Integrand size = 29, antiderivative size = 61 \[ \int (a+b \cos (c+d x)) (A+B \cos (c+d x)) \sec ^3(c+d x) \, dx=\frac {(a A+2 b B) \text {arctanh}(\sin (c+d x))}{2 d}+\frac {(A b+a B) \tan (c+d x)}{d}+\frac {a A \sec (c+d x) \tan (c+d x)}{2 d} \]
1/2*(A*a+2*B*b)*arctanh(sin(d*x+c))/d+(A*b+B*a)*tan(d*x+c)/d+1/2*a*A*sec(d *x+c)*tan(d*x+c)/d
Time = 0.01 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.23 \[ \int (a+b \cos (c+d x)) (A+B \cos (c+d x)) \sec ^3(c+d x) \, dx=\frac {a A \text {arctanh}(\sin (c+d x))}{2 d}+\frac {b B \text {arctanh}(\sin (c+d x))}{d}+\frac {A b \tan (c+d x)}{d}+\frac {a B \tan (c+d x)}{d}+\frac {a A \sec (c+d x) \tan (c+d x)}{2 d} \]
(a*A*ArcTanh[Sin[c + d*x]])/(2*d) + (b*B*ArcTanh[Sin[c + d*x]])/d + (A*b*T an[c + d*x])/d + (a*B*Tan[c + d*x])/d + (a*A*Sec[c + d*x]*Tan[c + d*x])/(2 *d)
Time = 0.58 (sec) , antiderivative size = 64, normalized size of antiderivative = 1.05, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.345, Rules used = {3042, 3447, 3042, 3500, 3042, 3227, 3042, 4254, 24, 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.
\(\displaystyle \int \sec ^3(c+d x) (a+b \cos (c+d x)) (A+B \cos (c+d x)) \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right ) \left (A+B \sin \left (c+d x+\frac {\pi }{2}\right )\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^3}dx\) |
\(\Big \downarrow \) 3447 |
\(\displaystyle \int \sec ^3(c+d x) \left ((a B+A b) \cos (c+d x)+a A+b B \cos ^2(c+d x)\right )dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {(a B+A b) \sin \left (c+d x+\frac {\pi }{2}\right )+a A+b B \sin \left (c+d x+\frac {\pi }{2}\right )^2}{\sin \left (c+d x+\frac {\pi }{2}\right )^3}dx\) |
\(\Big \downarrow \) 3500 |
\(\displaystyle \frac {1}{2} \int (2 (A b+a B)+(a A+2 b B) \cos (c+d x)) \sec ^2(c+d x)dx+\frac {a A \tan (c+d x) \sec (c+d x)}{2 d}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {1}{2} \int \frac {2 (A b+a B)+(a A+2 b B) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^2}dx+\frac {a A \tan (c+d x) \sec (c+d x)}{2 d}\) |
\(\Big \downarrow \) 3227 |
\(\displaystyle \frac {1}{2} \left (2 (a B+A b) \int \sec ^2(c+d x)dx+(a A+2 b B) \int \sec (c+d x)dx\right )+\frac {a A \tan (c+d x) \sec (c+d x)}{2 d}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {1}{2} \left ((a A+2 b B) \int \csc \left (c+d x+\frac {\pi }{2}\right )dx+2 (a B+A b) \int \csc \left (c+d x+\frac {\pi }{2}\right )^2dx\right )+\frac {a A \tan (c+d x) \sec (c+d x)}{2 d}\) |
\(\Big \downarrow \) 4254 |
\(\displaystyle \frac {1}{2} \left ((a A+2 b B) \int \csc \left (c+d x+\frac {\pi }{2}\right )dx-\frac {2 (a B+A b) \int 1d(-\tan (c+d x))}{d}\right )+\frac {a A \tan (c+d x) \sec (c+d x)}{2 d}\) |
\(\Big \downarrow \) 24 |
\(\displaystyle \frac {1}{2} \left ((a A+2 b B) \int \csc \left (c+d x+\frac {\pi }{2}\right )dx+\frac {2 (a B+A b) \tan (c+d x)}{d}\right )+\frac {a A \tan (c+d x) \sec (c+d x)}{2 d}\) |
\(\Big \downarrow \) 4257 |
\(\displaystyle \frac {1}{2} \left (\frac {(a A+2 b B) \text {arctanh}(\sin (c+d x))}{d}+\frac {2 (a B+A b) \tan (c+d x)}{d}\right )+\frac {a A \tan (c+d x) \sec (c+d x)}{2 d}\) |
(a*A*Sec[c + d*x]*Tan[c + d*x])/(2*d) + (((a*A + 2*b*B)*ArcTanh[Sin[c + d* x]])/d + (2*(A*b + a*B)*Tan[c + d*x])/d)/2
3.3.20.3.1 Defintions of rubi rules used
