Integrand size = 26, antiderivative size = 231 \[ \int \frac {\sec (c+d x)}{(a \cos (c+d x)+b \sin (c+d x))^4} \, dx=\frac {\text {arctanh}(\sin (c+d x))}{b^4 d}+\frac {a \text {arctanh}\left (\frac {b \cos (c+d x)-a \sin (c+d x)}{\sqrt {a^2+b^2}}\right )}{2 b^2 \left (a^2+b^2\right )^{3/2} d}+\frac {a \text {arctanh}\left (\frac {b \cos (c+d x)-a \sin (c+d x)}{\sqrt {a^2+b^2}}\right )}{b^4 \sqrt {a^2+b^2} d}-\frac {1}{3 b d (a \cos (c+d x)+b \sin (c+d x))^3}+\frac {a (b \cos (c+d x)-a \sin (c+d x))}{2 b^2 \left (a^2+b^2\right ) d (a \cos (c+d x)+b \sin (c+d x))^2}-\frac {1}{b^3 d (a \cos (c+d x)+b \sin (c+d x))} \]
arctanh(sin(d*x+c))/b^4/d+1/2*a*arctanh((b*cos(d*x+c)-a*sin(d*x+c))/(a^2+b ^2)^(1/2))/b^2/(a^2+b^2)^(3/2)/d-1/3/b/d/(a*cos(d*x+c)+b*sin(d*x+c))^3+1/2 *a*(b*cos(d*x+c)-a*sin(d*x+c))/b^2/(a^2+b^2)/d/(a*cos(d*x+c)+b*sin(d*x+c)) ^2-1/b^3/d/(a*cos(d*x+c)+b*sin(d*x+c))+a*arctanh((b*cos(d*x+c)-a*sin(d*x+c ))/(a^2+b^2)^(1/2))/b^4/d/(a^2+b^2)^(1/2)
Time = 4.26 (sec) , antiderivative size = 290, normalized size of antiderivative = 1.26 \[ \int \frac {\sec (c+d x)}{(a \cos (c+d x)+b \sin (c+d x))^4} \, dx=-\frac {\sec ^3(c+d x) (a \cos (c+d x)+b \sin (c+d x)) \left (2 b^3 \sec (c+d x)+3 b^2 (a \cos (c+d x)+b \sin (c+d x)) \tan (c+d x)+\frac {3 b \left (2 a^2+b^2\right ) \cos (c+d x) (a+b \tan (c+d x))^2}{a^2+b^2}+\frac {6 a \left (2 a^2+3 b^2\right ) \text {arctanh}\left (\frac {-b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2+b^2}}\right ) \cos ^2(c+d x) (a+b \tan (c+d x))^3}{\left (a^2+b^2\right )^{3/2}}+6 \cos ^2(c+d x) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right ) (a+b \tan (c+d x))^3-6 \cos ^2(c+d x) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right ) (a+b \tan (c+d x))^3\right )}{6 b^4 d (a+b \tan (c+d x))^4} \]
-1/6*(Sec[c + d*x]^3*(a*Cos[c + d*x] + b*Sin[c + d*x])*(2*b^3*Sec[c + d*x] + 3*b^2*(a*Cos[c + d*x] + b*Sin[c + d*x])*Tan[c + d*x] + (3*b*(2*a^2 + b^ 2)*Cos[c + d*x]*(a + b*Tan[c + d*x])^2)/(a^2 + b^2) + (6*a*(2*a^2 + 3*b^2) *ArcTanh[(-b + a*Tan[(c + d*x)/2])/Sqrt[a^2 + b^2]]*Cos[c + d*x]^2*(a + b* Tan[c + d*x])^3)/(a^2 + b^2)^(3/2) + 6*Cos[c + d*x]^2*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]]*(a + b*Tan[c + d*x])^3 - 6*Cos[c + d*x]^2*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]*(a + b*Tan[c + d*x])^3))/(b^4*d*(a + b*Tan[c + d*x])^4)
Time = 0.96 (sec) , antiderivative size = 235, normalized size of antiderivative = 1.02, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.462, Rules used = {3042, 3573, 3042, 3555, 3042, 3553, 219, 3573, 3042, 3553, 219, 4257}
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.
