Integrand size = 33, antiderivative size = 103 \[ \int (a+b \cos (c+d x))^2 \left (A+C \cos ^2(c+d x)\right ) \sec ^3(c+d x) \, dx=2 a b C x+\frac {\left (2 A b^2+a^2 (A+2 C)\right ) \text {arctanh}(\sin (c+d x))}{2 d}-\frac {b^2 (A-2 C) \sin (c+d x)}{2 d}+\frac {a A b \tan (c+d x)}{d}+\frac {A (a+b \cos (c+d x))^2 \sec (c+d x) \tan (c+d x)}{2 d} \] Output:
2*a*b*C*x+1/2*(2*A*b^2+a^2*(A+2*C))*arctanh(sin(d*x+c))/d-1/2*b^2*(A-2*C)* sin(d*x+c)/d+a*A*b*tan(d*x+c)/d+1/2*A*(a+b*cos(d*x+c))^2*sec(d*x+c)*tan(d* x+c)/d
Leaf count is larger than twice the leaf count of optimal. \(249\) vs. \(2(103)=206\).
Time = 2.78 (sec) , antiderivative size = 249, normalized size of antiderivative = 2.42 \[ \int (a+b \cos (c+d x))^2 \left (A+C \cos ^2(c+d x)\right ) \sec ^3(c+d x) \, dx=\frac {8 a b C (c+d x)-2 \left (2 A b^2+a^2 (A+2 C)\right ) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )+2 \left (2 A b^2+a^2 (A+2 C)\right ) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )+\frac {a^2 A}{\left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )^2}+\frac {8 a A b \sin \left (\frac {1}{2} (c+d x)\right )}{\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )}-\frac {a^2 A}{\left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^2}+\frac {8 a A b \sin \left (\frac {1}{2} (c+d x)\right )}{\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )}+4 b^2 C \sin (c+d x)}{4 d} \] Input:
Integrate[(a + b*Cos[c + d*x])^2*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^3,x]
Output:
(8*a*b*C*(c + d*x) - 2*(2*A*b^2 + a^2*(A + 2*C))*Log[Cos[(c + d*x)/2] - Si n[(c + d*x)/2]] + 2*(2*A*b^2 + a^2*(A + 2*C))*Log[Cos[(c + d*x)/2] + Sin[( c + d*x)/2]] + (a^2*A)/(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^2 + (8*a*A*b* Sin[(c + d*x)/2])/(Cos[(c + d*x)/2] - Sin[(c + d*x)/2]) - (a^2*A)/(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2 + (8*a*A*b*Sin[(c + d*x)/2])/(Cos[(c + d*x )/2] + Sin[(c + d*x)/2]) + 4*b^2*C*Sin[c + d*x])/(4*d)
Time = 0.88 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.01, number of steps used = 11, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {3042, 3527, 3042, 3510, 25, 3042, 3502, 3042, 3214, 3042, 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.
| \(\displaystyle \int \sec ^3(c+d x) (a+b \cos (c+d x))^2 \left (A+C \cos ^2(c+d x)\right ) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^2 \left (A+C \sin \left (c+d x+\frac {\pi }{2}\right )^2\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^3}dx\) |
| \(\Big \downarrow \) 3527 |
| \(\displaystyle \frac {1}{2} \int (a+b \cos (c+d x)) \left (-b (A-2 C) \cos ^2(c+d x)+a (A+2 C) \cos (c+d x)+2 A b\right ) \sec ^2(c+d x)dx+\frac {A \tan (c+d x) \sec (c+d x) (a+b \cos (c+d x))^2}{2 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{2} \int \frac {\left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right ) \left (-b (A-2 C) \sin \left (c+d x+\frac {\pi }{2}\right )^2+a (A+2 C) \sin \left (c+d x+\frac {\pi }{2}\right )+2 A b\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^2}dx+\frac {A \tan (c+d x) \sec (c+d x) (a+b \cos (c+d x))^2}{2 d}\) |
| \(\Big \downarrow \) 3510 |