Int[((b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_) + (d_.)*sin[(e_.) + (f_.)*(x _)]), x_Symbol] :> Simp[c Int[(b*Sin[e + f*x])^m, x], x] + Simp[d/b Int [(b*Sin[e + f*x])^(m + 1), x], x] /; FreeQ[{b, c, d, e, f, m}, x]
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Int[(a + b*Sin[e + f*x])^m*(A*c + (B*c + A*d)*Sin[e + f*x] + B*d*Sin[e + f*x]^2), x] /; FreeQ[{a, b, c, d, e, f, A, B, m}, x] && NeQ[b*c - a*d, 0]
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-(A*b^2 - a*b*B + a^2*C))*Cos[e + f*x]*((a + b*Sin[e + f*x])^(m + 1)/(b*f*(m + 1)* (a^2 - b^2))), x] + Simp[1/(b*(m + 1)*(a^2 - b^2)) Int[(a + b*Sin[e + f*x ])^(m + 1)*Simp[b*(a*A - b*B + a*C)*(m + 1) - (A*b^2 - a*b*B + a^2*C + b*(A *b - a*B + b*C)*(m + 1))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, e, f, A, B, C}, x] && LtQ[m, -1] && NeQ[a^2 - b^2, 0]
Int[csc[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> Simp[-d^(-1) Subst[Int[Exp andIntegrand[(1 + x^2)^(n/2 - 1), x], x], x, Cot[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[n/2, 0]
Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]
Time = 3.25 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.23
method | result | size |
derivativedivides | \(\frac {a A \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+B a \tan \left (d x +c \right )+A b \tan \left (d x +c \right )+B b \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}\) | \(75\) |
default | \(\frac {a A \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+B a \tan \left (d x +c \right )+A b \tan \left (d x +c \right )+B b \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}\) | \(75\) |
parts | \(\frac {a A \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )}{d}+\frac {\left (A b +B a \right ) \tan \left (d x +c \right )}{d}+\frac {B b \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}\) | \(76\) |
parallelrisch | \(\frac {-\left (1+\cos \left (2 d x +2 c \right )\right ) \left (a A +2 B b \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )+\left (1+\cos \left (2 d x +2 c \right )\right ) \left (a A +2 B b \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )+\left (2 A b +2 B a \right ) \sin \left (2 d x +2 c \right )+2 a A \sin \left (d x +c \right )}{2 d \left (1+\cos \left (2 d x +2 c \right )\right )}\) | \(114\) |
risch | \(-\frac {i \left (A a \,{\mathrm e}^{3 i \left (d x +c \right )}-2 A b \,{\mathrm e}^{2 i \left (d x +c \right )}-2 B a \,{\mathrm e}^{2 i \left (d x +c \right )}-a A \,{\mathrm e}^{i \left (d x +c \right )}-2 A b -2 B a \right )}{d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{2}}+\frac {a A \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{2 d}+\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) B b}{d}-\frac {a A \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{2 d}-\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) B b}{d}\) | \(160\) |
norman | \(\frac {\frac {\left (a A -2 A b -2 B a \right ) \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {\left (a A +2 A b +2 B a \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d}+\frac {\left (3 a A -2 A b -2 B a \right ) \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {\left (3 a A +2 A b +2 B a \right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2} \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}-\frac {\left (a A +2 B b \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{2 d}+\frac {\left (a A +2 B b \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{2 d}\) | \(192\) |