| \(\displaystyle \int \frac {\sec (c+d x)}{(a \cos (c+d x)+b \sin (c+d x))^4} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\cos (c+d x) (a \cos (c+d x)+b \sin (c+d x))^4}dx\) |
| \(\Big \downarrow \) 3573 |
| \(\displaystyle -\frac {a \int \frac {1}{(a \cos (c+d x)+b \sin (c+d x))^3}dx}{b^2}+\frac {\int \frac {\sec (c+d x)}{(a \cos (c+d x)+b \sin (c+d x))^2}dx}{b^2}-\frac {1}{3 b d (a \cos (c+d x)+b \sin (c+d x))^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {a \int \frac {1}{(a \cos (c+d x)+b \sin (c+d x))^3}dx}{b^2}+\frac {\int \frac {1}{\cos (c+d x) (a \cos (c+d x)+b \sin (c+d x))^2}dx}{b^2}-\frac {1}{3 b d (a \cos (c+d x)+b \sin (c+d x))^3}\) |
| \(\Big \downarrow \) 3555 |
| \(\displaystyle -\frac {a \left (\frac {\int \frac {1}{a \cos (c+d x)+b \sin (c+d x)}dx}{2 \left (a^2+b^2\right )}-\frac {b \cos (c+d x)-a \sin (c+d x)}{2 d \left (a^2+b^2\right ) (a \cos (c+d x)+b \sin (c+d x))^2}\right )}{b^2}+\frac {\int \frac {1}{\cos (c+d x) (a \cos (c+d x)+b \sin (c+d x))^2}dx}{b^2}-\frac {1}{3 b d (a \cos (c+d x)+b \sin (c+d x))^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {a \left (\frac {\int \frac {1}{a \cos (c+d x)+b \sin (c+d x)}dx}{2 \left (a^2+b^2\right )}-\frac {b \cos (c+d x)-a \sin (c+d x)}{2 d \left (a^2+b^2\right ) (a \cos (c+d x)+b \sin (c+d x))^2}\right )}{b^2}+\frac {\int \frac {1}{\cos (c+d x) (a \cos (c+d x)+b \sin (c+d x))^2}dx}{b^2}-\frac {1}{3 b d (a \cos (c+d x)+b \sin (c+d x))^3}\) |
| \(\Big \downarrow \) 3553 |
| \(\displaystyle -\frac {a \left (-\frac {\int \frac {1}{a^2+b^2-(b \cos (c+d x)-a \sin (c+d x))^2}d(b \cos (c+d x)-a \sin (c+d x))}{2 d \left (a^2+b^2\right )}-\frac {b \cos (c+d x)-a \sin (c+d x)}{2 d \left (a^2+b^2\right ) (a \cos (c+d x)+b \sin (c+d x))^2}\right )}{b^2}+\frac {\int \frac {1}{\cos (c+d x) (a \cos (c+d x)+b \sin (c+d x))^2}dx}{b^2}-\frac {1}{3 b d (a \cos (c+d x)+b \sin (c+d x))^3}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {\int \frac {1}{\cos (c+d x) (a \cos (c+d x)+b \sin (c+d x))^2}dx}{b^2}-\frac {a \left (-\frac {\text {arctanh}\left (\frac {b \cos (c+d x)-a \sin (c+d x)}{\sqrt {a^2+b^2}}\right )}{2 d \left (a^2+b^2\right )^{3/2}}-\frac {b \cos (c+d x)-a \sin (c+d x)}{2 d \left (a^2+b^2\right ) (a \cos (c+d x)+b \sin (c+d x))^2}\right )}{b^2}-\frac {1}{3 b d (a \cos (c+d x)+b \sin (c+d x))^3}\) |
| \(\Big \downarrow \) 3573 |
| \(\displaystyle \frac {-\frac {a \int \frac {1}{a \cos (c+d x)+b \sin (c+d x)}dx}{b^2}+\frac {\int \sec (c+d x)dx}{b^2}-\frac {1}{b d (a \cos (c+d x)+b \sin (c+d x))}}{b^2}-\frac {a \left (-\frac {\text {arctanh}\left (\frac {b \cos (c+d x)-a \sin (c+d x)}{\sqrt {a^2+b^2}}\right )}{2 d \left (a^2+b^2\right )^{3/2}}-\frac {b \cos (c+d x)-a \sin (c+d x)}{2 d \left (a^2+b^2\right ) (a \cos (c+d x)+b \sin (c+d x))^2}\right )}{b^2}-\frac {1}{3 b d (a \cos (c+d x)+b \sin (c+d x))^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {a \int \frac {1}{a \cos (c+d x)+b \sin (c+d x)}dx}{b^2}+\frac {\int \csc \left (c+d x+\frac {\pi }{2}\right )dx}{b^2}-\frac {1}{b d (a \cos (c+d x)+b \sin (c+d x))}}{b^2}-\frac {a \left (-\frac {\text {arctanh}\left (\frac {b \cos (c+d x)-a \sin (c+d x)}{\sqrt {a^2+b^2}}\right )}{2 d \left (a^2+b^2\right )^{3/2}}-\frac {b \cos (c+d x)-a \sin (c+d x)}{2 d \left (a^2+b^2\right ) (a \cos (c+d x)+b \sin (c+d x))^2}\right )}{b^2}-\frac {1}{3 b d (a \cos (c+d x)+b \sin (c+d x))^3}\) |