| \(\displaystyle \frac {1}{2} \left (\frac {2 a A b \tan (c+d x)}{d}-\int -\left (\left ((A+2 C) a^2+4 b C \cos (c+d x) a+2 A b^2-b^2 (A-2 C) \cos ^2(c+d x)\right ) \sec (c+d x)\right )dx\right )+\frac {A \tan (c+d x) \sec (c+d x) (a+b \cos (c+d x))^2}{2 d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{2} \left (\int \left ((A+2 C) a^2+4 b C \cos (c+d x) a+2 A b^2-b^2 (A-2 C) \cos ^2(c+d x)\right ) \sec (c+d x)dx+\frac {2 a A b \tan (c+d x)}{d}\right )+\frac {A \tan (c+d x) \sec (c+d x) (a+b \cos (c+d x))^2}{2 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{2} \left (\int \frac {(A+2 C) a^2+4 b C \sin \left (c+d x+\frac {\pi }{2}\right ) a+2 A b^2-b^2 (A-2 C) \sin \left (c+d x+\frac {\pi }{2}\right )^2}{\sin \left (c+d x+\frac {\pi }{2}\right )}dx+\frac {2 a A b \tan (c+d x)}{d}\right )+\frac {A \tan (c+d x) \sec (c+d x) (a+b \cos (c+d x))^2}{2 d}\) |
| \(\Big \downarrow \) 3502 |
| \(\displaystyle \frac {1}{2} \left (\int \left ((A+2 C) a^2+4 b C \cos (c+d x) a+2 A b^2\right ) \sec (c+d x)dx+\frac {2 a A b \tan (c+d x)}{d}-\frac {b^2 (A-2 C) \sin (c+d x)}{d}\right )+\frac {A \tan (c+d x) \sec (c+d x) (a+b \cos (c+d x))^2}{2 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{2} \left (\int \frac {(A+2 C) a^2+4 b C \sin \left (c+d x+\frac {\pi }{2}\right ) a+2 A b^2}{\sin \left (c+d x+\frac {\pi }{2}\right )}dx+\frac {2 a A b \tan (c+d x)}{d}-\frac {b^2 (A-2 C) \sin (c+d x)}{d}\right )+\frac {A \tan (c+d x) \sec (c+d x) (a+b \cos (c+d x))^2}{2 d}\) |
| \(\Big \downarrow \) 3214 |
| \(\displaystyle \frac {1}{2} \left (\left (a^2 (A+2 C)+2 A b^2\right ) \int \sec (c+d x)dx+\frac {2 a A b \tan (c+d x)}{d}+4 a b C x-\frac {b^2 (A-2 C) \sin (c+d x)}{d}\right )+\frac {A \tan (c+d x) \sec (c+d x) (a+b \cos (c+d x))^2}{2 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{2} \left (\left (a^2 (A+2 C)+2 A b^2\right ) \int \csc \left (c+d x+\frac {\pi }{2}\right )dx+\frac {2 a A b \tan (c+d x)}{d}+4 a b C x-\frac {b^2 (A-2 C) \sin (c+d x)}{d}\right )+\frac {A \tan (c+d x) \sec (c+d x) (a+b \cos (c+d x))^2}{2 d}\) |
| \(\Big \downarrow \) 4257 |
| \(\displaystyle \frac {1}{2} \left (\frac {\left (a^2 (A+2 C)+2 A b^2\right ) \text {arctanh}(\sin (c+d x))}{d}+\frac {2 a A b \tan (c+d x)}{d}+4 a b C x-\frac {b^2 (A-2 C) \sin (c+d x)}{d}\right )+\frac {A \tan (c+d x) \sec (c+d x) (a+b \cos (c+d x))^2}{2 d}\) |
Input:
Int[(a + b*Cos[c + d*x])^2*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^3,x]
Output:
(A*(a + b*Cos[c + d*x])^2*Sec[c + d*x]*Tan[c + d*x])/(2*d) + (4*a*b*C*x + ((2*A*b^2 + a^2*(A + 2*C))*ArcTanh[Sin[c + d*x]])/d - (b^2*(A - 2*C)*Sin[c + d*x])/d + (2*a*A*b*Tan[c + d*x])/d)/2
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])/((c_.) + (d_.)*sin[(e_.) + (f_. )*(x_)]), x_Symbol] :> Simp[b*(x/d), x] - Simp[(b*c - a*d)/d Int[1/(c + d *Sin[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f}, 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[(-C)*Co s[e + f*x]*((a + b*Sin[e + f*x])^(m + 1)/(b*f*(m + 2))), x] + Simp[1/(b*(m + 2)) Int[(a + b*Sin[e + f*x])^m*Simp[A*b*(m + 2) + b*C*(m + 1) + (b*B*(m + 2) - a*C)*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, e, f, A, B, C, m}, x] && !LtQ[m, -1]
