Integrand size = 28, antiderivative size = 232 \[ \int \frac {\sec ^4(c+d x)}{(a \cos (c+d x)+b \sin (c+d x))^4} \, dx=\frac {\left (a^2+b^2\right )^3}{3 a^3 b^4 d (b+a \cot (c+d x))^3}+\frac {2 a^6+3 a^4 b^2-b^6}{a^3 b^5 d (b+a \cot (c+d x))^2}+\frac {10 a^6+9 a^4 b^2+b^6}{a^3 b^6 d (b+a \cot (c+d x))}-\frac {4 a \left (5 a^2+3 b^2\right ) \log (b+a \cot (c+d x))}{b^7 d}-\frac {4 a \left (5 a^2+3 b^2\right ) \log (\tan (c+d x))}{b^7 d}+\frac {\left (10 a^2+3 b^2\right ) \tan (c+d x)}{b^6 d}-\frac {2 a \tan ^2(c+d x)}{b^5 d}+\frac {\tan ^3(c+d x)}{3 b^4 d} \]
1/3*(a^2+b^2)^3/a^3/b^4/d/(b+a*cot(d*x+c))^3+(2*a^6+3*a^4*b^2-b^6)/a^3/b^5 /d/(b+a*cot(d*x+c))^2+(10*a^6+9*a^4*b^2+b^6)/a^3/b^6/d/(b+a*cot(d*x+c))-4* a*(5*a^2+3*b^2)*ln(b+a*cot(d*x+c))/b^7/d-4*a*(5*a^2+3*b^2)*ln(tan(d*x+c))/ b^7/d+(10*a^2+3*b^2)*tan(d*x+c)/b^6/d-2*a*tan(d*x+c)^2/b^5/d+1/3*tan(d*x+c )^3/b^4/d
Time = 2.27 (sec) , antiderivative size = 295, normalized size of antiderivative = 1.27 \[ \int \frac {\sec ^4(c+d x)}{(a \cos (c+d x)+b \sin (c+d x))^4} \, dx=\frac {b^6 \sec ^6(c+d x)+3 b^4 \sec ^4(c+d x) \left (a^2+2 b^2-a b \tan (c+d x)\right )-2 \left (37 a^6+36 a^4 b^2+3 a^2 b^4+4 b^6+6 a^4 \left (5 a^2+3 b^2\right ) \log (a+b \tan (c+d x))+3 a b \left (27 a^4+30 a^2 b^2+b^4+6 a^2 \left (5 a^2+3 b^2\right ) \log (a+b \tan (c+d x))\right ) \tan (c+d x)+6 b^2 \left (6 a^4+11 a^2 b^2+2 b^4+3 a^2 \left (5 a^2+3 b^2\right ) \log (a+b \tan (c+d x))\right ) \tan ^2(c+d x)+6 a b^3 \left (-3 a^2+\left (5 a^2+3 b^2\right ) \log (a+b \tan (c+d x))\right ) \tan ^3(c+d x)-6 a^2 b^4 \tan ^4(c+d x)\right )}{3 b^7 d (a+b \tan (c+d x))^3} \]
(b^6*Sec[c + d*x]^6 + 3*b^4*Sec[c + d*x]^4*(a^2 + 2*b^2 - a*b*Tan[c + d*x] ) - 2*(37*a^6 + 36*a^4*b^2 + 3*a^2*b^4 + 4*b^6 + 6*a^4*(5*a^2 + 3*b^2)*Log [a + b*Tan[c + d*x]] + 3*a*b*(27*a^4 + 30*a^2*b^2 + b^4 + 6*a^2*(5*a^2 + 3 *b^2)*Log[a + b*Tan[c + d*x]])*Tan[c + d*x] + 6*b^2*(6*a^4 + 11*a^2*b^2 + 2*b^4 + 3*a^2*(5*a^2 + 3*b^2)*Log[a + b*Tan[c + d*x]])*Tan[c + d*x]^2 + 6* a*b^3*(-3*a^2 + (5*a^2 + 3*b^2)*Log[a + b*Tan[c + d*x]])*Tan[c + d*x]^3 - 6*a^2*b^4*Tan[c + d*x]^4))/(3*b^7*d*(a + b*Tan[c + d*x])^3)
Time = 0.46 (sec) , antiderivative size = 216, normalized size of antiderivative = 0.93, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {3042, 3567, 522, 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.
