Integrand size = 19, antiderivative size = 156 \[ \int \frac {1}{(a \cos (c+d x)+b \sin (c+d x))^5} \, dx=-\frac {3 \text {arctanh}\left (\frac {b \cos (c+d x)-a \sin (c+d x)}{\sqrt {a^2+b^2}}\right )}{8 \left (a^2+b^2\right )^{5/2} d}-\frac {b \cos (c+d x)-a \sin (c+d x)}{4 \left (a^2+b^2\right ) d (a \cos (c+d x)+b \sin (c+d x))^4}-\frac {3 (b \cos (c+d x)-a \sin (c+d x))}{8 \left (a^2+b^2\right )^2 d (a \cos (c+d x)+b \sin (c+d x))^2} \]
-3/8*arctanh((b*cos(d*x+c)-a*sin(d*x+c))/(a^2+b^2)^(1/2))/(a^2+b^2)^(5/2)/ d+1/4*(-b*cos(d*x+c)+a*sin(d*x+c))/(a^2+b^2)/d/(a*cos(d*x+c)+b*sin(d*x+c)) ^4-3/8*(b*cos(d*x+c)-a*sin(d*x+c))/(a^2+b^2)^2/d/(a*cos(d*x+c)+b*sin(d*x+c ))^2
Time = 1.03 (sec) , antiderivative size = 157, normalized size of antiderivative = 1.01 \[ \int \frac {1}{(a \cos (c+d x)+b \sin (c+d x))^5} \, dx=\frac {\frac {6 \text {arctanh}\left (\frac {-b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{5/2}}+\frac {-11 b \left (a^2+b^2\right ) \cos (c+d x)+\left (-9 a^2 b+3 b^3\right ) \cos (3 (c+d x))+2 a \left (7 a^2+b^2+3 \left (a^2-3 b^2\right ) \cos (2 (c+d x))\right ) \sin (c+d x)}{4 \left (a^2+b^2\right )^2 (a \cos (c+d x)+b \sin (c+d x))^4}}{8 d} \]
((6*ArcTanh[(-b + a*Tan[(c + d*x)/2])/Sqrt[a^2 + b^2]])/(a^2 + b^2)^(5/2) + (-11*b*(a^2 + b^2)*Cos[c + d*x] + (-9*a^2*b + 3*b^3)*Cos[3*(c + d*x)] + 2*a*(7*a^2 + b^2 + 3*(a^2 - 3*b^2)*Cos[2*(c + d*x)])*Sin[c + d*x])/(4*(a^2 + b^2)^2*(a*Cos[c + d*x] + b*Sin[c + d*x])^4))/(8*d)
Time = 0.47 (sec) , antiderivative size = 170, normalized size of antiderivative = 1.09, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.368, Rules used = {3042, 3555, 3042, 3555, 3042, 3553, 219}
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.
| \(\displaystyle \int \frac {1}{(a \cos (c+d x)+b \sin (c+d x))^5} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{(a \cos (c+d x)+b \sin (c+d x))^5}dx\) |
| \(\Big \downarrow \) 3555 |
| \(\displaystyle \frac {3 \int \frac {1}{(a \cos (c+d x)+b \sin (c+d x))^3}dx}{4 \left (a^2+b^2\right )}-\frac {b \cos (c+d x)-a \sin (c+d x)}{4 d \left (a^2+b^2\right ) (a \cos (c+d x)+b \sin (c+d x))^4}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {3 \int \frac {1}{(a \cos (c+d x)+b \sin (c+d x))^3}dx}{4 \left (a^2+b^2\right )}-\frac {b \cos (c+d x)-a \sin (c+d x)}{4 d \left (a^2+b^2\right ) (a \cos (c+d x)+b \sin (c+d x))^4}\) |
| \(\Big \downarrow \) 3555 |
