 \, \sin \left (d x + c\right )^{5} - 40 \, \sin \left (d x + c\right )^{3} + 33 \, \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{6} - 3 \, \sin \left (d x + c\right )^{4} + 3 \, \sin \left (d x + c\right )^{2} - 1} - 15 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 15 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + \frac {6912 \, a^{2} b}{\cos \left (d x + c\right )^{7}} - \frac {256 \, {\left (9 \, \cos \left (d x + c\right )^{2} - 7\right )} b^{3}}{\cos \left (d x + c\right )^{9}}}{16128 \, d} \] Input:
integrate(sec(d*x+c)^10*(a*cos(d*x+c)+b*sin(d*x+c))^3,x, algorithm="maxima ")
Output:
1/16128*(63*a*b^2*(2*(15*sin(d*x + c)^7 - 55*sin(d*x + c)^5 + 73*sin(d*x + c)^3 + 15*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) - 15*log(sin(d*x + c) + 1) + 15*log(sin(d*x + c) - 1)) - 168*a^3*(2*(15*sin(d*x + c)^5 - 40*sin(d*x + c)^3 + 33*sin(d*x + c))/(sin(d*x + c)^6 - 3*sin(d*x + c)^4 + 3*sin(d*x + c)^2 - 1) - 15*log (sin(d*x + c) + 1) + 15*log(sin(d*x + c) - 1)) + 6912*a^2*b/cos(d*x + c)^7 - 256*(9*cos(d*x + c)^2 - 7)*b^3/cos(d*x + c)^9)/d
Leaf count of result is larger than twice the leaf count of optimal. 597 vs. \(2 (235) = 470\).
Time = 0.19 (sec) , antiderivative size = 597, normalized size of antiderivative = 2.31 \[ \int \sec ^{10}(c+d x) (a \cos (c+d x)+b \sin (c+d x))^3 \, dx =\text {Too large to display} \] Input:
integrate(sec(d*x+c)^10*(a*cos(d*x+c)+b*sin(d*x+c))^3,x, algorithm="giac")
Output:
1/8064*(315*(8*a^3 - 3*a*b^2)*log(abs(tan(1/2*d*x + 1/2*c) + 1)) - 315*(8* a^3 - 3*a*b^2)*log(abs(tan(1/2*d*x + 1/2*c) - 1)) + 2*(5544*a^3*tan(1/2*d* x + 1/2*c)^17 + 945*a*b^2*tan(1/2*d*x + 1/2*c)^17 - 24192*a^2*b*tan(1/2*d* x + 1/2*c)^16 - 15792*a^3*tan(1/2*d*x + 1/2*c)^15 + 24066*a*b^2*tan(1/2*d* x + 1/2*c)^15 + 48384*a^2*b*tan(1/2*d*x + 1/2*c)^14 - 16128*b^3*tan(1/2*d* x + 1/2*c)^14 + 29232*a^3*tan(1/2*d*x + 1/2*c)^13 + 31374*a*b^2*tan(1/2*d* x + 1/2*c)^13 - 145152*a^2*b*tan(1/2*d*x + 1/2*c)^12 - 26880*b^3*tan(1/2*d *x + 1/2*c)^12 - 33264*a^3*tan(1/2*d*x + 1/2*c)^11 + 54810*a*b^2*tan(1/2*d *x + 1/2*c)^11 + 241920*a^2*b*tan(1/2*d*x + 1/2*c)^10 - 80640*b^3*tan(1/2* d*x + 1/2*c)^10 - 193536*a^2*b*tan(1/2*d*x + 1/2*c)^8 - 48384*b^3*tan(1/2* d*x + 1/2*c)^8 + 33264*a^3*tan(1/2*d*x + 1/2*c)^7 - 54810*a*b^2*tan(1/2*d* x + 1/2*c)^7 + 145152*a^2*b*tan(1/2*d*x + 1/2*c)^6 - 48384*b^3*tan(1/2*d*x + 1/2*c)^6 - 29232*a^3*tan(1/2*d*x + 1/2*c)^5 - 31374*a*b^2*tan(1/2*d*x + 1/2*c)^5 - 76032*a^2*b*tan(1/2*d*x + 1/2*c)^4 - 6912*b^3*tan(1/2*d*x + 1/ 2*c)^4 + 15792*a^3*tan(1/2*d*x + 1/2*c)^3 - 24066*a*b^2*tan(1/2*d*x + 1/2* c)^3 + 6912*a^2*b*tan(1/2*d*x + 1/2*c)^2 - 2304*b^3*tan(1/2*d*x + 1/2*c)^2 - 5544*a^3*tan(1/2*d*x + 1/2*c) - 945*a*b^2*tan(1/2*d*x + 1/2*c) - 3456*a ^2*b + 256*b^3)/(tan(1/2*d*x + 1/2*c)^2 - 1)^9)/d