| \(\displaystyle \int \frac {\sec ^4(c+d x)}{(a \cos (c+d x)+b \sin (c+d x))^4} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\cos (c+d x)^4 (a \cos (c+d x)+b \sin (c+d x))^4}dx\) |
| \(\Big \downarrow \) 3567 |
| \(\displaystyle -\frac {\int \frac {\left (\cot ^2(c+d x)+1\right )^3 \tan ^4(c+d x)}{(b+a \cot (c+d x))^4}d\cot (c+d x)}{d}\) |
| \(\Big \downarrow \) 522 |
| \(\displaystyle -\frac {\int \left (\frac {\tan ^4(c+d x)}{b^4}-\frac {4 a \tan ^3(c+d x)}{b^5}+\frac {\left (10 a^2+3 b^2\right ) \tan ^2(c+d x)}{b^6}-\frac {4 \left (5 a^3+3 b^2 a\right ) \tan (c+d x)}{b^7}+\frac {4 \left (5 a^4+3 b^2 a^2\right )}{b^7 (b+a \cot (c+d x))}+\frac {10 a^6+9 b^2 a^4+b^6}{a^2 b^6 (b+a \cot (c+d x))^2}+\frac {2 \left (2 a^6+3 b^2 a^4-b^6\right )}{a^2 b^5 (b+a \cot (c+d x))^3}+\frac {\left (a^2+b^2\right )^3}{a^2 b^4 (b+a \cot (c+d x))^4}\right )d\cot (c+d x)}{d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {-\frac {4 a \left (5 a^2+3 b^2\right ) \log (\cot (c+d x))}{b^7}+\frac {4 a \left (5 a^2+3 b^2\right ) \log (a \cot (c+d x)+b)}{b^7}-\frac {\left (10 a^2+3 b^2\right ) \tan (c+d x)}{b^6}-\frac {\left (a^2+b^2\right )^3}{3 a^3 b^4 (a \cot (c+d x)+b)^3}-\frac {10 a^6+9 a^4 b^2+b^6}{a^3 b^6 (a \cot (c+d x)+b)}-\frac {2 a^6+3 a^4 b^2-b^6}{a^3 b^5 (a \cot (c+d x)+b)^2}+\frac {2 a \tan ^2(c+d x)}{b^5}-\frac {\tan ^3(c+d x)}{3 b^4}}{d}\) |
-((-1/3*(a^2 + b^2)^3/(a^3*b^4*(b + a*Cot[c + d*x])^3) - (2*a^6 + 3*a^4*b^ 2 - b^6)/(a^3*b^5*(b + a*Cot[c + d*x])^2) - (10*a^6 + 9*a^4*b^2 + b^6)/(a^ 3*b^6*(b + a*Cot[c + d*x])) - (4*a*(5*a^2 + 3*b^2)*Log[Cot[c + d*x]])/b^7 + (4*a*(5*a^2 + 3*b^2)*Log[b + a*Cot[c + d*x]])/b^7 - ((10*a^2 + 3*b^2)*Ta n[c + d*x])/b^6 + (2*a*Tan[c + d*x]^2)/b^5 - Tan[c + d*x]^3/(3*b^4))/d)
3.2.49.3.1 Defintions of rubi rules used
Int[((e_.)*(x_))^(m_.)*((c_) + (d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_)^2)^(p_. ), x_Symbol] :> Int[ExpandIntegrand[(e*x)^m*(c + d*x)^n*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && IGtQ[p, 0]
Int[cos[(c_.) + (d_.)*(x_)]^(m_.)*(cos[(c_.) + (d_.)*(x_)]*(a_.) + (b_.)*si n[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[-d^(-1) Subst[Int[x^m*((b + a*x)^n/(1 + x^2)^((m + n + 2)/2)), x], x, Cot[c + d*x]], x] /; FreeQ[{a, b, c, d}, x] && IntegerQ[n] && IntegerQ[(m + n)/2] && NeQ[n, -1] && !(GtQ[ n, 0] && GtQ[m, 1])
Time = 5.42 (sec) , antiderivative size = 195, normalized size of antiderivative = 0.84
| method | result | size |
| derivativedivides | \(\frac {\frac {\frac {b^{2} \tan \left (d x +c \right )^{3}}{3}-2 a \tan \left (d x +c \right )^{2} b +10 \tan \left (d x +c \right ) a^{2}+3 \tan \left (d x +c \right ) b^{2}}{b^{6}}-\frac {4 a \left (5 a^{2}+3 b^{2}\right ) \ln \left (a +b \tan \left (d x +c \right )\right )}{b^{7}}-\frac {a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}}{3 b^{7} \left (a +b \tan \left (d x +c \right )\right )^{3}}-\frac {15 a^{4}+18 a^{2} b^{2}+3 b^{4}}{b^{7} \left (a +b \tan \left (d x +c \right )\right )}+\frac {3 a \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}{b^{7} \left (a +b \tan \left (d x +c \right )\right )^{2}}}{d}\) | \(195\) |