| \(\displaystyle \frac {3 \left (\frac {\int \frac {1}{a \cos (c+d x)+b \sin (c+d x)}dx}{2 \left (a^2+b^2\right )}-\frac {b \cos (c+d x)-a \sin (c+d x)}{2 d \left (a^2+b^2\right ) (a \cos (c+d x)+b \sin (c+d x))^2}\right )}{4 \left (a^2+b^2\right )}-\frac {b \cos (c+d x)-a \sin (c+d x)}{4 d \left (a^2+b^2\right ) (a \cos (c+d x)+b \sin (c+d x))^4}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {3 \left (\frac {\int \frac {1}{a \cos (c+d x)+b \sin (c+d x)}dx}{2 \left (a^2+b^2\right )}-\frac {b \cos (c+d x)-a \sin (c+d x)}{2 d \left (a^2+b^2\right ) (a \cos (c+d x)+b \sin (c+d x))^2}\right )}{4 \left (a^2+b^2\right )}-\frac {b \cos (c+d x)-a \sin (c+d x)}{4 d \left (a^2+b^2\right ) (a \cos (c+d x)+b \sin (c+d x))^4}\) |
| \(\Big \downarrow \) 3553 |
| \(\displaystyle \frac {3 \left (-\frac {\int \frac {1}{a^2+b^2-(b \cos (c+d x)-a \sin (c+d x))^2}d(b \cos (c+d x)-a \sin (c+d x))}{2 d \left (a^2+b^2\right )}-\frac {b \cos (c+d x)-a \sin (c+d x)}{2 d \left (a^2+b^2\right ) (a \cos (c+d x)+b \sin (c+d x))^2}\right )}{4 \left (a^2+b^2\right )}-\frac {b \cos (c+d x)-a \sin (c+d x)}{4 d \left (a^2+b^2\right ) (a \cos (c+d x)+b \sin (c+d x))^4}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {3 \left (-\frac {\text {arctanh}\left (\frac {b \cos (c+d x)-a \sin (c+d x)}{\sqrt {a^2+b^2}}\right )}{2 d \left (a^2+b^2\right )^{3/2}}-\frac {b \cos (c+d x)-a \sin (c+d x)}{2 d \left (a^2+b^2\right ) (a \cos (c+d x)+b \sin (c+d x))^2}\right )}{4 \left (a^2+b^2\right )}-\frac {b \cos (c+d x)-a \sin (c+d x)}{4 d \left (a^2+b^2\right ) (a \cos (c+d x)+b \sin (c+d x))^4}\) |
-1/4*(b*Cos[c + d*x] - a*Sin[c + d*x])/((a^2 + b^2)*d*(a*Cos[c + d*x] + b* Sin[c + d*x])^4) + (3*(-1/2*ArcTanh[(b*Cos[c + d*x] - a*Sin[c + d*x])/Sqrt [a^2 + b^2]]/((a^2 + b^2)^(3/2)*d) - (b*Cos[c + d*x] - a*Sin[c + d*x])/(2* (a^2 + b^2)*d*(a*Cos[c + d*x] + b*Sin[c + d*x])^2)))/(4*(a^2 + b^2))
3.3.30.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[(cos[(c_.) + (d_.)*(x_)]*(a_.) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(-1), x _Symbol] :> Simp[-d^(-1) Subst[Int[1/(a^2 + b^2 - x^2), x], x, b*Cos[c + d*x] - a*Sin[c + d*x]], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 + b^2, 0]
Int[(cos[(c_.) + (d_.)*(x_)]*(a_.) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x _Symbol] :> Simp[(b*Cos[c + d*x] - a*Sin[c + d*x])*((a*Cos[c + d*x] + b*Sin [c + d*x])^(n + 1)/(d*(n + 1)*(a^2 + b^2))), x] + Simp[(n + 2)/((n + 1)*(a^ 2 + b^2)) Int[(a*Cos[c + d*x] + b*Sin[c + d*x])^(n + 2), x], x] /; FreeQ[ {a, b, c, d}, x] && NeQ[a^2 + b^2, 0] && LtQ[n, -1] && NeQ[n, -2]
Result contains complex when optimal does not.