1/d*(a*A*(1/2*sec(d*x+c)*tan(d*x+c)+1/2*ln(sec(d*x+c)+tan(d*x+c)))+B*a*tan (d*x+c)+A*b*tan(d*x+c)+B*b*ln(sec(d*x+c)+tan(d*x+c)))
Time = 0.29 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.57 \[ \int (a+b \cos (c+d x)) (A+B \cos (c+d x)) \sec ^3(c+d x) \, dx=\frac {{\left (A a + 2 \, B b\right )} \cos \left (d x + c\right )^{2} \log \left (\sin \left (d x + c\right ) + 1\right ) - {\left (A a + 2 \, B b\right )} \cos \left (d x + c\right )^{2} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, {\left (A a + 2 \, {\left (B a + A b\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{4 \, d \cos \left (d x + c\right )^{2}} \]
1/4*((A*a + 2*B*b)*cos(d*x + c)^2*log(sin(d*x + c) + 1) - (A*a + 2*B*b)*co s(d*x + c)^2*log(-sin(d*x + c) + 1) + 2*(A*a + 2*(B*a + A*b)*cos(d*x + c)) *sin(d*x + c))/(d*cos(d*x + c)^2)
\[ \int (a+b \cos (c+d x)) (A+B \cos (c+d x)) \sec ^3(c+d x) \, dx=\int \left (A + B \cos {\left (c + d x \right )}\right ) \left (a + b \cos {\left (c + d x \right )}\right ) \sec ^{3}{\left (c + d x \right )}\, dx \]
Time = 0.21 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.56 \[ \int (a+b \cos (c+d x)) (A+B \cos (c+d x)) \sec ^3(c+d x) \, dx=-\frac {A a {\left (\frac {2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} - \log \left (\sin \left (d x + c\right ) + 1\right ) + \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 2 \, B b {\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 4 \, B a \tan \left (d x + c\right ) - 4 \, A b \tan \left (d x + c\right )}{4 \, d} \]
-1/4*(A*a*(2*sin(d*x + c)/(sin(d*x + c)^2 - 1) - log(sin(d*x + c) + 1) + l og(sin(d*x + c) - 1)) - 2*B*b*(log(sin(d*x + c) + 1) - log(sin(d*x + c) - 1)) - 4*B*a*tan(d*x + c) - 4*A*b*tan(d*x + c))/d
Leaf count of result is larger than twice the leaf count of optimal. 151 vs. \(2 (57) = 114\).
Time = 0.30 (sec) , antiderivative size = 151, normalized size of antiderivative = 2.48 \[ \int (a+b \cos (c+d x)) (A+B \cos (c+d x)) \sec ^3(c+d x) \, dx=\frac {{\left (A a + 2 \, B b\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - {\left (A a + 2 \, B b\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) + \frac {2 \, {\left (A a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 2 \, B a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 2 \, A b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + A a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 2 \, B a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 2 \, A b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{2}}}{2 \, d} \]
1/2*((A*a + 2*B*b)*log(abs(tan(1/2*d*x + 1/2*c) + 1)) - (A*a + 2*B*b)*log( abs(tan(1/2*d*x + 1/2*c) - 1)) + 2*(A*a*tan(1/2*d*x + 1/2*c)^3 - 2*B*a*tan (1/2*d*x + 1/2*c)^3 - 2*A*b*tan(1/2*d*x + 1/2*c)^3 + A*a*tan(1/2*d*x + 1/2 *c) + 2*B*a*tan(1/2*d*x + 1/2*c) + 2*A*b*tan(1/2*d*x + 1/2*c))/(tan(1/2*d* x + 1/2*c)^2 - 1)^2)/d
Time = 1.41 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.70 \[ \int (a+b \cos (c+d x)) (A+B \cos (c+d x)) \sec ^3(c+d x) \, dx=\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (A\,a+2\,A\,b+2\,B\,a\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (2\,A\,b-A\,a+2\,B\,a\right )}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}+\frac {\mathrm {atanh}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )\,\left (A\,a+2\,B\,b\right )}{d} \]