| default | \(\frac {\frac {\frac {b^{2} \tan \left (d x +c \right )^{3}}{3}-2 a \tan \left (d x +c \right )^{2} b +10 \tan \left (d x +c \right ) a^{2}+3 \tan \left (d x +c \right ) b^{2}}{b^{6}}-\frac {4 a \left (5 a^{2}+3 b^{2}\right ) \ln \left (a +b \tan \left (d x +c \right )\right )}{b^{7}}-\frac {a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}}{3 b^{7} \left (a +b \tan \left (d x +c \right )\right )^{3}}-\frac {15 a^{4}+18 a^{2} b^{2}+3 b^{4}}{b^{7} \left (a +b \tan \left (d x +c \right )\right )}+\frac {3 a \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}{b^{7} \left (a +b \tan \left (d x +c \right )\right )^{2}}}{d}\) | \(195\) |
| risch | \(\frac {8 i \left (-4 b^{5}+45 a^{4} b -3 a^{2} b^{3}+12 b^{5} {\mathrm e}^{4 i \left (d x +c \right )}-130 i a^{3} b^{2} {\mathrm e}^{6 i \left (d x +c \right )}+15 i a \,b^{4} {\mathrm e}^{2 i \left (d x +c \right )}+6 i a^{3} b^{2} {\mathrm e}^{10 i \left (d x +c \right )}+9 i a \,b^{4} {\mathrm e}^{10 i \left (d x +c \right )}-45 i a^{3} b^{2} {\mathrm e}^{8 i \left (d x +c \right )}-60 i a^{3} b^{2} {\mathrm e}^{4 i \left (d x +c \right )}-12 i a \,b^{4} {\mathrm e}^{4 i \left (d x +c \right )}+60 i a^{3} b^{2} {\mathrm e}^{2 i \left (d x +c \right )}-24 i a \,b^{4} {\mathrm e}^{6 i \left (d x +c \right )}-15 i a^{5}-15 i a^{5} {\mathrm e}^{10 i \left (d x +c \right )}-75 i a^{5} {\mathrm e}^{8 i \left (d x +c \right )}-150 i a^{5} {\mathrm e}^{6 i \left (d x +c \right )}-150 i a^{5} {\mathrm e}^{4 i \left (d x +c \right )}-75 i a^{5} {\mathrm e}^{2 i \left (d x +c \right )}+41 i a^{3} b^{2}+12 i a \,b^{4}-45 a^{2} b^{3} {\mathrm e}^{8 i \left (d x +c \right )}-75 a^{4} b \,{\mathrm e}^{8 i \left (d x +c \right )}-30 a^{4} b \,{\mathrm e}^{10 i \left (d x +c \right )}+30 a^{2} b^{3} {\mathrm e}^{2 i \left (d x +c \right )}+60 a^{2} b^{3} {\mathrm e}^{4 i \left (d x +c \right )}+150 a^{4} b \,{\mathrm e}^{2 i \left (d x +c \right )}+150 a^{4} b \,{\mathrm e}^{4 i \left (d x +c \right )}-18 a^{2} b^{3} {\mathrm e}^{10 i \left (d x +c \right )}\right )}{3 \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{3} \left (b \,{\mathrm e}^{2 i \left (d x +c \right )}+i a \,{\mathrm e}^{2 i \left (d x +c \right )}-b +i a \right )^{3} b^{6} d}-\frac {20 a^{3} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-\frac {i b +a}{i b -a}\right )}{b^{7} d}-\frac {12 a \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-\frac {i b +a}{i b -a}\right )}{b^{5} d}+\frac {20 a^{3} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}{b^{7} d}+\frac {12 a \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}{b^{5} d}\) | \(593\) |