Time = 2.22 (sec) , antiderivative size = 454, normalized size of antiderivative = 2.91
| method | result | size |
| risch | \(\frac {{\mathrm e}^{i \left (d x +c \right )} \left (3 i a^{3}-9 i a \,b^{2}-9 a^{2} b \,{\mathrm e}^{6 i \left (d x +c \right )}+3 b^{3} {\mathrm e}^{6 i \left (d x +c \right )}-3 i a^{3} {\mathrm e}^{6 i \left (d x +c \right )}-11 i a \,b^{2} {\mathrm e}^{4 i \left (d x +c \right )}-11 a^{2} b \,{\mathrm e}^{4 i \left (d x +c \right )}-11 b^{3} {\mathrm e}^{4 i \left (d x +c \right )}+11 i a \,b^{2} {\mathrm e}^{2 i \left (d x +c \right )}+11 i a^{3} {\mathrm e}^{2 i \left (d x +c \right )}-11 a^{2} b \,{\mathrm e}^{2 i \left (d x +c \right )}-11 b^{3} {\mathrm e}^{2 i \left (d x +c \right )}+9 i a \,b^{2} {\mathrm e}^{6 i \left (d x +c \right )}-11 i a^{3} {\mathrm e}^{4 i \left (d x +c \right )}-9 a^{2} b +3 b^{3}\right )}{4 \left (-i a +b \right )^{2} \left (b \,{\mathrm e}^{2 i \left (d x +c \right )}+i a \,{\mathrm e}^{2 i \left (d x +c \right )}-b +i a \right )^{4} d \left (i a +b \right )^{2}}+\frac {3 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i a^{5}+2 i a^{3} b^{2}+i a \,b^{4}-a^{4} b -2 a^{2} b^{3}-b^{5}}{\left (a^{2}+b^{2}\right )^{\frac {5}{2}}}\right )}{8 \left (a^{2}+b^{2}\right )^{\frac {5}{2}} d}-\frac {3 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-\frac {i a^{5}+2 i a^{3} b^{2}+i a \,b^{4}-a^{4} b -2 a^{2} b^{3}-b^{5}}{\left (a^{2}+b^{2}\right )^{\frac {5}{2}}}\right )}{8 \left (a^{2}+b^{2}\right )^{\frac {5}{2}} d}\) | \(454\) |
| derivativedivides | \(\frac {-\frac {2 \left (-\frac {\left (5 a^{4}+16 a^{2} b^{2}+8 b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{8 a \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}+\frac {3 b \left (a^{4}+16 a^{2} b^{2}+8 b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{8 a^{2} \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}-\frac {\left (3 a^{6}-36 a^{4} b^{2}+56 a^{2} b^{4}+32 b^{6}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{8 a^{3} \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}+\frac {b \left (15 a^{6}-114 a^{4} b^{2}-8 a^{2} b^{4}+16 b^{6}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{8 a^{4} \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}-\frac {\left (3 a^{6}+84 a^{4} b^{2}-56 a^{2} b^{4}-32 b^{6}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{8 a^{3} \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}-\frac {b \left (23 a^{4}-64 a^{2} b^{2}-24 b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{8 a^{2} \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}-\frac {\left (5 a^{4}-24 a^{2} b^{2}-8 b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 a \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}+\frac {b \left (5 a^{2}+2 b^{2}\right )}{8 a^{4}+16 a^{2} b^{2}+8 b^{4}}\right )}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -2 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-a \right )^{4}}+\frac {3 \,\operatorname {arctanh}\left (\frac {2 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-2 b}{2 \sqrt {a^{2}+b^{2}}}\right )}{4 \left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) \sqrt {a^{2}+b^{2}}}}{d}\) | \(514\) |