| \(\Big \downarrow \) 3553 |
| \(\displaystyle \frac {\frac {a \int \frac {1}{a^2+b^2-(b \cos (c+d x)-a \sin (c+d x))^2}d(b \cos (c+d x)-a \sin (c+d x))}{b^2 d}+\frac {\int \csc \left (c+d x+\frac {\pi }{2}\right )dx}{b^2}-\frac {1}{b d (a \cos (c+d x)+b \sin (c+d x))}}{b^2}-\frac {a \left (-\frac {\text {arctanh}\left (\frac {b \cos (c+d x)-a \sin (c+d x)}{\sqrt {a^2+b^2}}\right )}{2 d \left (a^2+b^2\right )^{3/2}}-\frac {b \cos (c+d x)-a \sin (c+d x)}{2 d \left (a^2+b^2\right ) (a \cos (c+d x)+b \sin (c+d x))^2}\right )}{b^2}-\frac {1}{3 b d (a \cos (c+d x)+b \sin (c+d x))^3}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {\frac {\int \csc \left (c+d x+\frac {\pi }{2}\right )dx}{b^2}+\frac {a \text {arctanh}\left (\frac {b \cos (c+d x)-a \sin (c+d x)}{\sqrt {a^2+b^2}}\right )}{b^2 d \sqrt {a^2+b^2}}-\frac {1}{b d (a \cos (c+d x)+b \sin (c+d x))}}{b^2}-\frac {a \left (-\frac {\text {arctanh}\left (\frac {b \cos (c+d x)-a \sin (c+d x)}{\sqrt {a^2+b^2}}\right )}{2 d \left (a^2+b^2\right )^{3/2}}-\frac {b \cos (c+d x)-a \sin (c+d x)}{2 d \left (a^2+b^2\right ) (a \cos (c+d x)+b \sin (c+d x))^2}\right )}{b^2}-\frac {1}{3 b d (a \cos (c+d x)+b \sin (c+d x))^3}\) |
| \(\Big \downarrow \) 4257 |
| \(\displaystyle -\frac {a \left (-\frac {\text {arctanh}\left (\frac {b \cos (c+d x)-a \sin (c+d x)}{\sqrt {a^2+b^2}}\right )}{2 d \left (a^2+b^2\right )^{3/2}}-\frac {b \cos (c+d x)-a \sin (c+d x)}{2 d \left (a^2+b^2\right ) (a \cos (c+d x)+b \sin (c+d x))^2}\right )}{b^2}+\frac {\frac {a \text {arctanh}\left (\frac {b \cos (c+d x)-a \sin (c+d x)}{\sqrt {a^2+b^2}}\right )}{b^2 d \sqrt {a^2+b^2}}-\frac {1}{b d (a \cos (c+d x)+b \sin (c+d x))}+\frac {\text {arctanh}(\sin (c+d x))}{b^2 d}}{b^2}-\frac {1}{3 b d (a \cos (c+d x)+b \sin (c+d x))^3}\) |
-1/3*1/(b*d*(a*Cos[c + d*x] + b*Sin[c + d*x])^3) - (a*(-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*Co s[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)))/b^2 + (ArcTanh[Sin[c + d*x]]/(b^2*d) + (a*ArcTanh[(b*Cos[c + d* x] - a*Sin[c + d*x])/Sqrt[a^2 + b^2]])/(b^2*Sqrt[a^2 + b^2]*d) - 1/(b*d*(a *Cos[c + d*x] + b*Sin[c + d*x])))/b^2
3.2.46.3.1 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]
Int[(cos[(c_.) + (d_.)*(x_)]*(a_.) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(n_)/co s[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[(a*Cos[c + d*x] + b*Sin[c + d*x])^ (n + 1)/(b*d*(n + 1)), x] + (Simp[1/b^2 Int[(a*Cos[c + d*x] + b*Sin[c + d *x])^(n + 2)/Cos[c + d*x], x], x] - Simp[a/b^2 Int[(a*Cos[c + d*x] + b*Si n[c + d*x])^(n + 1), x], x]) /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 + b^2, 0] && LtQ[n, -1]
Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]
Time = 2.58 (sec) , antiderivative size = 411, normalized size of antiderivative = 1.78
| method | result | size |