| norman | \(\frac {\frac {4 \left (100 a^{8}-160 a^{6} b^{2}-140 a^{4} b^{4}-a^{2} b^{6}-2 b^{8}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{a^{3} d \,b^{6}}-\frac {4 \left (100 a^{8}-160 a^{6} b^{2}-140 a^{4} b^{4}-a^{2} b^{6}-2 b^{8}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{a^{3} d \,b^{6}}-\frac {2 \left (300 a^{8}-260 a^{6} b^{2}-264 a^{4} b^{4}+3 a^{2} b^{6}-4 b^{8}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{3 a^{3} d \,b^{6}}+\frac {2 \left (300 a^{8}-260 a^{6} b^{2}-264 a^{4} b^{4}+3 a^{2} b^{6}-4 b^{8}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{3 a^{3} d \,b^{6}}-\frac {\left (800 a^{6}+400 a^{4} b^{2}-48 a^{2} b^{4}+16 b^{6}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{a^{2} b^{5} d}-\frac {\left (800 a^{6}+400 a^{4} b^{2}-48 a^{2} b^{4}+16 b^{6}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}}{a^{2} b^{5} d}+\frac {2 \left (100 a^{6}+60 a^{4} b^{2}+2 b^{6}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{a^{2} d \,b^{5}}+\frac {2 \left (100 a^{6}+60 a^{4} b^{2}+2 b^{6}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}}{a^{2} d \,b^{5}}+\frac {4 \left (900 a^{6}+420 a^{4} b^{2}-56 a^{2} b^{4}+18 b^{6}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{3 b^{5} d \,a^{2}}+\frac {2 \left (20 a^{6}+12 a^{4} b^{2}+b^{6}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{b^{6} d a}-\frac {2 \left (20 a^{6}+12 a^{4} b^{2}+b^{6}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{11}}{b^{6} d a}}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1\right )^{3} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -2 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-a \right )^{3}}+\frac {4 \left (5 a^{2}+3 b^{2}\right ) a \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{b^{7} d}+\frac {4 \left (5 a^{2}+3 b^{2}\right ) a \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{b^{7} d}-\frac {4 \left (5 a^{2}+3 b^{2}\right ) a \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -2 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-a \right )}{b^{7} d}\) | \(697\) |
| parallelrisch | \(\frac {-900 a^{2} \left (a^{2}+\frac {3 b^{2}}{5}\right ) \left (\left (a^{3}+\frac {1}{5} a \,b^{2}\right ) \cos \left (2 d x +2 c \right )+\frac {2 \left (a^{3}-a \,b^{2}\right ) \cos \left (4 d x +4 c \right )}{5}+\frac {\left (\frac {1}{3} a^{3}-a \,b^{2}\right ) \cos \left (6 d x +6 c \right )}{5}+\left (a^{2} b +\frac {1}{5} b^{3}\right ) \sin \left (2 d x +2 c \right )+\frac {\left (a^{2} b -\frac {1}{3} b^{3}\right ) \sin \left (6 d x +6 c \right )}{5}+\frac {4 a^{2} b \sin \left (4 d x +4 c \right )}{5}+\frac {2 a^{3}}{3}+\frac {2 a \,b^{2}}{5}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -2 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-a \right )+900 a^{2} \left (a^{2}+\frac {3 b^{2}}{5}\right ) \left (\left (a^{3}+\frac {1}{5} a \,b^{2}\right ) \cos \left (2 d x +2 c \right )+\frac {2 \left (a^{3}-a \,b^{2}\right ) \cos \left (4 d x +4 c \right )}{5}+\frac {\left (\frac {1}{3} a^{3}-a \,b^{2}\right ) \cos \left (6 d x +6 c \right )}{5}+\left (a^{2} b +\frac {1}{5} b^{3}\right ) \sin \left (2 d x +2 c \right )+\frac {\left (a^{2} b -\frac {1}{3} b^{3}\right ) \sin \left (6 d x +6 c \right )}{5}+\frac {4 a^{2} b \sin \left (4 d x +4 c \right )}{5}+\frac {2 a^{3}}{3}+\frac {2 a \,b^{2}}{5}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )+900 a^{2} \left (a^{2}+\frac {3 b^{2}}{5}\right ) \left (\left (a^{3}+\frac {1}{5} a \,b^{2}\right ) \cos \left (2 d x +2 c \right )+\frac {2 \left (a^{3}-a \,b^{2}\right ) \cos \left (4 d x +4 c \right )}{5}+\frac {\left (\frac {1}{3} a^{3}-a \,b^{2}\right ) \cos \left (6 d x +6 c \right )}{5}+\left (a^{2} b +\frac {1}{5} b^{3}\right ) \sin \left (2 d x +2 c \right )+\frac {\left (a^{2} b -\frac {1}{3} b^{3}\right ) \sin \left (6 d x +6 c \right )}{5}+\frac {4 a^{2} b \sin \left (4 d x +4 c \right )}{5}+\frac {2 a^{3}}{3}+\frac {2 a \,b^{2}}{5}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )+3 \left (-100 a^{7}-30 a^{5} b^{2}+13 a^{3} b^{4}-13 a \,b^{6}\right ) \cos \left (2 d x +2 c \right )+6 \left (-20 a^{7}-42 a^{5} b^{2}-23 a^{3} b^{4}-3 a \,b^{6}\right ) \cos \left (4 d x +4 c \right )+\left (-20 a^{7}-102 a^{5} b^{2}-39 a^{3} b^{4}+7 a \,b^{6}\right ) \cos \left (6 d x +6 c \right )+3 \left (90 a^{4} b^{3}+49 a^{2} b^{5}+3 b^{7}\right ) \sin \left (2 d x +2 c \right )+3 \left (-30 a^{4} b^{3}-19 a^{2} b^{5}-b^{7}\right ) \sin \left (6 d x +6 c \right )+12 a^{2} b^{5} \sin \left (4 d x +4 c \right )-200 a^{7}+60 a^{5} b^{2}+138 a^{3} b^{4}+18 a \,b^{6}}{3 b^{7} d \left (15 \left (a^{2}+\frac {b^{2}}{5}\right ) a^{2} \cos \left (2 d x +2 c \right )+a \left (6 a^{3} \cos \left (4 d x +4 c \right )+a^{3} \cos \left (6 d x +6 c \right )+12 a^{2} b \sin \left (4 d x +4 c \right )+3 a^{2} b \sin \left (6 d x +6 c \right )+15 a^{2} b \sin \left (2 d x +2 c \right )-6 a \,b^{2} \cos \left (4 d x +4 c \right )-3 a \,b^{2} \cos \left (6 d x +6 c \right )-b^{3} \sin \left (6 d x +6 c \right )+3 \sin \left (2 d x +2 c \right ) b^{3}+10 a^{3}+6 a \,b^{2}\right )\right )}\) | \(904\) |
1/d*(1/b^6*(1/3*b^2*tan(d*x+c)^3-2*a*tan(d*x+c)^2*b+10*tan(d*x+c)*a^2+3*ta n(d*x+c)*b^2)-4*a/b^7*(5*a^2+3*b^2)*ln(a+b*tan(d*x+c))-1/3/b^7*(a^6+3*a^4* b^2+3*a^2*b^4+b^6)/(a+b*tan(d*x+c))^3-(15*a^4+18*a^2*b^2+3*b^4)/b^7/(a+b*t an(d*x+c))+3*a/b^7*(a^4+2*a^2*b^2+b^4)/(a+b*tan(d*x+c))^2)
Leaf count of result is larger than twice the leaf count of optimal. 553 vs. \(2 (228) = 456\).