| default | \(\frac {-\frac {2 \left (-\frac {\left (5 a^{4}+16 a^{2} b^{2}+8 b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{8 a \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}+\frac {3 b \left (a^{4}+16 a^{2} b^{2}+8 b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{8 a^{2} \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}-\frac {\left (3 a^{6}-36 a^{4} b^{2}+56 a^{2} b^{4}+32 b^{6}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{8 a^{3} \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}+\frac {b \left (15 a^{6}-114 a^{4} b^{2}-8 a^{2} b^{4}+16 b^{6}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{8 a^{4} \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}-\frac {\left (3 a^{6}+84 a^{4} b^{2}-56 a^{2} b^{4}-32 b^{6}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{8 a^{3} \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}-\frac {b \left (23 a^{4}-64 a^{2} b^{2}-24 b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{8 a^{2} \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}-\frac {\left (5 a^{4}-24 a^{2} b^{2}-8 b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 a \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}+\frac {b \left (5 a^{2}+2 b^{2}\right )}{8 a^{4}+16 a^{2} b^{2}+8 b^{4}}\right )}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -2 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-a \right )^{4}}+\frac {3 \,\operatorname {arctanh}\left (\frac {2 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-2 b}{2 \sqrt {a^{2}+b^{2}}}\right )}{4 \left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) \sqrt {a^{2}+b^{2}}}}{d}\) | \(514\) |
1/4*exp(I*(d*x+c))*(3*I*a^3-9*I*a*b^2-9*a^2*b*exp(I*(d*x+c))^6+3*b^3*exp(I *(d*x+c))^6+11*I*a^3*exp(I*(d*x+c))^2+9*I*a*b^2*exp(I*(d*x+c))^6-11*a^2*b* exp(I*(d*x+c))^4-11*b^3*exp(I*(d*x+c))^4-11*I*a^3*exp(I*(d*x+c))^4+11*I*a* b^2*exp(I*(d*x+c))^2-11*a^2*b*exp(I*(d*x+c))^2-11*b^3*exp(I*(d*x+c))^2-3*I *a^3*exp(I*(d*x+c))^6-11*I*a*b^2*exp(I*(d*x+c))^4-9*a^2*b+3*b^3)/(b-I*a)^2 /(b*exp(I*(d*x+c))^2+I*a*exp(I*(d*x+c))^2-b+I*a)^4/d/(b+I*a)^2+3/8/(a^2+b^ 2)^(5/2)/d*ln(exp(I*(d*x+c))+(I*a^5+2*I*a^3*b^2+I*a*b^4-a^4*b-2*a^2*b^3-b^ 5)/(a^2+b^2)^(5/2))-3/8/(a^2+b^2)^(5/2)/d*ln(exp(I*(d*x+c))-(I*a^5+2*I*a^3 *b^2+I*a*b^4-a^4*b-2*a^2*b^3-b^5)/(a^2+b^2)^(5/2))
Leaf count of result is larger than twice the leaf count of optimal. 544 vs. \(2 (146) = 292\).
Time = 0.29 (sec) , antiderivative size = 544, normalized size of antiderivative = 3.49 \[ \int \frac {1}{(a \cos (c+d x)+b \sin (c+d x))^5} \, dx=-\frac {6 \, {\left (3 \, a^{4} b + 2 \, a^{2} b^{3} - b^{5}\right )} \cos \left (d x + c\right )^{3} - 3 \, {\left ({\left (a^{4} - 6 \, a^{2} b^{2} + b^{4}\right )} \cos \left (d x + c\right )^{4} + b^{4} + 2 \, {\left (3 \, a^{2} b^{2} - b^{4}\right )} \cos \left (d x + c\right )^{2} + 4 \, {\left (a b^{3} \cos \left (d x + c\right ) + {\left (a^{3} b - a b^{3}\right )} \cos \left (d x + c\right )^{3}\right )} \sin \left (d x + c\right )\right )} \sqrt {a^{2} + b^{2}} \log \left (-\frac {2 \, a b \cos \left (d x + c\right ) \sin \left (d x + c\right ) + {\left (a^{2} - b^{2}\right )} \cos \left (d x + c\right )^{2} - 2 \, a^{2} - b^{2} + 2 \, \sqrt {a^{2} + b^{2}} {\left (b \cos \left (d x + c\right ) - a \sin \left (d x + c\right )\right )}}{2 \, a b \cos \left (d x + c\right ) \sin \left (d x + c\right ) + {\left (a^{2} - b^{2}\right )} \cos \left (d x + c\right )^{2} + b^{2}}\right ) - 2 \, {\left (4 \, a^{4} b - a^{2} b^{3} - 5 \, b^{5}\right )} \cos \left (d x + c\right ) - 2 \, {\left (2 \, a^{5} + 7 \, a^{3} b^{2} + 5 \, a b^{4} + 3 \, {\left (a^{5} - 2 \, a^{3} b^{2} - 3 \, a b^{4}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )}{16 \, {\left ({\left (a^{10} - 3 \, a^{8} b^{2} - 14 \, a^{6} b^{4} - 14 \, a^{4} b^{6} - 3 \, a^{2} b^{8} + b^{10}\right )} d \cos \left (d x + c\right )^{4} + 2 \, {\left (3 \, a^{8} b^{2} + 8 \, a^{6} b^{4} + 6 \, a^{4} b^{6} - b^{10}\right )} d \cos \left (d x + c\right )^{2} + {\left (a^{6} b^{4} + 3 \, a^{4} b^{6} + 3 \, a^{2} b^{8} + b^{10}\right )} d + 4 \, {\left ({\left (a^{9} b + 2 \, a^{7} b^{3} - 2 \, a^{3} b^{7} - a b^{9}\right )} d \cos \left (d x + c\right )^{3} + {\left (a^{7} b^{3} + 3 \, a^{5} b^{5} + 3 \, a^{3} b^{7} + a b^{9}\right )} d \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )\right )}} \]
-1/16*(6*(3*a^4*b + 2*a^2*b^3 - b^5)*cos(d*x + c)^3 - 3*((a^4 - 6*a^2*b^2 + b^4)*cos(d*x + c)^4 + b^4 + 2*(3*a^2*b^2 - b^4)*cos(d*x + c)^2 + 4*(a*b^ 3*cos(d*x + c) + (a^3*b - a*b^3)*cos(d*x + c)^3)*sin(d*x + c))*sqrt(a^2 + b^2)*log(-(2*a*b*cos(d*x + c)*sin(d*x + c) + (a^2 - b^2)*cos(d*x + c)^2 - 2*a^2 - b^2 + 2*sqrt(a^2 + b^2)*(b*cos(d*x + c) - a*sin(d*x + c)))/(2*a*b* cos(d*x + c)*sin(d*x + c) + (a^2 - b^2)*cos(d*x + c)^2 + b^2)) - 2*(4*a^4* b - a^2*b^3 - 5*b^5)*cos(d*x + c) - 2*(2*a^5 + 7*a^3*b^2 + 5*a*b^4 + 3*(a^ 5 - 2*a^3*b^2 - 3*a*b^4)*cos(d*x + c)^2)*sin(d*x + c))/((a^10 - 3*a^8*b^2 - 14*a^6*b^4 - 14*a^4*b^6 - 3*a^2*b^8 + b^10)*d*cos(d*x + c)^4 + 2*(3*a^8* b^2 + 8*a^6*b^4 + 6*a^4*b^6 - b^10)*d*cos(d*x + c)^2 + (a^6*b^4 + 3*a^4*b^ 6 + 3*a^2*b^8 + b^10)*d + 4*((a^9*b + 2*a^7*b^3 - 2*a^3*b^7 - a*b^9)*d*cos (d*x + c)^3 + (a^7*b^3 + 3*a^5*b^5 + 3*a^3*b^7 + a*b^9)*d*cos(d*x + c))*si n(d*x + c))
Timed out. \[ \int \frac {1}{(a \cos (c+d x)+b \sin (c+d x))^5} \, dx=\text {Timed out} \]
Leaf count of result is larger than twice the leaf count of optimal. 822 vs. \(2 (146) = 292\).
Time = 0.33 (sec) , antiderivative size = 822, normalized size of antiderivative = 5.27 \[ \int \frac {1}{(a \cos (c+d x)+b \sin (c+d x))^5} \, dx=-\frac {\frac {2 \, {\left (5 \, a^{6} b + 2 \, a^{4} b^{3} - \frac {{\left (5 \, a^{7} - 24 \, a^{5} b^{2} - 8 \, a^{3} b^{4}\right )} \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {{\left (23 \, a^{6} b - 64 \, a^{4} b^{3} - 24 \, a^{2} b^{5}\right )} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {{\left (3 \, a^{7} + 84 \, a^{5} b^{2} - 56 \, a^{3} b^{4} - 32 \, a b^{6}\right )} \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {{\left (15 \, a^{6} b - 114 \, a^{4} b^{3} - 8 \, a^{2} b^{5} + 16 \, b^{7}\right )} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac {{\left (3 \, a^{7} - 36 \, a^{5} b^{2} + 56 \, a^{3} b^{4} + 32 \, a b^{6}\right )} \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {3 \, {\left (a^{6} b + 16 \, a^{4} b^{3} + 8 \, a^{2} b^{5}\right )} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} - \frac {{\left (5 \, a^{7} + 16 \, a^{5} b^{2} + 8 \, a^{3} b^{4}\right )} \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}}\right )}}{a^{12} + 2 \, a^{10} b^{2} + a^{8} b^{4} + \frac {8 \, {\left (a^{11} b + 2 \, a^{9} b^{3} + a^{7} b^{5}\right )} \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {4 \, {\left (a^{12} - 4 \, a^{10} b^{2} - 11 \, a^{8} b^{4} - 6 \, a^{6} b^{6}\right )} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {8 \, {\left (3 \, a^{11} b + 2 \, a^{9} b^{3} - 5 \, a^{7} b^{5} - 4 \, a^{5} b^{7}\right )} \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {2 \, {\left (3 \, a^{12} - 18 \, a^{10} b^{2} - 37 \, a^{8} b^{4} - 8 \, a^{6} b^{6} + 8 \, a^{4} b^{8}\right )} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {8 \, {\left (3 \, a^{11} b + 2 \, a^{9} b^{3} - 5 \, a^{7} b^{5} - 4 \, a^{5} b^{7}\right )} \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac {4 \, {\left (a^{12} - 4 \, a^{10} b^{2} - 11 \, a^{8} b^{4} - 6 \, a^{6} b^{6}\right )} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} - \frac {8 \, {\left (a^{11} b + 2 \, a^{9} b^{3} + a^{7} b^{5}\right )} \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} + \frac {{\left (a^{12} + 2 \, a^{10} b^{2} + a^{8} b^{4}\right )} \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}}} + \frac {3 \, \log \left (\frac {b - \frac {a \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \sqrt {a^{2} + b^{2}}}{b - \frac {a \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \sqrt {a^{2} + b^{2}}}\right )}{{\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} \sqrt {a^{2} + b^{2}}}}{8 \, d} \]
-1/8*(2*(5*a^6*b + 2*a^4*b^3 - (5*a^7 - 24*a^5*b^2 - 8*a^3*b^4)*sin(d*x + c)/(cos(d*x + c) + 1) - (23*a^6*b - 64*a^4*b^3 - 24*a^2*b^5)*sin(d*x + c)^ 2/(cos(d*x + c) + 1)^2 - (3*a^7 + 84*a^5*b^2 - 56*a^3*b^4 - 32*a*b^6)*sin( d*x + c)^3/(cos(d*x + c) + 1)^3 + (15*a^6*b - 114*a^4*b^3 - 8*a^2*b^5 + 16 *b^7)*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 - (3*a^7 - 36*a^5*b^2 + 56*a^3*b ^4 + 32*a*b^6)*sin(d*x + c)^5/(cos(d*x + c) + 1)^5 + 3*(a^6*b + 16*a^4*b^3 + 8*a^2*b^5)*sin(d*x + c)^6/(cos(d*x + c) + 1)^6 - (5*a^7 + 16*a^5*b^2 + 8*a^3*b^4)*sin(d*x + c)^7/(cos(d*x + c) + 1)^7)/(a^12 + 2*a^10*b^2 + a^8*b ^4 + 8*(a^11*b + 2*a^9*b^3 + a^7*b^5)*sin(d*x + c)/(cos(d*x + c) + 1) - 4* (a^12 - 4*a^10*b^2 - 11*a^8*b^4 - 6*a^6*b^6)*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 - 8*(3*a^11*b + 2*a^9*b^3 - 5*a^7*b^5 - 4*a^5*b^7)*sin(d*x + c)^3/( cos(d*x + c) + 1)^3 + 2*(3*a^12 - 18*a^10*b^2 - 37*a^8*b^4 - 8*a^6*b^6 + 8 *a^4*b^8)*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 + 8*(3*a^11*b + 2*a^9*b^3 - 5*a^7*b^5 - 4*a^5*b^7)*sin(d*x + c)^5/(cos(d*x + c) + 1)^5 - 4*(a^12 - 4*a ^10*b^2 - 11*a^8*b^4 - 6*a^6*b^6)*sin(d*x + c)^6/(cos(d*x + c) + 1)^6 - 8* (a^11*b + 2*a^9*b^3 + a^7*b^5)*sin(d*x + c)^7/(cos(d*x + c) + 1)^7 + (a^12 + 2*a^10*b^2 + a^8*b^4)*sin(d*x + c)^8/(cos(d*x + c) + 1)^8) + 3*log((b - a*sin(d*x + c)/(cos(d*x + c) + 1) + sqrt(a^2 + b^2))/(b - a*sin(d*x + c)/ (cos(d*x + c) + 1) - sqrt(a^2 + b^2)))/((a^4 + 2*a^2*b^2 + b^4)*sqrt(a^2 + b^2)))/d
Leaf count of result is larger than twice the leaf count of optimal. 588 vs. \(2 (146) = 292\).