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f _.)*(x_)]^2), x_Symbol] :> Simp[(-(b*c - a*d))*(A*b^2 - a*b*B + a^2*C)*Cos[ e + f*x]*((a + b*Sin[e + f*x])^(m + 1)/(b^2*f*(m + 1)*(a^2 - b^2))), x] - S imp[1/(b^2*(m + 1)*(a^2 - b^2)) Int[(a + b*Sin[e + f*x])^(m + 1)*Simp[b*( m + 1)*((b*B - a*C)*(b*c - a*d) - A*b*(a*c - b*d)) + (b*B*(a^2*d + b^2*d*(m + 1) - a*b*c*(m + 2)) + (b*c - a*d)*(A*b^2*(m + 2) + C*(a^2 + b^2*(m + 1)) ))*Sin[e + f*x] - b*C*d*(m + 1)*(a^2 - b^2)*Sin[e + f*x]^2, x], x], x] /; F reeQ[{a, b, c, d, e, f, A, B, C}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && LtQ[m, -1]
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-(c^2*C + A*d^2))*Cos[e + f*x]*(a + b*Sin[e + f*x])^m*((c + d*Sin[e + f*x])^(n + 1)/(d*f*(n + 1)*(c^2 - d^2))), x] + Simp[1/(d*(n + 1)*(c^2 - d^ 2)) Int[(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])^(n + 1)*Simp[A* d*(b*d*m + a*c*(n + 1)) + c*C*(b*c*m + a*d*(n + 1)) - (A*d*(a*d*(n + 2) - b *c*(n + 1)) - C*(b*c*d*(n + 1) - a*(c^2 + d^2*(n + 1))))*Sin[e + f*x] - b*( A*d^2*(m + n + 2) + C*(c^2*(m + 1) + d^2*(n + 1)))*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, C}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[m, 0] && LtQ[n, -1]
Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]
Time = 1.18 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.08
| method | result | size |
| parts | \(\frac {A \,a^{2} \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )}{d}+\frac {\left (A \,b^{2}+a^{2} C \right ) \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}+\frac {b^{2} C \sin \left (d x +c \right )}{d}+\frac {2 a A b \tan \left (d x +c \right )}{d}+\frac {2 a b C \left (d x +c \right )}{d}\) | \(111\) |
| derivativedivides | \(\frac {A \,a^{2} \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+a^{2} C \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+2 a A b \tan \left (d x +c \right )+2 a b C \left (d x +c \right )+A \,b^{2} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+C \,b^{2} \sin \left (d x +c \right )}{d}\) | \(112\) |
| default | \(\frac {A \,a^{2} \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+a^{2} C \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+2 a A b \tan \left (d x +c \right )+2 a b C \left (d x +c \right )+A \,b^{2} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+C \,b^{2} \sin \left (d x +c \right )}{d}\) | \(112\) |
| parallelrisch | \(\frac {-\left (1+\cos \left (2 d x +2 c \right )\right ) \left (2 A \,b^{2}+a^{2} \left (A +2 C \right )\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )+\left (1+\cos \left (2 d x +2 c \right )\right ) \left (2 A \,b^{2}+a^{2} \left (A +2 C \right )\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )+4 a b x d C \cos \left (2 d x +2 c \right )+4 a A b \sin \left (2 d x +2 c \right )+C \,b^{2} \sin \left (3 d x +3 c \right )+\left (2 A \,a^{2}+C \,b^{2}\right ) \sin \left (d x +c \right )+4 a b x d C}{2 d \left (1+\cos \left (2 d x +2 c \right )\right )}\) | \(171\) |
| risch | \(2 a b C x -\frac {i {\mathrm e}^{i \left (d x +c \right )} C \,b^{2}}{2 d}+\frac {i {\mathrm e}^{-i \left (d x +c \right )} C \,b^{2}}{2 d}-\frac {i A a \left (a \,{\mathrm e}^{3 i \left (d x +c \right )}-4 b \,{\mathrm e}^{2 i \left (d x +c \right )}-a \,{\mathrm e}^{i \left (d x +c \right )}-4 b \right )}{d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{2}}+\frac {A \,a^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{2 d}+\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) A \,b^{2}}{d}+\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) a^{2} C}{d}-\frac {A \,a^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{2 d}-\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) A \,b^{2}}{d}-\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) a^{2} C}{d}\) | \(236\) |