| derivativedivides | \(\frac {\frac {\frac {2 \left (\frac {b^{2} \left (a^{4}+2 a^{2} b^{2}+2 b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{2 \left (a^{2}+b^{2}\right ) a}+\frac {b \left (2 a^{6}-3 a^{4} b^{2}-4 a^{2} b^{4}-4 b^{6}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{2 \left (a^{2}+b^{2}\right ) a^{2}}-\frac {b^{2} \left (18 a^{6}+3 a^{4} b^{2}-4 a^{2} b^{4}-4 b^{6}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{3 a^{3} \left (a^{2}+b^{2}\right )}-\frac {b \left (2 a^{6}-8 a^{4} b^{2}-7 a^{2} b^{4}-2 b^{6}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{a^{2} \left (a^{2}+b^{2}\right )}+\frac {b^{2} \left (11 a^{4}+8 a^{2} b^{2}+2 b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 a \left (a^{2}+b^{2}\right )}+\frac {b \left (6 a^{4}+5 a^{2} b^{2}+2 b^{4}\right )}{6 a^{2}+6 b^{2}}\right )}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -2 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-a \right )^{3}}-\frac {a \left (2 a^{2}+3 b^{2}\right ) \operatorname {arctanh}\left (\frac {2 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-2 b}{2 \sqrt {a^{2}+b^{2}}}\right )}{\left (a^{2}+b^{2}\right )^{\frac {3}{2}}}}{b^{4}}-\frac {\ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{b^{4}}+\frac {\ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{b^{4}}}{d}\) | \(411\) |
| default | \(\frac {\frac {\frac {2 \left (\frac {b^{2} \left (a^{4}+2 a^{2} b^{2}+2 b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{2 \left (a^{2}+b^{2}\right ) a}+\frac {b \left (2 a^{6}-3 a^{4} b^{2}-4 a^{2} b^{4}-4 b^{6}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{2 \left (a^{2}+b^{2}\right ) a^{2}}-\frac {b^{2} \left (18 a^{6}+3 a^{4} b^{2}-4 a^{2} b^{4}-4 b^{6}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{3 a^{3} \left (a^{2}+b^{2}\right )}-\frac {b \left (2 a^{6}-8 a^{4} b^{2}-7 a^{2} b^{4}-2 b^{6}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{a^{2} \left (a^{2}+b^{2}\right )}+\frac {b^{2} \left (11 a^{4}+8 a^{2} b^{2}+2 b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 a \left (a^{2}+b^{2}\right )}+\frac {b \left (6 a^{4}+5 a^{2} b^{2}+2 b^{4}\right )}{6 a^{2}+6 b^{2}}\right )}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -2 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-a \right )^{3}}-\frac {a \left (2 a^{2}+3 b^{2}\right ) \operatorname {arctanh}\left (\frac {2 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-2 b}{2 \sqrt {a^{2}+b^{2}}}\right )}{\left (a^{2}+b^{2}\right )^{\frac {3}{2}}}}{b^{4}}-\frac {\ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{b^{4}}+\frac {\ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{b^{4}}}{d}\) | \(411\) |
| risch | \(-\frac {-6 b^{4} {\mathrm e}^{i \left (d x +c \right )}-15 i a^{3} b \,{\mathrm e}^{5 i \left (d x +c \right )}+6 a^{4} {\mathrm e}^{5 i \left (d x +c \right )}+12 a^{4} {\mathrm e}^{3 i \left (d x +c \right )}+20 b^{4} {\mathrm e}^{3 i \left (d x +c \right )}-6 b^{4} {\mathrm e}^{5 i \left (d x +c \right )}+32 a^{2} b^{2} {\mathrm e}^{3 i \left (d x +c \right )}-6 a^{2} b^{2} {\mathrm e}^{i \left (d x +c \right )}-6 a^{2} b^{2} {\mathrm e}^{5 i \left (d x +c \right )}+9 i a \,b^{3} {\mathrm e}^{i \left (d x +c \right )}-9 i a \,b^{3} {\mathrm e}^{5 i \left (d x +c \right )}+15 i a^{3} b \,{\mathrm e}^{i \left (d x +c \right )}+6 a^{4} {\mathrm e}^{i \left (d x +c \right )}}{3 \left (i b +a \right ) b^{3} \left (-i b \,{\mathrm e}^{2 i \left (d x +c \right )}+{\mathrm e}^{2 i \left (d x +c \right )} a +i b +a \right )^{3} \left (-i b +a \right ) d}+\frac {a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-\frac {i a^{3}+i a \,b^{2}-a^{2} b -b^{3}}{\left (a^{2}+b^{2}\right )^{\frac {3}{2}}}\right )}{\left (a^{2}+b^{2}\right )^{\frac {3}{2}} d \,b^{4}}+\frac {3 a \ln \left ({\mathrm e}^{i \left (d x +c \right )}-\frac {i a^{3}+i a \,b^{2}-a^{2} b -b^{3}}{\left (a^{2}+b^{2}\right )^{\frac {3}{2}}}\right )}{2 \left (a^{2}+b^{2}\right )^{\frac {3}{2}} d \,b^{2}}-\frac {a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i a^{3}+i a \,b^{2}-a^{2} b -b^{3}}{\left (a^{2}+b^{2}\right )^{\frac {3}{2}}}\right )}{\left (a^{2}+b^{2}\right )^{\frac {3}{2}} d \,b^{4}}-\frac {3 a \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i a^{3}+i a \,b^{2}-a^{2} b -b^{3}}{\left (a^{2}+b^{2}\right )^{\frac {3}{2}}}\right )}{2 \left (a^{2}+b^{2}\right )^{\frac {3}{2}} d \,b^{2}}-\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{b^{4} d}+\frac {\ln \left (i+{\mathrm e}^{i \left (d x +c \right )}\right )}{b^{4} d}\) | \(560\) |
1/d*(2/b^4*((1/2*b^2*(a^4+2*a^2*b^2+2*b^4)/(a^2+b^2)/a*tan(1/2*d*x+1/2*c)^ 5+1/2*b*(2*a^6-3*a^4*b^2-4*a^2*b^4-4*b^6)/(a^2+b^2)/a^2*tan(1/2*d*x+1/2*c) ^4-1/3/a^3*b^2*(18*a^6+3*a^4*b^2-4*a^2*b^4-4*b^6)/(a^2+b^2)*tan(1/2*d*x+1/ 2*c)^3-1/a^2*b*(2*a^6-8*a^4*b^2-7*a^2*b^4-2*b^6)/(a^2+b^2)*tan(1/2*d*x+1/2 *c)^2+1/2/a*b^2*(11*a^4+8*a^2*b^2+2*b^4)/(a^2+b^2)*tan(1/2*d*x+1/2*c)+1/6* b*(6*a^4+5*a^2*b^2+2*b^4)/(a^2+b^2))/(tan(1/2*d*x+1/2*c)^2*a-2*b*tan(1/2*d *x+1/2*c)-a)^3-1/2*a*(2*a^2+3*b^2)/(a^2+b^2)^(3/2)*arctanh(1/2*(2*a*tan(1/ 2*d*x+1/2*c)-2*b)/(a^2+b^2)^(1/2)))-1/b^4*ln(tan(1/2*d*x+1/2*c)-1)+1/b^4*l n(tan(1/2*d*x+1/2*c)+1))
Leaf count of result is larger than twice the leaf count of optimal. 745 vs. \(2 (217) = 434\).
Time = 0.39 (sec) , antiderivative size = 745, normalized size of antiderivative = 3.23 \[ \int \frac {\sec (c+d x)}{(a \cos (c+d x)+b \sin (c+d x))^4} \, dx=-\frac {22 \, a^{4} b^{3} + 38 \, a^{2} b^{5} + 16 \, b^{7} + 12 \, {\left (a^{6} b - 2 \, a^{2} b^{5} - b^{7}\right )} \cos \left (d x + c\right )^{2} + 6 \, {\left (5 \, a^{5} b^{2} + 8 \, a^{3} b^{4} + 3 \, a b^{6}\right )} \cos \left (d x + c\right ) \sin \left (d x + c\right ) - 3 \, {\left ({\left (2 \, a^{6} - 3 \, a^{4} b^{2} - 9 \, a^{2} b^{4}\right )} \cos \left (d x + c\right )^{3} + 3 \, {\left (2 \, a^{4} b^{2} + 3 \, a^{2} b^{4}\right )} \cos \left (d x + c\right ) + {\left (2 \, a^{3} b^{3} + 3 \, a b^{5} + {\left (6 \, a^{5} b + 7 \, a^{3} b^{3} - 3 \, a b^{5}\right )} \cos \left (d x + c\right )^{2}\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 ) - 6 \, {\left ({\left (a^{7} - a^{5} b^{2} - 5 \, a^{3} b^{4} - 3 \, a b^{6}\right )} \cos \left (d x + c\right )^{3} + 3 \, {\left (a^{5} b^{2} + 2 \, a^{3} b^{4} + a b^{6}\right )} \cos \left (d x + c\right ) + {\left (a^{4} b^{3} + 2 \, a^{2} b^{5} + b^{7} + {\left (3 \, a^{6} b + 5 \, a^{4} b^{3} + a^{2} b^{5} - b^{7}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) + 6 \, {\left ({\left (a^{7} - a^{5} b^{2} - 5 \, a^{3} b^{4} - 3 \, a b^{6}\right )} \cos \left (d x + c\right )^{3} + 3 \, {\left (a^{5} b^{2} + 2 \, a^{3} b^{4} + a b^{6}\right )} \cos \left (d x + c\right ) + {\left (a^{4} b^{3} + 2 \, a^{2} b^{5} + b^{7} + {\left (3 \, a^{6} b + 5 \, a^{4} b^{3} + a^{2} b^{5} - b^{7}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )\right )} \log \left (-\sin \left (d x + c\right ) + 1\right )}{12 \, {\left ({\left (a^{7} b^{4} - a^{5} b^{6} - 5 \, a^{3} b^{8} - 3 \, a b^{10}\right )} d \cos \left (d x + c\right )^{3} + 3 \, {\left (a^{5} b^{6} + 2 \, a^{3} b^{8} + a b^{10}\right )} d \cos \left (d x + c\right ) + {\left ({\left (3 \, a^{6} b^{5} + 5 \, a^{4} b^{7} + a^{2} b^{9} - b^{11}\right )} d \cos \left (d x + c\right )^{2} + {\left (a^{4} b^{7} + 2 \, a^{2} b^{9} + b^{11}\right )} d\right )} \sin \left (d x + c\right )\right )}} \]
-1/12*(22*a^4*b^3 + 38*a^2*b^5 + 16*b^7 + 12*(a^6*b - 2*a^2*b^5 - b^7)*cos (d*x + c)^2 + 6*(5*a^5*b^2 + 8*a^3*b^4 + 3*a*b^6)*cos(d*x + c)*sin(d*x + c ) - 3*((2*a^6 - 3*a^4*b^2 - 9*a^2*b^4)*cos(d*x + c)^3 + 3*(2*a^4*b^2 + 3*a ^2*b^4)*cos(d*x + c) + (2*a^3*b^3 + 3*a*b^5 + (6*a^5*b + 7*a^3*b^3 - 3*a*b ^5)*cos(d*x + c)^2)*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)) - 6*((a^7 - a^5*b^2 - 5*a^3*b^4 - 3*a*b^6 )*cos(d*x + c)^3 + 3*(a^5*b^2 + 2*a^3*b^4 + a*b^6)*cos(d*x + c) + (a^4*b^3 + 2*a^2*b^5 + b^7 + (3*a^6*b + 5*a^4*b^3 + a^2*b^5 - b^7)*cos(d*x + c)^2) *sin(d*x + c))*log(sin(d*x + c) + 1) + 6*((a^7 - a^5*b^2 - 5*a^3*b^4 - 3*a *b^6)*cos(d*x + c)^3 + 3*(a^5*b^2 + 2*a^3*b^4 + a*b^6)*cos(d*x + c) + (a^4 *b^3 + 2*a^2*b^5 + b^7 + (3*a^6*b + 5*a^4*b^3 + a^2*b^5 - b^7)*cos(d*x + c )^2)*sin(d*x + c))*log(-sin(d*x + c) + 1))/((a^7*b^4 - a^5*b^6 - 5*a^3*b^8 - 3*a*b^10)*d*cos(d*x + c)^3 + 3*(a^5*b^6 + 2*a^3*b^8 + a*b^10)*d*cos(d*x + c) + ((3*a^6*b^5 + 5*a^4*b^7 + a^2*b^9 - b^11)*d*cos(d*x + c)^2 + (a^4* b^7 + 2*a^2*b^9 + b^11)*d)*sin(d*x + c))
\[ \int \frac {\sec (c+d x)}{(a \cos (c+d x)+b \sin (c+d x))^4} \, dx=\int \frac {\sec {\left (c + d x \right )}}{\left (a \cos {\left (c + d x \right )} + b \sin {\left (c + d x \right )}\right )^{4}}\, dx \]
Leaf count of result is larger than twice the leaf count of optimal. 661 vs. \(2 (217) = 434\).
Time = 0.35 (sec) , antiderivative size = 661, normalized size of antiderivative = 2.86 \[ \int \frac {\sec (c+d x)}{(a \cos (c+d x)+b \sin (c+d x))^4} \, dx=\frac {\frac {3 \, {\left (2 \, a^{2} + 3 \, b^{2}\right )} a \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^{2} b^{4} + b^{6}\right )} \sqrt {a^{2} + b^{2}}} - \frac {2 \, {\left (6 \, a^{7} + 5 \, a^{5} b^{2} + 2 \, a^{3} b^{4} + \frac {3 \, {\left (11 \, a^{6} b + 8 \, a^{4} b^{3} + 2 \, a^{2} b^{5}\right )} \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {6 \, {\left (2 \, a^{7} - 8 \, a^{5} b^{2} - 7 \, a^{3} b^{4} - 2 \, a b^{6}\right )} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {2 \, {\left (18 \, a^{6} b + 3 \, a^{4} b^{3} - 4 \, a^{2} b^{5} - 4 \, b^{7}\right )} \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {3 \, {\left (2 \, a^{7} - 3 \, a^{5} b^{2} - 4 \, a^{3} b^{4} - 4 \, a b^{6}\right )} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {3 \, {\left (a^{6} b + 2 \, a^{4} b^{3} + 2 \, a^{2} b^{5}\right )} \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}}\right )}}{a^{8} b^{3} + a^{6} b^{5} + \frac {6 \, {\left (a^{7} b^{4} + a^{5} b^{6}\right )} \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {3 \, {\left (a^{8} b^{3} - 3 \, a^{6} b^{5} - 4 \, a^{4} b^{7}\right )} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {4 \, {\left (3 \, a^{7} b^{4} + a^{5} b^{6} - 2 \, a^{3} b^{8}\right )} \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {3 \, {\left (a^{8} b^{3} - 3 \, a^{6} b^{5} - 4 \, a^{4} b^{7}\right )} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {6 \, {\left (a^{7} b^{4} + a^{5} b^{6}\right )} \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac {{\left (a^{8} b^{3} + a^{6} b^{5}\right )} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}}} + \frac {6 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}{b^{4}} - \frac {6 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - 1\right )}{b^{4}}}{6 \, d} \]
1/6*(3*(2*a^2 + 3*b^2)*a*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^2*b^4 + b^6)*sqrt(a^2 + b^2)) - 2*(6*a^7 + 5*a^5*b^2 + 2*a^3*b^4 + 3*(11 *a^6*b + 8*a^4*b^3 + 2*a^2*b^5)*sin(d*x + c)/(cos(d*x + c) + 1) - 6*(2*a^7 - 8*a^5*b^2 - 7*a^3*b^4 - 2*a*b^6)*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 - 2*(18*a^6*b + 3*a^4*b^3 - 4*a^2*b^5 - 4*b^7)*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 + 3*(2*a^7 - 3*a^5*b^2 - 4*a^3*b^4 - 4*a*b^6)*sin(d*x + c)^4/(cos(d *x + c) + 1)^4 + 3*(a^6*b + 2*a^4*b^3 + 2*a^2*b^5)*sin(d*x + c)^5/(cos(d*x + c) + 1)^5)/(a^8*b^3 + a^6*b^5 + 6*(a^7*b^4 + a^5*b^6)*sin(d*x + c)/(cos (d*x + c) + 1) - 3*(a^8*b^3 - 3*a^6*b^5 - 4*a^4*b^7)*sin(d*x + c)^2/(cos(d *x + c) + 1)^2 - 4*(3*a^7*b^4 + a^5*b^6 - 2*a^3*b^8)*sin(d*x + c)^3/(cos(d *x + c) + 1)^3 + 3*(a^8*b^3 - 3*a^6*b^5 - 4*a^4*b^7)*sin(d*x + c)^4/(cos(d *x + c) + 1)^4 + 6*(a^7*b^4 + a^5*b^6)*sin(d*x + c)^5/(cos(d*x + c) + 1)^5 - (a^8*b^3 + a^6*b^5)*sin(d*x + c)^6/(cos(d*x + c) + 1)^6) + 6*log(sin(d* x + c)/(cos(d*x + c) + 1) + 1)/b^4 - 6*log(sin(d*x + c)/(cos(d*x + c) + 1) - 1)/b^4)/d
Leaf count of result is larger than twice the leaf count of optimal. 527 vs. \(2 (217) = 434\).
Time = 0.39 (sec) , antiderivative size = 527, normalized size of antiderivative = 2.28 \[ \int \frac {\sec (c+d x)}{(a \cos (c+d x)+b \sin (c+d x))^4} \, dx=\frac {\frac {3 \, {\left (2 \, a^{3} + 3 \, a b^{2}\right )} \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^{2} b^{4} + b^{6}\right )} \sqrt {a^{2} + b^{2}}} + \frac {2 \, {\left (3 \, a^{6} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 6 \, a^{4} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 6 \, a^{2} b^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 6 \, a^{7} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 9 \, a^{5} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 12 \, a^{3} b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 12 \, a b^{6} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 36 \, a^{6} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 6 \, a^{4} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 8 \, a^{2} b^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 8 \, b^{7} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 12 \, a^{7} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 48 \, a^{5} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 42 \, a^{3} b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 12 \, a b^{6} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 33 \, a^{6} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 24 \, a^{4} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 6 \, a^{2} b^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 6 \, a^{7} + 5 \, a^{5} b^{2} + 2 \, a^{3} b^{4}\right )}}{{\left (a^{5} b^{3} + a^{3} b^{5}\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 )}^{3}} + \frac {6 \, \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right )}{b^{4}} - \frac {6 \, \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right )}{b^{4}}}{6 \, d} \]
1/6*(3*(2*a^3 + 3*a*b^2)*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^2* b^4 + b^6)*sqrt(a^2 + b^2)) + 2*(3*a^6*b*tan(1/2*d*x + 1/2*c)^5 + 6*a^4*b^ 3*tan(1/2*d*x + 1/2*c)^5 + 6*a^2*b^5*tan(1/2*d*x + 1/2*c)^5 + 6*a^7*tan(1/ 2*d*x + 1/2*c)^4 - 9*a^5*b^2*tan(1/2*d*x + 1/2*c)^4 - 12*a^3*b^4*tan(1/2*d *x + 1/2*c)^4 - 12*a*b^6*tan(1/2*d*x + 1/2*c)^4 - 36*a^6*b*tan(1/2*d*x + 1 /2*c)^3 - 6*a^4*b^3*tan(1/2*d*x + 1/2*c)^3 + 8*a^2*b^5*tan(1/2*d*x + 1/2*c )^3 + 8*b^7*tan(1/2*d*x + 1/2*c)^3 - 12*a^7*tan(1/2*d*x + 1/2*c)^2 + 48*a^ 5*b^2*tan(1/2*d*x + 1/2*c)^2 + 42*a^3*b^4*tan(1/2*d*x + 1/2*c)^2 + 12*a*b^ 6*tan(1/2*d*x + 1/2*c)^2 + 33*a^6*b*tan(1/2*d*x + 1/2*c) + 24*a^4*b^3*tan( 1/2*d*x + 1/2*c) + 6*a^2*b^5*tan(1/2*d*x + 1/2*c) + 6*a^7 + 5*a^5*b^2 + 2* a^3*b^4)/((a^5*b^3 + a^3*b^5)*(a*tan(1/2*d*x + 1/2*c)^2 - 2*b*tan(1/2*d*x + 1/2*c) - a)^3) + 6*log(abs(tan(1/2*d*x + 1/2*c) + 1))/b^4 - 6*log(abs(ta n(1/2*d*x + 1/2*c) - 1))/b^4)/d
Time = 27.55 (sec) , antiderivative size = 2848, normalized size of antiderivative = 12.33 \[ \int \frac {\sec (c+d x)}{(a \cos (c+d x)+b \sin (c+d x))^4} \, dx=\text {Too large to display} \]
(2*atanh((64*a*b^5*tan(c/2 + (d*x)/2))/((176*a^3*b^15)/(b^12 + 2*a^2*b^10 + a^4*b^8) + (160*a^5*b^13)/(b^12 + 2*a^2*b^10 + a^4*b^8) + (48*a^7*b^11)/ (b^12 + 2*a^2*b^10 + a^4*b^8) + (64*a*b^17)/(b^12 + 2*a^2*b^10 + a^4*b^8)) + (48*a^3*b^3*tan(c/2 + (d*x)/2))/((176*a^3*b^15)/(b^12 + 2*a^2*b^10 + a^ 4*b^8) + (160*a^5*b^13)/(b^12 + 2*a^2*b^10 + a^4*b^8) + (48*a^7*b^11)/(b^1 2 + 2*a^2*b^10 + a^4*b^8) + (64*a*b^17)/(b^12 + 2*a^2*b^10 + a^4*b^8))))/( b^4*d) - ((6*a^4 + 2*b^4 + 5*a^2*b^2)/(3*b^3*(a^2 + b^2)) + (tan(c/2 + (d* x)/2)*(11*a^4 + 2*b^4 + 8*a^2*b^2))/(a*b^2*(a^2 + b^2)) + (tan(c/2 + (d*x) /2)^5*(a^4 + 2*b^4 + 2*a^2*b^2))/(a*b^2*(a^2 + b^2)) - (tan(c/2 + (d*x)/2) ^4*(4*b^6 - 2*a^6 + 4*a^2*b^4 + 3*a^4*b^2))/(a^2*b^3*(a^2 + b^2)) + (2*tan (c/2 + (d*x)/2)^2*(2*b^6 - 2*a^6 + 7*a^2*b^4 + 8*a^4*b^2))/(a^2*b^3*(a^2 + b^2)) - (2*tan(c/2 + (d*x)/2)^3*(3*a^2 - 2*b^2)*(6*a^4 + 2*b^4 + 5*a^2*b^ 2))/(3*a^3*b^2*(a^2 + b^2)))/(d*(tan(c/2 + (d*x)/2)^2*(12*a*b^2 - 3*a^3) - a^3*tan(c/2 + (d*x)/2)^6 - tan(c/2 + (d*x)/2)^4*(12*a*b^2 - 3*a^3) - tan( c/2 + (d*x)/2)^3*(12*a^2*b - 8*b^3) + a^3 + 6*a^2*b*tan(c/2 + (d*x)/2) + 6 *a^2*b*tan(c/2 + (d*x)/2)^5)) - (a*atan(((a*((a^2 + b^2)^3)^(1/2)*(2*a^2 + 3*b^2)*((8*(4*a^2*b^7 + 8*a^4*b^5 + 4*a^6*b^3))/(b^12 + 2*a^2*b^10 + a^4* b^8) + (8*tan(c/2 + (d*x)/2)*(8*a*b^9 + 29*a^3*b^7 + 28*a^5*b^5 + 8*a^7*b^ 3))/(b^13 + 2*a^2*b^11 + a^4*b^9) - (a*((a^2 + b^2)^3)^(1/2)*(2*a^2 + 3*b^ 2)*((8*tan(c/2 + (d*x)/2)*(12*a^2*b^12 + 20*a^4*b^10 + 8*a^6*b^8))/(b^1...