Time = 0.35 (sec) , antiderivative size = 588, normalized size of antiderivative = 3.77 \[ \int \frac {1}{(a \cos (c+d x)+b \sin (c+d x))^5} \, dx=-\frac {\frac {3 \, \log \left (\frac {{\left | 2 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 2 \, b - 2 \, \sqrt {a^{2} + b^{2}} \right |}}{{\left | 2 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 2 \, b + 2 \, \sqrt {a^{2} + b^{2}} \right |}}\right )}{{\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} \sqrt {a^{2} + b^{2}}} - \frac {2 \, {\left (5 \, a^{7} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 16 \, a^{5} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 8 \, a^{3} b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 3 \, a^{6} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 48 \, a^{4} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 24 \, a^{2} b^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 3 \, a^{7} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 36 \, a^{5} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 56 \, a^{3} b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 32 \, a b^{6} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 15 \, a^{6} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 114 \, a^{4} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 8 \, a^{2} b^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 16 \, b^{7} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 3 \, a^{7} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 84 \, a^{5} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 56 \, a^{3} b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 32 \, a b^{6} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 23 \, a^{6} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 64 \, a^{4} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 24 \, a^{2} b^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 5 \, a^{7} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 24 \, a^{5} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 8 \, a^{3} b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 5 \, a^{6} b - 2 \, a^{4} b^{3}\right )}}{{\left (a^{8} + 2 \, a^{6} b^{2} + a^{4} b^{4}\right )} {\left (a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 2 \, b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - a\right )}^{4}}}{8 \, d} \]
-1/8*(3*log(abs(2*a*tan(1/2*d*x + 1/2*c) - 2*b - 2*sqrt(a^2 + b^2))/abs(2* a*tan(1/2*d*x + 1/2*c) - 2*b + 2*sqrt(a^2 + b^2)))/((a^4 + 2*a^2*b^2 + b^4 )*sqrt(a^2 + b^2)) - 2*(5*a^7*tan(1/2*d*x + 1/2*c)^7 + 16*a^5*b^2*tan(1/2* d*x + 1/2*c)^7 + 8*a^3*b^4*tan(1/2*d*x + 1/2*c)^7 - 3*a^6*b*tan(1/2*d*x + 1/2*c)^6 - 48*a^4*b^3*tan(1/2*d*x + 1/2*c)^6 - 24*a^2*b^5*tan(1/2*d*x + 1/ 2*c)^6 + 3*a^7*tan(1/2*d*x + 1/2*c)^5 - 36*a^5*b^2*tan(1/2*d*x + 1/2*c)^5 + 56*a^3*b^4*tan(1/2*d*x + 1/2*c)^5 + 32*a*b^6*tan(1/2*d*x + 1/2*c)^5 - 15 *a^6*b*tan(1/2*d*x + 1/2*c)^4 + 114*a^4*b^3*tan(1/2*d*x + 1/2*c)^4 + 8*a^2 *b^5*tan(1/2*d*x + 1/2*c)^4 - 16*b^7*tan(1/2*d*x + 1/2*c)^4 + 3*a^7*tan(1/ 2*d*x + 1/2*c)^3 + 84*a^5*b^2*tan(1/2*d*x + 1/2*c)^3 - 56*a^3*b^4*tan(1/2* d*x + 1/2*c)^3 - 32*a*b^6*tan(1/2*d*x + 1/2*c)^3 + 23*a^6*b*tan(1/2*d*x + 1/2*c)^2 - 64*a^4*b^3*tan(1/2*d*x + 1/2*c)^2 - 24*a^2*b^5*tan(1/2*d*x + 1/ 2*c)^2 + 5*a^7*tan(1/2*d*x + 1/2*c) - 24*a^5*b^2*tan(1/2*d*x + 1/2*c) - 8* a^3*b^4*tan(1/2*d*x + 1/2*c) - 5*a^6*b - 2*a^4*b^3)/((a^8 + 2*a^6*b^2 + a^ 4*b^4)*(a*tan(1/2*d*x + 1/2*c)^2 - 2*b*tan(1/2*d*x + 1/2*c) - a)^4))/d