| norman | \(\frac {\frac {\left (A \,a^{2}-4 a A b +2 C \,b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{11}}{d}+\frac {\left (A \,a^{2}+4 a A b +2 C \,b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d}+\frac {\left (5 A \,a^{2}-12 a A b +2 C \,b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{d}+\frac {\left (5 A \,a^{2}+12 a A b +2 C \,b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{d}+\frac {2 \left (5 A \,a^{2}-4 a A b -2 C \,b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{d}+\frac {2 \left (5 A \,a^{2}+4 a A b -2 C \,b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{d}+2 a b C x +4 a b C x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-2 a b C x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}-8 a b C x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}-2 a b C x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}+4 a b C x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}+2 a b C x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{12}}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{4} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1\right )^{2}}-\frac {\left (A \,a^{2}+2 A \,b^{2}+2 a^{2} C \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{2 d}+\frac {\left (A \,a^{2}+2 A \,b^{2}+2 a^{2} C \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{2 d}\) | \(408\) |
Input:
int((a+b*cos(d*x+c))^2*(A+C*cos(d*x+c)^2)*sec(d*x+c)^3,x,method=_RETURNVER BOSE)
Output:
A*a^2/d*(1/2*sec(d*x+c)*tan(d*x+c)+1/2*ln(sec(d*x+c)+tan(d*x+c)))+(A*b^2+C *a^2)/d*ln(sec(d*x+c)+tan(d*x+c))+b^2*C*sin(d*x+c)/d+2*a*A*b*tan(d*x+c)/d+ 2*a*b*C/d*(d*x+c)
Time = 0.10 (sec) , antiderivative size = 139, normalized size of antiderivative = 1.35 \[ \int (a+b \cos (c+d x))^2 \left (A+C \cos ^2(c+d x)\right ) \sec ^3(c+d x) \, dx=\frac {8 \, C a b d x \cos \left (d x + c\right )^{2} + {\left ({\left (A + 2 \, C\right )} a^{2} + 2 \, A b^{2}\right )} \cos \left (d x + c\right )^{2} \log \left (\sin \left (d x + c\right ) + 1\right ) - {\left ({\left (A + 2 \, C\right )} a^{2} + 2 \, A b^{2}\right )} \cos \left (d x + c\right )^{2} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, {\left (2 \, C b^{2} \cos \left (d x + c\right )^{2} + 4 \, A a b \cos \left (d x + c\right ) + A a^{2}\right )} \sin \left (d x + c\right )}{4 \, d \cos \left (d x + c\right )^{2}} \] Input:
integrate((a+b*cos(d*x+c))^2*(A+C*cos(d*x+c)^2)*sec(d*x+c)^3,x, algorithm= "fricas")
Output:
1/4*(8*C*a*b*d*x*cos(d*x + c)^2 + ((A + 2*C)*a^2 + 2*A*b^2)*cos(d*x + c)^2 *log(sin(d*x + c) + 1) - ((A + 2*C)*a^2 + 2*A*b^2)*cos(d*x + c)^2*log(-sin (d*x + c) + 1) + 2*(2*C*b^2*cos(d*x + c)^2 + 4*A*a*b*cos(d*x + c) + A*a^2) *sin(d*x + c))/(d*cos(d*x + c)^2)
\[ \int (a+b \cos (c+d x))^2 \left (A+C \cos ^2(c+d x)\right ) \sec ^3(c+d x) \, dx=\int \left (A + C \cos ^{2}{\left (c + d x \right )}\right ) \left (a + b \cos {\left (c + d x \right )}\right )^{2} \sec ^{3}{\left (c + d x \right )}\, dx \] Input:
integrate((a+b*cos(d*x+c))**2*(A+C*cos(d*x+c)**2)*sec(d*x+c)**3,x)
Output:
Integral((A + C*cos(c + d*x)**2)*(a + b*cos(c + d*x))**2*sec(c + d*x)**3, x)