Time = 19.88 (sec) , antiderivative size = 547, normalized size of antiderivative = 2.11 \[ \int \sec ^{10}(c+d x) (a \cos (c+d x)+b \sin (c+d x))^3 \, dx =\text {Too large to display} \] Input:
int((a*cos(c + d*x) + b*sin(c + d*x))^3/cos(c + d*x)^10,x)
Output:
- (atanh(tan(c/2 + (d*x)/2))*((15*a*b^2)/64 - (5*a^3)/8))/d - (tan(c/2 + ( d*x)/2)*((15*a*b^2)/64 + (11*a^3)/8) + (6*a^2*b)/7 - tan(c/2 + (d*x)/2)^17 *((15*a*b^2)/64 + (11*a^3)/8) + tan(c/2 + (d*x)/2)^3*((191*a*b^2)/32 - (47 *a^3)/12) - tan(c/2 + (d*x)/2)^15*((191*a*b^2)/32 - (47*a^3)/12) + tan(c/2 + (d*x)/2)^5*((249*a*b^2)/32 + (29*a^3)/4) - tan(c/2 + (d*x)/2)^13*((249* a*b^2)/32 + (29*a^3)/4) + tan(c/2 + (d*x)/2)^7*((435*a*b^2)/32 - (33*a^3)/ 4) - tan(c/2 + (d*x)/2)^11*((435*a*b^2)/32 - (33*a^3)/4) - tan(c/2 + (d*x) /2)^14*(12*a^2*b - 4*b^3) - tan(c/2 + (d*x)/2)^2*((12*a^2*b)/7 - (4*b^3)/7 ) - tan(c/2 + (d*x)/2)^6*(36*a^2*b - 12*b^3) + tan(c/2 + (d*x)/2)^8*(48*a^ 2*b + 12*b^3) + tan(c/2 + (d*x)/2)^12*(36*a^2*b + (20*b^3)/3) - tan(c/2 + (d*x)/2)^10*(60*a^2*b - 20*b^3) + tan(c/2 + (d*x)/2)^4*((132*a^2*b)/7 + (1 2*b^3)/7) - (4*b^3)/63 + 6*a^2*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 = 0.16 (sec) , antiderivative size = 1023, normalized size of antiderivative = 3.95 \[ \int \sec ^{10}(c+d x) (a \cos (c+d x)+b \sin (c+d x))^3 \, dx =\text {Too large to display} \] Input:
int(sec(d*x+c)^10*(a*cos(d*x+c)+b*sin(d*x+c))^3,x)
Output:
( - 2520*cos(c + d*x)*log(tan((c + d*x)/2) - 1)*sin(c + d*x)**8*a**3 + 945 *cos(c + d*x)*log(tan((c + d*x)/2) - 1)*sin(c + d*x)**8*a*b**2 + 10080*cos (c + d*x)*log(tan((c + d*x)/2) - 1)*sin(c + d*x)**6*a**3 - 3780*cos(c + d* x)*log(tan((c + d*x)/2) - 1)*sin(c + d*x)**6*a*b**2 - 15120*cos(c + d*x)*l og(tan((c + d*x)/2) - 1)*sin(c + d*x)**4*a**3 + 5670*cos(c + d*x)*log(tan( (c + d*x)/2) - 1)*sin(c + d*x)**4*a*b**2 + 10080*cos(c + d*x)*log(tan((c + d*x)/2) - 1)*sin(c + d*x)**2*a**3 - 3780*cos(c + d*x)*log(tan((c + d*x)/2 ) - 1)*sin(c + d*x)**2*a*b**2 - 2520*cos(c + d*x)*log(tan((c + d*x)/2) - 1 )*a**3 + 945*cos(c + d*x)*log(tan((c + d*x)/2) - 1)*a*b**2 + 2520*cos(c + d*x)*log(tan((c + d*x)/2) + 1)*sin(c + d*x)**8*a**3 - 945*cos(c + d*x)*log (tan((c + d*x)/2) + 1)*sin(c + d*x)**8*a*b**2 - 10080*cos(c + d*x)*log(tan ((c + d*x)/2) + 1)*sin(c + d*x)**6*a**3 + 3780*cos(c + d*x)*log(tan((c + d *x)/2) + 1)*sin(c + d*x)**6*a*b**2 + 15120*cos(c + d*x)*log(tan((c + d*x)/ 2) + 1)*sin(c + d*x)**4*a**3 - 5670*cos(c + d*x)*log(tan((c + d*x)/2) + 1) *sin(c + d*x)**4*a*b**2 - 10080*cos(c + d*x)*log(tan((c + d*x)/2) + 1)*sin (c + d*x)**2*a**3 + 3780*cos(c + d*x)*log(tan((c + d*x)/2) + 1)*sin(c + d* x)**2*a*b**2 + 2520*cos(c + d*x)*log(tan((c + d*x)/2) + 1)*a**3 - 945*cos( c + d*x)*log(tan((c + d*x)/2) + 1)*a*b**2 - 3456*cos(c + d*x)*sin(c + d*x) **8*a**2*b + 256*cos(c + d*x)*sin(c + d*x)**8*b**3 - 2520*cos(c + d*x)*sin (c + d*x)**7*a**3 + 945*cos(c + d*x)*sin(c + d*x)**7*a*b**2 + 13824*cos...