Time = 32.33 (sec) , antiderivative size = 719, normalized size of antiderivative = 4.61 \[ \int \frac {1}{(a \cos (c+d x)+b \sin (c+d x))^5} \, dx=-\frac {\frac {5\,a^2\,b+2\,b^3}{4\,\left (a^4+2\,a^2\,b^2+b^4\right )}+\frac {3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,\left (a^4\,b+16\,a^2\,b^3+8\,b^5\right )}{4\,a^2\,\left (a^4+2\,a^2\,b^2+b^4\right )}+\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (-23\,a^4\,b+64\,a^2\,b^3+24\,b^5\right )}{4\,a^2\,\left (a^4+2\,a^2\,b^2+b^4\right )}+\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (-5\,a^4+24\,a^2\,b^2+8\,b^4\right )}{4\,a\,\left (a^4+2\,a^2\,b^2+b^4\right )}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,\left (3\,a^6-36\,a^4\,b^2+56\,a^2\,b^4+32\,b^6\right )}{4\,a^3\,\left (a^4+2\,a^2\,b^2+b^4\right )}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (3\,a^6+84\,a^4\,b^2-56\,a^2\,b^4-32\,b^6\right )}{4\,a^3\,\left (a^4+2\,a^2\,b^2+b^4\right )}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7\,\left (5\,a^4+16\,a^2\,b^2+8\,b^4\right )}{4\,a\,\left (a^4+2\,a^2\,b^2+b^4\right )}+\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\left (5\,a^2\,b+2\,b^3\right )\,\left (3\,a^4-24\,a^2\,b^2+8\,b^4\right )}{4\,a^4\,\left (a^4+2\,a^2\,b^2+b^4\right )}}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\left (6\,a^4-48\,a^2\,b^2+16\,b^4\right )+a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+a^4-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (4\,a^4-24\,a^2\,b^2\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,\left (4\,a^4-24\,a^2\,b^2\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (32\,a\,b^3-24\,a^3\,b\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,\left (32\,a\,b^3-24\,a^3\,b\right )+8\,a^3\,b\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )-8\,a^3\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7\right )}+\frac {\mathrm {atan}\left (\frac {-1{}\mathrm {i}\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,a^5+a^4\,b\,1{}\mathrm {i}-2{}\mathrm {i}\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,a^3\,b^2+a^2\,b^3\,2{}\mathrm {i}-1{}\mathrm {i}\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,a\,b^4+b^5\,1{}\mathrm {i}}{{\left (a^2+b^2\right )}^{5/2}}\right )\,3{}\mathrm {i}}{4\,d\,{\left (a^2+b^2\right )}^{5/2}} \]
(atan((a^4*b*1i + b^5*1i + a^2*b^3*2i - a^5*tan(c/2 + (d*x)/2)*1i - a*b^4* tan(c/2 + (d*x)/2)*1i - a^3*b^2*tan(c/2 + (d*x)/2)*2i)/(a^2 + b^2)^(5/2))* 3i)/(4*d*(a^2 + b^2)^(5/2)) - ((5*a^2*b + 2*b^3)/(4*(a^4 + b^4 + 2*a^2*b^2 )) + (3*tan(c/2 + (d*x)/2)^6*(a^4*b + 8*b^5 + 16*a^2*b^3))/(4*a^2*(a^4 + b ^4 + 2*a^2*b^2)) + (tan(c/2 + (d*x)/2)^2*(24*b^5 - 23*a^4*b + 64*a^2*b^3)) /(4*a^2*(a^4 + b^4 + 2*a^2*b^2)) + (tan(c/2 + (d*x)/2)*(8*b^4 - 5*a^4 + 24 *a^2*b^2))/(4*a*(a^4 + b^4 + 2*a^2*b^2)) - (tan(c/2 + (d*x)/2)^5*(3*a^6 + 32*b^6 + 56*a^2*b^4 - 36*a^4*b^2))/(4*a^3*(a^4 + b^4 + 2*a^2*b^2)) - (tan( c/2 + (d*x)/2)^3*(3*a^6 - 32*b^6 - 56*a^2*b^4 + 84*a^4*b^2))/(4*a^3*(a^4 + b^4 + 2*a^2*b^2)) - (tan(c/2 + (d*x)/2)^7*(5*a^4 + 8*b^4 + 16*a^2*b^2))/( 4*a*(a^4 + b^4 + 2*a^2*b^2)) + (tan(c/2 + (d*x)/2)^4*(5*a^2*b + 2*b^3)*(3* a^4 + 8*b^4 - 24*a^2*b^2))/(4*a^4*(a^4 + b^4 + 2*a^2*b^2)))/(d*(tan(c/2 + (d*x)/2)^4*(6*a^4 + 16*b^4 - 48*a^2*b^2) + a^4*tan(c/2 + (d*x)/2)^8 + a^4 - tan(c/2 + (d*x)/2)^2*(4*a^4 - 24*a^2*b^2) - tan(c/2 + (d*x)/2)^6*(4*a^4 - 24*a^2*b^2) + tan(c/2 + (d*x)/2)^3*(32*a*b^3 - 24*a^3*b) - tan(c/2 + (d* x)/2)^5*(32*a*b^3 - 24*a^3*b) + 8*a^3*b*tan(c/2 + (d*x)/2) - 8*a^3*b*tan(c /2 + (d*x)/2)^7))