Time = 0.04 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.36 \[ \int (a+b \cos (c+d x))^2 \left (A+C \cos ^2(c+d x)\right ) \sec ^3(c+d x) \, dx=\frac {8 \, {\left (d x + c\right )} C a b - A a^{2} {\left (\frac {2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} - \log \left (\sin \left (d x + c\right ) + 1\right ) + \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 2 \, C a^{2} {\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 2 \, A b^{2} {\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 4 \, C b^{2} \sin \left (d x + c\right ) + 8 \, A a b \tan \left (d x + c\right )}{4 \, d} \] Input:
integrate((a+b*cos(d*x+c))^2*(A+C*cos(d*x+c)^2)*sec(d*x+c)^3,x, algorithm= "maxima")
Output:
1/4*(8*(d*x + c)*C*a*b - A*a^2*(2*sin(d*x + c)/(sin(d*x + c)^2 - 1) - log( sin(d*x + c) + 1) + log(sin(d*x + c) - 1)) + 2*C*a^2*(log(sin(d*x + c) + 1 ) - log(sin(d*x + c) - 1)) + 2*A*b^2*(log(sin(d*x + c) + 1) - log(sin(d*x + c) - 1)) + 4*C*b^2*sin(d*x + c) + 8*A*a*b*tan(d*x + c))/d
Time = 0.19 (sec) , antiderivative size = 189, normalized size of antiderivative = 1.83 \[ \int (a+b \cos (c+d x))^2 \left (A+C \cos ^2(c+d x)\right ) \sec ^3(c+d x) \, dx=\frac {4 \, {\left (d x + c\right )} C a b + \frac {4 \, C b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1} + {\left (A a^{2} + 2 \, C a^{2} + 2 \, A b^{2}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - {\left (A a^{2} + 2 \, C a^{2} + 2 \, A b^{2}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) + \frac {2 \, {\left (A a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 4 \, A a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + A a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 4 \, A a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{2}}}{2 \, d} \] Input:
integrate((a+b*cos(d*x+c))^2*(A+C*cos(d*x+c)^2)*sec(d*x+c)^3,x, algorithm= "giac")
Output:
1/2*(4*(d*x + c)*C*a*b + 4*C*b^2*tan(1/2*d*x + 1/2*c)/(tan(1/2*d*x + 1/2*c )^2 + 1) + (A*a^2 + 2*C*a^2 + 2*A*b^2)*log(abs(tan(1/2*d*x + 1/2*c) + 1)) - (A*a^2 + 2*C*a^2 + 2*A*b^2)*log(abs(tan(1/2*d*x + 1/2*c) - 1)) + 2*(A*a^ 2*tan(1/2*d*x + 1/2*c)^3 - 4*A*a*b*tan(1/2*d*x + 1/2*c)^3 + A*a^2*tan(1/2* d*x + 1/2*c) + 4*A*a*b*tan(1/2*d*x + 1/2*c))/(tan(1/2*d*x + 1/2*c)^2 - 1)^ 2)/d
Time = 0.77 (sec) , antiderivative size = 188, normalized size of antiderivative = 1.83 \[ \int (a+b \cos (c+d x))^2 \left (A+C \cos ^2(c+d x)\right ) \sec ^3(c+d x) \, dx=\frac {2\,\left (\frac {A\,a^2\,\mathrm {atanh}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{2}+A\,b^2\,\mathrm {atanh}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )+C\,a^2\,\mathrm {atanh}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )+2\,C\,a\,b\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )\right )}{d}+\frac {\frac {C\,b^2\,\sin \left (3\,c+3\,d\,x\right )}{4}+\frac {A\,a^2\,\sin \left (c+d\,x\right )}{2}+\frac {C\,b^2\,\sin \left (c+d\,x\right )}{4}+A\,a\,b\,\sin \left (2\,c+2\,d\,x\right )}{d\,\left (\frac {\cos \left (2\,c+2\,d\,x\right )}{2}+\frac {1}{2}\right )} \] Input:
int(((A + C*cos(c + d*x)^2)*(a + b*cos(c + d*x))^2)/cos(c + d*x)^3,x)
Output:
(2*((A*a^2*atanh(sin(c/2 + (d*x)/2)/cos(c/2 + (d*x)/2)))/2 + A*b^2*atanh(s in(c/2 + (d*x)/2)/cos(c/2 + (d*x)/2)) + C*a^2*atanh(sin(c/2 + (d*x)/2)/cos (c/2 + (d*x)/2)) + 2*C*a*b*atan(sin(c/2 + (d*x)/2)/cos(c/2 + (d*x)/2))))/d + ((C*b^2*sin(3*c + 3*d*x))/4 + (A*a^2*sin(c + d*x))/2 + (C*b^2*sin(c + d *x))/4 + A*a*b*sin(2*c + 2*d*x))/(d*(cos(2*c + 2*d*x)/2 + 1/2))
Time = 0.18 (sec) , antiderivative size = 471, normalized size of antiderivative = 4.57 \[ \int (a+b \cos (c+d x))^2 \left (A+C \cos ^2(c+d x)\right ) \sec ^3(c+d x) \, dx=\frac {-\cos \left (d x +c \right ) \mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right ) \sin \left (d x +c \right )^{2} a^{3}-2 \cos \left (d x +c \right ) \mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right ) \sin \left (d x +c \right )^{2} a^{2} c -2 \cos \left (d x +c \right ) \mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right ) \sin \left (d x +c \right )^{2} a \,b^{2}+\cos \left (d x +c \right ) \mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right ) a^{3}+2 \cos \left (d x +c \right ) \mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right ) a^{2} c +2 \cos \left (d x +c \right ) \mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right ) a \,b^{2}+\cos \left (d x +c \right ) \mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right ) \sin \left (d x +c \right )^{2} a^{3}+2 \cos \left (d x +c \right ) \mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right ) \sin \left (d x +c \right )^{2} a^{2} c +2 \cos \left (d x +c \right ) \mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right ) \sin \left (d x +c \right )^{2} a \,b^{2}-\cos \left (d x +c \right ) \mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right ) a^{3}-2 \cos \left (d x +c \right ) \mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right ) a^{2} c -2 \cos \left (d x +c \right ) \mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right ) a \,b^{2}+2 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{3} b^{2} c +4 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{2} a b c d x -\cos \left (d x +c \right ) \sin \left (d x +c \right ) a^{3}-2 \cos \left (d x +c \right ) \sin \left (d x +c \right ) b^{2} c -4 \cos \left (d x +c \right ) a b c d x +4 \sin \left (d x +c \right )^{3} a^{2} b -4 \sin \left (d x +c \right ) a^{2} b}{2 \cos \left (d x +c \right ) d \left (\sin \left (d x +c \right )^{2}-1\right )} \] Input:
int((a+b*cos(d*x+c))^2*(A+C*cos(d*x+c)^2)*sec(d*x+c)^3,x)
Output:
( - cos(c + d*x)*log(tan((c + d*x)/2) - 1)*sin(c + d*x)**2*a**3 - 2*cos(c + d*x)*log(tan((c + d*x)/2) - 1)*sin(c + d*x)**2*a**2*c - 2*cos(c + d*x)*l og(tan((c + d*x)/2) - 1)*sin(c + d*x)**2*a*b**2 + cos(c + d*x)*log(tan((c + d*x)/2) - 1)*a**3 + 2*cos(c + d*x)*log(tan((c + d*x)/2) - 1)*a**2*c + 2* cos(c + d*x)*log(tan((c + d*x)/2) - 1)*a*b**2 + cos(c + d*x)*log(tan((c + d*x)/2) + 1)*sin(c + d*x)**2*a**3 + 2*cos(c + d*x)*log(tan((c + d*x)/2) + 1)*sin(c + d*x)**2*a**2*c + 2*cos(c + d*x)*log(tan((c + d*x)/2) + 1)*sin(c + d*x)**2*a*b**2 - cos(c + d*x)*log(tan((c + d*x)/2) + 1)*a**3 - 2*cos(c + d*x)*log(tan((c + d*x)/2) + 1)*a**2*c - 2*cos(c + d*x)*log(tan((c + d*x) /2) + 1)*a*b**2 + 2*cos(c + d*x)*sin(c + d*x)**3*b**2*c + 4*cos(c + d*x)*s in(c + d*x)**2*a*b*c*d*x - cos(c + d*x)*sin(c + d*x)*a**3 - 2*cos(c + d*x) *sin(c + d*x)*b**2*c - 4*cos(c + d*x)*a*b*c*d*x + 4*sin(c + d*x)**3*a**2*b - 4*sin(c + d*x)*a**2*b)/(2*cos(c + d*x)*d*(sin(c + d*x)**2 - 1))