Time = 0.29 (sec) , antiderivative size = 553, normalized size of antiderivative = 2.38 \[ \int \frac {\sec ^4(c+d x)}{(a \cos (c+d x)+b \sin (c+d x))^4} \, dx=-\frac {4 \, {\left (45 \, a^{4} b^{2} - 3 \, a^{2} b^{4} - 4 \, b^{6}\right )} \cos \left (d x + c\right )^{6} - b^{6} - 6 \, {\left (25 \, a^{4} b^{2} - 5 \, a^{2} b^{4} - 4 \, b^{6}\right )} \cos \left (d x + c\right )^{4} - 3 \, {\left (5 \, a^{2} b^{4} + 2 \, b^{6}\right )} \cos \left (d x + c\right )^{2} + 6 \, {\left ({\left (5 \, a^{6} - 12 \, a^{4} b^{2} - 9 \, a^{2} b^{4}\right )} \cos \left (d x + c\right )^{6} + 3 \, {\left (5 \, a^{4} b^{2} + 3 \, a^{2} b^{4}\right )} \cos \left (d x + c\right )^{4} + {\left ({\left (15 \, a^{5} b + 4 \, a^{3} b^{3} - 3 \, a b^{5}\right )} \cos \left (d x + c\right )^{5} + {\left (5 \, a^{3} b^{3} + 3 \, a b^{5}\right )} \cos \left (d x + c\right )^{3}\right )} \sin \left (d x + c\right )\right )} \log \left (2 \, a b \cos \left (d x + c\right ) \sin \left (d x + c\right ) + {\left (a^{2} - b^{2}\right )} \cos \left (d x + c\right )^{2} + b^{2}\right ) - 6 \, {\left ({\left (5 \, a^{6} - 12 \, a^{4} b^{2} - 9 \, a^{2} b^{4}\right )} \cos \left (d x + c\right )^{6} + 3 \, {\left (5 \, a^{4} b^{2} + 3 \, a^{2} b^{4}\right )} \cos \left (d x + c\right )^{4} + {\left ({\left (15 \, a^{5} b + 4 \, a^{3} b^{3} - 3 \, a b^{5}\right )} \cos \left (d x + c\right )^{5} + {\left (5 \, a^{3} b^{3} + 3 \, a b^{5}\right )} \cos \left (d x + c\right )^{3}\right )} \sin \left (d x + c\right )\right )} \log \left (\cos \left (d x + c\right )^{2}\right ) + {\left (3 \, a b^{5} \cos \left (d x + c\right ) - 4 \, {\left (15 \, a^{5} b - 41 \, a^{3} b^{3} - 12 \, a b^{5}\right )} \cos \left (d x + c\right )^{5} - 2 \, {\left (55 \, a^{3} b^{3} + 21 \, a b^{5}\right )} \cos \left (d x + c\right )^{3}\right )} \sin \left (d x + c\right )}{3 \, {\left (3 \, a b^{9} d \cos \left (d x + c\right )^{4} + {\left (a^{3} b^{7} - 3 \, a b^{9}\right )} d \cos \left (d x + c\right )^{6} + {\left (b^{10} d \cos \left (d x + c\right )^{3} + {\left (3 \, a^{2} b^{8} - b^{10}\right )} d \cos \left (d x + c\right )^{5}\right )} \sin \left (d x + c\right )\right )}} \]
-1/3*(4*(45*a^4*b^2 - 3*a^2*b^4 - 4*b^6)*cos(d*x + c)^6 - b^6 - 6*(25*a^4* b^2 - 5*a^2*b^4 - 4*b^6)*cos(d*x + c)^4 - 3*(5*a^2*b^4 + 2*b^6)*cos(d*x + c)^2 + 6*((5*a^6 - 12*a^4*b^2 - 9*a^2*b^4)*cos(d*x + c)^6 + 3*(5*a^4*b^2 + 3*a^2*b^4)*cos(d*x + c)^4 + ((15*a^5*b + 4*a^3*b^3 - 3*a*b^5)*cos(d*x + c )^5 + (5*a^3*b^3 + 3*a*b^5)*cos(d*x + c)^3)*sin(d*x + c))*log(2*a*b*cos(d* x + c)*sin(d*x + c) + (a^2 - b^2)*cos(d*x + c)^2 + b^2) - 6*((5*a^6 - 12*a ^4*b^2 - 9*a^2*b^4)*cos(d*x + c)^6 + 3*(5*a^4*b^2 + 3*a^2*b^4)*cos(d*x + c )^4 + ((15*a^5*b + 4*a^3*b^3 - 3*a*b^5)*cos(d*x + c)^5 + (5*a^3*b^3 + 3*a* b^5)*cos(d*x + c)^3)*sin(d*x + c))*log(cos(d*x + c)^2) + (3*a*b^5*cos(d*x + c) - 4*(15*a^5*b - 41*a^3*b^3 - 12*a*b^5)*cos(d*x + c)^5 - 2*(55*a^3*b^3 + 21*a*b^5)*cos(d*x + c)^3)*sin(d*x + c))/(3*a*b^9*d*cos(d*x + c)^4 + (a^ 3*b^7 - 3*a*b^9)*d*cos(d*x + c)^6 + (b^10*d*cos(d*x + c)^3 + (3*a^2*b^8 - b^10)*d*cos(d*x + c)^5)*sin(d*x + c))
\[ \int \frac {\sec ^4(c+d x)}{(a \cos (c+d x)+b \sin (c+d x))^4} \, dx=\int \frac {\sec ^{4}{\left (c + d x \right )}}{\left (a \cos {\left (c + d x \right )} + b \sin {\left (c + d x \right )}\right )^{4}}\, dx \]
Time = 0.25 (sec) , antiderivative size = 217, normalized size of antiderivative = 0.94 \[ \int \frac {\sec ^4(c+d x)}{(a \cos (c+d x)+b \sin (c+d x))^4} \, dx=-\frac {\frac {37 \, a^{6} + 39 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6} + 9 \, {\left (5 \, a^{4} b^{2} + 6 \, a^{2} b^{4} + b^{6}\right )} \tan \left (d x + c\right )^{2} + 9 \, {\left (9 \, a^{5} b + 10 \, a^{3} b^{3} + a b^{5}\right )} \tan \left (d x + c\right )}{b^{10} \tan \left (d x + c\right )^{3} + 3 \, a b^{9} \tan \left (d x + c\right )^{2} + 3 \, a^{2} b^{8} \tan \left (d x + c\right ) + a^{3} b^{7}} - \frac {b^{2} \tan \left (d x + c\right )^{3} - 6 \, a b \tan \left (d x + c\right )^{2} + 3 \, {\left (10 \, a^{2} + 3 \, b^{2}\right )} \tan \left (d x + c\right )}{b^{6}} + \frac {12 \, {\left (5 \, a^{3} + 3 \, a b^{2}\right )} \log \left (b \tan \left (d x + c\right ) + a\right )}{b^{7}}}{3 \, d} \]
-1/3*((37*a^6 + 39*a^4*b^2 + 3*a^2*b^4 + b^6 + 9*(5*a^4*b^2 + 6*a^2*b^4 + b^6)*tan(d*x + c)^2 + 9*(9*a^5*b + 10*a^3*b^3 + a*b^5)*tan(d*x + c))/(b^10 *tan(d*x + c)^3 + 3*a*b^9*tan(d*x + c)^2 + 3*a^2*b^8*tan(d*x + c) + a^3*b^ 7) - (b^2*tan(d*x + c)^3 - 6*a*b*tan(d*x + c)^2 + 3*(10*a^2 + 3*b^2)*tan(d *x + c))/b^6 + 12*(5*a^3 + 3*a*b^2)*log(b*tan(d*x + c) + a)/b^7)/d
Time = 0.37 (sec) , antiderivative size = 249, normalized size of antiderivative = 1.07 \[ \int \frac {\sec ^4(c+d x)}{(a \cos (c+d x)+b \sin (c+d x))^4} \, dx=-\frac {\frac {12 \, {\left (5 \, a^{3} + 3 \, a b^{2}\right )} \log \left ({\left | b \tan \left (d x + c\right ) + a \right |}\right )}{b^{7}} - \frac {110 \, a^{3} b^{3} \tan \left (d x + c\right )^{3} + 66 \, a b^{5} \tan \left (d x + c\right )^{3} + 285 \, a^{4} b^{2} \tan \left (d x + c\right )^{2} + 144 \, a^{2} b^{4} \tan \left (d x + c\right )^{2} - 9 \, b^{6} \tan \left (d x + c\right )^{2} + 249 \, a^{5} b \tan \left (d x + c\right ) + 108 \, a^{3} b^{3} \tan \left (d x + c\right ) - 9 \, a b^{5} \tan \left (d x + c\right ) + 73 \, a^{6} + 27 \, a^{4} b^{2} - 3 \, a^{2} b^{4} - b^{6}}{{\left (b \tan \left (d x + c\right ) + a\right )}^{3} b^{7}} - \frac {b^{8} \tan \left (d x + c\right )^{3} - 6 \, a b^{7} \tan \left (d x + c\right )^{2} + 30 \, a^{2} b^{6} \tan \left (d x + c\right ) + 9 \, b^{8} \tan \left (d x + c\right )}{b^{12}}}{3 \, d} \]
-1/3*(12*(5*a^3 + 3*a*b^2)*log(abs(b*tan(d*x + c) + a))/b^7 - (110*a^3*b^3 *tan(d*x + c)^3 + 66*a*b^5*tan(d*x + c)^3 + 285*a^4*b^2*tan(d*x + c)^2 + 1 44*a^2*b^4*tan(d*x + c)^2 - 9*b^6*tan(d*x + c)^2 + 249*a^5*b*tan(d*x + c) + 108*a^3*b^3*tan(d*x + c) - 9*a*b^5*tan(d*x + c) + 73*a^6 + 27*a^4*b^2 - 3*a^2*b^4 - b^6)/((b*tan(d*x + c) + a)^3*b^7) - (b^8*tan(d*x + c)^3 - 6*a* b^7*tan(d*x + c)^2 + 30*a^2*b^6*tan(d*x + c) + 9*b^8*tan(d*x + c))/b^12)/d
Time = 32.43 (sec) , antiderivative size = 1599, normalized size of antiderivative = 6.89 \[ \int \frac {\sec ^4(c+d x)}{(a \cos (c+d x)+b \sin (c+d x))^4} \, dx=\text {Too large to display} \]
((4*tan(c/2 + (d*x)/2)^2*(50*a^6 + b^6 + 30*a^4*b^2))/(a^2*b^5) - (16*tan( c/2 + (d*x)/2)^8*(50*a^6 + b^6 - 3*a^2*b^4 + 25*a^4*b^2))/(a^2*b^5) - (2*t an(c/2 + (d*x)/2)^11*(20*a^6 + b^6 + 12*a^4*b^2))/(a*b^6) - (16*tan(c/2 + (d*x)/2)^4*(50*a^6 + b^6 - 3*a^2*b^4 + 25*a^4*b^2))/(a^2*b^5) + (4*tan(c/2 + (d*x)/2)^10*(50*a^6 + b^6 + 30*a^4*b^2))/(a^2*b^5) - (4*tan(c/2 + (d*x) /2)^5*(2*b^8 - 100*a^8 + a^2*b^6 + 140*a^4*b^4 + 160*a^6*b^2))/(a^3*b^6) + (4*tan(c/2 + (d*x)/2)^7*(2*b^8 - 100*a^8 + a^2*b^6 + 140*a^4*b^4 + 160*a^ 6*b^2))/(a^3*b^6) + (2*tan(c/2 + (d*x)/2)^3*(4*b^8 - 300*a^8 - 3*a^2*b^6 + 264*a^4*b^4 + 260*a^6*b^2))/(3*a^3*b^6) - (2*tan(c/2 + (d*x)/2)^9*(4*b^8 - 300*a^8 - 3*a^2*b^6 + 264*a^4*b^4 + 260*a^6*b^2))/(3*a^3*b^6) + (8*tan(c /2 + (d*x)/2)^6*(450*a^6 + 9*b^6 - 28*a^2*b^4 + 210*a^4*b^2))/(3*a^2*b^5) + (2*tan(c/2 + (d*x)/2)*(20*a^6 + b^6 + 12*a^4*b^2))/(a*b^6))/(d*(a^3*tan( c/2 + (d*x)/2)^12 + tan(c/2 + (d*x)/2)^2*(12*a*b^2 - 6*a^3) + tan(c/2 + (d *x)/2)^10*(12*a*b^2 - 6*a^3) - tan(c/2 + (d*x)/2)^4*(48*a*b^2 - 15*a^3) - tan(c/2 + (d*x)/2)^8*(48*a*b^2 - 15*a^3) + tan(c/2 + (d*x)/2)^6*(72*a*b^2 - 20*a^3) - tan(c/2 + (d*x)/2)^3*(30*a^2*b - 8*b^3) + tan(c/2 + (d*x)/2)^9 *(30*a^2*b - 8*b^3) + tan(c/2 + (d*x)/2)^5*(60*a^2*b - 24*b^3) - tan(c/2 + (d*x)/2)^7*(60*a^2*b - 24*b^3) + a^3 + 6*a^2*b*tan(c/2 + (d*x)/2) - 6*a^2 *b*tan(c/2 + (d*x)/2)^11)) + (a*atan(((a*(5*a^2 + 3*b^2)*((16*tan(c/2 + (d *x)/2)*(20*a^5 + 12*a^3*b^2))/b^6 - (4*(24*a^2*b^9 + 40*a^4*b^7))/b^12 ...